Research ArticlePata-Type Fixed-Point Theorems inKalevandashSeikkalarsquos Type FuzzyMetric Space

Ao-Lei Sima Fei He and Ning Lu

School of Mathematical Sciences Inner Mongolia University Hohhot 010021 China

Correspondence should be addressed to Fei He hefeiimueducn

Received 26 September 2019 Revised 7 January 2020 Accepted 29 January 2020 Published 28 February 2020

Academic Editor Giuseppe Marino

Copyright copy 2020 Ao-Lei Sima et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e purpose of this paper is to generalize the fixed-point theorems for BanachndashPata-type contraction and KannanndashPata-typecontraction frommetric spaces to KalevandashSeikkalarsquos type fuzzymetric spaces Moreover two examples are given for the support ofour results

1 Introduction and Preliminaries

In 1984 Kaleva and Seikkala [1] introduced the concept offuzzy metric spaces for the first time and they establishedsome fixed-point theorems in a complete fuzzy metric spaceis result was investigated by many authors from differentpoints of view see [2ndash10] and the references therein

Definition 1 (see [11]) A mapping η R⟶ [0 1] is calleda fuzzy real number whose αminus level set is denoted by [η]α

t isin R η(t)ge α1113864 1113865 if it satisfies two axioms

(1) ere exists t0 isin R such that η(t0) 1(2) For each α isin (0 1] [η]α [λα ρα] is a closed interval

of R where minus infinlt λα le ρα lt +infin

e set of all such fuzzy real numbers is denoted byF Ifη isin F and η(q) 0 whenever qlt 0 then η is called anonnegative fuzzy real number and byF+ we mean the setof all nonnegative fuzzy real numbers e notation 0 standsfor the fuzzy number satisfying 0(t) 1 and 0(t) 0 if tne 0Clearly 0 isinF+ R can be embedded in F if a isinF satisfiesa(q) 0(q minus a)

Definition 2 (see [1]) Let X be a nonempty set and themappings L R [0 1] times [0 1]⟶ [0 1] be symmetricnondecreasing in both arguments and satisfy L(0 0) 0

R(1 1) 1 Let d be the mapping X times X⟶ F+ and write

[d(x y)]α [λα(x y) ρα(x y)] for all x y isin X and allα isin (0 1] e quadruple (X d L R) is called a fuzzy metricspace if the following axioms are satisfied

(D1) d(x y) 0⟺x y(D2) for all x y isin X d(x y) d(y x)(D3) for all x y z isin X(D3L) whenever ple λ1(x z) qle λ1(z y) and p + qleλ1(x y) d(x y)(p + q)geL(d(x z)(p) d(z y)(q))(D3R) whenever pge λ1(x z) qge λ1(z y) and p + qgeλ1(x y) d(x y)(p + q)leR(d(x z)(p) d(z y)(q))

Lemma 1 (see [6]) Let (X d L R) be a fuzzy metric spaceand [d(x y)]t [λt(x y) ρt(x y)] for all t isin (0 1] wherex y isin X are any two fixed elements -en

(1) limq⟶minus infind(x y)(q) limq⟶+infind(x y)(q) 0(2) d(x y)(q) is a left continuous and nonincreasing

function for q isin (λ1(x y) +infin)

(3) ρt(x y) is a left continuous and nonincreasingfunction for t isin (0 1]

Lemma 2 (see [12]) Let (X d L R) be a fuzzy metric spaceand suppose that

(R-1) R(a b)lemax a b

HindawiJournal of Function SpacesVolume 2020 Article ID 6185894 9 pageshttpsdoiorg10115520206185894

(R-2) for each t isin (0 1] there exists s isin (0 t] such thatR(s r)lt t for all r isin (0 t)

(R-3) limt⟶0+ R(t t) 0

-en (R-1)rArr (R-2)rArr (R-3)

Lemma 3 (see [6]) Let (X d L R) be a fuzzy metric space-en

(1) (R-1) implies that for each t isin (0 1]

ρt(x y)le ρt(x z) + ρt(z y) (1)

for all x y z isin X

(2) (R-2) implies that for each t isin (0 1] there exists s

s(t) isin (0 t] such that

ρt(x y)le ρs(x z) + ρt(z y) (2)

for all x y z isin X

(3) (R-3) implies that for each t isin (0 1] there exists s

s(t) isin (0 t] such that

ρt(x y)le ρs(x z) + ρs(z y) (3)

for all x y z isin X

Definition 3 (see [6]) Let (X d L R) be a fuzzy metricspace and xn1113864 1113865 sub X x isin X en

(1) xn1113864 1113865 is said to be convergent to x iflimn⟶infind(xn x) 0 ie limn⟶infinρt(xn x) 0 forall t isin (0 1]

(2) xn1113864 1113865 is called a Cauchy sequence in X iflimnm⟶infind(xn xm) 0 equivalently for any givenεgt 0 and t isin (0 1] there exists N N(ε t) isin N suchthat ρt(xn xm)lt ε whenever n mgeN

(3) (X d L R) is said to be complete if each Cauchysequence in X is convergent to some point in X

Lemma 4 (see [6]) Let (X d L R) be a fuzzy metric spacewith (R-2) -en for each t isin (0 1] ρt(x y) is continuous at(x y) isin X times X

In 2011 Pata [13] extended the Banach contractionprinciple with weaker hypotheses than those of the Banachcontraction principle in the complete metric space Sincethen several other fixed point results in the spirit of Patahave appeared see [14ndash18] In particular Chakraborty andSamanta [14] proved a generalization of Kannanrsquos fixed-point theorem based on the result of Pata

roughout the following (X d) will be a completemetric space and (X d L R) will be a complete fuzzy metricspace Fix an arbitrary point x0 isin X and we denote x

d(x x0) and xt ρt(x x0) for all x isin X and t isin (0 1]Also ψ [0 1 ]⟶ [0infin) is an increasing function con-tinuous at zero with ψ(0) 0 Given a function f X⟶ X

Theorem 1 (see [13]) Let (X d) be a complete metric spaceLet Λge 0αge 1 and β isin [0 α] be fixed constants If theinequality

d(fx fy)le (1 minus ε)d(x y) + Λεαψ(ε)[1 +x +||y||]β

(4)

is satisfied for every ε isin [0 1] and all x y isin X then f has aunique fixed point z isin X

Theorem 2 (see [14]) Let (X d) be a complete metric spaceLet Λge 0 αge 1 and β isin [0 α] be fixed constants If theinequality

d(fx fy)le1 minus ε2

[d(x fx) + d(y fy)] + Λεαψ(ε)[1 +x

+ y +fx + fy]β

(5)

is satisfied for every εisin[0 1] and all x yisinX then f has aunique fixed point zisinX

In this paper we prove two further extensions of Pata-type fixed theorems in complete KalevandashSeikkalarsquos typefuzzy metric space using contractive condition of Banachtype and Kannan type Afterwards the fixed-point theoremsfor the corresponding linear contraction are given as cor-ollaries Our theorems extend the main results of [13 14]Moreover two nontrivial examples are given to illustrate ourtwo theorems and our examples show that these two the-orems are independent to each other

2 Main Results

Our results of this paper are stated as follows

Theorem 3 Let (X d L R) be a complete fuzzy metric spacewith (R-2) Let Λge0 and αge1 be fixed constants Letf X⟶X be a mapping such that

ρt(fx fy)le (1 minus ε)ρt(x y) + Λεαψ(ε) 1 +xt +yt1113858

+fxt +fyt1113859α

(6)

for every ε isin [0 1] t isin (0 1] and all x y isin X -en f has aunique fixed point z isin X

Proof Starting from x0 construct a sequence xn1113864 1113865 such thatxn fxnminus 1 fnx0 For all t isin (0 1] we denote Cn(t)

xnt If xn0 xn0+1 for some n0 then xn0

is a fixed point of fus we always assume that xn nexn+1 for all n isin N

In order to prove this theorem we divide into the fol-lowing five steps

Step 1 We show that the sequence ρt(xn+1 xn)1113864 1113865 is de-creasing Clearly suppose that t isin (0 1] and ε 0 in (6) weobtain

2 Journal of Function Spaces

ρt xn+1 xn( 1113857le ρt xn xnminus 1( 1113857le middot middot middot le ρt x1 x0( 1113857 (7)

Step 2 We prove that the sequence Cn(t)1113864 1113865 is bounded Forany t isin (0 1] by Lemma 3 there exists s s(t) isin (0 t] suchthat

ρt(x y)le ρs(x z) + ρt(z y) (8)

for all x y z isin X en from Step 1 we have

Cn(t) ρt xn x0( 1113857

le ρs xn xn+1( 1113857 + ρs x1 x0( 1113857 + ρt xn+1 x1( 1113857

le ρt xn+1 x1( 1113857 + 2ρs x1 x0( 1113857

le (1 minus ε)ρt xn x0( 1113857 + Λεαψ(ε) 1 + xn

t

+ x0

t1113960

+ xn+1

t+ x1

t

1113961α+2ρs x1 x0( 1113857

(1 minus ε)ρt xn x0( 1113857 + Λεαψ(ε) 1 + Cn(t) + Cn+1(t)1113858

+ C1(t)1113859α+2ρs x1 x0( 1113857

(9)

Since Cn+1(t)le ρs(xn+1 xn) + ρt(xn x0)le ρs(x1 x0) +

ρt(xn x0) we have

Cn(t)le (1 minus ε)Cn(t) + Λεαψ(ε) 1 + 2Cn(t) + C1(t)1113858

+ ρs x1 x0( 11138571113859α

+ 2ρs x1 x0( 1113857

le (1 minus ε)Cn(t) + Λεαψ(ε) 1 + 2Cn(t) + 2ρs x1 x0( 11138571113858 1113859α

+ 2ρs x1 x0( 1113857

(10)

Hence

εCn(t) minus 2ρs x1 x0( 1113857leΛεαψ(ε) 1 + 2Cn(t) + 2ρs x1 x0( 11138571113858 1113859α

(11)

Suppose that Cn(t)1113864 1113865 is unbounded en there existt0 isin (0 1] s0 isin (0 t0] and a subsequence Cnk

(t0)1113966 1113967 ofCn(t0)1113864 1113865 such that Cnk

(t0)⟶infin (k⟶infin) andCnk

(t0)ge 1 + 2ρs0(x0 x1) Let

ε εk 1 + 2ρs0

x1 x0( 1113857

Cnkt0( 1113857

(12)

en we get

1 + 2ρs0x1 x0( 1113857

Cnkt0( 1113857

Cnkt0( 1113857 minus 2ρs0

x1 x0( 1113857

leΛεαkψ εk( 1113857 1 + 2Cnkt0( 1113857 + 2ρs0

x1 x0( 11138571113960 1113961α

Λ 1 + 2ρs0x1 x0( 11138571113872 1113873

1 + 2Cnkt0( 1113857 + 2ρs0

x1 x0( 1113857

Cnkt0( 1113857

1113890 1113891

α

ψ εk( 1113857

(13)

which implies that

1leΛ 1 + 2ρs0x1 x0( 11138571113872 1113873

1 + 2Cnkt0( 1113857 + 2ρs0

x1 x0( 1113857

Cnkt0( 1113857

1113890 1113891

α

ψ εk( 1113857

(14)

Letting k⟶infin in (14) we have ψ(εk)⟶ 0 and

Λ 1 + 2ρs0x1 x0( 11138571113872 1113873

α 1 + C1 t0( 1113857 + 2ρs0x1 x0( 1113857

Cnkt0( 1113857

+ 21113890 1113891

α

ψ εk( 1113857⟶ 0

(15)

which contradict (14) us Cn(t)1113864 1113865 is bounded that is forany t isin (0 1] there exists a constant Mt gt 0 such thatxnt Cn(t)leMt

Step 3 We shall show that

limn⟶infin

ρt xn+1 xn( 1113857 0 (t isin (0 1]) (16)

Note that ρt(xn+1 xn)1113864 1113865 is a decreasing and boundedsequence So assume that limn⟶infinρt0

(xn+1 xn) a forsome agt 0 and t0 isin (0 1] By (6) we obtain that

ρt0xn+1 xn( 1113857 ρt0

fxn fxnminus 1( 1113857

le (1 minus ε)ρt0xn xnminus 1( 1113857 + Λεαψ(ε) 1 + xn

t0

1113876

+ xnminus 1

t0+ xn+1

t0

+ xn

t0

1113877α

le (1 minus ε)ρt0xn xnminus 1( 1113857 + Λεαψ(ε) 1 + 4Mt0

1113960 1113961α

(17)

Letting n⟶infin we have

aleΛεαminus 1ψ(ε) 1 + 4Mt01113960 1113961

α (18)

Note that

Λεαminus 1ψ(ε) 1 + 4Mt01113960 1113961

α⟶ 0 as ε⟶ 0 (19)

en we can see ale 0 which contradicts ourassumption

Step 4 We show that xn1113864 1113865 is a Cauchy sequence Supposenot choose δ gt 0 and t0 isin (0 1] and then there exist sub-sequences xnk

1113966 1113967 and xmk1113966 1113967 of xn1113864 1113865 with kltmk lt nk such that

ρt0xmk

xnk1113872 1113873gt δ and ρt0

xmk xnkminus 11113872 1113873le δ (20)

By Lemma 3 there exists s0 isin (0 t0] such that

δ lt ρt0xmk

xnk1113872 1113873le ρt0

xmk xnkminus 11113872 1113873 + ρs0

xnkminus 1 xnk1113872 1113873

le ρs0xnkminus 1 xnk

1113872 1113873 + δ(21)

Putting n⟶infin we get ρt0(xmk

xnk)⟶ δ Similarly

we can see that

Journal of Function Spaces 3

ρt0xnk+1 xmk+11113872 1113873le ρt0

xnk xmk

1113872 1113873 + ρs0xnk

xnk+11113872 1113873

+ ρs0xmk

xmk+11113872 1113873(22)

ρt0xnk+1 xmk+11113872 1113873ge ρt0

xnk xmk

1113872 1113873 minus ρs0xnk

xnk+11113872 1113873

minus ρs0xmk

xmk+11113872 1113873(23)

Passing to the limit as n⟶infin in (22) and (23) we haveρt0

(xnk+1 xmk+1)⟶ δ By (6) we get

ρt0xnk+1 xmk+11113872 1113873 ρt0

fxnk fxmk

1113872 1113873

le (1 minus ε)ρt0xnk

xmk1113872 1113873 + Λεαψ(ε) 1 + xnk

t01113876

+ xmk

t0+ xnk+1

t0+ xmk+1

t01113877α

le (1 minus ε)ρt0xnk

xmk1113872 1113873 + Λεαψ(ε) 1 + 4Mt0

1113960 1113961α

(24)

Taking n⟶infin in (2) we obtain that

δ le (1 minus ε)δ + Λεαψ(ε) 1 + 4Mt01113960 1113961

α

δ leΛεαminus 1ψ(ε) 1 + 4Mt01113960 1113961

α

(25)

Note that Λεαminus 1ψ(ε)[1 + 4Mt0]α⟶ 0 (ε⟶ 0) a

contradiction us xn1113864 1113865 is a Cauchy sequence Since X iscomplete there exists z isin X such that xn⟶ z (n⟶infin)

Step 5 We prove that z is the unique fixed point formapping f

First we show that fz z By (6) for all t isin (0 1] thereexists s isin (0 t] and we have

ρt(fz z)le ρs xn+1 z( 1113857 + ρt fz xn+1( 1113857

le (1 minus ε)ρt z xn( 1113857 + Λεαψ(ε) 1 +zt + xn

t

1113872

+fzt + xn+1

t1113873α+ρs xn+1 z( 1113857

le (1 minus ε)ρt z xn( 1113857 + Λεαψ(ε) 1 + 4Mt( 1113857α

+ ρs xn+1 z( 1113857

(26)

Letting n⟶infin we get ρt(fz z) 0 Hence fz zNext we prove the uniqueness of z Assume that zprime is

another fixed point for f For each t isin (0 1] there existss isin (0 t] such that

ρt z zprime( 1113857 ρt fz fzprime( 1113857

le (1 minus ε)ρt z zprime( 1113857 + Λϵαψ(ε) 1 +zt + zprime

t1113960

+fzt + fzprime

t1113961α

le (1 minus ε)ρt z zprime( 1113857 + Λϵαψ(ε) 1 + 4Mt1113858 1113859α

(27)

us we have ρt(z zprime)leΛεαminus 1ψ(ε)[1 + 4Mt]α⟶ 0 as

ε⟶ 0 which implies z zprime erefore f has a unique fixedpoint z isin X

Remark 1 From the proof of eorem 3 we can see that tokeep the sequence xn1113864 1113865 converge to the fixed point the rangeof ε in (6) can be limited from [0 1] to [0 c] for some givenconstant c isin (0 1]

By Remark 1 letting c λ we can deduce the followingcorollary which is the Banach contraction principle in fuzzymetric spaces

Corollary 1 (see Corollary 53 [6]) Let (X d L R) be acomplete fuzzy metric space with (R-2) and f X⟶ X be amapping If there exists λ isin [0 1) such that

ρt(fx fy)le λρt(x y) (28)

for all t isin (0 1] and x y isin X then f has a unique fixed pointz isin X

eorem 4 generalizes the result in [14] to fuzzy metricspaces

Theorem 4 Let (X d L R) be a complete fuzzy metric spacewith (R-2) Let Λge 0 and αge 1 be fixed constants Letf X⟶ X be a mapping such that

ρt(fx fy)le1 minus ε2

ρt(x fx) + ρt(y fy)1113858 1113859 + Λεαψ(ε) 1[

+xt +yt +fxt +fyt1113859α

(29)

for every ε isin [0 1] t isin (0 1] and all x y isin X -en f has aunique fixed point z isin X

Proof Starting from x0 construct a sequence xn1113864 1113865 such thatxn fxnminus 1 fnx0 For any t isin (0 1] we denote Cn(t)

xnt If xn0 xn0+1 for some n0 then xn0

is a fixed point of fus we always assume that xn nexn+1 for all n isin Nen theproof is divided into the following five steps

Step 1 We show that the sequence ρt(xn xn+1)1113864 1113865 is de-creasing Suppose that t isin (0 1] and ε 0 in (29) we obtain

ρt xn xn+1( 1113857le12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857 (30)

us we get

ρt xn+1 xn( 1113857le ρt xn xnminus 1( 1113857le middot middot middot le ρt x1 x0( 1113857 (31)

Step 2 e sequence Cn(t)1113864 1113865 is bounded For each t isin (0 1]there exists s s(t) isin (0 t] and

4 Journal of Function Spaces

Cn(t) ρt xn x0( 1113857

le ρs xn xn+1( 1113857 + ρs x1 x0( 1113857 + ρt xn+1 x1( 1113857

le ρt xn+1 x1( 1113857 + 2ρs x1 x0( 1113857

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt x1 x0( 1113857( 1113857

+ Λεαψ(ε) 1 + xn

t

+ x0

t+ xn+1

t

+ x1

t1113960 1113961

α

+ 2ρs x1 x0( 1113857

le (1 minus ε)ρt x1 x0( 1113857 + Λεαψ(ε) 1 + Cn(t)1113858

+ Cn+1(t) + C1(t)1113859α+2ρs x1 x0( 1113857

(32)

Letting ε 0 in (32) there exists a constant Mt gt 0 suchthat Cn(t)le ρt(x0 x1) Mt

Step 3 We shall prove that limn⟶infinρt(xn xn+1) 0 foreach t isin (0 1] For any t isin (0 1] we obtain

ρt xn xn+1( 1113857 ρt fxnminus 1 fxn( 1113857

le (1 minus ε)12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 + xnminus 1

t+ xn

t

+ xn

t

+ xn+1

t1113960 1113961

α

le (1 minus ε)12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 + 4Mt1113858 1113859α

(33)

Since ρt(xn xn+1)1113864 1113865 is a decreasing and bounded se-quence assume that limn⟶infinρt(xn xn+1) a for some agt 0Letting n⟶infin in (2) we have

ale (1 minus ε)12

middot 2a + Λεαψ(ε) 1 + 4Mt1113858 1113859α

le (1 minus ε)a + Λεαψ(ε) 1 + 4Mt1113858 1113859α

aleΛεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α

(34)

From

Λεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α⟶ 0 (ε⟶ 0) (35)

we see ale 0 a contradiction For each t isin (0 1] we getρt(xn xn+1)⟶ 0 (n⟶infin)

Step 4 We show that limn⟶infinρt(xn xm) 0 for eacht isin (0 1] and all n m isin N For any t isin (0 1] there existss s(t) isin (0 t] and we have

ρt xn xm( 1113857 ρt fxnminus 1 fxmminus 1( 1113857

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt xm xm+1( 1113857( 1113857

+ Λεαminus 1ψ(ε) 1 + xnminus 1

t+ xmminus 1

t

+ xn

t

1113960

+ xm

t

1113961α

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt xm xm+1( 1113857( 1113857

+ Λεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α

(36)

Putting n⟶infin and ε⟶ 0 in (36) we getρt(xn xm)⟶ 0

erefore xn1113864 1113865 is a Cauchy sequence Since X is com-plete there exists z isin X such that xn⟶ z (n⟶infin)

Step 5 We prove that z is the unique fixed point formapping f

First we show that fz z Using (29) for any t isin (0 1]there exists s isin (0 t] and we have

ρt(fz z)le ρs xn+1 z( 1113857 + ρt fz xn+1( 1113857

le (1 minus ε)12

ρt(z fz) + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 +zt + xn

t

+fzt + xn+1

t1113872 1113873

α

+ ρs xn+1 z( 1113857

le (1 minus ε)12

ρt(z fz) + ρt xn xn+1( 1113857( 1113857 + Λεαψ(ε)

1 + 4Mt( 1113857α

+ ρs xn+1 z( 1113857

(37)

By taking limits as n⟶infin and ε⟶ 0 in the inequalityabove we get ρt(fz z)le (12)ρt(fz z) Hence ρt(fz z)

0 which implies fz zFinally we prove the uniqueness of z Assume that zprime is

another fixed point of f For any t isin (0 1] there existss isin (0 t] and we getρt z zprime( 1113857 ρt fz fzprime( 1113857

le (1 minus ε)12

ρt(z fz) + ρt zprime fzprime( 1113857( 1113857 + Λϵαψ(ε)

middot 1 +zt + zprime

t+fzt + fzprime

t

1113960 1113961α

le (1 minus ε)12

ρt(z z) + ρt zprime zprime( 1113857( 1113857 + Λϵαψ(ε) 1 + 4Mt1113858 1113859α

leΛϵαψ(ε) 1 + 4Mt1113858 1113859α

(38)

Passing to the limit as ε⟶ 0 in (38) we haveρt(z zprime) 0 us z zprime erefore f has a unique fixedpoint z isin X

Journal of Function Spaces 5

Following the arguments in Remark 1 from eorem 4we can obtain the Corollary 2

Corollary 2 (see Corollary 53 [6]) Let (X d L R) be acomplete fuzzy metric space with (R-2) and f X⟶ X be amapping If there exists λ isin [0 (12)) such that

ρt(fx fy) le λ ρt(x fx) + ρt(y fy)1113858 1113859 (39)

for all t isin (0 1] and x y isin X then f has a unique fixed pointz isin X

Now we construct two examples to illustrateeorems 3and 4 respectively

Example 1 Let X (12n) n isin Ncup 0 cup 0 L(a b)

min a b and R(a b) max a b Define the fuzzy metricd X times X⟶ F+ by

d(x y) 0(q minus |x minus y|) 1 q |x minus y|

0 qne |x minus y|1113896 (40)

Let T X⟶ X be a map defined by Tx (12)x forx isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (6) is satisfied for x0 1Λ 1 α 1 and ψ(ε) 2ε(c) T has the unique fixed point x 0(d) (29) is not satisfied Soeorem 4 cannot be verified

by this example

Proof First it is easy to see that (X d L R) is a completefuzzy metric space and R(a b) max a b satisfies (R-2)Note that Tx x implies that x 0 isin X then conclusion(c) is true Next we prove conclusions (b) and (d)respectively

(b) Letting x0 1Λ 1 α 1 β 1 and ψ(ε) 2ε weshow that (6) holds for all ε isin [0 1] t isin (0 1] andx y isin X

For any t isin (0 1] we can see that ρt(x y) |x minus y|en we consider the following two cases

Case 1 If εle (12) then 1 minus εge (12) For any x y isin X wehave

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386812

|x minus y|le (1 minus ε)ρt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+ Txt +Tyt1113857

(41)

Case 2 If εgt (12) then ψ(ε)gt 1 Note that

ρt(x y)le ρt x x0( 1113857 + ρ0 x0 y( 1113857 xt +yt (42)

en for any x y isin X we have

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le |x minus y| (1 minus ε)ρt(x y) + ερt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) xt +yt( 1113857

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+Txt +Tyt1113857

(43)

From Cases 1 and 2 we see that (6) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 3 is verified

(d) We show that (29) is not satisfied

Letting ε 0 by (29) we deduce that

ρt(Tx Ty)leρt(x Tx) + ρt(y Ty)

2 (44)

However let x0 0 and y0 1 For any t isin (0 1] wehave

ρt Tx0 Ty0( 1113857 0 minus12

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868gt12

middot12

ρt x0 Tx0( 1113857 + ρt y0 Ty0( 1113857

2

(45)

So this example cannot verify eorem 4 and we proveconclusion (d)

Example 2 Let X 6 minus 5 4 minus 3 2 minus 1 0 L(a b) mina b and R(a b) max a b Let d X times X⟶ F+ be afuzzy metric such that

d(x y)(q)

0 qlt 0

1 q 0

|x minus y|

|x minus y| + q qgt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)

Let T X⟶ X be a map such that

Tx

minus |x| + 1 xgt 0

|x| minus 1 xlt 0

0 x 0

⎧⎪⎪⎨

⎪⎪⎩(47)

for x isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (29) is satisfied for x0 1 Λ 1 α 1 and

ψ(ε) 6ε(c) T has the unique fixed point x 0(d) (6) is not satisfied So this example cannot verify

eorem 3

Proof Clearly we can see that x 0 is the unique fixedpoint for map T and conclusion (c) is true Now we proveconclusions (a) (b) and (d) respectively

6 Journal of Function Spaces

(a) We show that (X d L R) is a complete fuzzy metricspace with (R-2)

Clearly for any x y isin X d(x y)(q) is a fuzzy numberand (D1) and (D2) in Definition 2 are satisfied Note thatR(a b) max a b satisfies (R-2) So it is sufficient to provethat (D3) holds

For any x y z isin X it is obvious that λ1(x z)

λ1(z y) λ1(x y) 0 To see (D3L) let us consider thefollowing two cases

(i) If plt 0 or qlt 0 then d(x z)(p) 0 ord(z y)(q) 0 leading to that

min(d(x z)(p) d(z y)(q)) 0 (48)

So (D3L) is satisfied

(ii) If p 0 and q 0 then p + q 0 Sod(x z)(p) d(z y)(q) d(x y)(p + q) 1 lead-ing to that

d(x y)(p + q) min d(x z)(p) d(z y)(q)1113864 1113865 (49)

us (D3L) holds To see (D3R) we consider the fol-lowing two cases

Case a1 Suppose that there are at least two points inx y z1113864 1113865 which are equal For any pge 0 and qge 0

(i) If z x or z y without loss of generality let z xthen d(z y) d(x y) Since d(x y)(q) is decreasingas qge 0 we have

max d(x z)(p) d(z y)(q)1113864 1113865ged(z y)(q)

d(x y)(q)ge d(x y)(p + q)(50)

(ii) If x y then we have

p + q 0rArrp 0 q 0rArr d(x y)(p + q)

1 max d(x z)(p) d(z y)(q)1113864 1113865

p + qgt 0rArr d(x y)(p + q) 0

lemax d(x z)(p) d(z y)(q)1113864 1113865

(51)

Case a2 Suppose that xney xne z and yne z For anypge 0 and qge 0 denote s d(x z)(p) andt d(z y)(q) Without loss of generality let 0lt sle ten we have

p |x minus z|1 minus s

s

q |z minus y|1 minus t

t

(52)

Since d(x y)(q) is decreasing as qge 0 and (1 minus tt) isdecreasing we deduce that

max d(x z)(p) d(z y)(q)1113864 1113865 t d(x y) |x minus y|1 minus t

t1113874 1113875

ged(x y) (|x minus z| +|z minus y|)1 minus t

t1113876 1113877

ged(x y) |x minus z|1 minus s

s+|z minus y|

1 minus t

t1113876 1113877

d(x y)(p + q)

(53)

From Cases a1 and a2 we conclude that (D3R) holds forall x y z isin X

Note that X is a finite set en (X d L R) is completeand the proof of conclusion (a) is completed

(b) We prove (29) for x0 1 Λ 1 α 1 β 1 andψ(ε) 6ε

For any t isin (0 1] we can see that ρt(x y)

|x minus y|(1 minus tt) en we consider the following two cases

Case b1 If εle (112) it is easy to see that for any xne 0we have |Tx| |x| minus 1 and x middot Txle 0 So we can obtainthat |x minus Tx| |x| + |Tx| and

|Tx minus Ty|le |Tx| +|Ty| |Tx| +|x| minus 1

2+

|Ty| +|y| minus 12

|x minus Tx| +|y minus Ty|

2minus 1

(54)

for all x yne 0 If y 0 then we have

|Tx minus Ty| |Tx| |Tx| +|x| minus 1

2

|x minus Tx| +|y minus Ty|

2minus12

(55)

From the above two equalities by calculation we obtainthat for any x y isin X

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle1011

middot|x minus Tx|(1 minus tt) +|y minus Ty|(1 minus tt)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +||Ty||t( 1113857

(56)

Journal of Function Spaces 7

Case b2 If εgt (112) then we have ψ(ε)gt (12) Notethat

ρt(x Tx) + ρt(y Ty)le ρt x x0( 1113857 + ρt x0 Tx( 1113857

+ ρt y x0( 1113857 + ρt x0 Ty( 1113857

xt +||y||t +Txt +Tyt

(57)

en for any x y isin X we have

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle

|x minus Tx|1 minus tt +|y minus Ty|1 minus tt2

(1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

12

xt +yt +Txt +Tyt( 1113857

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +Tyt( 1113857

(58)

From Cases b1 and b2 we see that (29) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 4 is verified

(d) We show that (6) is not satisfied for map T in(X d L R)

Finally we show that (6) is not satisfied for map T in(X d L R) If ρt(Tx Ty) ρt(x y) for some x y isin X andsome t isin (0 1] then by (6) we deduce that for any εgt 0

ρt(x y)leΛεαψ(ε) 1 +xt +yt +fxt +fyt1113858 1113859β

(59)

Letting εrarr 0 we have ψ(ε)rarr 0 which implies thatx y However for x0 6 and y0 4 we have Tx0 minus 5and Ty0 minus 3 and

ρt Tx0 Ty0( 1113857 2 middot1 minus t

t ρt x0 y0( 1113857 (60)

for all t isin (0 1] which is a contradiction

Remark 2 From Examples 1 and 2 we can see that e-orems 3 and 4 are independent to each other

Recently Jamshaid et al [19] investigated the fuzzy fixedpoints of fuzzy mappings via F-contractions On the basis oftheir work a natural question for multivalued mappings canbe raised as follows

Question 1 Is the multivalued case of eorem 3 true

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally and significantly in writingthis article All authors read and approved the finalmanuscript

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (11561049)

References

[1] O Kaleva and S Seikkala ldquoOn fuzzymetric spacesrdquo Fuzzy Setsand Systems vol 12 no 3 pp 215ndash229 1984

[2] O Hadzic and E Pap ldquoA fixed point theorem for multivaluedmappings in probaailistic metric spaces and an application infuzzy metric spacesrdquo Fuzzy Sets and Systems vol 127 no 3pp 333ndash444 2002

[3] N Lu F He and J Fan ldquoFixed point theorems of cyclic-contractive mapping in fuzzy metric spacesrdquo Mathematics inPractice and -eory vol 22 pp 237ndash248 2018

[4] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1pp 59ndash62 2014

[5] G Yun S Hwang and J Chang ldquoFuzzy lipschitz maps andfixed point theorems in fuzzy metric spacesrdquo Fuzzy Sets andSystems vol 161 no 8 pp 1117ndash1130 2010

[6] J-Z Xiao X-H Zhu and X Jin ldquoFixed point theorems fornonlinear contractions in Kaleva-Seikkalarsquos type fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 200 pp 65ndash83 2012

[7] S Phiangsungnoen P ounthong and P Kumam ldquoFixedpoint results in fuzzy metric spaces via α and βκ-admissiblemappings with application to integral typesrdquo Journal of In-telligent amp Fuzzy Systems vol 34 no 1 pp 467ndash475 2018

[8] G-J Yun J Chang and S Hwang ldquoFuzzy isometries andnon-existence of fuzzy contractive maps on fuzzy metricspacesrdquo International Journal of Fuzzy Systems vol 13 no 3pp 206ndash217 2011

8 Journal of Function Spaces

[9] V Gregori J-J Mintildeana and D Miravet ldquoContractive se-quences in fuzzy metric spacesrdquo Fuzzy Sets and Systemsvol 379 pp 125ndash133 2020

[10] S S Chauhan M Imdad G Kaur and A Sharma ldquoSomefixed point theorems for SF-contraction in complete fuzzymetric spacesrdquo Afrika Matematika vol 30 no 3-4pp 651ndash662 2019

[11] J Xiao and X Zhu ldquoOn linearly topological structure andproperty of fuzzy normed linear spacerdquo Fuzzy Sets and Sys-tems vol 125 no 2 pp 153ndash161 2002

[12] J-z Xiao and X-h Zhu ldquoTopological degree theory and fixedpoint theorems in fuzzy normed spacerdquo Fuzzy Sets andSystems vol 147 no 3 pp 437ndash452 2004

[13] V Pata ldquoA fixed point theorem in metric spacesrdquo Journal ofFixed Point -eory and Applications vol 10 no 2 pp 299ndash305 2011

[14] M Chakraborty and S K Samanta ldquoA fixed point theorem forKannan-type maps in metric spacesrdquo 2012 httpsarxivorgabs1211 7331v2

[15] Z Kadelburg and S Radenovic ldquoFixed point theorems underPata-type conditions in metric spacesrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 77ndash82 2016

[16] Z Kadelburg and S Radenovic ldquoFixed point and tripled fixedpoint theorems under pata-type conditions in ordered metricspacesrdquo International Journal of Analysis and Applicationsvol 6 pp 113ndash122 2014

[17] S Balasubramanian ldquoA pata-type fixed point theoremrdquoMathematical Sciences vol 8 no 3 pp 65ndash69 2014

[18] G K Jacob M S Khan C Park and S Jun ldquoOn generalizedpata type contractionsrdquoMathematics vol 6 no 2 p 25 2018

[19] J Ahmad H Aydi and N Mlaiki ldquoFuzzy fixed points of fuzzymappings via F-contractions and an applicationrdquo Journal ofIntelligent amp Fuzzy Systems vol 37 no 4 pp 5487ndash54932019

Journal of Function Spaces 9

ρt xn+1 xn( 1113857le ρt xn xnminus 1( 1113857le middot middot middot le ρt x1 x0( 1113857 (7)

Step 2 We prove that the sequence Cn(t)1113864 1113865 is bounded Forany t isin (0 1] by Lemma 3 there exists s s(t) isin (0 t] suchthat

ρt(x y)le ρs(x z) + ρt(z y) (8)

for all x y z isin X en from Step 1 we have

Cn(t) ρt xn x0( 1113857

le ρs xn xn+1( 1113857 + ρs x1 x0( 1113857 + ρt xn+1 x1( 1113857

le ρt xn+1 x1( 1113857 + 2ρs x1 x0( 1113857

le (1 minus ε)ρt xn x0( 1113857 + Λεαψ(ε) 1 + xn

t

+ x0

t1113960

+ xn+1

t+ x1

t

1113961α+2ρs x1 x0( 1113857

(1 minus ε)ρt xn x0( 1113857 + Λεαψ(ε) 1 + Cn(t) + Cn+1(t)1113858

+ C1(t)1113859α+2ρs x1 x0( 1113857

(9)

Since Cn+1(t)le ρs(xn+1 xn) + ρt(xn x0)le ρs(x1 x0) +

ρt(xn x0) we have

Cn(t)le (1 minus ε)Cn(t) + Λεαψ(ε) 1 + 2Cn(t) + C1(t)1113858

+ ρs x1 x0( 11138571113859α

+ 2ρs x1 x0( 1113857

le (1 minus ε)Cn(t) + Λεαψ(ε) 1 + 2Cn(t) + 2ρs x1 x0( 11138571113858 1113859α

+ 2ρs x1 x0( 1113857

(10)

Hence

εCn(t) minus 2ρs x1 x0( 1113857leΛεαψ(ε) 1 + 2Cn(t) + 2ρs x1 x0( 11138571113858 1113859α

(11)

Suppose that Cn(t)1113864 1113865 is unbounded en there existt0 isin (0 1] s0 isin (0 t0] and a subsequence Cnk

(t0)1113966 1113967 ofCn(t0)1113864 1113865 such that Cnk

(t0)⟶infin (k⟶infin) andCnk

(t0)ge 1 + 2ρs0(x0 x1) Let

ε εk 1 + 2ρs0

x1 x0( 1113857

Cnkt0( 1113857

(12)

en we get

1 + 2ρs0x1 x0( 1113857

Cnkt0( 1113857

Cnkt0( 1113857 minus 2ρs0

x1 x0( 1113857

leΛεαkψ εk( 1113857 1 + 2Cnkt0( 1113857 + 2ρs0

x1 x0( 11138571113960 1113961α

Λ 1 + 2ρs0x1 x0( 11138571113872 1113873

1 + 2Cnkt0( 1113857 + 2ρs0

x1 x0( 1113857

Cnkt0( 1113857

1113890 1113891

α

ψ εk( 1113857

(13)

which implies that

1leΛ 1 + 2ρs0x1 x0( 11138571113872 1113873

1 + 2Cnkt0( 1113857 + 2ρs0

x1 x0( 1113857

Cnkt0( 1113857

1113890 1113891

α

ψ εk( 1113857

(14)

Letting k⟶infin in (14) we have ψ(εk)⟶ 0 and

Λ 1 + 2ρs0x1 x0( 11138571113872 1113873

α 1 + C1 t0( 1113857 + 2ρs0x1 x0( 1113857

Cnkt0( 1113857

+ 21113890 1113891

α

ψ εk( 1113857⟶ 0

(15)

which contradict (14) us Cn(t)1113864 1113865 is bounded that is forany t isin (0 1] there exists a constant Mt gt 0 such thatxnt Cn(t)leMt

Step 3 We shall show that

limn⟶infin

ρt xn+1 xn( 1113857 0 (t isin (0 1]) (16)

Note that ρt(xn+1 xn)1113864 1113865 is a decreasing and boundedsequence So assume that limn⟶infinρt0

(xn+1 xn) a forsome agt 0 and t0 isin (0 1] By (6) we obtain that

ρt0xn+1 xn( 1113857 ρt0

fxn fxnminus 1( 1113857

le (1 minus ε)ρt0xn xnminus 1( 1113857 + Λεαψ(ε) 1 + xn

t0

1113876

+ xnminus 1

t0+ xn+1

t0

+ xn

t0

1113877α

le (1 minus ε)ρt0xn xnminus 1( 1113857 + Λεαψ(ε) 1 + 4Mt0

1113960 1113961α

(17)

Letting n⟶infin we have

aleΛεαminus 1ψ(ε) 1 + 4Mt01113960 1113961

α (18)

Note that

Λεαminus 1ψ(ε) 1 + 4Mt01113960 1113961

α⟶ 0 as ε⟶ 0 (19)

en we can see ale 0 which contradicts ourassumption

Step 4 We show that xn1113864 1113865 is a Cauchy sequence Supposenot choose δ gt 0 and t0 isin (0 1] and then there exist sub-sequences xnk

1113966 1113967 and xmk1113966 1113967 of xn1113864 1113865 with kltmk lt nk such that

ρt0xmk

xnk1113872 1113873gt δ and ρt0

xmk xnkminus 11113872 1113873le δ (20)

By Lemma 3 there exists s0 isin (0 t0] such that

δ lt ρt0xmk

xnk1113872 1113873le ρt0

xmk xnkminus 11113872 1113873 + ρs0

xnkminus 1 xnk1113872 1113873

le ρs0xnkminus 1 xnk

1113872 1113873 + δ(21)

Putting n⟶infin we get ρt0(xmk

xnk)⟶ δ Similarly

we can see that

Journal of Function Spaces 3

ρt0xnk+1 xmk+11113872 1113873le ρt0

xnk xmk

1113872 1113873 + ρs0xnk

xnk+11113872 1113873

+ ρs0xmk

xmk+11113872 1113873(22)

ρt0xnk+1 xmk+11113872 1113873ge ρt0

xnk xmk

1113872 1113873 minus ρs0xnk

xnk+11113872 1113873

minus ρs0xmk

xmk+11113872 1113873(23)

Passing to the limit as n⟶infin in (22) and (23) we haveρt0

(xnk+1 xmk+1)⟶ δ By (6) we get

ρt0xnk+1 xmk+11113872 1113873 ρt0

fxnk fxmk

1113872 1113873

le (1 minus ε)ρt0xnk

xmk1113872 1113873 + Λεαψ(ε) 1 + xnk

t01113876

+ xmk

t0+ xnk+1

t0+ xmk+1

t01113877α

le (1 minus ε)ρt0xnk

xmk1113872 1113873 + Λεαψ(ε) 1 + 4Mt0

1113960 1113961α

(24)

Taking n⟶infin in (2) we obtain that

δ le (1 minus ε)δ + Λεαψ(ε) 1 + 4Mt01113960 1113961

α

δ leΛεαminus 1ψ(ε) 1 + 4Mt01113960 1113961

α

(25)

Note that Λεαminus 1ψ(ε)[1 + 4Mt0]α⟶ 0 (ε⟶ 0) a

contradiction us xn1113864 1113865 is a Cauchy sequence Since X iscomplete there exists z isin X such that xn⟶ z (n⟶infin)

Step 5 We prove that z is the unique fixed point formapping f

First we show that fz z By (6) for all t isin (0 1] thereexists s isin (0 t] and we have

ρt(fz z)le ρs xn+1 z( 1113857 + ρt fz xn+1( 1113857

le (1 minus ε)ρt z xn( 1113857 + Λεαψ(ε) 1 +zt + xn

t

1113872

+fzt + xn+1

t1113873α+ρs xn+1 z( 1113857

le (1 minus ε)ρt z xn( 1113857 + Λεαψ(ε) 1 + 4Mt( 1113857α

+ ρs xn+1 z( 1113857

(26)

Letting n⟶infin we get ρt(fz z) 0 Hence fz zNext we prove the uniqueness of z Assume that zprime is

another fixed point for f For each t isin (0 1] there existss isin (0 t] such that

ρt z zprime( 1113857 ρt fz fzprime( 1113857

le (1 minus ε)ρt z zprime( 1113857 + Λϵαψ(ε) 1 +zt + zprime

t1113960

+fzt + fzprime

t1113961α

le (1 minus ε)ρt z zprime( 1113857 + Λϵαψ(ε) 1 + 4Mt1113858 1113859α

(27)

us we have ρt(z zprime)leΛεαminus 1ψ(ε)[1 + 4Mt]α⟶ 0 as

ε⟶ 0 which implies z zprime erefore f has a unique fixedpoint z isin X

Remark 1 From the proof of eorem 3 we can see that tokeep the sequence xn1113864 1113865 converge to the fixed point the rangeof ε in (6) can be limited from [0 1] to [0 c] for some givenconstant c isin (0 1]

By Remark 1 letting c λ we can deduce the followingcorollary which is the Banach contraction principle in fuzzymetric spaces

Corollary 1 (see Corollary 53 [6]) Let (X d L R) be acomplete fuzzy metric space with (R-2) and f X⟶ X be amapping If there exists λ isin [0 1) such that

ρt(fx fy)le λρt(x y) (28)

for all t isin (0 1] and x y isin X then f has a unique fixed pointz isin X

eorem 4 generalizes the result in [14] to fuzzy metricspaces

Theorem 4 Let (X d L R) be a complete fuzzy metric spacewith (R-2) Let Λge 0 and αge 1 be fixed constants Letf X⟶ X be a mapping such that

ρt(fx fy)le1 minus ε2

ρt(x fx) + ρt(y fy)1113858 1113859 + Λεαψ(ε) 1[

+xt +yt +fxt +fyt1113859α

(29)

for every ε isin [0 1] t isin (0 1] and all x y isin X -en f has aunique fixed point z isin X

Proof Starting from x0 construct a sequence xn1113864 1113865 such thatxn fxnminus 1 fnx0 For any t isin (0 1] we denote Cn(t)

xnt If xn0 xn0+1 for some n0 then xn0

is a fixed point of fus we always assume that xn nexn+1 for all n isin Nen theproof is divided into the following five steps

Step 1 We show that the sequence ρt(xn xn+1)1113864 1113865 is de-creasing Suppose that t isin (0 1] and ε 0 in (29) we obtain

ρt xn xn+1( 1113857le12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857 (30)

us we get

ρt xn+1 xn( 1113857le ρt xn xnminus 1( 1113857le middot middot middot le ρt x1 x0( 1113857 (31)

Step 2 e sequence Cn(t)1113864 1113865 is bounded For each t isin (0 1]there exists s s(t) isin (0 t] and

4 Journal of Function Spaces

Cn(t) ρt xn x0( 1113857

le ρs xn xn+1( 1113857 + ρs x1 x0( 1113857 + ρt xn+1 x1( 1113857

le ρt xn+1 x1( 1113857 + 2ρs x1 x0( 1113857

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt x1 x0( 1113857( 1113857

+ Λεαψ(ε) 1 + xn

t

+ x0

t+ xn+1

t

+ x1

t1113960 1113961

α

+ 2ρs x1 x0( 1113857

le (1 minus ε)ρt x1 x0( 1113857 + Λεαψ(ε) 1 + Cn(t)1113858

+ Cn+1(t) + C1(t)1113859α+2ρs x1 x0( 1113857

(32)

Letting ε 0 in (32) there exists a constant Mt gt 0 suchthat Cn(t)le ρt(x0 x1) Mt

Step 3 We shall prove that limn⟶infinρt(xn xn+1) 0 foreach t isin (0 1] For any t isin (0 1] we obtain

ρt xn xn+1( 1113857 ρt fxnminus 1 fxn( 1113857

le (1 minus ε)12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 + xnminus 1

t+ xn

t

+ xn

t

+ xn+1

t1113960 1113961

α

le (1 minus ε)12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 + 4Mt1113858 1113859α

(33)

Since ρt(xn xn+1)1113864 1113865 is a decreasing and bounded se-quence assume that limn⟶infinρt(xn xn+1) a for some agt 0Letting n⟶infin in (2) we have

ale (1 minus ε)12

middot 2a + Λεαψ(ε) 1 + 4Mt1113858 1113859α

le (1 minus ε)a + Λεαψ(ε) 1 + 4Mt1113858 1113859α

aleΛεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α

(34)

From

Λεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α⟶ 0 (ε⟶ 0) (35)

we see ale 0 a contradiction For each t isin (0 1] we getρt(xn xn+1)⟶ 0 (n⟶infin)

Step 4 We show that limn⟶infinρt(xn xm) 0 for eacht isin (0 1] and all n m isin N For any t isin (0 1] there existss s(t) isin (0 t] and we have

ρt xn xm( 1113857 ρt fxnminus 1 fxmminus 1( 1113857

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt xm xm+1( 1113857( 1113857

+ Λεαminus 1ψ(ε) 1 + xnminus 1

t+ xmminus 1

t

+ xn

t

1113960

+ xm

t

1113961α

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt xm xm+1( 1113857( 1113857

+ Λεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α

(36)

Putting n⟶infin and ε⟶ 0 in (36) we getρt(xn xm)⟶ 0

erefore xn1113864 1113865 is a Cauchy sequence Since X is com-plete there exists z isin X such that xn⟶ z (n⟶infin)

Step 5 We prove that z is the unique fixed point formapping f

First we show that fz z Using (29) for any t isin (0 1]there exists s isin (0 t] and we have

ρt(fz z)le ρs xn+1 z( 1113857 + ρt fz xn+1( 1113857

le (1 minus ε)12

ρt(z fz) + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 +zt + xn

t

+fzt + xn+1

t1113872 1113873

α

+ ρs xn+1 z( 1113857

le (1 minus ε)12

ρt(z fz) + ρt xn xn+1( 1113857( 1113857 + Λεαψ(ε)

1 + 4Mt( 1113857α

+ ρs xn+1 z( 1113857

(37)

By taking limits as n⟶infin and ε⟶ 0 in the inequalityabove we get ρt(fz z)le (12)ρt(fz z) Hence ρt(fz z)

0 which implies fz zFinally we prove the uniqueness of z Assume that zprime is

another fixed point of f For any t isin (0 1] there existss isin (0 t] and we getρt z zprime( 1113857 ρt fz fzprime( 1113857

le (1 minus ε)12

ρt(z fz) + ρt zprime fzprime( 1113857( 1113857 + Λϵαψ(ε)

middot 1 +zt + zprime

t+fzt + fzprime

t

1113960 1113961α

le (1 minus ε)12

ρt(z z) + ρt zprime zprime( 1113857( 1113857 + Λϵαψ(ε) 1 + 4Mt1113858 1113859α

leΛϵαψ(ε) 1 + 4Mt1113858 1113859α

(38)

Passing to the limit as ε⟶ 0 in (38) we haveρt(z zprime) 0 us z zprime erefore f has a unique fixedpoint z isin X

Journal of Function Spaces 5

Following the arguments in Remark 1 from eorem 4we can obtain the Corollary 2

Corollary 2 (see Corollary 53 [6]) Let (X d L R) be acomplete fuzzy metric space with (R-2) and f X⟶ X be amapping If there exists λ isin [0 (12)) such that

ρt(fx fy) le λ ρt(x fx) + ρt(y fy)1113858 1113859 (39)

for all t isin (0 1] and x y isin X then f has a unique fixed pointz isin X

Now we construct two examples to illustrateeorems 3and 4 respectively

Example 1 Let X (12n) n isin Ncup 0 cup 0 L(a b)

min a b and R(a b) max a b Define the fuzzy metricd X times X⟶ F+ by

d(x y) 0(q minus |x minus y|) 1 q |x minus y|

0 qne |x minus y|1113896 (40)

Let T X⟶ X be a map defined by Tx (12)x forx isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (6) is satisfied for x0 1Λ 1 α 1 and ψ(ε) 2ε(c) T has the unique fixed point x 0(d) (29) is not satisfied Soeorem 4 cannot be verified

by this example

Proof First it is easy to see that (X d L R) is a completefuzzy metric space and R(a b) max a b satisfies (R-2)Note that Tx x implies that x 0 isin X then conclusion(c) is true Next we prove conclusions (b) and (d)respectively

(b) Letting x0 1Λ 1 α 1 β 1 and ψ(ε) 2ε weshow that (6) holds for all ε isin [0 1] t isin (0 1] andx y isin X

For any t isin (0 1] we can see that ρt(x y) |x minus y|en we consider the following two cases

Case 1 If εle (12) then 1 minus εge (12) For any x y isin X wehave

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386812

|x minus y|le (1 minus ε)ρt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+ Txt +Tyt1113857

(41)

Case 2 If εgt (12) then ψ(ε)gt 1 Note that

ρt(x y)le ρt x x0( 1113857 + ρ0 x0 y( 1113857 xt +yt (42)

en for any x y isin X we have

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le |x minus y| (1 minus ε)ρt(x y) + ερt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) xt +yt( 1113857

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+Txt +Tyt1113857

(43)

From Cases 1 and 2 we see that (6) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 3 is verified

(d) We show that (29) is not satisfied

Letting ε 0 by (29) we deduce that

ρt(Tx Ty)leρt(x Tx) + ρt(y Ty)

2 (44)

However let x0 0 and y0 1 For any t isin (0 1] wehave

ρt Tx0 Ty0( 1113857 0 minus12

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868gt12

middot12

ρt x0 Tx0( 1113857 + ρt y0 Ty0( 1113857

2

(45)

So this example cannot verify eorem 4 and we proveconclusion (d)

Example 2 Let X 6 minus 5 4 minus 3 2 minus 1 0 L(a b) mina b and R(a b) max a b Let d X times X⟶ F+ be afuzzy metric such that

d(x y)(q)

0 qlt 0

1 q 0

|x minus y|

|x minus y| + q qgt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)

Let T X⟶ X be a map such that

Tx

minus |x| + 1 xgt 0

|x| minus 1 xlt 0

0 x 0

⎧⎪⎪⎨

⎪⎪⎩(47)

for x isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (29) is satisfied for x0 1 Λ 1 α 1 and

ψ(ε) 6ε(c) T has the unique fixed point x 0(d) (6) is not satisfied So this example cannot verify

eorem 3

Proof Clearly we can see that x 0 is the unique fixedpoint for map T and conclusion (c) is true Now we proveconclusions (a) (b) and (d) respectively

6 Journal of Function Spaces

(a) We show that (X d L R) is a complete fuzzy metricspace with (R-2)

Clearly for any x y isin X d(x y)(q) is a fuzzy numberand (D1) and (D2) in Definition 2 are satisfied Note thatR(a b) max a b satisfies (R-2) So it is sufficient to provethat (D3) holds

For any x y z isin X it is obvious that λ1(x z)

λ1(z y) λ1(x y) 0 To see (D3L) let us consider thefollowing two cases

(i) If plt 0 or qlt 0 then d(x z)(p) 0 ord(z y)(q) 0 leading to that

min(d(x z)(p) d(z y)(q)) 0 (48)

So (D3L) is satisfied

(ii) If p 0 and q 0 then p + q 0 Sod(x z)(p) d(z y)(q) d(x y)(p + q) 1 lead-ing to that

d(x y)(p + q) min d(x z)(p) d(z y)(q)1113864 1113865 (49)

us (D3L) holds To see (D3R) we consider the fol-lowing two cases

Case a1 Suppose that there are at least two points inx y z1113864 1113865 which are equal For any pge 0 and qge 0

(i) If z x or z y without loss of generality let z xthen d(z y) d(x y) Since d(x y)(q) is decreasingas qge 0 we have

max d(x z)(p) d(z y)(q)1113864 1113865ged(z y)(q)

d(x y)(q)ge d(x y)(p + q)(50)

(ii) If x y then we have

p + q 0rArrp 0 q 0rArr d(x y)(p + q)

1 max d(x z)(p) d(z y)(q)1113864 1113865

p + qgt 0rArr d(x y)(p + q) 0

lemax d(x z)(p) d(z y)(q)1113864 1113865

(51)

Case a2 Suppose that xney xne z and yne z For anypge 0 and qge 0 denote s d(x z)(p) andt d(z y)(q) Without loss of generality let 0lt sle ten we have

p |x minus z|1 minus s

s

q |z minus y|1 minus t

t

(52)

Since d(x y)(q) is decreasing as qge 0 and (1 minus tt) isdecreasing we deduce that

max d(x z)(p) d(z y)(q)1113864 1113865 t d(x y) |x minus y|1 minus t

t1113874 1113875

ged(x y) (|x minus z| +|z minus y|)1 minus t

t1113876 1113877

ged(x y) |x minus z|1 minus s

s+|z minus y|

1 minus t

t1113876 1113877

d(x y)(p + q)

(53)

From Cases a1 and a2 we conclude that (D3R) holds forall x y z isin X

Note that X is a finite set en (X d L R) is completeand the proof of conclusion (a) is completed

(b) We prove (29) for x0 1 Λ 1 α 1 β 1 andψ(ε) 6ε

For any t isin (0 1] we can see that ρt(x y)

|x minus y|(1 minus tt) en we consider the following two cases

Case b1 If εle (112) it is easy to see that for any xne 0we have |Tx| |x| minus 1 and x middot Txle 0 So we can obtainthat |x minus Tx| |x| + |Tx| and

|Tx minus Ty|le |Tx| +|Ty| |Tx| +|x| minus 1

2+

|Ty| +|y| minus 12

|x minus Tx| +|y minus Ty|

2minus 1

(54)

for all x yne 0 If y 0 then we have

|Tx minus Ty| |Tx| |Tx| +|x| minus 1

2

|x minus Tx| +|y minus Ty|

2minus12

(55)

From the above two equalities by calculation we obtainthat for any x y isin X

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle1011

middot|x minus Tx|(1 minus tt) +|y minus Ty|(1 minus tt)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +||Ty||t( 1113857

(56)

Journal of Function Spaces 7

Case b2 If εgt (112) then we have ψ(ε)gt (12) Notethat

ρt(x Tx) + ρt(y Ty)le ρt x x0( 1113857 + ρt x0 Tx( 1113857

+ ρt y x0( 1113857 + ρt x0 Ty( 1113857

xt +||y||t +Txt +Tyt

(57)

en for any x y isin X we have

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle

|x minus Tx|1 minus tt +|y minus Ty|1 minus tt2

(1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

12

xt +yt +Txt +Tyt( 1113857

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +Tyt( 1113857

(58)

From Cases b1 and b2 we see that (29) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 4 is verified

(d) We show that (6) is not satisfied for map T in(X d L R)

Finally we show that (6) is not satisfied for map T in(X d L R) If ρt(Tx Ty) ρt(x y) for some x y isin X andsome t isin (0 1] then by (6) we deduce that for any εgt 0

ρt(x y)leΛεαψ(ε) 1 +xt +yt +fxt +fyt1113858 1113859β

(59)

Letting εrarr 0 we have ψ(ε)rarr 0 which implies thatx y However for x0 6 and y0 4 we have Tx0 minus 5and Ty0 minus 3 and

ρt Tx0 Ty0( 1113857 2 middot1 minus t

t ρt x0 y0( 1113857 (60)

for all t isin (0 1] which is a contradiction

Remark 2 From Examples 1 and 2 we can see that e-orems 3 and 4 are independent to each other

Recently Jamshaid et al [19] investigated the fuzzy fixedpoints of fuzzy mappings via F-contractions On the basis oftheir work a natural question for multivalued mappings canbe raised as follows

Question 1 Is the multivalued case of eorem 3 true

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally and significantly in writingthis article All authors read and approved the finalmanuscript

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (11561049)

References

[1] O Kaleva and S Seikkala ldquoOn fuzzymetric spacesrdquo Fuzzy Setsand Systems vol 12 no 3 pp 215ndash229 1984

[2] O Hadzic and E Pap ldquoA fixed point theorem for multivaluedmappings in probaailistic metric spaces and an application infuzzy metric spacesrdquo Fuzzy Sets and Systems vol 127 no 3pp 333ndash444 2002

[3] N Lu F He and J Fan ldquoFixed point theorems of cyclic-contractive mapping in fuzzy metric spacesrdquo Mathematics inPractice and -eory vol 22 pp 237ndash248 2018

[4] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1pp 59ndash62 2014

[5] G Yun S Hwang and J Chang ldquoFuzzy lipschitz maps andfixed point theorems in fuzzy metric spacesrdquo Fuzzy Sets andSystems vol 161 no 8 pp 1117ndash1130 2010

[6] J-Z Xiao X-H Zhu and X Jin ldquoFixed point theorems fornonlinear contractions in Kaleva-Seikkalarsquos type fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 200 pp 65ndash83 2012

[7] S Phiangsungnoen P ounthong and P Kumam ldquoFixedpoint results in fuzzy metric spaces via α and βκ-admissiblemappings with application to integral typesrdquo Journal of In-telligent amp Fuzzy Systems vol 34 no 1 pp 467ndash475 2018

[8] G-J Yun J Chang and S Hwang ldquoFuzzy isometries andnon-existence of fuzzy contractive maps on fuzzy metricspacesrdquo International Journal of Fuzzy Systems vol 13 no 3pp 206ndash217 2011

8 Journal of Function Spaces

[9] V Gregori J-J Mintildeana and D Miravet ldquoContractive se-quences in fuzzy metric spacesrdquo Fuzzy Sets and Systemsvol 379 pp 125ndash133 2020

[10] S S Chauhan M Imdad G Kaur and A Sharma ldquoSomefixed point theorems for SF-contraction in complete fuzzymetric spacesrdquo Afrika Matematika vol 30 no 3-4pp 651ndash662 2019

[11] J Xiao and X Zhu ldquoOn linearly topological structure andproperty of fuzzy normed linear spacerdquo Fuzzy Sets and Sys-tems vol 125 no 2 pp 153ndash161 2002

[12] J-z Xiao and X-h Zhu ldquoTopological degree theory and fixedpoint theorems in fuzzy normed spacerdquo Fuzzy Sets andSystems vol 147 no 3 pp 437ndash452 2004

[13] V Pata ldquoA fixed point theorem in metric spacesrdquo Journal ofFixed Point -eory and Applications vol 10 no 2 pp 299ndash305 2011

[14] M Chakraborty and S K Samanta ldquoA fixed point theorem forKannan-type maps in metric spacesrdquo 2012 httpsarxivorgabs1211 7331v2

[15] Z Kadelburg and S Radenovic ldquoFixed point theorems underPata-type conditions in metric spacesrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 77ndash82 2016

[16] Z Kadelburg and S Radenovic ldquoFixed point and tripled fixedpoint theorems under pata-type conditions in ordered metricspacesrdquo International Journal of Analysis and Applicationsvol 6 pp 113ndash122 2014

[17] S Balasubramanian ldquoA pata-type fixed point theoremrdquoMathematical Sciences vol 8 no 3 pp 65ndash69 2014

[18] G K Jacob M S Khan C Park and S Jun ldquoOn generalizedpata type contractionsrdquoMathematics vol 6 no 2 p 25 2018

[19] J Ahmad H Aydi and N Mlaiki ldquoFuzzy fixed points of fuzzymappings via F-contractions and an applicationrdquo Journal ofIntelligent amp Fuzzy Systems vol 37 no 4 pp 5487ndash54932019

Journal of Function Spaces 9

ρt0xnk+1 xmk+11113872 1113873le ρt0

xnk xmk

1113872 1113873 + ρs0xnk

xnk+11113872 1113873

+ ρs0xmk

xmk+11113872 1113873(22)

ρt0xnk+1 xmk+11113872 1113873ge ρt0

xnk xmk

1113872 1113873 minus ρs0xnk

xnk+11113872 1113873

minus ρs0xmk

xmk+11113872 1113873(23)

Passing to the limit as n⟶infin in (22) and (23) we haveρt0

(xnk+1 xmk+1)⟶ δ By (6) we get

ρt0xnk+1 xmk+11113872 1113873 ρt0

fxnk fxmk

1113872 1113873

le (1 minus ε)ρt0xnk

xmk1113872 1113873 + Λεαψ(ε) 1 + xnk

t01113876

+ xmk

t0+ xnk+1

t0+ xmk+1

t01113877α

le (1 minus ε)ρt0xnk

xmk1113872 1113873 + Λεαψ(ε) 1 + 4Mt0

1113960 1113961α

(24)

Taking n⟶infin in (2) we obtain that

δ le (1 minus ε)δ + Λεαψ(ε) 1 + 4Mt01113960 1113961

α

δ leΛεαminus 1ψ(ε) 1 + 4Mt01113960 1113961

α

(25)

Note that Λεαminus 1ψ(ε)[1 + 4Mt0]α⟶ 0 (ε⟶ 0) a

contradiction us xn1113864 1113865 is a Cauchy sequence Since X iscomplete there exists z isin X such that xn⟶ z (n⟶infin)

Step 5 We prove that z is the unique fixed point formapping f

First we show that fz z By (6) for all t isin (0 1] thereexists s isin (0 t] and we have

ρt(fz z)le ρs xn+1 z( 1113857 + ρt fz xn+1( 1113857

le (1 minus ε)ρt z xn( 1113857 + Λεαψ(ε) 1 +zt + xn

t

1113872

+fzt + xn+1

t1113873α+ρs xn+1 z( 1113857

le (1 minus ε)ρt z xn( 1113857 + Λεαψ(ε) 1 + 4Mt( 1113857α

+ ρs xn+1 z( 1113857

(26)

Letting n⟶infin we get ρt(fz z) 0 Hence fz zNext we prove the uniqueness of z Assume that zprime is

another fixed point for f For each t isin (0 1] there existss isin (0 t] such that

ρt z zprime( 1113857 ρt fz fzprime( 1113857

le (1 minus ε)ρt z zprime( 1113857 + Λϵαψ(ε) 1 +zt + zprime

t1113960

+fzt + fzprime

t1113961α

le (1 minus ε)ρt z zprime( 1113857 + Λϵαψ(ε) 1 + 4Mt1113858 1113859α

(27)

us we have ρt(z zprime)leΛεαminus 1ψ(ε)[1 + 4Mt]α⟶ 0 as

ε⟶ 0 which implies z zprime erefore f has a unique fixedpoint z isin X

Remark 1 From the proof of eorem 3 we can see that tokeep the sequence xn1113864 1113865 converge to the fixed point the rangeof ε in (6) can be limited from [0 1] to [0 c] for some givenconstant c isin (0 1]

By Remark 1 letting c λ we can deduce the followingcorollary which is the Banach contraction principle in fuzzymetric spaces

Corollary 1 (see Corollary 53 [6]) Let (X d L R) be acomplete fuzzy metric space with (R-2) and f X⟶ X be amapping If there exists λ isin [0 1) such that

ρt(fx fy)le λρt(x y) (28)

for all t isin (0 1] and x y isin X then f has a unique fixed pointz isin X

eorem 4 generalizes the result in [14] to fuzzy metricspaces

Theorem 4 Let (X d L R) be a complete fuzzy metric spacewith (R-2) Let Λge 0 and αge 1 be fixed constants Letf X⟶ X be a mapping such that

ρt(fx fy)le1 minus ε2

ρt(x fx) + ρt(y fy)1113858 1113859 + Λεαψ(ε) 1[

+xt +yt +fxt +fyt1113859α

(29)

for every ε isin [0 1] t isin (0 1] and all x y isin X -en f has aunique fixed point z isin X

Proof Starting from x0 construct a sequence xn1113864 1113865 such thatxn fxnminus 1 fnx0 For any t isin (0 1] we denote Cn(t)

xnt If xn0 xn0+1 for some n0 then xn0

is a fixed point of fus we always assume that xn nexn+1 for all n isin Nen theproof is divided into the following five steps

Step 1 We show that the sequence ρt(xn xn+1)1113864 1113865 is de-creasing Suppose that t isin (0 1] and ε 0 in (29) we obtain

ρt xn xn+1( 1113857le12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857 (30)

us we get

ρt xn+1 xn( 1113857le ρt xn xnminus 1( 1113857le middot middot middot le ρt x1 x0( 1113857 (31)

Step 2 e sequence Cn(t)1113864 1113865 is bounded For each t isin (0 1]there exists s s(t) isin (0 t] and

4 Journal of Function Spaces

Cn(t) ρt xn x0( 1113857

le ρs xn xn+1( 1113857 + ρs x1 x0( 1113857 + ρt xn+1 x1( 1113857

le ρt xn+1 x1( 1113857 + 2ρs x1 x0( 1113857

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt x1 x0( 1113857( 1113857

+ Λεαψ(ε) 1 + xn

t

+ x0

t+ xn+1

t

+ x1

t1113960 1113961

α

+ 2ρs x1 x0( 1113857

le (1 minus ε)ρt x1 x0( 1113857 + Λεαψ(ε) 1 + Cn(t)1113858

+ Cn+1(t) + C1(t)1113859α+2ρs x1 x0( 1113857

(32)

Letting ε 0 in (32) there exists a constant Mt gt 0 suchthat Cn(t)le ρt(x0 x1) Mt

Step 3 We shall prove that limn⟶infinρt(xn xn+1) 0 foreach t isin (0 1] For any t isin (0 1] we obtain

ρt xn xn+1( 1113857 ρt fxnminus 1 fxn( 1113857

le (1 minus ε)12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 + xnminus 1

t+ xn

t

+ xn

t

+ xn+1

t1113960 1113961

α

le (1 minus ε)12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 + 4Mt1113858 1113859α

(33)

Since ρt(xn xn+1)1113864 1113865 is a decreasing and bounded se-quence assume that limn⟶infinρt(xn xn+1) a for some agt 0Letting n⟶infin in (2) we have

ale (1 minus ε)12

middot 2a + Λεαψ(ε) 1 + 4Mt1113858 1113859α

le (1 minus ε)a + Λεαψ(ε) 1 + 4Mt1113858 1113859α

aleΛεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α

(34)

From

Λεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α⟶ 0 (ε⟶ 0) (35)

we see ale 0 a contradiction For each t isin (0 1] we getρt(xn xn+1)⟶ 0 (n⟶infin)

Step 4 We show that limn⟶infinρt(xn xm) 0 for eacht isin (0 1] and all n m isin N For any t isin (0 1] there existss s(t) isin (0 t] and we have

ρt xn xm( 1113857 ρt fxnminus 1 fxmminus 1( 1113857

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt xm xm+1( 1113857( 1113857

+ Λεαminus 1ψ(ε) 1 + xnminus 1

t+ xmminus 1

t

+ xn

t

1113960

+ xm

t

1113961α

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt xm xm+1( 1113857( 1113857

+ Λεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α

(36)

Putting n⟶infin and ε⟶ 0 in (36) we getρt(xn xm)⟶ 0

erefore xn1113864 1113865 is a Cauchy sequence Since X is com-plete there exists z isin X such that xn⟶ z (n⟶infin)

Step 5 We prove that z is the unique fixed point formapping f

First we show that fz z Using (29) for any t isin (0 1]there exists s isin (0 t] and we have

ρt(fz z)le ρs xn+1 z( 1113857 + ρt fz xn+1( 1113857

le (1 minus ε)12

ρt(z fz) + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 +zt + xn

t

+fzt + xn+1

t1113872 1113873

α

+ ρs xn+1 z( 1113857

le (1 minus ε)12

ρt(z fz) + ρt xn xn+1( 1113857( 1113857 + Λεαψ(ε)

1 + 4Mt( 1113857α

+ ρs xn+1 z( 1113857

(37)

By taking limits as n⟶infin and ε⟶ 0 in the inequalityabove we get ρt(fz z)le (12)ρt(fz z) Hence ρt(fz z)

0 which implies fz zFinally we prove the uniqueness of z Assume that zprime is

another fixed point of f For any t isin (0 1] there existss isin (0 t] and we getρt z zprime( 1113857 ρt fz fzprime( 1113857

le (1 minus ε)12

ρt(z fz) + ρt zprime fzprime( 1113857( 1113857 + Λϵαψ(ε)

middot 1 +zt + zprime

t+fzt + fzprime

t

1113960 1113961α

le (1 minus ε)12

ρt(z z) + ρt zprime zprime( 1113857( 1113857 + Λϵαψ(ε) 1 + 4Mt1113858 1113859α

leΛϵαψ(ε) 1 + 4Mt1113858 1113859α

(38)

Passing to the limit as ε⟶ 0 in (38) we haveρt(z zprime) 0 us z zprime erefore f has a unique fixedpoint z isin X

Journal of Function Spaces 5

Following the arguments in Remark 1 from eorem 4we can obtain the Corollary 2

Corollary 2 (see Corollary 53 [6]) Let (X d L R) be acomplete fuzzy metric space with (R-2) and f X⟶ X be amapping If there exists λ isin [0 (12)) such that

ρt(fx fy) le λ ρt(x fx) + ρt(y fy)1113858 1113859 (39)

for all t isin (0 1] and x y isin X then f has a unique fixed pointz isin X

Now we construct two examples to illustrateeorems 3and 4 respectively

Example 1 Let X (12n) n isin Ncup 0 cup 0 L(a b)

min a b and R(a b) max a b Define the fuzzy metricd X times X⟶ F+ by

d(x y) 0(q minus |x minus y|) 1 q |x minus y|

0 qne |x minus y|1113896 (40)

Let T X⟶ X be a map defined by Tx (12)x forx isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (6) is satisfied for x0 1Λ 1 α 1 and ψ(ε) 2ε(c) T has the unique fixed point x 0(d) (29) is not satisfied Soeorem 4 cannot be verified

by this example

Proof First it is easy to see that (X d L R) is a completefuzzy metric space and R(a b) max a b satisfies (R-2)Note that Tx x implies that x 0 isin X then conclusion(c) is true Next we prove conclusions (b) and (d)respectively

(b) Letting x0 1Λ 1 α 1 β 1 and ψ(ε) 2ε weshow that (6) holds for all ε isin [0 1] t isin (0 1] andx y isin X

For any t isin (0 1] we can see that ρt(x y) |x minus y|en we consider the following two cases

Case 1 If εle (12) then 1 minus εge (12) For any x y isin X wehave

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386812

|x minus y|le (1 minus ε)ρt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+ Txt +Tyt1113857

(41)

Case 2 If εgt (12) then ψ(ε)gt 1 Note that

ρt(x y)le ρt x x0( 1113857 + ρ0 x0 y( 1113857 xt +yt (42)

en for any x y isin X we have

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le |x minus y| (1 minus ε)ρt(x y) + ερt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) xt +yt( 1113857

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+Txt +Tyt1113857

(43)

From Cases 1 and 2 we see that (6) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 3 is verified

(d) We show that (29) is not satisfied

Letting ε 0 by (29) we deduce that

ρt(Tx Ty)leρt(x Tx) + ρt(y Ty)

2 (44)

However let x0 0 and y0 1 For any t isin (0 1] wehave

ρt Tx0 Ty0( 1113857 0 minus12

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868gt12

middot12

ρt x0 Tx0( 1113857 + ρt y0 Ty0( 1113857

2

(45)

So this example cannot verify eorem 4 and we proveconclusion (d)

Example 2 Let X 6 minus 5 4 minus 3 2 minus 1 0 L(a b) mina b and R(a b) max a b Let d X times X⟶ F+ be afuzzy metric such that

d(x y)(q)

0 qlt 0

1 q 0

|x minus y|

|x minus y| + q qgt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)

Let T X⟶ X be a map such that

Tx

minus |x| + 1 xgt 0

|x| minus 1 xlt 0

0 x 0

⎧⎪⎪⎨

⎪⎪⎩(47)

for x isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (29) is satisfied for x0 1 Λ 1 α 1 and

ψ(ε) 6ε(c) T has the unique fixed point x 0(d) (6) is not satisfied So this example cannot verify

eorem 3

Proof Clearly we can see that x 0 is the unique fixedpoint for map T and conclusion (c) is true Now we proveconclusions (a) (b) and (d) respectively

6 Journal of Function Spaces

(a) We show that (X d L R) is a complete fuzzy metricspace with (R-2)

Clearly for any x y isin X d(x y)(q) is a fuzzy numberand (D1) and (D2) in Definition 2 are satisfied Note thatR(a b) max a b satisfies (R-2) So it is sufficient to provethat (D3) holds

For any x y z isin X it is obvious that λ1(x z)

λ1(z y) λ1(x y) 0 To see (D3L) let us consider thefollowing two cases

(i) If plt 0 or qlt 0 then d(x z)(p) 0 ord(z y)(q) 0 leading to that

min(d(x z)(p) d(z y)(q)) 0 (48)

So (D3L) is satisfied

(ii) If p 0 and q 0 then p + q 0 Sod(x z)(p) d(z y)(q) d(x y)(p + q) 1 lead-ing to that

d(x y)(p + q) min d(x z)(p) d(z y)(q)1113864 1113865 (49)

us (D3L) holds To see (D3R) we consider the fol-lowing two cases

Case a1 Suppose that there are at least two points inx y z1113864 1113865 which are equal For any pge 0 and qge 0

(i) If z x or z y without loss of generality let z xthen d(z y) d(x y) Since d(x y)(q) is decreasingas qge 0 we have

max d(x z)(p) d(z y)(q)1113864 1113865ged(z y)(q)

d(x y)(q)ge d(x y)(p + q)(50)

(ii) If x y then we have

p + q 0rArrp 0 q 0rArr d(x y)(p + q)

1 max d(x z)(p) d(z y)(q)1113864 1113865

p + qgt 0rArr d(x y)(p + q) 0

lemax d(x z)(p) d(z y)(q)1113864 1113865

(51)

Case a2 Suppose that xney xne z and yne z For anypge 0 and qge 0 denote s d(x z)(p) andt d(z y)(q) Without loss of generality let 0lt sle ten we have

p |x minus z|1 minus s

s

q |z minus y|1 minus t

t

(52)

Since d(x y)(q) is decreasing as qge 0 and (1 minus tt) isdecreasing we deduce that

max d(x z)(p) d(z y)(q)1113864 1113865 t d(x y) |x minus y|1 minus t

t1113874 1113875

ged(x y) (|x minus z| +|z minus y|)1 minus t

t1113876 1113877

ged(x y) |x minus z|1 minus s

s+|z minus y|

1 minus t

t1113876 1113877

d(x y)(p + q)

(53)

From Cases a1 and a2 we conclude that (D3R) holds forall x y z isin X

Note that X is a finite set en (X d L R) is completeand the proof of conclusion (a) is completed

(b) We prove (29) for x0 1 Λ 1 α 1 β 1 andψ(ε) 6ε

For any t isin (0 1] we can see that ρt(x y)

|x minus y|(1 minus tt) en we consider the following two cases

Case b1 If εle (112) it is easy to see that for any xne 0we have |Tx| |x| minus 1 and x middot Txle 0 So we can obtainthat |x minus Tx| |x| + |Tx| and

|Tx minus Ty|le |Tx| +|Ty| |Tx| +|x| minus 1

2+

|Ty| +|y| minus 12

|x minus Tx| +|y minus Ty|

2minus 1

(54)

for all x yne 0 If y 0 then we have

|Tx minus Ty| |Tx| |Tx| +|x| minus 1

2

|x minus Tx| +|y minus Ty|

2minus12

(55)

From the above two equalities by calculation we obtainthat for any x y isin X

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle1011

middot|x minus Tx|(1 minus tt) +|y minus Ty|(1 minus tt)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +||Ty||t( 1113857

(56)

Journal of Function Spaces 7

Case b2 If εgt (112) then we have ψ(ε)gt (12) Notethat

ρt(x Tx) + ρt(y Ty)le ρt x x0( 1113857 + ρt x0 Tx( 1113857

+ ρt y x0( 1113857 + ρt x0 Ty( 1113857

xt +||y||t +Txt +Tyt

(57)

en for any x y isin X we have

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle

|x minus Tx|1 minus tt +|y minus Ty|1 minus tt2

(1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

12

xt +yt +Txt +Tyt( 1113857

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +Tyt( 1113857

(58)

From Cases b1 and b2 we see that (29) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 4 is verified

(d) We show that (6) is not satisfied for map T in(X d L R)

Finally we show that (6) is not satisfied for map T in(X d L R) If ρt(Tx Ty) ρt(x y) for some x y isin X andsome t isin (0 1] then by (6) we deduce that for any εgt 0

ρt(x y)leΛεαψ(ε) 1 +xt +yt +fxt +fyt1113858 1113859β

(59)

Letting εrarr 0 we have ψ(ε)rarr 0 which implies thatx y However for x0 6 and y0 4 we have Tx0 minus 5and Ty0 minus 3 and

ρt Tx0 Ty0( 1113857 2 middot1 minus t

t ρt x0 y0( 1113857 (60)

for all t isin (0 1] which is a contradiction

Remark 2 From Examples 1 and 2 we can see that e-orems 3 and 4 are independent to each other

Recently Jamshaid et al [19] investigated the fuzzy fixedpoints of fuzzy mappings via F-contractions On the basis oftheir work a natural question for multivalued mappings canbe raised as follows

Question 1 Is the multivalued case of eorem 3 true

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally and significantly in writingthis article All authors read and approved the finalmanuscript

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (11561049)

References

[1] O Kaleva and S Seikkala ldquoOn fuzzymetric spacesrdquo Fuzzy Setsand Systems vol 12 no 3 pp 215ndash229 1984

[2] O Hadzic and E Pap ldquoA fixed point theorem for multivaluedmappings in probaailistic metric spaces and an application infuzzy metric spacesrdquo Fuzzy Sets and Systems vol 127 no 3pp 333ndash444 2002

[3] N Lu F He and J Fan ldquoFixed point theorems of cyclic-contractive mapping in fuzzy metric spacesrdquo Mathematics inPractice and -eory vol 22 pp 237ndash248 2018

[4] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1pp 59ndash62 2014

[5] G Yun S Hwang and J Chang ldquoFuzzy lipschitz maps andfixed point theorems in fuzzy metric spacesrdquo Fuzzy Sets andSystems vol 161 no 8 pp 1117ndash1130 2010

[6] J-Z Xiao X-H Zhu and X Jin ldquoFixed point theorems fornonlinear contractions in Kaleva-Seikkalarsquos type fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 200 pp 65ndash83 2012

[7] S Phiangsungnoen P ounthong and P Kumam ldquoFixedpoint results in fuzzy metric spaces via α and βκ-admissiblemappings with application to integral typesrdquo Journal of In-telligent amp Fuzzy Systems vol 34 no 1 pp 467ndash475 2018

[8] G-J Yun J Chang and S Hwang ldquoFuzzy isometries andnon-existence of fuzzy contractive maps on fuzzy metricspacesrdquo International Journal of Fuzzy Systems vol 13 no 3pp 206ndash217 2011

8 Journal of Function Spaces

[9] V Gregori J-J Mintildeana and D Miravet ldquoContractive se-quences in fuzzy metric spacesrdquo Fuzzy Sets and Systemsvol 379 pp 125ndash133 2020

[10] S S Chauhan M Imdad G Kaur and A Sharma ldquoSomefixed point theorems for SF-contraction in complete fuzzymetric spacesrdquo Afrika Matematika vol 30 no 3-4pp 651ndash662 2019

[11] J Xiao and X Zhu ldquoOn linearly topological structure andproperty of fuzzy normed linear spacerdquo Fuzzy Sets and Sys-tems vol 125 no 2 pp 153ndash161 2002

[12] J-z Xiao and X-h Zhu ldquoTopological degree theory and fixedpoint theorems in fuzzy normed spacerdquo Fuzzy Sets andSystems vol 147 no 3 pp 437ndash452 2004

[13] V Pata ldquoA fixed point theorem in metric spacesrdquo Journal ofFixed Point -eory and Applications vol 10 no 2 pp 299ndash305 2011

[14] M Chakraborty and S K Samanta ldquoA fixed point theorem forKannan-type maps in metric spacesrdquo 2012 httpsarxivorgabs1211 7331v2

[15] Z Kadelburg and S Radenovic ldquoFixed point theorems underPata-type conditions in metric spacesrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 77ndash82 2016

[16] Z Kadelburg and S Radenovic ldquoFixed point and tripled fixedpoint theorems under pata-type conditions in ordered metricspacesrdquo International Journal of Analysis and Applicationsvol 6 pp 113ndash122 2014

[17] S Balasubramanian ldquoA pata-type fixed point theoremrdquoMathematical Sciences vol 8 no 3 pp 65ndash69 2014

[18] G K Jacob M S Khan C Park and S Jun ldquoOn generalizedpata type contractionsrdquoMathematics vol 6 no 2 p 25 2018

[19] J Ahmad H Aydi and N Mlaiki ldquoFuzzy fixed points of fuzzymappings via F-contractions and an applicationrdquo Journal ofIntelligent amp Fuzzy Systems vol 37 no 4 pp 5487ndash54932019

Journal of Function Spaces 9

Cn(t) ρt xn x0( 1113857

le ρs xn xn+1( 1113857 + ρs x1 x0( 1113857 + ρt xn+1 x1( 1113857

le ρt xn+1 x1( 1113857 + 2ρs x1 x0( 1113857

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt x1 x0( 1113857( 1113857

+ Λεαψ(ε) 1 + xn

t

+ x0

t+ xn+1

t

+ x1

t1113960 1113961

α

+ 2ρs x1 x0( 1113857

le (1 minus ε)ρt x1 x0( 1113857 + Λεαψ(ε) 1 + Cn(t)1113858

+ Cn+1(t) + C1(t)1113859α+2ρs x1 x0( 1113857

(32)

Letting ε 0 in (32) there exists a constant Mt gt 0 suchthat Cn(t)le ρt(x0 x1) Mt

Step 3 We shall prove that limn⟶infinρt(xn xn+1) 0 foreach t isin (0 1] For any t isin (0 1] we obtain

ρt xn xn+1( 1113857 ρt fxnminus 1 fxn( 1113857

le (1 minus ε)12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 + xnminus 1

t+ xn

t

+ xn

t

+ xn+1

t1113960 1113961

α

le (1 minus ε)12

ρt xnminus 1 xn( 1113857 + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 + 4Mt1113858 1113859α

(33)

Since ρt(xn xn+1)1113864 1113865 is a decreasing and bounded se-quence assume that limn⟶infinρt(xn xn+1) a for some agt 0Letting n⟶infin in (2) we have

ale (1 minus ε)12

middot 2a + Λεαψ(ε) 1 + 4Mt1113858 1113859α

le (1 minus ε)a + Λεαψ(ε) 1 + 4Mt1113858 1113859α

aleΛεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α

(34)

From

Λεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α⟶ 0 (ε⟶ 0) (35)

we see ale 0 a contradiction For each t isin (0 1] we getρt(xn xn+1)⟶ 0 (n⟶infin)

Step 4 We show that limn⟶infinρt(xn xm) 0 for eacht isin (0 1] and all n m isin N For any t isin (0 1] there existss s(t) isin (0 t] and we have

ρt xn xm( 1113857 ρt fxnminus 1 fxmminus 1( 1113857

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt xm xm+1( 1113857( 1113857

+ Λεαminus 1ψ(ε) 1 + xnminus 1

t+ xmminus 1

t

+ xn

t

1113960

+ xm

t

1113961α

le (1 minus ε)12

ρt xn xn+1( 1113857 + ρt xm xm+1( 1113857( 1113857

+ Λεαminus 1ψ(ε) 1 + 4Mt1113858 1113859α

(36)

Putting n⟶infin and ε⟶ 0 in (36) we getρt(xn xm)⟶ 0

erefore xn1113864 1113865 is a Cauchy sequence Since X is com-plete there exists z isin X such that xn⟶ z (n⟶infin)

Step 5 We prove that z is the unique fixed point formapping f

First we show that fz z Using (29) for any t isin (0 1]there exists s isin (0 t] and we have

ρt(fz z)le ρs xn+1 z( 1113857 + ρt fz xn+1( 1113857

le (1 minus ε)12

ρt(z fz) + ρt xn xn+1( 1113857( 1113857

+ Λεαψ(ε) 1 +zt + xn

t

+fzt + xn+1

t1113872 1113873

α

+ ρs xn+1 z( 1113857

le (1 minus ε)12

ρt(z fz) + ρt xn xn+1( 1113857( 1113857 + Λεαψ(ε)

1 + 4Mt( 1113857α

+ ρs xn+1 z( 1113857

(37)

By taking limits as n⟶infin and ε⟶ 0 in the inequalityabove we get ρt(fz z)le (12)ρt(fz z) Hence ρt(fz z)

0 which implies fz zFinally we prove the uniqueness of z Assume that zprime is

another fixed point of f For any t isin (0 1] there existss isin (0 t] and we getρt z zprime( 1113857 ρt fz fzprime( 1113857

le (1 minus ε)12

ρt(z fz) + ρt zprime fzprime( 1113857( 1113857 + Λϵαψ(ε)

middot 1 +zt + zprime

t+fzt + fzprime

t

1113960 1113961α

le (1 minus ε)12

ρt(z z) + ρt zprime zprime( 1113857( 1113857 + Λϵαψ(ε) 1 + 4Mt1113858 1113859α

leΛϵαψ(ε) 1 + 4Mt1113858 1113859α

(38)

Passing to the limit as ε⟶ 0 in (38) we haveρt(z zprime) 0 us z zprime erefore f has a unique fixedpoint z isin X

Journal of Function Spaces 5

Following the arguments in Remark 1 from eorem 4we can obtain the Corollary 2

Corollary 2 (see Corollary 53 [6]) Let (X d L R) be acomplete fuzzy metric space with (R-2) and f X⟶ X be amapping If there exists λ isin [0 (12)) such that

ρt(fx fy) le λ ρt(x fx) + ρt(y fy)1113858 1113859 (39)

for all t isin (0 1] and x y isin X then f has a unique fixed pointz isin X

Now we construct two examples to illustrateeorems 3and 4 respectively

Example 1 Let X (12n) n isin Ncup 0 cup 0 L(a b)

min a b and R(a b) max a b Define the fuzzy metricd X times X⟶ F+ by

d(x y) 0(q minus |x minus y|) 1 q |x minus y|

0 qne |x minus y|1113896 (40)

Let T X⟶ X be a map defined by Tx (12)x forx isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (6) is satisfied for x0 1Λ 1 α 1 and ψ(ε) 2ε(c) T has the unique fixed point x 0(d) (29) is not satisfied Soeorem 4 cannot be verified

by this example

Proof First it is easy to see that (X d L R) is a completefuzzy metric space and R(a b) max a b satisfies (R-2)Note that Tx x implies that x 0 isin X then conclusion(c) is true Next we prove conclusions (b) and (d)respectively

(b) Letting x0 1Λ 1 α 1 β 1 and ψ(ε) 2ε weshow that (6) holds for all ε isin [0 1] t isin (0 1] andx y isin X

For any t isin (0 1] we can see that ρt(x y) |x minus y|en we consider the following two cases

Case 1 If εle (12) then 1 minus εge (12) For any x y isin X wehave

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386812

|x minus y|le (1 minus ε)ρt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+ Txt +Tyt1113857

(41)

Case 2 If εgt (12) then ψ(ε)gt 1 Note that

ρt(x y)le ρt x x0( 1113857 + ρ0 x0 y( 1113857 xt +yt (42)

en for any x y isin X we have

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le |x minus y| (1 minus ε)ρt(x y) + ερt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) xt +yt( 1113857

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+Txt +Tyt1113857

(43)

From Cases 1 and 2 we see that (6) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 3 is verified

(d) We show that (29) is not satisfied

Letting ε 0 by (29) we deduce that

ρt(Tx Ty)leρt(x Tx) + ρt(y Ty)

2 (44)

However let x0 0 and y0 1 For any t isin (0 1] wehave

ρt Tx0 Ty0( 1113857 0 minus12

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868gt12

middot12

ρt x0 Tx0( 1113857 + ρt y0 Ty0( 1113857

2

(45)

So this example cannot verify eorem 4 and we proveconclusion (d)

Example 2 Let X 6 minus 5 4 minus 3 2 minus 1 0 L(a b) mina b and R(a b) max a b Let d X times X⟶ F+ be afuzzy metric such that

d(x y)(q)

0 qlt 0

1 q 0

|x minus y|

|x minus y| + q qgt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)

Let T X⟶ X be a map such that

Tx

minus |x| + 1 xgt 0

|x| minus 1 xlt 0

0 x 0

⎧⎪⎪⎨

⎪⎪⎩(47)

for x isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (29) is satisfied for x0 1 Λ 1 α 1 and

ψ(ε) 6ε(c) T has the unique fixed point x 0(d) (6) is not satisfied So this example cannot verify

eorem 3

Proof Clearly we can see that x 0 is the unique fixedpoint for map T and conclusion (c) is true Now we proveconclusions (a) (b) and (d) respectively

6 Journal of Function Spaces

(a) We show that (X d L R) is a complete fuzzy metricspace with (R-2)

Clearly for any x y isin X d(x y)(q) is a fuzzy numberand (D1) and (D2) in Definition 2 are satisfied Note thatR(a b) max a b satisfies (R-2) So it is sufficient to provethat (D3) holds

For any x y z isin X it is obvious that λ1(x z)

λ1(z y) λ1(x y) 0 To see (D3L) let us consider thefollowing two cases

(i) If plt 0 or qlt 0 then d(x z)(p) 0 ord(z y)(q) 0 leading to that

min(d(x z)(p) d(z y)(q)) 0 (48)

So (D3L) is satisfied

(ii) If p 0 and q 0 then p + q 0 Sod(x z)(p) d(z y)(q) d(x y)(p + q) 1 lead-ing to that

d(x y)(p + q) min d(x z)(p) d(z y)(q)1113864 1113865 (49)

us (D3L) holds To see (D3R) we consider the fol-lowing two cases

Case a1 Suppose that there are at least two points inx y z1113864 1113865 which are equal For any pge 0 and qge 0

(i) If z x or z y without loss of generality let z xthen d(z y) d(x y) Since d(x y)(q) is decreasingas qge 0 we have

max d(x z)(p) d(z y)(q)1113864 1113865ged(z y)(q)

d(x y)(q)ge d(x y)(p + q)(50)

(ii) If x y then we have

p + q 0rArrp 0 q 0rArr d(x y)(p + q)

1 max d(x z)(p) d(z y)(q)1113864 1113865

p + qgt 0rArr d(x y)(p + q) 0

lemax d(x z)(p) d(z y)(q)1113864 1113865

(51)

Case a2 Suppose that xney xne z and yne z For anypge 0 and qge 0 denote s d(x z)(p) andt d(z y)(q) Without loss of generality let 0lt sle ten we have

p |x minus z|1 minus s

s

q |z minus y|1 minus t

t

(52)

Since d(x y)(q) is decreasing as qge 0 and (1 minus tt) isdecreasing we deduce that

max d(x z)(p) d(z y)(q)1113864 1113865 t d(x y) |x minus y|1 minus t

t1113874 1113875

ged(x y) (|x minus z| +|z minus y|)1 minus t

t1113876 1113877

ged(x y) |x minus z|1 minus s

s+|z minus y|

1 minus t

t1113876 1113877

d(x y)(p + q)

(53)

From Cases a1 and a2 we conclude that (D3R) holds forall x y z isin X

Note that X is a finite set en (X d L R) is completeand the proof of conclusion (a) is completed

(b) We prove (29) for x0 1 Λ 1 α 1 β 1 andψ(ε) 6ε

For any t isin (0 1] we can see that ρt(x y)

|x minus y|(1 minus tt) en we consider the following two cases

Case b1 If εle (112) it is easy to see that for any xne 0we have |Tx| |x| minus 1 and x middot Txle 0 So we can obtainthat |x minus Tx| |x| + |Tx| and

|Tx minus Ty|le |Tx| +|Ty| |Tx| +|x| minus 1

2+

|Ty| +|y| minus 12

|x minus Tx| +|y minus Ty|

2minus 1

(54)

for all x yne 0 If y 0 then we have

|Tx minus Ty| |Tx| |Tx| +|x| minus 1

2

|x minus Tx| +|y minus Ty|

2minus12

(55)

From the above two equalities by calculation we obtainthat for any x y isin X

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle1011

middot|x minus Tx|(1 minus tt) +|y minus Ty|(1 minus tt)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +||Ty||t( 1113857

(56)

Journal of Function Spaces 7

Case b2 If εgt (112) then we have ψ(ε)gt (12) Notethat

ρt(x Tx) + ρt(y Ty)le ρt x x0( 1113857 + ρt x0 Tx( 1113857

+ ρt y x0( 1113857 + ρt x0 Ty( 1113857

xt +||y||t +Txt +Tyt

(57)

en for any x y isin X we have

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle

|x minus Tx|1 minus tt +|y minus Ty|1 minus tt2

(1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

12

xt +yt +Txt +Tyt( 1113857

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +Tyt( 1113857

(58)

From Cases b1 and b2 we see that (29) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 4 is verified

(d) We show that (6) is not satisfied for map T in(X d L R)

Finally we show that (6) is not satisfied for map T in(X d L R) If ρt(Tx Ty) ρt(x y) for some x y isin X andsome t isin (0 1] then by (6) we deduce that for any εgt 0

ρt(x y)leΛεαψ(ε) 1 +xt +yt +fxt +fyt1113858 1113859β

(59)

Letting εrarr 0 we have ψ(ε)rarr 0 which implies thatx y However for x0 6 and y0 4 we have Tx0 minus 5and Ty0 minus 3 and

ρt Tx0 Ty0( 1113857 2 middot1 minus t

t ρt x0 y0( 1113857 (60)

for all t isin (0 1] which is a contradiction

Remark 2 From Examples 1 and 2 we can see that e-orems 3 and 4 are independent to each other

Recently Jamshaid et al [19] investigated the fuzzy fixedpoints of fuzzy mappings via F-contractions On the basis oftheir work a natural question for multivalued mappings canbe raised as follows

Question 1 Is the multivalued case of eorem 3 true

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally and significantly in writingthis article All authors read and approved the finalmanuscript

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (11561049)

References

[1] O Kaleva and S Seikkala ldquoOn fuzzymetric spacesrdquo Fuzzy Setsand Systems vol 12 no 3 pp 215ndash229 1984

[2] O Hadzic and E Pap ldquoA fixed point theorem for multivaluedmappings in probaailistic metric spaces and an application infuzzy metric spacesrdquo Fuzzy Sets and Systems vol 127 no 3pp 333ndash444 2002

[3] N Lu F He and J Fan ldquoFixed point theorems of cyclic-contractive mapping in fuzzy metric spacesrdquo Mathematics inPractice and -eory vol 22 pp 237ndash248 2018

[4] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1pp 59ndash62 2014

[5] G Yun S Hwang and J Chang ldquoFuzzy lipschitz maps andfixed point theorems in fuzzy metric spacesrdquo Fuzzy Sets andSystems vol 161 no 8 pp 1117ndash1130 2010

[6] J-Z Xiao X-H Zhu and X Jin ldquoFixed point theorems fornonlinear contractions in Kaleva-Seikkalarsquos type fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 200 pp 65ndash83 2012

[7] S Phiangsungnoen P ounthong and P Kumam ldquoFixedpoint results in fuzzy metric spaces via α and βκ-admissiblemappings with application to integral typesrdquo Journal of In-telligent amp Fuzzy Systems vol 34 no 1 pp 467ndash475 2018

[8] G-J Yun J Chang and S Hwang ldquoFuzzy isometries andnon-existence of fuzzy contractive maps on fuzzy metricspacesrdquo International Journal of Fuzzy Systems vol 13 no 3pp 206ndash217 2011

8 Journal of Function Spaces

[9] V Gregori J-J Mintildeana and D Miravet ldquoContractive se-quences in fuzzy metric spacesrdquo Fuzzy Sets and Systemsvol 379 pp 125ndash133 2020

[10] S S Chauhan M Imdad G Kaur and A Sharma ldquoSomefixed point theorems for SF-contraction in complete fuzzymetric spacesrdquo Afrika Matematika vol 30 no 3-4pp 651ndash662 2019

[11] J Xiao and X Zhu ldquoOn linearly topological structure andproperty of fuzzy normed linear spacerdquo Fuzzy Sets and Sys-tems vol 125 no 2 pp 153ndash161 2002

[12] J-z Xiao and X-h Zhu ldquoTopological degree theory and fixedpoint theorems in fuzzy normed spacerdquo Fuzzy Sets andSystems vol 147 no 3 pp 437ndash452 2004

[13] V Pata ldquoA fixed point theorem in metric spacesrdquo Journal ofFixed Point -eory and Applications vol 10 no 2 pp 299ndash305 2011

[14] M Chakraborty and S K Samanta ldquoA fixed point theorem forKannan-type maps in metric spacesrdquo 2012 httpsarxivorgabs1211 7331v2

[15] Z Kadelburg and S Radenovic ldquoFixed point theorems underPata-type conditions in metric spacesrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 77ndash82 2016

[16] Z Kadelburg and S Radenovic ldquoFixed point and tripled fixedpoint theorems under pata-type conditions in ordered metricspacesrdquo International Journal of Analysis and Applicationsvol 6 pp 113ndash122 2014

[17] S Balasubramanian ldquoA pata-type fixed point theoremrdquoMathematical Sciences vol 8 no 3 pp 65ndash69 2014

[18] G K Jacob M S Khan C Park and S Jun ldquoOn generalizedpata type contractionsrdquoMathematics vol 6 no 2 p 25 2018

[19] J Ahmad H Aydi and N Mlaiki ldquoFuzzy fixed points of fuzzymappings via F-contractions and an applicationrdquo Journal ofIntelligent amp Fuzzy Systems vol 37 no 4 pp 5487ndash54932019

Journal of Function Spaces 9

Following the arguments in Remark 1 from eorem 4we can obtain the Corollary 2

Corollary 2 (see Corollary 53 [6]) Let (X d L R) be acomplete fuzzy metric space with (R-2) and f X⟶ X be amapping If there exists λ isin [0 (12)) such that

ρt(fx fy) le λ ρt(x fx) + ρt(y fy)1113858 1113859 (39)

for all t isin (0 1] and x y isin X then f has a unique fixed pointz isin X

Now we construct two examples to illustrateeorems 3and 4 respectively

Example 1 Let X (12n) n isin Ncup 0 cup 0 L(a b)

min a b and R(a b) max a b Define the fuzzy metricd X times X⟶ F+ by

d(x y) 0(q minus |x minus y|) 1 q |x minus y|

0 qne |x minus y|1113896 (40)

Let T X⟶ X be a map defined by Tx (12)x forx isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (6) is satisfied for x0 1Λ 1 α 1 and ψ(ε) 2ε(c) T has the unique fixed point x 0(d) (29) is not satisfied Soeorem 4 cannot be verified

by this example

Proof First it is easy to see that (X d L R) is a completefuzzy metric space and R(a b) max a b satisfies (R-2)Note that Tx x implies that x 0 isin X then conclusion(c) is true Next we prove conclusions (b) and (d)respectively

(b) Letting x0 1Λ 1 α 1 β 1 and ψ(ε) 2ε weshow that (6) holds for all ε isin [0 1] t isin (0 1] andx y isin X

For any t isin (0 1] we can see that ρt(x y) |x minus y|en we consider the following two cases

Case 1 If εle (12) then 1 minus εge (12) For any x y isin X wehave

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

111386811138681113868111386811138681113868111386812

|x minus y|le (1 minus ε)ρt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+ Txt +Tyt1113857

(41)

Case 2 If εgt (12) then ψ(ε)gt 1 Note that

ρt(x y)le ρt x x0( 1113857 + ρ0 x0 y( 1113857 xt +yt (42)

en for any x y isin X we have

ρt(Tx Ty) 12

x minus12

y

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868le |x minus y| (1 minus ε)ρt(x y) + ερt(x y)

le (1 minus ε)ρt(x y) + εψ(ε) xt +yt( 1113857

le (1 minus ε)ρt(x y) + εψ(ε) 1 +xt +yt(

+Txt +Tyt1113857

(43)

From Cases 1 and 2 we see that (6) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 3 is verified

(d) We show that (29) is not satisfied

Letting ε 0 by (29) we deduce that

ρt(Tx Ty)leρt(x Tx) + ρt(y Ty)

2 (44)

However let x0 0 and y0 1 For any t isin (0 1] wehave

ρt Tx0 Ty0( 1113857 0 minus12

1113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868gt12

middot12

ρt x0 Tx0( 1113857 + ρt y0 Ty0( 1113857

2

(45)

So this example cannot verify eorem 4 and we proveconclusion (d)

Example 2 Let X 6 minus 5 4 minus 3 2 minus 1 0 L(a b) mina b and R(a b) max a b Let d X times X⟶ F+ be afuzzy metric such that

d(x y)(q)

0 qlt 0

1 q 0

|x minus y|

|x minus y| + q qgt 0

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(46)

Let T X⟶ X be a map such that

Tx

minus |x| + 1 xgt 0

|x| minus 1 xlt 0

0 x 0

⎧⎪⎪⎨

⎪⎪⎩(47)

for x isin X en the following hold

(a) (X d L R) is a complete fuzzy metric with (R-2)(b) (29) is satisfied for x0 1 Λ 1 α 1 and

ψ(ε) 6ε(c) T has the unique fixed point x 0(d) (6) is not satisfied So this example cannot verify

eorem 3

Proof Clearly we can see that x 0 is the unique fixedpoint for map T and conclusion (c) is true Now we proveconclusions (a) (b) and (d) respectively

6 Journal of Function Spaces

(a) We show that (X d L R) is a complete fuzzy metricspace with (R-2)

Clearly for any x y isin X d(x y)(q) is a fuzzy numberand (D1) and (D2) in Definition 2 are satisfied Note thatR(a b) max a b satisfies (R-2) So it is sufficient to provethat (D3) holds

For any x y z isin X it is obvious that λ1(x z)

λ1(z y) λ1(x y) 0 To see (D3L) let us consider thefollowing two cases

(i) If plt 0 or qlt 0 then d(x z)(p) 0 ord(z y)(q) 0 leading to that

min(d(x z)(p) d(z y)(q)) 0 (48)

So (D3L) is satisfied

(ii) If p 0 and q 0 then p + q 0 Sod(x z)(p) d(z y)(q) d(x y)(p + q) 1 lead-ing to that

d(x y)(p + q) min d(x z)(p) d(z y)(q)1113864 1113865 (49)

us (D3L) holds To see (D3R) we consider the fol-lowing two cases

Case a1 Suppose that there are at least two points inx y z1113864 1113865 which are equal For any pge 0 and qge 0

(i) If z x or z y without loss of generality let z xthen d(z y) d(x y) Since d(x y)(q) is decreasingas qge 0 we have

max d(x z)(p) d(z y)(q)1113864 1113865ged(z y)(q)

d(x y)(q)ge d(x y)(p + q)(50)

(ii) If x y then we have

p + q 0rArrp 0 q 0rArr d(x y)(p + q)

1 max d(x z)(p) d(z y)(q)1113864 1113865

p + qgt 0rArr d(x y)(p + q) 0

lemax d(x z)(p) d(z y)(q)1113864 1113865

(51)

Case a2 Suppose that xney xne z and yne z For anypge 0 and qge 0 denote s d(x z)(p) andt d(z y)(q) Without loss of generality let 0lt sle ten we have

p |x minus z|1 minus s

s

q |z minus y|1 minus t

t

(52)

Since d(x y)(q) is decreasing as qge 0 and (1 minus tt) isdecreasing we deduce that

max d(x z)(p) d(z y)(q)1113864 1113865 t d(x y) |x minus y|1 minus t

t1113874 1113875

ged(x y) (|x minus z| +|z minus y|)1 minus t

t1113876 1113877

ged(x y) |x minus z|1 minus s

s+|z minus y|

1 minus t

t1113876 1113877

d(x y)(p + q)

(53)

From Cases a1 and a2 we conclude that (D3R) holds forall x y z isin X

Note that X is a finite set en (X d L R) is completeand the proof of conclusion (a) is completed

(b) We prove (29) for x0 1 Λ 1 α 1 β 1 andψ(ε) 6ε

For any t isin (0 1] we can see that ρt(x y)

|x minus y|(1 minus tt) en we consider the following two cases

Case b1 If εle (112) it is easy to see that for any xne 0we have |Tx| |x| minus 1 and x middot Txle 0 So we can obtainthat |x minus Tx| |x| + |Tx| and

|Tx minus Ty|le |Tx| +|Ty| |Tx| +|x| minus 1

2+

|Ty| +|y| minus 12

|x minus Tx| +|y minus Ty|

2minus 1

(54)

for all x yne 0 If y 0 then we have

|Tx minus Ty| |Tx| |Tx| +|x| minus 1

2

|x minus Tx| +|y minus Ty|

2minus12

(55)

From the above two equalities by calculation we obtainthat for any x y isin X

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle1011

middot|x minus Tx|(1 minus tt) +|y minus Ty|(1 minus tt)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +||Ty||t( 1113857

(56)

Journal of Function Spaces 7

Case b2 If εgt (112) then we have ψ(ε)gt (12) Notethat

ρt(x Tx) + ρt(y Ty)le ρt x x0( 1113857 + ρt x0 Tx( 1113857

+ ρt y x0( 1113857 + ρt x0 Ty( 1113857

xt +||y||t +Txt +Tyt

(57)

en for any x y isin X we have

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle

|x minus Tx|1 minus tt +|y minus Ty|1 minus tt2

(1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

12

xt +yt +Txt +Tyt( 1113857

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +Tyt( 1113857

(58)

From Cases b1 and b2 we see that (29) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 4 is verified

(d) We show that (6) is not satisfied for map T in(X d L R)

Finally we show that (6) is not satisfied for map T in(X d L R) If ρt(Tx Ty) ρt(x y) for some x y isin X andsome t isin (0 1] then by (6) we deduce that for any εgt 0

ρt(x y)leΛεαψ(ε) 1 +xt +yt +fxt +fyt1113858 1113859β

(59)

Letting εrarr 0 we have ψ(ε)rarr 0 which implies thatx y However for x0 6 and y0 4 we have Tx0 minus 5and Ty0 minus 3 and

ρt Tx0 Ty0( 1113857 2 middot1 minus t

t ρt x0 y0( 1113857 (60)

for all t isin (0 1] which is a contradiction

Remark 2 From Examples 1 and 2 we can see that e-orems 3 and 4 are independent to each other

Recently Jamshaid et al [19] investigated the fuzzy fixedpoints of fuzzy mappings via F-contractions On the basis oftheir work a natural question for multivalued mappings canbe raised as follows

Question 1 Is the multivalued case of eorem 3 true

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally and significantly in writingthis article All authors read and approved the finalmanuscript

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (11561049)

References

[1] O Kaleva and S Seikkala ldquoOn fuzzymetric spacesrdquo Fuzzy Setsand Systems vol 12 no 3 pp 215ndash229 1984

[2] O Hadzic and E Pap ldquoA fixed point theorem for multivaluedmappings in probaailistic metric spaces and an application infuzzy metric spacesrdquo Fuzzy Sets and Systems vol 127 no 3pp 333ndash444 2002

[3] N Lu F He and J Fan ldquoFixed point theorems of cyclic-contractive mapping in fuzzy metric spacesrdquo Mathematics inPractice and -eory vol 22 pp 237ndash248 2018

[4] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1pp 59ndash62 2014

[5] G Yun S Hwang and J Chang ldquoFuzzy lipschitz maps andfixed point theorems in fuzzy metric spacesrdquo Fuzzy Sets andSystems vol 161 no 8 pp 1117ndash1130 2010

[6] J-Z Xiao X-H Zhu and X Jin ldquoFixed point theorems fornonlinear contractions in Kaleva-Seikkalarsquos type fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 200 pp 65ndash83 2012

[7] S Phiangsungnoen P ounthong and P Kumam ldquoFixedpoint results in fuzzy metric spaces via α and βκ-admissiblemappings with application to integral typesrdquo Journal of In-telligent amp Fuzzy Systems vol 34 no 1 pp 467ndash475 2018

[8] G-J Yun J Chang and S Hwang ldquoFuzzy isometries andnon-existence of fuzzy contractive maps on fuzzy metricspacesrdquo International Journal of Fuzzy Systems vol 13 no 3pp 206ndash217 2011

8 Journal of Function Spaces

[9] V Gregori J-J Mintildeana and D Miravet ldquoContractive se-quences in fuzzy metric spacesrdquo Fuzzy Sets and Systemsvol 379 pp 125ndash133 2020

[10] S S Chauhan M Imdad G Kaur and A Sharma ldquoSomefixed point theorems for SF-contraction in complete fuzzymetric spacesrdquo Afrika Matematika vol 30 no 3-4pp 651ndash662 2019

[11] J Xiao and X Zhu ldquoOn linearly topological structure andproperty of fuzzy normed linear spacerdquo Fuzzy Sets and Sys-tems vol 125 no 2 pp 153ndash161 2002

[12] J-z Xiao and X-h Zhu ldquoTopological degree theory and fixedpoint theorems in fuzzy normed spacerdquo Fuzzy Sets andSystems vol 147 no 3 pp 437ndash452 2004

[13] V Pata ldquoA fixed point theorem in metric spacesrdquo Journal ofFixed Point -eory and Applications vol 10 no 2 pp 299ndash305 2011

[14] M Chakraborty and S K Samanta ldquoA fixed point theorem forKannan-type maps in metric spacesrdquo 2012 httpsarxivorgabs1211 7331v2

[15] Z Kadelburg and S Radenovic ldquoFixed point theorems underPata-type conditions in metric spacesrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 77ndash82 2016

[16] Z Kadelburg and S Radenovic ldquoFixed point and tripled fixedpoint theorems under pata-type conditions in ordered metricspacesrdquo International Journal of Analysis and Applicationsvol 6 pp 113ndash122 2014

[17] S Balasubramanian ldquoA pata-type fixed point theoremrdquoMathematical Sciences vol 8 no 3 pp 65ndash69 2014

[18] G K Jacob M S Khan C Park and S Jun ldquoOn generalizedpata type contractionsrdquoMathematics vol 6 no 2 p 25 2018

[19] J Ahmad H Aydi and N Mlaiki ldquoFuzzy fixed points of fuzzymappings via F-contractions and an applicationrdquo Journal ofIntelligent amp Fuzzy Systems vol 37 no 4 pp 5487ndash54932019

Journal of Function Spaces 9

(a) We show that (X d L R) is a complete fuzzy metricspace with (R-2)

Clearly for any x y isin X d(x y)(q) is a fuzzy numberand (D1) and (D2) in Definition 2 are satisfied Note thatR(a b) max a b satisfies (R-2) So it is sufficient to provethat (D3) holds

For any x y z isin X it is obvious that λ1(x z)

λ1(z y) λ1(x y) 0 To see (D3L) let us consider thefollowing two cases

(i) If plt 0 or qlt 0 then d(x z)(p) 0 ord(z y)(q) 0 leading to that

min(d(x z)(p) d(z y)(q)) 0 (48)

So (D3L) is satisfied

(ii) If p 0 and q 0 then p + q 0 Sod(x z)(p) d(z y)(q) d(x y)(p + q) 1 lead-ing to that

d(x y)(p + q) min d(x z)(p) d(z y)(q)1113864 1113865 (49)

us (D3L) holds To see (D3R) we consider the fol-lowing two cases

Case a1 Suppose that there are at least two points inx y z1113864 1113865 which are equal For any pge 0 and qge 0

(i) If z x or z y without loss of generality let z xthen d(z y) d(x y) Since d(x y)(q) is decreasingas qge 0 we have

max d(x z)(p) d(z y)(q)1113864 1113865ged(z y)(q)

d(x y)(q)ge d(x y)(p + q)(50)

(ii) If x y then we have

p + q 0rArrp 0 q 0rArr d(x y)(p + q)

1 max d(x z)(p) d(z y)(q)1113864 1113865

p + qgt 0rArr d(x y)(p + q) 0

lemax d(x z)(p) d(z y)(q)1113864 1113865

(51)

Case a2 Suppose that xney xne z and yne z For anypge 0 and qge 0 denote s d(x z)(p) andt d(z y)(q) Without loss of generality let 0lt sle ten we have

p |x minus z|1 minus s

s

q |z minus y|1 minus t

t

(52)

Since d(x y)(q) is decreasing as qge 0 and (1 minus tt) isdecreasing we deduce that

max d(x z)(p) d(z y)(q)1113864 1113865 t d(x y) |x minus y|1 minus t

t1113874 1113875

ged(x y) (|x minus z| +|z minus y|)1 minus t

t1113876 1113877

ged(x y) |x minus z|1 minus s

s+|z minus y|

1 minus t

t1113876 1113877

d(x y)(p + q)

(53)

From Cases a1 and a2 we conclude that (D3R) holds forall x y z isin X

Note that X is a finite set en (X d L R) is completeand the proof of conclusion (a) is completed

(b) We prove (29) for x0 1 Λ 1 α 1 β 1 andψ(ε) 6ε

For any t isin (0 1] we can see that ρt(x y)

|x minus y|(1 minus tt) en we consider the following two cases

Case b1 If εle (112) it is easy to see that for any xne 0we have |Tx| |x| minus 1 and x middot Txle 0 So we can obtainthat |x minus Tx| |x| + |Tx| and

|Tx minus Ty|le |Tx| +|Ty| |Tx| +|x| minus 1

2+

|Ty| +|y| minus 12

|x minus Tx| +|y minus Ty|

2minus 1

(54)

for all x yne 0 If y 0 then we have

|Tx minus Ty| |Tx| |Tx| +|x| minus 1

2

|x minus Tx| +|y minus Ty|

2minus12

(55)

From the above two equalities by calculation we obtainthat for any x y isin X

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle1011

middot|x minus Tx|(1 minus tt) +|y minus Ty|(1 minus tt)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +||Ty||t( 1113857

(56)

Journal of Function Spaces 7

Case b2 If εgt (112) then we have ψ(ε)gt (12) Notethat

ρt(x Tx) + ρt(y Ty)le ρt x x0( 1113857 + ρt x0 Tx( 1113857

+ ρt y x0( 1113857 + ρt x0 Ty( 1113857

xt +||y||t +Txt +Tyt

(57)

en for any x y isin X we have

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle

|x minus Tx|1 minus tt +|y minus Ty|1 minus tt2

(1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

12

xt +yt +Txt +Tyt( 1113857

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +Tyt( 1113857

(58)

From Cases b1 and b2 we see that (29) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 4 is verified

(d) We show that (6) is not satisfied for map T in(X d L R)

Finally we show that (6) is not satisfied for map T in(X d L R) If ρt(Tx Ty) ρt(x y) for some x y isin X andsome t isin (0 1] then by (6) we deduce that for any εgt 0

ρt(x y)leΛεαψ(ε) 1 +xt +yt +fxt +fyt1113858 1113859β

(59)

Letting εrarr 0 we have ψ(ε)rarr 0 which implies thatx y However for x0 6 and y0 4 we have Tx0 minus 5and Ty0 minus 3 and

ρt Tx0 Ty0( 1113857 2 middot1 minus t

t ρt x0 y0( 1113857 (60)

for all t isin (0 1] which is a contradiction

Remark 2 From Examples 1 and 2 we can see that e-orems 3 and 4 are independent to each other

Recently Jamshaid et al [19] investigated the fuzzy fixedpoints of fuzzy mappings via F-contractions On the basis oftheir work a natural question for multivalued mappings canbe raised as follows

Question 1 Is the multivalued case of eorem 3 true

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally and significantly in writingthis article All authors read and approved the finalmanuscript

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (11561049)

References

[1] O Kaleva and S Seikkala ldquoOn fuzzymetric spacesrdquo Fuzzy Setsand Systems vol 12 no 3 pp 215ndash229 1984

[2] O Hadzic and E Pap ldquoA fixed point theorem for multivaluedmappings in probaailistic metric spaces and an application infuzzy metric spacesrdquo Fuzzy Sets and Systems vol 127 no 3pp 333ndash444 2002

[3] N Lu F He and J Fan ldquoFixed point theorems of cyclic-contractive mapping in fuzzy metric spacesrdquo Mathematics inPractice and -eory vol 22 pp 237ndash248 2018

[4] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1pp 59ndash62 2014

[5] G Yun S Hwang and J Chang ldquoFuzzy lipschitz maps andfixed point theorems in fuzzy metric spacesrdquo Fuzzy Sets andSystems vol 161 no 8 pp 1117ndash1130 2010

[6] J-Z Xiao X-H Zhu and X Jin ldquoFixed point theorems fornonlinear contractions in Kaleva-Seikkalarsquos type fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 200 pp 65ndash83 2012

[7] S Phiangsungnoen P ounthong and P Kumam ldquoFixedpoint results in fuzzy metric spaces via α and βκ-admissiblemappings with application to integral typesrdquo Journal of In-telligent amp Fuzzy Systems vol 34 no 1 pp 467ndash475 2018

[8] G-J Yun J Chang and S Hwang ldquoFuzzy isometries andnon-existence of fuzzy contractive maps on fuzzy metricspacesrdquo International Journal of Fuzzy Systems vol 13 no 3pp 206ndash217 2011

8 Journal of Function Spaces

[9] V Gregori J-J Mintildeana and D Miravet ldquoContractive se-quences in fuzzy metric spacesrdquo Fuzzy Sets and Systemsvol 379 pp 125ndash133 2020

[10] S S Chauhan M Imdad G Kaur and A Sharma ldquoSomefixed point theorems for SF-contraction in complete fuzzymetric spacesrdquo Afrika Matematika vol 30 no 3-4pp 651ndash662 2019

[11] J Xiao and X Zhu ldquoOn linearly topological structure andproperty of fuzzy normed linear spacerdquo Fuzzy Sets and Sys-tems vol 125 no 2 pp 153ndash161 2002

[12] J-z Xiao and X-h Zhu ldquoTopological degree theory and fixedpoint theorems in fuzzy normed spacerdquo Fuzzy Sets andSystems vol 147 no 3 pp 437ndash452 2004

[13] V Pata ldquoA fixed point theorem in metric spacesrdquo Journal ofFixed Point -eory and Applications vol 10 no 2 pp 299ndash305 2011

[14] M Chakraborty and S K Samanta ldquoA fixed point theorem forKannan-type maps in metric spacesrdquo 2012 httpsarxivorgabs1211 7331v2

[15] Z Kadelburg and S Radenovic ldquoFixed point theorems underPata-type conditions in metric spacesrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 77ndash82 2016

[16] Z Kadelburg and S Radenovic ldquoFixed point and tripled fixedpoint theorems under pata-type conditions in ordered metricspacesrdquo International Journal of Analysis and Applicationsvol 6 pp 113ndash122 2014

[17] S Balasubramanian ldquoA pata-type fixed point theoremrdquoMathematical Sciences vol 8 no 3 pp 65ndash69 2014

[18] G K Jacob M S Khan C Park and S Jun ldquoOn generalizedpata type contractionsrdquoMathematics vol 6 no 2 p 25 2018

[19] J Ahmad H Aydi and N Mlaiki ldquoFuzzy fixed points of fuzzymappings via F-contractions and an applicationrdquo Journal ofIntelligent amp Fuzzy Systems vol 37 no 4 pp 5487ndash54932019

Journal of Function Spaces 9

Case b2 If εgt (112) then we have ψ(ε)gt (12) Notethat

ρt(x Tx) + ρt(y Ty)le ρt x x0( 1113857 + ρt x0 Tx( 1113857

+ ρt y x0( 1113857 + ρt x0 Ty( 1113857

xt +||y||t +Txt +Tyt

(57)

en for any x y isin X we have

ρt(Tx Ty) |Tx minus Ty|1 minus t

tle

|x minus Tx|1 minus tt +|y minus Ty|1 minus tt2

(1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

ρt(x Tx) + ρt(y Ty)

2

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ ε

12

xt +yt +Txt +Tyt( 1113857

le (1 minus ε)ρt(x Tx) + ρt(y Ty)

2+ εψ(ε) 1 +xt +yt +Txt +Tyt( 1113857

(58)

From Cases b1 and b2 we see that (29) holds for allε isin [0 1] In fact x 0 is the unique fixed point for map TSo eorem 4 is verified

(d) We show that (6) is not satisfied for map T in(X d L R)

Finally we show that (6) is not satisfied for map T in(X d L R) If ρt(Tx Ty) ρt(x y) for some x y isin X andsome t isin (0 1] then by (6) we deduce that for any εgt 0

ρt(x y)leΛεαψ(ε) 1 +xt +yt +fxt +fyt1113858 1113859β

(59)

Letting εrarr 0 we have ψ(ε)rarr 0 which implies thatx y However for x0 6 and y0 4 we have Tx0 minus 5and Ty0 minus 3 and

ρt Tx0 Ty0( 1113857 2 middot1 minus t

t ρt x0 y0( 1113857 (60)

for all t isin (0 1] which is a contradiction

Remark 2 From Examples 1 and 2 we can see that e-orems 3 and 4 are independent to each other

Recently Jamshaid et al [19] investigated the fuzzy fixedpoints of fuzzy mappings via F-contractions On the basis oftheir work a natural question for multivalued mappings canbe raised as follows

Question 1 Is the multivalued case of eorem 3 true

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Authorsrsquo Contributions

All authors contributed equally and significantly in writingthis article All authors read and approved the finalmanuscript

Acknowledgments

is research was supported by the National Natural ScienceFoundation of China (11561049)

References

[1] O Kaleva and S Seikkala ldquoOn fuzzymetric spacesrdquo Fuzzy Setsand Systems vol 12 no 3 pp 215ndash229 1984

[2] O Hadzic and E Pap ldquoA fixed point theorem for multivaluedmappings in probaailistic metric spaces and an application infuzzy metric spacesrdquo Fuzzy Sets and Systems vol 127 no 3pp 333ndash444 2002

[3] N Lu F He and J Fan ldquoFixed point theorems of cyclic-contractive mapping in fuzzy metric spacesrdquo Mathematics inPractice and -eory vol 22 pp 237ndash248 2018

[4] M A Ahmed ldquoFixed point theorems in fuzzy metric spacesrdquoJournal of the Egyptian Mathematical Society vol 22 no 1pp 59ndash62 2014

[5] G Yun S Hwang and J Chang ldquoFuzzy lipschitz maps andfixed point theorems in fuzzy metric spacesrdquo Fuzzy Sets andSystems vol 161 no 8 pp 1117ndash1130 2010

[6] J-Z Xiao X-H Zhu and X Jin ldquoFixed point theorems fornonlinear contractions in Kaleva-Seikkalarsquos type fuzzy metricspacesrdquo Fuzzy Sets and Systems vol 200 pp 65ndash83 2012

[7] S Phiangsungnoen P ounthong and P Kumam ldquoFixedpoint results in fuzzy metric spaces via α and βκ-admissiblemappings with application to integral typesrdquo Journal of In-telligent amp Fuzzy Systems vol 34 no 1 pp 467ndash475 2018

[8] G-J Yun J Chang and S Hwang ldquoFuzzy isometries andnon-existence of fuzzy contractive maps on fuzzy metricspacesrdquo International Journal of Fuzzy Systems vol 13 no 3pp 206ndash217 2011

8 Journal of Function Spaces

[9] V Gregori J-J Mintildeana and D Miravet ldquoContractive se-quences in fuzzy metric spacesrdquo Fuzzy Sets and Systemsvol 379 pp 125ndash133 2020

[10] S S Chauhan M Imdad G Kaur and A Sharma ldquoSomefixed point theorems for SF-contraction in complete fuzzymetric spacesrdquo Afrika Matematika vol 30 no 3-4pp 651ndash662 2019

[11] J Xiao and X Zhu ldquoOn linearly topological structure andproperty of fuzzy normed linear spacerdquo Fuzzy Sets and Sys-tems vol 125 no 2 pp 153ndash161 2002

[12] J-z Xiao and X-h Zhu ldquoTopological degree theory and fixedpoint theorems in fuzzy normed spacerdquo Fuzzy Sets andSystems vol 147 no 3 pp 437ndash452 2004

[13] V Pata ldquoA fixed point theorem in metric spacesrdquo Journal ofFixed Point -eory and Applications vol 10 no 2 pp 299ndash305 2011

[14] M Chakraborty and S K Samanta ldquoA fixed point theorem forKannan-type maps in metric spacesrdquo 2012 httpsarxivorgabs1211 7331v2

[15] Z Kadelburg and S Radenovic ldquoFixed point theorems underPata-type conditions in metric spacesrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 77ndash82 2016

[16] Z Kadelburg and S Radenovic ldquoFixed point and tripled fixedpoint theorems under pata-type conditions in ordered metricspacesrdquo International Journal of Analysis and Applicationsvol 6 pp 113ndash122 2014

[17] S Balasubramanian ldquoA pata-type fixed point theoremrdquoMathematical Sciences vol 8 no 3 pp 65ndash69 2014

[18] G K Jacob M S Khan C Park and S Jun ldquoOn generalizedpata type contractionsrdquoMathematics vol 6 no 2 p 25 2018

[19] J Ahmad H Aydi and N Mlaiki ldquoFuzzy fixed points of fuzzymappings via F-contractions and an applicationrdquo Journal ofIntelligent amp Fuzzy Systems vol 37 no 4 pp 5487ndash54932019

Journal of Function Spaces 9

[9] V Gregori J-J Mintildeana and D Miravet ldquoContractive se-quences in fuzzy metric spacesrdquo Fuzzy Sets and Systemsvol 379 pp 125ndash133 2020

[10] S S Chauhan M Imdad G Kaur and A Sharma ldquoSomefixed point theorems for SF-contraction in complete fuzzymetric spacesrdquo Afrika Matematika vol 30 no 3-4pp 651ndash662 2019

[11] J Xiao and X Zhu ldquoOn linearly topological structure andproperty of fuzzy normed linear spacerdquo Fuzzy Sets and Sys-tems vol 125 no 2 pp 153ndash161 2002

[12] J-z Xiao and X-h Zhu ldquoTopological degree theory and fixedpoint theorems in fuzzy normed spacerdquo Fuzzy Sets andSystems vol 147 no 3 pp 437ndash452 2004

[13] V Pata ldquoA fixed point theorem in metric spacesrdquo Journal ofFixed Point -eory and Applications vol 10 no 2 pp 299ndash305 2011

[14] M Chakraborty and S K Samanta ldquoA fixed point theorem forKannan-type maps in metric spacesrdquo 2012 httpsarxivorgabs1211 7331v2

[15] Z Kadelburg and S Radenovic ldquoFixed point theorems underPata-type conditions in metric spacesrdquo Journal of the EgyptianMathematical Society vol 24 no 1 pp 77ndash82 2016

[16] Z Kadelburg and S Radenovic ldquoFixed point and tripled fixedpoint theorems under pata-type conditions in ordered metricspacesrdquo International Journal of Analysis and Applicationsvol 6 pp 113ndash122 2014

[17] S Balasubramanian ldquoA pata-type fixed point theoremrdquoMathematical Sciences vol 8 no 3 pp 65ndash69 2014

[18] G K Jacob M S Khan C Park and S Jun ldquoOn generalizedpata type contractionsrdquoMathematics vol 6 no 2 p 25 2018

[19] J Ahmad H Aydi and N Mlaiki ldquoFuzzy fixed points of fuzzymappings via F-contractions and an applicationrdquo Journal ofIntelligent amp Fuzzy Systems vol 37 no 4 pp 5487ndash54932019

Journal of Function Spaces 9