ResearchArticle Nemytzki-Edelstein-Meir-Keeler Type Results in …downloads.hindawi.com/journals/ddns/2018/4745764.pdf · DiscreteDynamicsinNatureandSociety Ontheotherhand,Bakhtin[]andCzerwik[]intro-duced

Research ArticleNemytzki-Edelstein-Meir-Keeler Type Results in 119887-Metric Spaces

Hassen Aydi 12 Radoje BankoviT3 IvanMitroviT4 andMuhammad Nazam 5

1Department of Mathematics College of Education of Jubail Imam Abdulrahman Bin Faisal University PO 12020Industrial Jubail 31961 Saudi Arabia2Department of Medical Research China Medical University Hospital China Medical University Taichung Taiwan3Vojnogeografski Institut Beograd Vojska Srbije Serbia4First Technical School 35 000 Jagodina Serbia5Department of Mathematics and Statistics International Islamic University Islamabad Pakistan

Correspondence should be addressed to Hassen Aydi hmaydiuodedusa

Received 12 April 2018 Revised 5 June 2018 Accepted 19 June 2018 Published 4 July 2018

Academic Editor Pasquale Candito

Copyright copy 2018 Hassen Aydi et al This 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

We consider someNemytzki-Edelstein-Meir-Keeler type results in the context of b-metric spaces In some cases we assume that theb-metric is continuous Our results generalize several known ones in existing literatureWe also present some examples to illustratethe usability of our results

1 Definitions Notations and Preliminaries

Let (119883 119889) be a metric space and 119879 119883 997888rarr 119883 be a self-mapping The following Meir-Keeler conditions are wellknown For each 120576 gt 0 there exists 120575 = 120575(120576) gt 0 such that

120576 le 119889 (119909 119910) lt 120576 + 120575 implies 119889 (119879119909 119879119910) lt 120576 (1)

or

120576 lt 119889 (119909 119910) lt 120576 + 120575 implies 119889 (119879119909 119879119910) le 120576 (2)

or 119879 is contractive and

120576 le 119889 (119909 119910) lt 120576 + 120575 implies 119889 (119879119909 119879119910) le 120576 (3)

It is clear that (1) implies (2) and (2) implies (3) while theconverse is not true One says that the mapping 119879 defined onthe metric space (119883 119889) is contractive if 119889(119879119909 119879119910) lt 119889(119909 119910)holds whenever 119909 = 119910 For more details see [1] (pages 30-33 and 56-58) and [2] In 1969 Meir-Keeler [2] proved thefollowing

Theorem 1 ([2] Theorem) Let (119883 119889) be a complete metricspace and let 119879 be a self-mapping on 119883 satisfying (1) Then 119879has a unique fixed point say 119906 isin 119883 and for each 119909 isin 119883lim119899997888rarrinfin119879119899119909 = 119906

For other fixed point results via generalized Meir-Keelercontractions see [3ndash5] Inspired from Meir-Keeler theoremCiric proved the next slightly more general result

Theorem2 ([1]Theorem 25) Let (119883 119889) be a complete metricspace and let 119879 be a self-mapping on 119883 satisfying (2) Then 119879has a unique fixed point say 119906 isin 119883 and for each 119909 isin 119883lim119899997888rarrinfin119879119899119909 = 119906

The following example shows that Ciric result is a propergeneralization of Meir-Keeler theorem

Example 3 Let 119883 = [0 1] cup 2 3 + 13 sdot sdot sdot 3119899 minus 1 3119899 +13119899 sdot sdot sdot be a subset of reals with the Euclidean metric andlet 119879 be a self-mapping on119883 defined by

119879119909 = 0 if 0 le 119909 le 1 119909 = 2 sdot sdot sdot 3119899 minus 1 sdot sdot sdot (4)

119879119909 = 1 if 119909 = 3 + 13 6 +

16 sdot sdot sdot 3119899 + 1

3119899 sdot sdot sdot (5)

Then one can verify that 119879 satisfies (2) while it does notsatisfy the Meir-Keeler condition (1) For all details see [1]

Remark 4 Theorems 1 and 2 are true if the self-mapping119879 119883 997888rarr 119883 satisfies condition (3) For more details see[1] pages 30-33

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 4745764 7 pageshttpsdoiorg10115520184745764

2 Discrete Dynamics in Nature and Society

On the other hand Bakhtin [6] and Czerwik [7] intro-duced the concept of b-metric spaces (a generalization ofmetric spaces) and proved the Banach contraction principleThe definition of a b-metric space is the following

Definition 5 (Bakhtin [6] and Czerwik [7]) Let 119883 be anonempty set and let 119904 ge 1 be a given real number A function119889 119883times119883 997888rarr [0infin) is said to be a b-metric if and only if forall 119909 119910 119911 isin 119883 the following conditions are satisfied

(b1) 119889(119909 119910) = 0 if and only if 119909 = 119910(b2) 119889(119909 119910) = 119889(119910 119909)(b3) 119889(119909 119911) le 119904[119889(119909 119910) + 119889(119910 119911)]The triplet (119883 119889 119904 ge 1) is called a b-metric space with

coefficient 119904In the last period many authors obtained several fixed

point results for single-valued or set-valued mappings in thecontext of b-metric spaces For more details see [5 8ndash35] Itshould be noted that the class of b-metric spaces is effectivelylarger than that of standard metric spaces since a b-metricis a metric when 119904 = 1 The following example shows thatin general a b-metric does not necessarily need to be ametric

Example 6 Let (119883 120588) be a metric space and 119889(119909 119910) = (120588(119909119910))119901 where 119901 gt 1 is a real number Then 119889 is a b-metric with119904 = 2119901minus1 but 119889 is not a metric on119883

The concepts of b-convergence b-completeness b-Cauchy and b-closed set in b-metric spaces have beeninitiated in [6 7]

The following two lemmas are very significant in the classof b-metric spaces

Lemma 7 ([21] Lemma 31) Let 119910119899 be a sequence in a b-metric space (119883 119889 119904 ge 1) such that

119889 (119910119899 119910119899+1) le 120582119889 (119910119899minus1 119910119899) (6)

for some 120582 isin [0 1119904) and each 119899 = 1 2 sdot sdot sdot Then 119910119899 is ab-Cauchy sequence in a b-metric space (119883 119889)Lemma 8 ([30] Lemma 22) Let 119910119899 be a sequence in a b-metric space (119883 119889 119904 ge 1) such that

119889 (119910119899 119910119899+1) le 120582119889 (119910119899minus1 119910119899) (7)

for some 120582 isin [0 1) and each 119899 = 1 2 sdot sdot sdot Then 119910119899 is a b-Cauchy sequence in a b-metric space (119883 119889)

Since in general a b-metric is not continuous we need thefollowing two lemmas

Lemma 9 ([36] Lemma 21) Let (119883 119889 119904 ge 1) be a b-metricspace with 119904 ge 1 Suppose that 119909119899 and 119910119899 are b-convergentto 119909 and 119910 respectively Then

11199042 119889 (119909 119910) le lim inf

119899997888rarrinfin119889 (119909119899 119910119899) le lim sup

119899997888rarrinfin119889 (119909119899 119910119899)

le 1199042119889 (119909 119910) (8)

In particular if 119909 = 119910 then we have lim119899997888rarrinfin119889(119909119899 119910119899) = 0Moreover for each 119911 isin 119883 we have

1119904 119889 (119909 119911) le lim inf


119899997888rarrinfin119889 (119909119899 119911)

le 1199042119889 (119909 119911) (9)

Lemma 10 (see [37]) Let (119883 119889 119904 ge 1) be a b-metric space and119909119899 be a sequence in 119883 such that

lim119899997888rarrinfin

119889 (119909119899 119909119899+1) = 0 (10)

If 119909119899 is not b-Cauchy then there exist 120576 gt 0 and two sequen-ces 119898(119896) and 119899(119896) of positive integers such that for thefollowing four sequences

119889 (119909119898(119896) 119909119899(119896)) 119889 (119909119898(119896) 119909119899(119896)+1) 119889 (119909119898(119896)+1 119909119899(119896)) 119889 (119909119898(119896)+1 119909119899(119896)+1)

(11)

we have120576 le lim inf

119896997888rarrinfin119889 (119909119898(119896) 119909119899(119896))

le lim sup119896997888rarrinfin

119889 (119909119898(119896) 119909119899(119896)) le 120576119904

120576119904 le lim inf

119896997888rarrinfin119889 (119909119898(119896) 119909119899(119896)+1)


119889 (119909119898(119896) 119909119899(119896)+1) le 1205761199042

120576119904 le lim inf

119896997888rarrinfin119889 (119909119898(119896)+1 119909119899(119896))


119889 (119909119898(119896)+1 119909119899(119896)) le 1205761199042

1205761199042 le lim inf

119896997888rarrinfin119889 (119909119898(119896)+1 119909119899(119896)+1)


119889 (119909119898(119896)+1 119909119899(119896)+1) le 1205761199043

(12)

Essential to the proofs of fixed point theorems for themost contractive conditions in the context of b-metric spacesare the above two lemmas (see for example [3 5 9 13 17 2325 28]) However it is not hard to show that the proofs of themost fixed point theorems in the context of b-metric spacesbecome simpler and shorter if they are based on Lemma 8

2 Main Result

To our knowledge it is not known whether Meir-Keeler andCiric theorems hold in the context of b-metric spaces Alsoit is not known that if there are examples such that condition(1) or (2) or (3) holds in the context of b-metric spaces but 119879has no fixed point

Our first result generalizes Lemma 1 of [2] For someresults also see recent paper [38]

Discrete Dynamics in Nature and Society 3

Lemma 11 Let (119883 119889 119904 gt 1) be a b-complete b-metric spaceand 119879 119883 997888rarr 119883 such that condition (1) holds If 119879119899119909 is ab-Cauchy sequence for each 119909 isin 119883 then 119879 has a unique fixedpoint say 119906 isin 119883 and 119879119899119909 997888rarr 119906Proof Since (119883 119889 119904 gt 1) is b-complete each 119879119899119909 has a limitpoint say 120578(119909) Since condition (1) implies the continuity of119879 we have

119879 (120578 (119909)) = 119879 ( lim119899997888rarrinfin

119879119899 (119909)) = lim119899997888rarrinfin

119879119899+1 (119909)

= 120578 (119909) (13)

Thus 120578(119909) is a fixed point and therefore all 120578(119909) are equal

Remark 12 If condition (1) holds on 119887-metric spaces (119883119889 119904 gt 1) we do not know whether every sequence 119879119899119909 isb-Cauchy

However with a stronger condition than (1) we have apositive response It will be the subject of Theorem 13

Now we announce a Meir-Keeler type result in thecontext of b-metric spaces

Theorem 13 Let (119883 119889) be a complete b-metric space and let 119879be a self-mapping on 119883 satisfying the following condition

Given 120576 gt 0 there exists 120575 gt 0 such that120576 le 119889 (119909 119910) lt 120576 + 120575 implies 119904119886119889 (119879119909 119879119910) lt 120576 (14)

where 119886 gt 0 is givenThen 119879 has a unique fixed point say 119906 isin 119883 and for each

119909 isin 119883 lim119899997888rarrinfin119879119899119909 = 119906Proof It is clear that for all 119909 119910 isin 119883 with 119889(119909 119910) gt 0 weobtain

119889 (119879119909 119879119910) le 120582119889 (119909 119910) (15)

where 120582 = 1119904119886 isin [0 1)Let1199090 isin 119883 be an arbitrary point Define the sequence 119909119899

by 119909119899+1 = 119879119909119899 for all 119899 ge 0 If for some 119899 119909119899 = 119909119899+1 then 119909119899is a fixed point of119879 From now on suppose that 119909119899 = 119909119899+1 forall 119899 ge 0 From condition (15) we obtain

119889 (119909119899 119909119899+1) le 120582119889 (119909119899minus1 119909119899) (16)

Further according to ([30] Lemma 22) the sequence 119909119899 isb-Cauchy in the b-metric space (119883 119889) By b-completeness of(119883 119889) there exists 119906 isin 119883 such that


119909119899 = 119906 (17)

Finally (15) and (17) imply that119879119906 = 119906 that is 119906 is the uniquefixed point of 119879 in119883

Example 14 Let 119883 = 0 1 2 and define 119889 119883 times 119883 997888rarr[0 +infin) as follows 119889(119909 119909) = 0 119889(119909 119910) = 119889(119910 119909) for all119909 119910 isin 119883 119889(0 1) = 1 119889(0 2) = 22 and 119889(1 2) = 11 Then

(119883 119889 2221 gt 1) is a b-complete b-metric space but it is nota metric space Let 119879 119883 997888rarr 119883 be defined by

119879119909 =

0 if 119909 = 21 if 119909 = 2

(18)

We shall check that for all 119909 119910 isin 119883 the contractive condition(15) holds For this we distinguish three cases

(a) 119909 = 0 119910 = 1 997904rArr 119889(1198790 1198791) = 119889(0 0) = 0 Obviouslycondition (15) holds

(b) 119909 = 0 119910 = 2 997904rArr 119889(1198790 1198792) = 119889(0 1) Since(2221)119886119889(0 1) le 119889(0 2) ie (2221)119886 sdot 1 le 22 119886 gt 0 whichis true hence again (15) holds

(c) 119909 = 1 119910 = 2 997904rArr 119889(1198791 1198792) = 119889(0 1) = 1 Now wehave (2221)119886 sdot 1 le 11 ie (2221)119886 le 11 which is also truebecause 119886 gt 0

Therefore condition (15) holds for each 119886 gt 0 Howevercondition (14) is not true for 119886 = 1 Indeed for 119909 = 0 and119910 = 2 it becomes

120576 le 119889 (0 2) lt 120576 + 120575 implies 2221 lt 120576 (19)

or equivalently

120576 le 22 lt 120576 + 120575 implies 2221 lt 120576 (20)

Take 120576 = 12Then there exists 120575 = 120575(12) gt 0 such that 12 le22 lt 12 + 120575 (for example any 120575 gt 1710) But 2221 lt 12is false

Now we give an example supportingTheorem 13

Example 15 Let119883 = [0 1] 119889(119909 119910) = (119909 minus 119910)2Then (119883 119889 2)is a b-complete b-metric space Let 119879 119883 997888rarr 119883 be definedas 119879119909 = (14)1199092 Taking 120575 = 120576 we get for 119909 and 119910 satisfying120576 le 119889(119909 119910) lt 120576 + 120575 = 2120576119904 sdot 119889 (119879119909 119879119910) = 2 sdot 119889 (119879119909 119879119910)

= 2 sdot 116 (119909 minus 119910)2 (119909 + 119910)2 le 1

2 (119909 minus 119910)2

lt 120576

(21)

Hence all the conditions of Theorem 13 are satisfied Themapping 119879 has a unique fixed point which is 119906 = 0

LetG119904 be the class of all mappings 119892 [0infin) 997888rarr [0 1119904)which satisfy the condition 119892(119905119899) 997888rarr 1119904 whenever 119905119899 997888rarr0 Note that G119904 = 0 As an example consider the mapping119892 [0infin) 997888rarr [0 1119904) given by 119892(119905) = (1119904)119890minus119905 for 119905 gt 0 and119892(0) isin [0 1119904)

The following is Geraghty type result in the context ofb-metric spaces (see for instance [17] where authors useLemma 14)

Theorem 16 Let (119883 119889 119904 gt 1) be a complete b-metric spaceSuppose that the mapping 119879 119883 997888rarr 119883 satisfies the condition

119889 (119879119909 119879119910) le 119892 (119889 (119909 119910)) 119889 (119909 119910) (22)


for all 119909 119910 isin 119883 and some 119892 isin G119904 Then 119879 has a unique fixedpoint 119906 isin 119883 and for each 119909 isin 119883 the Picard sequence 119879119899119909converges to 119906 in (119883 119889 119904 gt 1)Proof Since 119892 [0infin) 997888rarr [0 1119904) we get

119889 (119879119909 119879119910) le 1119904 119889 (119909 119910) = 120582119889 (119909 119910) (23)

In view of 120582 isin (0 1) the result follows according to Lemma 8and condition (15)

It is well known that in compact metric spaces fixedpoint results can be obtained under the strict contractivecondition (119889(119879119909 119879119910) lt 119889(119909 119910) whenever 119909 = 119910) In the caseof b-metric spaces with a continuous b-metric the followingresults of Nemytzki and Edelstein can be obtained in the sameway as in the metric case (see [1] pages 56-58)

Theorem 17 Let (119883 119889 119904 gt 1) be a compact b-metric spacewith continuous b-metric 119889 and let 119879 119883 997888rarr 119883 be a self-mapping Suppose that the following condition holds

119889 (119879119909 119879119910) lt 119889 (119909 119910) for 119909 = 119910 (24)

Then 119879 has a unique fixed point say 119906 isin 119883 and for each 119909 isin119883 lim119899997888rarrinfin119879119899119909 = 119906Proof Define a function 119891 119883 997888rarr [0 +infin) by

119891 (119909) = 119889 (119909 119879119909) (25)

Since119879 is continuous119891 is also continuous So as (119883 119889 119904 gt 1)is a compact b-metric space there exists a point 119906 isin 119883 suchthat

119891 (119906) = 119889 (119906 119879119906) = min119909isin119883

119889 (119909 119879119909) (26)

If we assume that 119906 = 119879119906 then as 119879 is contractive (119889(119879119909119879119910) lt 119889(119909 119910) whether 119909 = 119910) one writes

119891 (119879119906) = 119889 (119879119906 119879119879119906) lt 119889 (119906 119879119906) = 119891 (119906) (27)

which is a contradictionTherefore 119906 is a fixed point of119879Theuniqueness is obvious

Theorem 18 Let (119883 119889 119904 gt 1) be a b-metric space with acontinuous b-metric 119889 and let 119879 119883 997888rarr 119883 be a self-mapSuppose that the following condition holds

119889 (119879119909 119879119910) lt 119889 (119909 119910) for 119909 = 119910 (28)

If there exists a point 1199090 isin 119883 such that the sequence 1198791198991199090contains a convergent subsequence 1198791198991198941199090 to 119906 then 119906 is theunique fixed point of 119879

Proof Consider the real sequence 119889(1198791198991199090 119879119899+11199090) If119879119896+11199090 = 1198791198961199090 for some 119896 isin N then 1198791198991199090 for 119899 ge 119896 is astationary sequence and so 1198791198961199090 = 119906 Thus 119879119896+11199090 = 1198791198961199090implies 119879119906 = 119906 Assume now that 119879119899+11199090 = 1198791198991199090 for all119899 isin NThen as 119879 is contractive 119889(1198791198991199090 119879119899+11199090) is a strictly

decreasing sequence of positive realsTherefore it convergesSince 1198791198991198941199090 997888rarr 119906 as 119894 997888rarr infin and 119879 is continuous we have


119879119899119894+11199090 = lim119894997888rarrinfin

1198791198791198991198941199090 = 119879119906


119879119899119894+21199090 = 1198792119906(29)

Thus


119889 (1198791198991198941199090 119879119899119894+11199090) = 119889 (119906 119879119906)


119889 (119879119899119894+11199090 119879119899119894+21199090) = 119889 (119879119906 1198792119906) (30)

Since 119889(1198791198991198941199090 119879119899119894+11199090) and 119889(119879119899119894+11199090 119879119899119894+21199090) are subse-quences of the convergent sequence 119889(1198791198991199090 119879119899+11199090) theyhave the same limit Therefore

119889 (119906 119879119906) = 119889 (119879119906 1198792119906) (31)

Hence 119879119906 = 119906 If not as 119879 is contractive we have

119889 (119879119906 1198792119906) lt 119889 (119906 119879119906) (32)

which is a contradiction

Remark 19 The two previous theorems are known in litera-ture as Nemytzki and Edelstein theorems respectively It isclear that Edelstein theorem extends the result of Nemytzki

In the sequel we consider 120576minuscontractive mappings in thecontext of b-metric spaces Namely we first introduce thefollowing

Definition 20 A mapping 119879 of a b-metric space (119883 119889 119904 ge 1)into itself is said to be 120576minuscontractive if and only if there exists120576 gt 0 such that

0 lt 119889 (119909 119910) lt 120576 implies 119889 (119879119909 119879119910) lt 119889 (119909 119910) (33)

The following results extend ones from standard metricspaces to b-metric spaces with a continuous b-metric 119889Theorem 21 Let (119883 119889 119904 ge 1) be a b-metric space withcontinuous b-metric 119889 and 119879 an 120576minuscontractive self-mapping on119883 If for some 1199090 isin 119883 the sequence 1198791198991199090 has a convergentsubsequence 1198791198991198941199090 to 119906 isin 119883 then 119906 is a periodic point that isthere exists a positive integer 119896 such that 119879119896119906 = 119906Proof Since 1198791198991199090 has a cluster point there exist positiveintegers 119901 and 119902 with 119901 lt 119902 such that 119889(1198791199011199090 1198791199021199090) lt 120576 thatis 119889(1198791199011199090 1198791198961198791199011199090) lt 120576 where 119896 = 119902 minus 119901Then the sequence119889(1198791198991199090 119879119899+1198961199090)+infin119899=119901 is nonincreasing due to the fact that119879 is120576minuscontractive Thus this sequence converges and so


119889 (1198791198991199090 119879119899+1198961199090) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)

= lim119894997888rarrinfin

119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (34)


Since 119879 and the b-metric 119889 are both continuous we have

119889 (119906 119879119896119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(35)

and

119889 (119879119906 119879119879119896119906) = lim119894997888rarrinfin

119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (36)

Thus

119889 (119879119906 119879119879119896119906) = 119889 (119906 119879119896119906) lt 120576 (37)

Hence 119879119896119906 = 119906 Otherwise as 119879 is 120576minuscontractive we wouldhave 119889(119879119906 119879119879119896119906) lt 119889(119906 119879119896119906)

Now consider a class of mappings 119879 of a b-metric space(119883 119889 119904 ge 1) into itself which satisfy the following condition

For every 119909 119910 isin 119883 there exists a positive integer 119896(119909 119910)such that

0 lt 119889 (119909 119910) implies 119889 (119879119899119909 119879119899119910) lt 119889 (119909 119910)forall119899 ge 119896 (119909 119910)

(38)

A mapping 119879 satisfying (38) is called an eventually contrac-tive mapping

It is obvious that any contractive mapping is eventuallycontractive (it satisfies (38) with 119896(119909 119910) = 1) but theimplication is not reverse

Contractive and 120576minuscontractive mappings are continuousHowever eventually contractive mappings need not be con-tinuous nor orbitally continuous Recall that a mapping 119879 issaid to be orbitally continuous if for each 119909 isin 119883 119879119899119894119909 997888rarr119906 isin 119883 implies 119879119879119899119894119909 997888rarr 119879119906

Now we announce the next result

Theorem 22 Let (119883 119889 119904 ge 1) be a b-metric space withcontinuous b-metric 119889 and 119879 119883 997888rarr 119883 be an eventuallycontractive and orbitally continuous mapping If for some 1199090 isin119883 the sequence of iterates 1198791198991199090 has a subsequence 1198791198991198941199090converging to 119906 isin 119883 then 119906 is the unique fixed point of 119879 andlim119899997888rarrinfin1198791198991199090 = 119906Proof If 119889(11987911989901199090 1198791198990+11199090) = 0 for some 1198990 isin N then 119879119899119909 =11987911989901199090 for all 119899 ge 1198990Thus 11987911989901199090 = 119906 and so 1198791198990+11199090 = 11987911989901199090implies that 119879119906 = 119906

Assume now that 119889(11987911989901199090 1198791198990+11199090) gt 0 for all 119899 isin NConsider 119906 = lim119894997888rarrinfin1198791198991198941199090 Since 119879 is orbitally continuousfor any fixed positive integer 119901 we have


119889 (119879119899119894+1199011199090 119879119899119894+119901+11199090) = 119889 (119879119901119906 119879119901+1119906) (39)

Assume that 119879119906 = 119906 As 119879 is eventually contractive thereexists 119903 isin N such that

119889 (119879119903119906 119879119903119879119906) lt 119889 (119906 119879119906) (40)

Hence 1205760 = (12)[119889(119906 119879119906) minus 119889(119879119903119906 119879119903119879119906)] gt 0 Sincelim119894997888rarrinfin

119889 (119879119899119894+1199031199090 119879119899119894+119903+11199090) = 119889 (119879119903119906 119879119879119906) (41)

for arbitrary 120576 gt 0 there exists a sufficiently large 119899119894 = 119902 suchthat

119889 (119879119902+1199031199090 119879119902+119903+11199090) lt 119889 (119879119903119906119879r119879119906) + 120576 (42)

For 120576 = 1205760 we have119889 (119879119902+1199031199090 119879119902+119903+11199090)

lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(43)

Since 119889(11987911989901199090 1198791198990+11199090) gt 0 for all 119899 isin N and 119879 is even-tually contractive there exists some positive integer 119896 =119896(119879119902+1199031199090 119879119902+119903+11199090) such that

119889 (119879119899119879119902+1199031199090 119879119899119879119902+119903+11199090) lt 119889 (119879119902+1199031199090 119879119902+119903+11199090)forall119899 ge 119896

(44)

that is 119889(1198791198991199090 119879119899+11199090) lt 119889(119879119902+1199031199090 119879119902+119903+11199090) for all 119899 ge 119896 +119902 + 119903 So we obtain that

119889 (1198791198991199090 119879119899+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899 ge 119896 + 119902 + 119903(45)

Hence

119889 (1198791198991198941199090 119879119899119894+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899119894 ge 119896 + 119902 + 119903(46)

Thus we get

119889 (119906 119879119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+11199090)

le 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(47)

Hence

119889 (119906 119879119906) le 119889 (119879119903119906 119879119903119879119906) (48)

which contradicts the choice of 119903 Therefore 119889(119906 119879119906) = 0that is 119879119906 = 119906

Now we show that lim119899infin1198791198991199090 = 119906 Let 120576 gt 0 be arbitrarySince 1198791198991198941199090 997888rarr 119906 as 119894 997888rarr infin there exists a sufficientlylarge 119899119894 = 119897 such that 119889(119906 1198791198971199090) lt 120576 If 1198791198991198941199090 = 119906 then119879119897+11199090 = 119879119906 = 119906 and hence 1198791198991199090 = 119906 for all 119899 ge 119897 Assumethat 119889(119906 1198791198971199090) gt 0 As 119879 is eventually 120576minuscontractive and 119889(1199061198791198971199090) lt 120576 there is 119896 = 119896(119906 1198791198971199090) isin N such that

119889 (119879119899119906 1198791198991198791198971199090) lt 119889 (119906 1198791198971199090) forall119899 ge 119896 (49)

Since 119879119906 = 119906 we have119889 (119906 1198791198991199090) = 119889 (119879119899minus119897119906 119879119899minus1198971198791198971199090) lt 119889 (119906 119879119897119906) lt 120576

forall119899 ge 119896 + 119897(50)

Therefore lim119899997888rarrinfin1198791198991199090 = 119906


Corollary 23 If (119883 119889 119904 ge 1) is a compact b-metric space witha continuous b-metric 119889 and 119879 is a continuous and eventuallycontractive self-mapping on (119883 119889) then 119879 has a unique fixedpoint say 119906 isin 119883 Also for every 119909 isin 119883 lim119899997888rarrinfin119879119899119909 = 119906

Amapping 119879 of a b-metric space (119883 119889 119904 ge 1) into itself issaid to be an eventually 120576minuscontractive mapping if there exists120576 gt 0 such that for every 119909 119910 isin 119883 there is a positive integer119896(119909 119910) such that0 lt 119889 (119909 119910) lt 120576 implies 119889 (119879119899119909 119879119899119910) lt 119889 (119909 119910)

forall119899 ge 119896 (119909 119910) (51)

For eventually 120576minuscontractive mappings in the context of b-metric spaces we announce the next result

Theorem 24 Let (119883 119889 119904 ge 1) be a b-metric space with acontinuous b-metric 119889 and 119879 119883 997888rarr 119883 be an eventually120576minuscontractive and orbitally continuous self-map on (119883 119889) Iffor some 1199090 isin 119883 the sequence 1198791198991199090 has a convergentsubsequence 1198791198991198941199090 to 119906 isin 119883 then 119906 is a periodic point of119879Proof The proof is very similar to the ones in the previoustheorem Therefore it is omitted

Corollary 25 If (119883 119889 119904 ge 1) is a compact b-metric space witha continuous b-metric 119889 and 119879 119883 997888rarr 119883 is an eventually120576minuscontractive and continuous self-map on (119883 119889) then the setof periodic points of 119879 is not empty

The following two results extend ones from standardmetric spaces to b-metric spaces (see [15])

Theorem 26 Let (119883 119889 119904 ge 1) be a compact b-metric spacewith continuous b-metric 119889 and 119879 a continuous self-map on119883 such that for every 119909 119910 isin 119883 there exists a positive integer119899(119909 119910) such that

0 lt 119889 (119909 119910) implies 119889 (119879119899(119909119910)119909 119879119899(119909119910)119910) lt 119889 (119909 119910) (52)Then 119879 has a unique fixed point

Proof Define 119891 119883 997888rarr [0 +infin) by 119891(119909) = 119889(119909 119879119909) Since119889 and 119879 are continuous 119891 is also continuous Therefore 119891attains its minimum on 119883 in some point say 119906 isin 119883 Assumethat 119891(119906) = 119889(119906 119879119906) gt 0 Then by hypothesis there exists apositive integer 119899(119906 119879119906) such that

119889 (119879119899(119906119879119906)119906 119879119899(119906119879119906)119879119906) lt 119889 (119906 119879119906) (53)

that is 119891(119879119899(119906119879119906)119906) lt 119891(119906) which contradicts the choice of119906Therefore 119879119906 = 119906Theorem 27 Let (119883 119889 119904 ge 1) be a compact b-metric spacewith a continuous b-metric 119889 and 119879 be a continuous self-mapon119883 such that for every 119909 119910 isin 119883

0 lt 119889 (119909 119910) lt 120576 implies 119889 (119879119899(119909119910)119909 119879119899(119909119910)119910)lt 119889 (119909 119910) 119899 (119909 119910) isin N

(54)

Then the set of periodic points of 119879 is not empty

Proof Let 1199090 isin 119883 Since (119883 119889) is compact 1198791198991199090 has acluster point Therefore there exist positive integers 119901 and 119896such that

119889 (1198791199011199090 1198791198961198791199011199090) = 119889 (1198791199011199090 119879119896+1199011199090) lt 120576 (55)

Define 119891(1199090) = 119889(1199090 1198791198961199090) Since 119879 is continuous 119879119896 iscontinuous and so 119891 is continuous Therefore there is some119906 isin 119883 such that 119891(119906) = min119891(119909) 119909 isin 119883Then

119891 (119906) = 119889 (119906 119879119896119906) = min 119889 (119909 119879119896119909) 119909 isin 119883

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(56)

Assume that 119879119896119906 = 119906 Then as 119889(119906 119879119896119906) lt 120576 by hypothesisthere exists 119899 = 119899(119906 119879119896119906) isin N such that

119889 (119879119899119906 119879119899119879119896119906) lt 119889 (119906 119879119896119906) (57)

Thus 119891(119879119899119906) lt 119891(119906) which contradicts the choice of 119906Therefore 119879119896119906 = 119906

Data Availability

No data is used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interestregarding the publication of this paper

References

[1] L Ciric Some recent results in metrical fixed point theoryBeograd 2003

[2] A Meir and E Keeler ldquoA theorem on contraction mappingsrdquoJournal of Mathematical Analysis and Applications vol 28 pp326ndash329 1969

[3] T Abdeljawad H Aydi and E Karapinar ldquoCoupled fixed pointsfor Meir-Keeler contractions in ordered partial metric spacesrdquoMathematical Problems in Engineering vol 2012 Article ID327273 20 pages 2012

[4] H Aydi E Karapınar and C Vetro ldquoMeir-Keeler type contrac-tions for tripled fixed pointsrdquo Acta Mathematica Scientia vol32 no 6 pp 2119ndash2130 2012

[5] Z Mustafa H Aydi and E Karapınar ldquoGeneralized Meir-Keeler type contractions on 119866-metric spacesrdquo Applied Mathe-matics and Computation vol 219 no 21 pp 10441ndash10447 2013

[6] I A Bakhtin ldquoThe contraction mapping principle in almostmetric spacerdquo Functional Analysis and its Applications vol 30pp 26ndash37 1989

[7] S Czerwik ldquoContraction mappings in 119887-metric spacesrdquo ActaMathematica et Informatica Universitatis Ostraviensis vol 1 pp5ndash11 1993

[8] M Abbas I Z Chema and A Razani ldquoExistence of commonfixed point for 119887-metric rational type contractionrdquo Filomat vol30 no 6 pp 1413ndash1439 2016

[9] H Afshari S Kalantari and H Aydi ldquoFixed point results forgeneralized (120572-120595)-Suzuki-contractions in quasi b-metric-likespacesrdquo Asian-European Journal of Mathematics vol 11 no 1article 1850012 12 pages 2018


[10] A H Ansari M A Barakat and H Aydi ldquoNew approach forcommon fixed point theorems via C-class functions in Gp-metric spacesrdquo Journal of Functions Spaces vol 2017 Article ID2624569 9 pages 2017

[11] H AydiM-F Bota E Karapinar and SMitrovic ldquoA fixed pointtheorem for set-valued quasi-contractions in b-metric spacesrdquoFixed Point Theory and Applications vol 2012 article 88 2012

[12] H Aydi M-F Bota E Karapinar and S Moradi ldquoA commonfixed point for weak f-contractions on b-metric spacesrdquo FixedPoint Theory and Applications vol 13 no 2 pp 337ndash346 2012

[13] H Aydi A Felhi and S Sahmim ldquoCommon fixed pointsvia implicit contractions on b-metric-like spacesrdquo Journal ofNonlinear Sciences and Applications vol 10 no 4 pp 1524ndash15372017

[14] G V Babu and T M Dula ldquoFixed points of generalized TAC-contractive mappings in b-metric spacesrdquo Matematicki Vesnikvol 69 no 2 pp 75ndash88 2017

[15] D F Bailey ldquoSome theorems on contractive mappingsrdquo JournalOf The London Mathematical Society-Second Series vol 41 pp101ndash106 1966

[16] S ChandokM Jovanovic and S Radenovic ldquoOrdered b-metricspaces and Geraghty type contractive mappingsrdquo Vojnotehnickiglasnik vol 65 no 2 pp 331ndash345 2017

[17] C Chifu and G Petrusel ldquoFixed point results for multivaluedhardyndashrogers contractions in b-metric spacesrdquo Filomat vol 31no 8 pp 2499ndash3507 2017

[18] T Dosenovic M Pavlovic and S Radenovic ldquoContractiveconditions in b-metric spacesrdquoVojnotehnicki glasnik vol 65 no4 pp 851ndash865 2017

[19] A K Dubey R Shukla and R P Dubey ldquoSome fixed pointresults in b-metric spacesrdquoAsian Journal ofMathematics articleama0147 2014

[20] H Faraji and K Nourouzi ldquoA generalization of Kannan andChatterjea fixed point theorem on complete b-metric spacesrdquoSahand Communications in Mathematical Analysis (SCMA)vol 6 no 1 pp 77ndash86 2017

[21] M Jovanovic Z Kadelburg and S Radenovic ldquoCommon fixedpoint results in metric-type spacesrdquo Fixed Point Theory andApplications vol 2010 Article ID 978121 15 pages 2010

[22] M Jovanovic Contribution to the theory of abstract metricspaces Doctoral Dissertation [Doctoral thesis] Belgrade 2016

[23] H Huang J Vujakovic and S Radenovic ldquoA note on com-mon fixed point theorems for isotone increasing mappings inordered b-metric spacesrdquo Journal of Nonlinear Sciences andApplications vol 8 no 5 pp 808ndash815 2015

[24] N Hussain and Z D Mitrovic ldquoOn multi-valued weak quasi-contractions in b-metric spacesrdquo Journal of Nonlinear Sciencesand Applications JNSA vol 10 no 7 pp 3815ndash3823 2017

[25] N Hussain Z D Mitrovic and S Radenovic ldquoA commonfixed point theorem of Fisher in b-metric spacesrdquo Revista de laReal Academia de Ciencias Exactas Fısicas y Naturales Serie AMatematicas pp 1ndash8 2018

[26] E Karap nar S Czerwik and H Aydi ldquo(120572120595)-Meir-KeelerContraction Mappings in Generalized -Metric Spacesrdquo Journalof Function spaces vol 201 Article ID 3264620 4 pages 2018

[27] M Kir and H Kizitune ldquoOn some well known fixed pointtheorems inb-metric spacesrdquo Turkish Journal of Analysis andNumber Theory vol 1 pp 13ndash16 2013

[28] W A Kirk and N Shahzad Fixed Point Theory in DistanceSpaces Springer International Publishing Switzerland 2014

[29] R Koleva and B Zlatanov ldquoOn fixed points for Chatterjeasmaps in b-metric spacesrdquo Turkish Journal of Analysis and vol4 no 2 pp 31ndash34 2016

[30] R Miculescu and A Mihail ldquoNew fixed point theorems for set-valued contractions in 119887-metric spacesrdquo Journal of Fixed PointTheory and Applications vol 19 no 3 pp 2153ndash2163 2017

[31] P Kumar Mishra S Sachdeva and S K Banerjee ldquoSome FixedPoint Theorems in b-metric Spacerdquo Turkish Journal of Analysisand Number Theory vol 2 no 1 pp 19ndash22 2014

[32] Z Mustafa M M Jaradat H Aydi and A Alrhayyel ldquoSomecommon fixed points of six mappings on 119866119887-metric spacesusing (119864119860) propertyrdquo European Journal of Pure and AppliedMathematics vol 11 no 1 pp 90ndash109 2018

[33] S L Singh S Czerwik K Krol and A Singh ldquoCoincidencesand fixed points of hybrid contractionsrdquo Tamsui Oxford Journalof Information andMathematical Sciences vol 24 no 4 pp 401ndash416 2008

[34] T Suzuki ldquoBasic inequality on a 119887-metric space and its applica-tionsrdquo Journal of Inequalities and Applications article 256 2017

[35] K Zare and R Arab ldquoCommon fixed point results for infinitefamilies in partially ordered 119887-metric spaces and applicationsrdquoElectronic Journal of Mathematical Analysis and ApplicationsEJMAA vol 4 no 2 pp 56ndash67 2016

[36] A Aghajani M Abbas and J R Roshan ldquoCommon fixed pointof generalized weak contractive mappings in partially orderedb-metric spacesrdquo Mathematica Slovaca vol 64 no 4 pp 941ndash960 2014

[37] H-SDingM Imdad S Radenovic and J Vujakovic ldquoOn somefixed point results in b-metric rectangular and b-rectangularmetric spacesrdquo Arab Journal of Mathematical Sciences vol 22pp 151ndash164 2016

[38] R K Bisht and V Rakocevicrsquo ldquoGeneralized Meir-Keeler typecontractions and discontinuity at fixed pointrdquo Fixed PointTheory An International Journal on Fixed Point Theory Com-putation and Applications vol 19 no 1 pp 57ndash64 2018

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Submit your manuscripts atwwwhindawicom


On the other hand Bakhtin [6] and Czerwik [7] intro-duced the concept of b-metric spaces (a generalization ofmetric spaces) and proved the Banach contraction principleThe definition of a b-metric space is the following

Definition 5 (Bakhtin [6] and Czerwik [7]) Let 119883 be anonempty set and let 119904 ge 1 be a given real number A function119889 119883times119883 997888rarr [0infin) is said to be a b-metric if and only if forall 119909 119910 119911 isin 119883 the following conditions are satisfied

(b1) 119889(119909 119910) = 0 if and only if 119909 = 119910(b2) 119889(119909 119910) = 119889(119910 119909)(b3) 119889(119909 119911) le 119904[119889(119909 119910) + 119889(119910 119911)]The triplet (119883 119889 119904 ge 1) is called a b-metric space with

coefficient 119904In the last period many authors obtained several fixed

point results for single-valued or set-valued mappings in thecontext of b-metric spaces For more details see [5 8ndash35] Itshould be noted that the class of b-metric spaces is effectivelylarger than that of standard metric spaces since a b-metricis a metric when 119904 = 1 The following example shows thatin general a b-metric does not necessarily need to be ametric

Example 6 Let (119883 120588) be a metric space and 119889(119909 119910) = (120588(119909119910))119901 where 119901 gt 1 is a real number Then 119889 is a b-metric with119904 = 2119901minus1 but 119889 is not a metric on119883

The concepts of b-convergence b-completeness b-Cauchy and b-closed set in b-metric spaces have beeninitiated in [6 7]

The following two lemmas are very significant in the classof b-metric spaces

Lemma 7 ([21] Lemma 31) Let 119910119899 be a sequence in a b-metric space (119883 119889 119904 ge 1) such that

119889 (119910119899 119910119899+1) le 120582119889 (119910119899minus1 119910119899) (6)

for some 120582 isin [0 1119904) and each 119899 = 1 2 sdot sdot sdot Then 119910119899 is ab-Cauchy sequence in a b-metric space (119883 119889)Lemma 8 ([30] Lemma 22) Let 119910119899 be a sequence in a b-metric space (119883 119889 119904 ge 1) such that

119889 (119910119899 119910119899+1) le 120582119889 (119910119899minus1 119910119899) (7)

for some 120582 isin [0 1) and each 119899 = 1 2 sdot sdot sdot Then 119910119899 is a b-Cauchy sequence in a b-metric space (119883 119889)

Since in general a b-metric is not continuous we need thefollowing two lemmas

Lemma 9 ([36] Lemma 21) Let (119883 119889 119904 ge 1) be a b-metricspace with 119904 ge 1 Suppose that 119909119899 and 119910119899 are b-convergentto 119909 and 119910 respectively Then

11199042 119889 (119909 119910) le lim inf


119899997888rarrinfin119889 (119909119899 119910119899)

le 1199042119889 (119909 119910) (8)

In particular if 119909 = 119910 then we have lim119899997888rarrinfin119889(119909119899 119910119899) = 0Moreover for each 119911 isin 119883 we have

1119904 119889 (119909 119911) le lim inf


119899997888rarrinfin119889 (119909119899 119911)

le 1199042119889 (119909 119911) (9)

Lemma 10 (see [37]) Let (119883 119889 119904 ge 1) be a b-metric space and119909119899 be a sequence in 119883 such that


119889 (119909119899 119909119899+1) = 0 (10)

If 119909119899 is not b-Cauchy then there exist 120576 gt 0 and two sequen-ces 119898(119896) and 119899(119896) of positive integers such that for thefollowing four sequences

119889 (119909119898(119896) 119909119899(119896)) 119889 (119909119898(119896) 119909119899(119896)+1) 119889 (119909119898(119896)+1 119909119899(119896)) 119889 (119909119898(119896)+1 119909119899(119896)+1)

(11)

we have120576 le lim inf

119896997888rarrinfin119889 (119909119898(119896) 119909119899(119896))


119889 (119909119898(119896) 119909119899(119896)) le 120576119904

120576119904 le lim inf

119896997888rarrinfin119889 (119909119898(119896) 119909119899(119896)+1)


119889 (119909119898(119896) 119909119899(119896)+1) le 1205761199042

120576119904 le lim inf

119896997888rarrinfin119889 (119909119898(119896)+1 119909119899(119896))


119889 (119909119898(119896)+1 119909119899(119896)) le 1205761199042

1205761199042 le lim inf

119896997888rarrinfin119889 (119909119898(119896)+1 119909119899(119896)+1)


119889 (119909119898(119896)+1 119909119899(119896)+1) le 1205761199043

(12)

Essential to the proofs of fixed point theorems for themost contractive conditions in the context of b-metric spacesare the above two lemmas (see for example [3 5 9 13 17 2325 28]) However it is not hard to show that the proofs of themost fixed point theorems in the context of b-metric spacesbecome simpler and shorter if they are based on Lemma 8

2 Main Result

To our knowledge it is not known whether Meir-Keeler andCiric theorems hold in the context of b-metric spaces Alsoit is not known that if there are examples such that condition(1) or (2) or (3) holds in the context of b-metric spaces but 119879has no fixed point

Our first result generalizes Lemma 1 of [2] For someresults also see recent paper [38]



119879 (120578 (119909)) = 119879 ( lim119899997888rarrinfin

119879119899 (119909)) = lim119899997888rarrinfin

119879119899+1 (119909)

= 120578 (119909) (13)









119889 (119879119909 119879119910) le 120582119889 (119909 119910) (15)



119889 (119909119899 119909119899+1) le 120582119889 (119909119899minus1 119909119899) (16)



119909119899 = 119906 (17)




119879119909 =

0 if 119909 = 21 if 119909 = 2

(18)






120576 le 119889 (0 2) lt 120576 + 120575 implies 2221 lt 120576 (19)

or equivalently

120576 le 22 lt 120576 + 120575 implies 2221 lt 120576 (20)




= 2 sdot 116 (119909 minus 119910)2 (119909 + 119910)2 le 1

2 (119909 minus 119910)2

lt 120576

(21)





119889 (119879119909 119879119910) le 119892 (119889 (119909 119910)) 119889 (119909 119910) (22)



119889 (119879119909 119879119910) le 1119904 119889 (119909 119910) = 120582119889 (119909 119910) (23)




119889 (119879119909 119879119910) lt 119889 (119909 119910) for 119909 = 119910 (24)


119891 (119909) = 119889 (119909 119879119909) (25)


119891 (119906) = 119889 (119906 119879119906) = min119909isin119883

119889 (119909 119879119909) (26)


119891 (119879119906) = 119889 (119879119906 119879119879119906) lt 119889 (119906 119879119906) = 119891 (119906) (27)



119889 (119879119909 119879119910) lt 119889 (119909 119910) for 119909 = 119910 (28)





119879119899119894+11199090 = lim119894997888rarrinfin

1198791198791198991198941199090 = 119879119906


119879119899119894+21199090 = 1198792119906(29)

Thus


119889 (1198791198991198941199090 119879119899119894+11199090) = 119889 (119906 119879119906)


119889 (119879119899119894+11199090 119879119899119894+21199090) = 119889 (119879119906 1198792119906) (30)


119889 (119906 119879119906) = 119889 (119879119906 1198792119906) (31)


119889 (119879119906 1198792119906) lt 119889 (119906 119879119906) (32)





0 lt 119889 (119909 119910) lt 120576 implies 119889 (119879119909 119879119910) lt 119889 (119909 119910) (33)



119889 (1198791198991199090 119879119899+1198961199090) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)


119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (34)



119889 (119906 119879119896119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(35)

and

119889 (119879119906 119879119879119896119906) = lim119894997888rarrinfin

119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (36)

Thus

119889 (119879119906 119879119879119896119906) = 119889 (119906 119879119896119906) lt 120576 (37)





(38)








119889 (119879119899119894+1199011199090 119879119899119894+119901+11199090) = 119889 (119879119901119906 119879119901+1119906) (39)


119889 (119879119903119906 119879119903119879119906) lt 119889 (119906 119879119906) (40)


119889 (119879119899119894+1199031199090 119879119899119894+119903+11199090) = 119889 (119879119903119906 119879119879119906) (41)


119889 (119879119902+1199031199090 119879119902+119903+11199090) lt 119889 (119879119903119906119879r119879119906) + 120576 (42)

For 120576 = 1205760 we have119889 (119879119902+1199031199090 119879119902+119903+11199090)

lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(43)


119889 (119879119899119879119902+1199031199090 119879119899119879119902+119903+11199090) lt 119889 (119879119902+1199031199090 119879119902+119903+11199090)forall119899 ge 119896

(44)


119889 (1198791198991199090 119879119899+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899 ge 119896 + 119902 + 119903(45)

Hence

119889 (1198791198991198941199090 119879119899119894+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899119894 ge 119896 + 119902 + 119903(46)

Thus we get

119889 (119906 119879119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+11199090)

le 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(47)

Hence

119889 (119906 119879119906) le 119889 (119879119903119906 119879119903119879119906) (48)



119889 (119879119899119906 1198791198991198791198971199090) lt 119889 (119906 1198791198971199090) forall119899 ge 119896 (49)


forall119899 ge 119896 + 119897(50)





forall119899 ge 119896 (119909 119910) (51)








119889 (119879119899(119906119879119906)119906 119879119899(119906119879119906)119879119906) lt 119889 (119906 119879119906) (53)



(54)



119889 (1198791199011199090 1198791198961198791199011199090) = 119889 (1198791199011199090 119879119896+1199011199090) lt 120576 (55)


119891 (119906) = 119889 (119906 119879119896119906) = min 119889 (119909 119879119896119909) 119909 isin 119883

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(56)


119889 (119879119899119906 119879119899119879119896119906) lt 119889 (119906 119879119896119906) (57)


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018










119879 (120578 (119909)) = 119879 ( lim119899997888rarrinfin

119879119899 (119909)) = lim119899997888rarrinfin

119879119899+1 (119909)

= 120578 (119909) (13)









119889 (119879119909 119879119910) le 120582119889 (119909 119910) (15)



119889 (119909119899 119909119899+1) le 120582119889 (119909119899minus1 119909119899) (16)



119909119899 = 119906 (17)




119879119909 =

0 if 119909 = 21 if 119909 = 2

(18)






120576 le 119889 (0 2) lt 120576 + 120575 implies 2221 lt 120576 (19)

or equivalently

120576 le 22 lt 120576 + 120575 implies 2221 lt 120576 (20)




= 2 sdot 116 (119909 minus 119910)2 (119909 + 119910)2 le 1

2 (119909 minus 119910)2

lt 120576

(21)





119889 (119879119909 119879119910) le 119892 (119889 (119909 119910)) 119889 (119909 119910) (22)



119889 (119879119909 119879119910) le 1119904 119889 (119909 119910) = 120582119889 (119909 119910) (23)




119889 (119879119909 119879119910) lt 119889 (119909 119910) for 119909 = 119910 (24)


119891 (119909) = 119889 (119909 119879119909) (25)


119891 (119906) = 119889 (119906 119879119906) = min119909isin119883

119889 (119909 119879119909) (26)


119891 (119879119906) = 119889 (119879119906 119879119879119906) lt 119889 (119906 119879119906) = 119891 (119906) (27)



119889 (119879119909 119879119910) lt 119889 (119909 119910) for 119909 = 119910 (28)





119879119899119894+11199090 = lim119894997888rarrinfin

1198791198791198991198941199090 = 119879119906


119879119899119894+21199090 = 1198792119906(29)

Thus


119889 (1198791198991198941199090 119879119899119894+11199090) = 119889 (119906 119879119906)


119889 (119879119899119894+11199090 119879119899119894+21199090) = 119889 (119879119906 1198792119906) (30)


119889 (119906 119879119906) = 119889 (119879119906 1198792119906) (31)


119889 (119879119906 1198792119906) lt 119889 (119906 119879119906) (32)





0 lt 119889 (119909 119910) lt 120576 implies 119889 (119879119909 119879119910) lt 119889 (119909 119910) (33)



119889 (1198791198991199090 119879119899+1198961199090) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)


119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (34)



119889 (119906 119879119896119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(35)

and

119889 (119879119906 119879119879119896119906) = lim119894997888rarrinfin

119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (36)

Thus

119889 (119879119906 119879119879119896119906) = 119889 (119906 119879119896119906) lt 120576 (37)





(38)








119889 (119879119899119894+1199011199090 119879119899119894+119901+11199090) = 119889 (119879119901119906 119879119901+1119906) (39)


119889 (119879119903119906 119879119903119879119906) lt 119889 (119906 119879119906) (40)


119889 (119879119899119894+1199031199090 119879119899119894+119903+11199090) = 119889 (119879119903119906 119879119879119906) (41)


119889 (119879119902+1199031199090 119879119902+119903+11199090) lt 119889 (119879119903119906119879r119879119906) + 120576 (42)

For 120576 = 1205760 we have119889 (119879119902+1199031199090 119879119902+119903+11199090)

lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(43)


119889 (119879119899119879119902+1199031199090 119879119899119879119902+119903+11199090) lt 119889 (119879119902+1199031199090 119879119902+119903+11199090)forall119899 ge 119896

(44)


119889 (1198791198991199090 119879119899+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899 ge 119896 + 119902 + 119903(45)

Hence

119889 (1198791198991198941199090 119879119899119894+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899119894 ge 119896 + 119902 + 119903(46)

Thus we get

119889 (119906 119879119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+11199090)

le 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(47)

Hence

119889 (119906 119879119906) le 119889 (119879119903119906 119879119903119879119906) (48)



119889 (119879119899119906 1198791198991198791198971199090) lt 119889 (119906 1198791198971199090) forall119899 ge 119896 (49)


forall119899 ge 119896 + 119897(50)





forall119899 ge 119896 (119909 119910) (51)








119889 (119879119899(119906119879119906)119906 119879119899(119906119879119906)119879119906) lt 119889 (119906 119879119906) (53)



(54)



119889 (1198791199011199090 1198791198961198791199011199090) = 119889 (1198791199011199090 119879119896+1199011199090) lt 120576 (55)


119891 (119906) = 119889 (119906 119879119896119906) = min 119889 (119909 119879119896119909) 119909 isin 119883

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(56)


119889 (119879119899119906 119879119899119879119896119906) lt 119889 (119906 119879119896119906) (57)


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018










119889 (119879119909 119879119910) le 1119904 119889 (119909 119910) = 120582119889 (119909 119910) (23)




119889 (119879119909 119879119910) lt 119889 (119909 119910) for 119909 = 119910 (24)


119891 (119909) = 119889 (119909 119879119909) (25)


119891 (119906) = 119889 (119906 119879119906) = min119909isin119883

119889 (119909 119879119909) (26)


119891 (119879119906) = 119889 (119879119906 119879119879119906) lt 119889 (119906 119879119906) = 119891 (119906) (27)



119889 (119879119909 119879119910) lt 119889 (119909 119910) for 119909 = 119910 (28)





119879119899119894+11199090 = lim119894997888rarrinfin

1198791198791198991198941199090 = 119879119906


119879119899119894+21199090 = 1198792119906(29)

Thus


119889 (1198791198991198941199090 119879119899119894+11199090) = 119889 (119906 119879119906)


119889 (119879119899119894+11199090 119879119899119894+21199090) = 119889 (119879119906 1198792119906) (30)


119889 (119906 119879119906) = 119889 (119879119906 1198792119906) (31)


119889 (119879119906 1198792119906) lt 119889 (119906 119879119906) (32)





0 lt 119889 (119909 119910) lt 120576 implies 119889 (119879119909 119879119910) lt 119889 (119909 119910) (33)



119889 (1198791198991199090 119879119899+1198961199090) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)


119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (34)



119889 (119906 119879119896119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(35)

and

119889 (119879119906 119879119879119896119906) = lim119894997888rarrinfin

119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (36)

Thus

119889 (119879119906 119879119879119896119906) = 119889 (119906 119879119896119906) lt 120576 (37)





(38)








119889 (119879119899119894+1199011199090 119879119899119894+119901+11199090) = 119889 (119879119901119906 119879119901+1119906) (39)


119889 (119879119903119906 119879119903119879119906) lt 119889 (119906 119879119906) (40)


119889 (119879119899119894+1199031199090 119879119899119894+119903+11199090) = 119889 (119879119903119906 119879119879119906) (41)


119889 (119879119902+1199031199090 119879119902+119903+11199090) lt 119889 (119879119903119906119879r119879119906) + 120576 (42)

For 120576 = 1205760 we have119889 (119879119902+1199031199090 119879119902+119903+11199090)

lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(43)


119889 (119879119899119879119902+1199031199090 119879119899119879119902+119903+11199090) lt 119889 (119879119902+1199031199090 119879119902+119903+11199090)forall119899 ge 119896

(44)


119889 (1198791198991199090 119879119899+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899 ge 119896 + 119902 + 119903(45)

Hence

119889 (1198791198991198941199090 119879119899119894+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899119894 ge 119896 + 119902 + 119903(46)

Thus we get

119889 (119906 119879119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+11199090)

le 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(47)

Hence

119889 (119906 119879119906) le 119889 (119879119903119906 119879119903119879119906) (48)



119889 (119879119899119906 1198791198991198791198971199090) lt 119889 (119906 1198791198971199090) forall119899 ge 119896 (49)


forall119899 ge 119896 + 119897(50)





forall119899 ge 119896 (119909 119910) (51)








119889 (119879119899(119906119879119906)119906 119879119899(119906119879119906)119879119906) lt 119889 (119906 119879119906) (53)



(54)



119889 (1198791199011199090 1198791198961198791199011199090) = 119889 (1198791199011199090 119879119896+1199011199090) lt 120576 (55)


119891 (119906) = 119889 (119906 119879119896119906) = min 119889 (119909 119879119896119909) 119909 isin 119883

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(56)


119889 (119879119899119906 119879119899119879119896119906) lt 119889 (119906 119879119896119906) (57)


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018










119889 (119906 119879119896119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+1198961199090)

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(35)

and

119889 (119879119906 119879119879119896119906) = lim119894997888rarrinfin

119889 (119879119899119894+11199090 119879119899119894+119896+11199090) (36)

Thus

119889 (119879119906 119879119879119896119906) = 119889 (119906 119879119896119906) lt 120576 (37)





(38)








119889 (119879119899119894+1199011199090 119879119899119894+119901+11199090) = 119889 (119879119901119906 119879119901+1119906) (39)


119889 (119879119903119906 119879119903119879119906) lt 119889 (119906 119879119906) (40)


119889 (119879119899119894+1199031199090 119879119899119894+119903+11199090) = 119889 (119879119903119906 119879119879119906) (41)


119889 (119879119902+1199031199090 119879119902+119903+11199090) lt 119889 (119879119903119906119879r119879119906) + 120576 (42)

For 120576 = 1205760 we have119889 (119879119902+1199031199090 119879119902+119903+11199090)

lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(43)


119889 (119879119899119879119902+1199031199090 119879119899119879119902+119903+11199090) lt 119889 (119879119902+1199031199090 119879119902+119903+11199090)forall119899 ge 119896

(44)


119889 (1198791198991199090 119879119899+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899 ge 119896 + 119902 + 119903(45)

Hence

119889 (1198791198991198941199090 119879119899119894+11199090) lt 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

forall119899119894 ge 119896 + 119902 + 119903(46)

Thus we get

119889 (119906 119879119906) = lim119894997888rarrinfin

119889 (1198791198991198941199090 119879119899119894+11199090)

le 12 [119889 (119906 119879119906) + 119889 (119879119903119906 119879119903119879119906)]

(47)

Hence

119889 (119906 119879119906) le 119889 (119879119903119906 119879119903119879119906) (48)



119889 (119879119899119906 1198791198991198791198971199090) lt 119889 (119906 1198791198971199090) forall119899 ge 119896 (49)


forall119899 ge 119896 + 119897(50)





forall119899 ge 119896 (119909 119910) (51)








119889 (119879119899(119906119879119906)119906 119879119899(119906119879119906)119879119906) lt 119889 (119906 119879119906) (53)



(54)



119889 (1198791199011199090 1198791198961198791199011199090) = 119889 (1198791199011199090 119879119896+1199011199090) lt 120576 (55)


119891 (119906) = 119889 (119906 119879119896119906) = min 119889 (119909 119879119896119909) 119909 isin 119883

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(56)


119889 (119879119899119906 119879119899119879119896119906) lt 119889 (119906 119879119896119906) (57)


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018











forall119899 ge 119896 (119909 119910) (51)








119889 (119879119899(119906119879119906)119906 119879119899(119906119879119906)119879119906) lt 119889 (119906 119879119906) (53)



(54)



119889 (1198791199011199090 1198791198961198791199011199090) = 119889 (1198791199011199090 119879119896+1199011199090) lt 120576 (55)


119891 (119906) = 119889 (119906 119879119896119906) = min 119889 (119909 119879119896119909) 119909 isin 119883

le 119889 (1198791199011199090 1198791198961198791199011199090) lt 120576(56)


119889 (119879119899119906 119879119899119879119896119906) lt 119889 (119906 119879119896119906) (57)


Data Availability




References















































Journal of












Journal of







Volume 2018






Volume 2018













































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

ResearchArticle Nemytzki-Edelstein-Meir-Keeler Type Results in …downloads.hindawi.com/journals/ddns/2018/4745764.pdf · DiscreteDynamicsinNatureandSociety Ontheotherhand,Bakhtin[]andCzerwik[]intro-duced