A new extension to the controlled metric type spaces

Mlaiki et al. Advances in Difference Equations (2021) 2021:94 https://doi.org/10.1186/s13662-021-03252-9

R E S E A R C H Open Access

A new extension to the controlled metrictype spaces endowed with a graphNabil Mlaiki1, Nizar Souayah2, Thabet Abdeljawad1,3,4* and Hassen Aydi5,3,6*

*Correspondence:[email protected];[email protected] of Mathematics andGeneral Sciences, Prince SultanUniversity, P.O. Box 66833, Riyadh11586, Saudi Arabia5Institut Supérieur d’Informatique etdes Techniques de Communication,Université de Sousse, H. Sousse4000, TunisiaFull list of author information isavailable at the end of the article

AbstractIn this paper, we initiate a new extension of b-metric spaces, called controlledmetric-like spaces, by changing the condition

[℘(s, r) = 0⇔ s = r

]by

[℘(s, r) = 0 ⇒ s = r

]

and that means basically we may have a non-zero self-distance. We prove some fixedpoint theorems which generalize many results in the literature. Also, we present aninteresting application to illustrate our results by considering controlled metric-likespaces endowed with a graph.

MSC: 47H10; 54H25

Keywords: b-metric spaces; Extended b-metric space; Controlled metric type space;Controlled metric-like space; Fixed point

1 IntroductionBanach [1] in 1922 started a new field in mathematics, so called fixed point theory. Thatwas the starting point for researchers around the globe to start generalize his resultwhether by changing the contractions or by generalizing the type of metric spaces, soit can cover a larger class of metrics; see [2–23]. Lately, in [24], an extension of b-metricspaces to extended b-metric spaces was given by Kamran et al. For related work, see [25–30]. Not much later, Mlaiki et al. in [31], introduced another generalization to the b-metricspaces, so called controlled metric type spaces.

In this manuscript, we introduce the concept of controlled metric-like spaces, whichinvolves a generalization of the controlled metric type spaces, that is, by assuming thatthe “self-distance” of an element is not necessary zero. However, if the value of the metricbetween two elements is zero, then these two elements are equal. One may ask: what isthe point of all these generalizations? In fact the answer to that is quite simple: the largerthe class of functions or metrics, the more fields that results can be applied to, such ascomputer science and engineering. Inspired by the work of Jachymski in [32], where heintroduced the concept of metric spaces endowed with a graph, we present in the lastsection some fixed point results on a controlled metric-like space endowed with a graph.

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Mlaiki et al. Advances in Difference Equations (2021) 2021:94 Page 2 of 13

2 PreliminariesIn 2017, Kamran et al. [24] introduced the notion of extended b-metric spaces.

Definition 2.1 ([24]) Given F �= ∅ and θ : F ×F → [1,∞). The function � : F ×F → [0,∞)is called an extended b-metric if

(1) �(s, r) = 0 ⇐⇒ s = r;(2) �(s, r) = �(r, s);(3) �(s, r) ≤ θ (s, r)[�(s, z) + �(z, r)],

for all s, r, z ∈ F . In 2018, Mlaiki et al. in [31] introduced the following generalization ofextended b-metric spaces, called controlled metric type spaces.

Definition 2.2 ([31]) Given F �= ∅ and η : F ×F → [1,∞). The function � : F ×F → [0,∞)is said to be of controlled metric type if

(1) �(s, r) = 0 ⇔ s = r;(2) �(s, r) = �(r, s);(3) �(s, r) ≤ η(s, z)�(s, z) + η(z, r)�(z, r),

for all s, r, z ∈ F . The pair (F ,�) is called a controlled metric type space.

Now, we present the definition of controlled metric-like spaces.

Definition 2.3 Given F �= ∅ and η : F × F → [1,∞). The function � : F × F → [0,∞)verifying:

(CL1) ℘(s, r) = 0 ⇒ s = r;(CL2) ℘(s, r) = ℘(r, s);(CL3) ℘(s, r) ≤ η(s, z)℘(s, z) + η(z, r)℘(z, r),

for all s, r, z ∈ F , is called a controlled metric-like space on F .

Notice that every space called a controlled metric type is a controlled metric-like space,but the converse is not always true, and to prove it we present the following example of acontrolled metric-like space, that is not a controlled metric type space.

Example 2.4 Choose F = {1, 2, . . .}. Take ℘ : F × F → [0,∞) as

℘(s, r) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, if r = s and both even,1

s+r , if s = r and both odd,1s , if s is even and r is odd,1r , if s is odd and r is even,

1, otherwise.

Given η : F × F → [1,∞) as

η(s, r) =

⎧⎪⎪⎨

⎪⎪⎩

s, if s is even and r is odd,

r, if s is odd and r is even,

1, otherwise.

(CL1) and (CL2) are obvious. We claim that (CL3) is satisfied.


Case 1: If z = s or z = r, (CL3) is satisfied.Case 2: If z �= s and z �= r, (CL3) holds if s = r. Now, assume that s �= r, so s �= r �= z. Again,

(CL3) is verified in the following subcases:(s1): s, z are even and r is odd;(s2): s is even and r, z are odd;(s3): s, z are odd and r is even;(s4): s, z are even and r is odd;(s5): s, r, z are even;(s6): s, r are even and z is odd;(s7): s, r are odd and z is even;(s8): s, r, z are odd.

Thus, (F ,℘) is a controlled metric-like space. It is not difficult to see that (F ,℘) is not acontrolled metric type space.

Example 2.5 Take F = {0, 1, 2}. Define the function ℘ by

℘(0, 0) = ℘(1, 1) = 0, ℘(2, 2) =1

10

and

℘(0, 1) = ℘(1, 0) = 1, ℘(0, 2) = ℘(2, 0) =12

, ℘(1, 2) = ℘(2, 1) =25

.

Take η : F × F → [1,∞) to be symmetric and to be defined by

η(0, 0) = η(1, 1) = η(2, 2) = η(0, 2) = 1, η(1, 2) =54

, η(0, 1) =1110

.

Here, ℘ is controlled metric-like on F .We have ℘(2, 2) = 1

10 �= 0, which implies that (F ,℘) is not a controlled metric type space.

For work in the same direction, see [33]. The definition of being of Cauchy type and ofconvergence in controlled metric-like spaces is given as follows.

Definition 2.6 Let (F ,℘) be a controlled metric-like space and {sn}n≥0 be a sequence in F .(1) {sn} is convergent to s in F , if and only if

limn→∞℘(sn, s) = ℘(s, s).

Here, one writes limn→∞ sn = s.(2) {sn} is Cauchy, if and only if limn,m→∞ ℘(sn, sm) exists and is finite.(3) (F ,℘) is called complete if, for each Cauchy sequence {sn}, there is some s ∈ F so that

limn→∞℘(sn, s) = ℘(s, s) = lim

n,m→∞℘(sn, sm).

Definition 2.7 Let (F ,℘) be a controlled metric-like space. Let s ∈ F and τ > 0.(i) The open ball B(s, τ ) is

B(s, τ ) ={

y ∈ F ,∣∣℘(s, r) – ℘(s, s)

∣∣ < τ

}.


(ii) The mapping ζ : F → F is called continuous at s ∈ F if for υ > 0, there is ν > 0 so thatζ (B(s,ν)) ⊆ B(ζ (s),υ). Thus, if ζ is continuous at s, then, for any sequence {sn}converging to s, we have limn→∞ ζ sn = ζ s, that is,

limn→∞℘(ζ sn, ζ s) = ℘(ζ s, ζ s).

Notice that each controlled metric-like space is a controlled metric space. But the con-verse is not always true (see Example 2.4 and Example 2.5).

3 Main resultsThe following result corresponds to the Banach contraction principle on controlledmetric-like spaces.

Theorem 3.1 Let ζ be a self-mapping on a complete controlled metric-like space (F ,℘) sothat

℘(ζ (s), ζ (r)

) ≤ k℘(s, r), (3.1)

for all s, r ∈ F , where k ∈ (0, 1). For s0 ∈ F , take sn = ζ n(s0). Suppose that

supm≥1

limi→∞

η(si+1, si+2)η(si, si+1)

η(si+1, sm) <1k

. (3.2)

Also, assume for each s ∈ F ,

limn→∞η(sn, s) and lim

n→∞η(s, sn) exist and are finite. (3.3)

Then ζ possesses a unique fixed point, say τ ∈ F . We have ℘(τ , τ ) = 0.

Proof Take the sequence {sn = ζ n(s0)}. By using (3.1), we get ℘(sn, sn+1) ≤ kn℘(s0, s1) for alln ≥ 0. For all integers n < m, one writes

℘(sn, sm) ≤ η(sn, sn+1)℘(sn, sn+1) + η(sn+1, sm)℘(sn+1, sm)

≤ η(sn, sn+1)℘(sn, sn+1) + η(sn+1, sm)η(sn+1, sn+2)℘(sn+1, sn+2)

+ η(sn+1, sm)η(sn+2, sm)℘(sn+2, sm)

≤ η(sn, sn+1)℘(sn, sn+1) + η(sn+1, sm)η(sn+1, sn+2)℘(sn+1, sn+2)

+ η(sn+1, sm)η(sn+2, sm)η(sn+2, sn+3)℘(sn+2, sn+3)

+ η(sn+1, sm)η(sn+2, sm)η(sn+3, sm)℘(sn+3, sm)

≤ · · ·

≤ η(sn, sn+1)℘(sn, sn+1) +m–2∑

i=n+1

( i∏

j=n+1

η(sj, sm)

)

η(si, si+1)℘(si, si+1)

+m–1∏

k=n+1

η(sk , sm)℘(sm–1, sm)


≤ η(sn, sn+1)kn℘(s0, s1) +m–2∑

i=n+1

( i∏

j=n+1

η(sj, sm)

)

η(si, si+1)ki℘(s0, s1)

+m–1∏

i=n+1

η(si, sm)km–1℘(s0, s1)

≤ η(sn, sn+1)kn℘(s0, s1) +m–2∑

i=n+1

( i∏

j=n+1

η(sj, sm)

)

η(si, si+1)ki℘(s0, s1)

+

( m–1∏

i=n+1

η(si, sm)

)

km–1η(sm–1, sm)℘(s0, s1)

= η(sn, sn+1)kn℘(s0, s1) +m–1∑

i=n+1

( i∏

j=n+1

η(sj, sm)

)

η(si, si+1)ki℘(s0, s1).

Thus, due to η(s, r) ≥ 1,

℘(sn, sm) ≤ η(sn, sn+1)kn℘(s0, s1) +m–1∑

i=n+1

( i∏

j=0

η(sj, sm)

)

η(si, si+1)ki℘(s0, s1).

Let

ϒp =p∑

i=0

( i∏

j=0

η(sj, sm)

)

η(si, si+1)ki.

Hence, we have

℘(sn, sm) ≤ ℘(s0, s1)[knη(sn, sn+1) + (ϒm–1 – ϒn)

]. (3.4)

Condition (3.2), by using the ratio test, implies that limn→∞ ϒn exists. Hence, {ϒn} is a realCauchy sequence. Letting n, m → ∞ in the inequality (3.4), we get

limn,m→∞℘(sn, sm) = 0, (3.5)

that is, the sequence {sn} is Cauchy in (F ,℘), which is a complete controlled metric-likespace, so {sn} converges to some s ∈ F . We have

limn→∞℘(sn, s) = ℘(s, s) = lim

n,m→∞℘(sn, sm) = 0. (3.6)

Then ℘(s, s) = 0. We claim that ζ s = s. Applying (CL3), one writes

℘(s, sn+1) ≤ η(s, sn)℘(u, sn) + η(sn, sn+1)℘(sn, sn+1).

Using (3.2), (3.3), (3.5) and (3.6),

limn→∞℘(s, sn+1) = 0. (3.7)


Again, in view of (CL3) and using (3.1),

℘(s, ζ (s)

) ≤ η(s, sn+1)℘(s, sn+1) + η(sn+1, ζ (s)

)℘

(sn+1, ζ (s)

)

≤ η(s, sn+1)℘(s, sn+1) + kη(sn+1, ζ (s)

)℘(sn, s).

Letting n → ∞ and inserting (3.3) and (3.7), we conclude that ℘(s, ζ (s)) = 0, i.e., ζ (s) = s.Assume that ζσ = σ and ζς = ς . Here,

℘(σ ,ς ) = ℘(ζ (σ ), ζ (ς )

) ≤ k℘(σ ,ς ).

It holds unless ℘(σ ,ς ) = 0, so σ = ς . �

Theorem 3.1 is supported by the two following examples.

Example 3.2 Choose F = {0, 1, 2}. Take ℘ to be symmetric and to be defined by

℘(0, 0) = 0, ℘(1, 1) = 0, ℘(2, 2) =1

100,

and

℘(0, 1) = 1, ℘(0, 2) =1120

, ℘(1, 2) =25

.

Given η : F × F → [1,∞) as

η(0, 0) = η(1, 1) = η(2, 2) = η(0, 2) = η(2, 0) = 1, η(1, 2) = η(2, 1) =4940

,

and

η(0, 1) = η(1, 0) =1110

.

Clearly, ℘ is controlled metric-like (℘ is not a controlled metric type on F). Define theself-mapping ζ on F by

ζ (0) = 2 and ζ (1) = ζ (2) = 1.

Set k = 45 . It is clear that (3.1) is verified. For each s0 ∈ F , (3.2) holds. All hypotheses of

Theorem 3.1 are fulfilled, and hence ζ possesses a unique fixed point, which is s = 1.

Example 3.3 Let F = [0,∞). Define ℘ : F2 → [0,∞) by

℘(x, y) =

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

0 if x = y = 0,y

1+y if x = 0, y �= 0,x

1+x if y = 0, x �= 0,

x + y if x �= 0, y �= 0.

Consider η : F2 → [1,∞) as η(x, y) = 2 + 2x + 2y. Note that ℘ is controlled metric-like on F .


First, (CL1) and (CL2) are obvious. We need to prove (CL3). For this, let x, y and z in F .We state the following cases:

Case 1: x = z = 0. Here, (CL3) holds.Case 2: x = 0 and z �= 0.Subcase 1: y = 0. We have

℘(x, z) =z

1 + z

≤ (2)(0) + (2 + 2z)z

1 + z

= η(x, y)℘(x, y) + η(y, z)℘(y, z).

Subcase 2: y �= 0. We have

℘(x, z) =z

1 + z

≤ (2 + 2y)y

1 + y+ (2 + 2z)(y + z)

= η(x, y)℘(x, y) + η(y, z)℘(y, z).

Case 3: z = 0 and x �= 0. Proceeding similarly to Case 2, (CL3) holds.Case 4: x �= 0 and z �= 0.Subcase 1: y = 0. We have

℘(x, z) = x + z

≤ (2 + 2x)x

1 + x+ (2 + 2z)

z1 + z

= η(x, y)℘(x, y) + η(y, z)℘(y, z).

Subcase 2: y �= 0. We have

℘(x, z) = x + z

≤ (2 + 2x + 2y)(x + y) + (2 + 2y + 2z)(y + z)

= η(x, y)℘(x, y) + η(y, z)℘(y, z).

On the other hand, (F ,℘) is not a b-metric-like space. We argue by contradiction byassuming that (F ,℘) is a b-metric-like space with a coefficient s ≥ 1 (a constant). Then,for any real y > 0, we have

℘(y, y + 1) ≤ s[℘(y, 0) + ℘(0, y + 1)

].

That is,

2y + 1 ≤ s[

y1 + y

+y + 12 + y

].

Letting y → ∞, we get +∞ ≤ 2s, which is a contradiction.


Consider the self-mapping ζ on F defined by ζ (x) = x3 . It is clear that (3.1) holds for all

x, y ∈ F , with k = 25 . Take s0 = 1. The sequence {sn} given as sn = ζ n(s0) is written as

sn =13n , n = 0, 1, . . . .

A simple calculation yields

supm≥1

limi→∞


η(si+1, sm) =(

2 +13

)=

73

<52

=1k

.

That is, (3.2) is verified. Moreover, we have, for each s ∈ F ,

limn→∞η(sn, s) = lim

n→∞η(s, sn) = (2 + 2s) exists.

All hypotheses of Theorem 3.1 hold. Hence, ζ admits a unique fixed point, which is τ = 0.It verifies ℘(τ , τ ) = 0.

Definition 3.4 Let ζ : F −→ F . For some s0 ∈ F , let O(s0) = {s0, ζ s0, ζ 2s0, . . .} be the orbitof s0. Given P : F −→ R. Such a P is called ζ -orbitally lower semi-continuous at ∈ F iffor {sn} ⊂ O(s0) so that sn −→ , we have P( ) ≤ limn→∞ inf P(sn).

As in [24], using Definition 3.4, a consequence of Theorem 3.1 (it is a generalization ofTheorem 1 in [13]) is as follows.

Corollary 3.5 Let ζ : F → F be a self-mapping on a complete controlled metric-like space(F ,℘). Given s0 ∈ F . Assume there is k ∈ (0, 1) so that

℘(ζ (z), ζ 2(z)

) ≤ k℘(z, ζ (z)

), for each z ∈ O(s0). (3.8)

Take sn = ζ n(s0). Suppose that

supm≥1

limi→∞


η(si+1, sm) <1k

. (3.9)

Then sn → l ∈ F as n → ∞. Further, if the functional δ �→ ℘(s, ζ (δ)) is ζ -orbitally lowersemi-continuous at l, we get ζ (l) = l.

4 Controlled metric-like spaces endowed with a graphThroughout this section, we denote controlled metric-like spaces by (CMLS). Now, wepresent (CMLS) endowed with a graph. Figure 1 is an example of a controlled metric-likespace endowed with a graph.

Consider the (CMLS) (F ,℘), let � be the diagonal of F2. A graph G is defined by thepair (V , E) where V is a set of vertices coinciding with F and E is the set of its edges with� ⊂ E. From now on, assume that G has no parallel edges.

Definition 4.1 ([32]) Let t and s be two vertices of a graph G. We define q ∈N∪ {0} to bethe length of the path between t and s in G by a sequence (ki)

qi=0 of q + 1 distinct vertices

where k0 = t, kn = s and (ki, ki+1) ∈ E(G) for i = 1, 2, . . . , q.


Figure 1 Controlled metric-like space as in Example 2.5

Note that the graph G can be converted to a weighted graph and associate to each edgethe value of (CMLS).

Notation Let Fg = {x ∈ F/(x, gx) ∈ E(G) or (gx, x) ∈ E(G)}.

Definition 4.2 Let (F ,℘) be a complete (CMLS) endowed with a graph G. The mappingζ : F → F is said to be a Gφ-contraction if

•for all t, s ∈ F , (t, s) ∈ E(G) �⇒ (ζ t, ζ s) ∈ E(G); (4.1)

• there is φ : [0,∞) −→ [0,∞) so that

ξ(ζ t, ζ 2t

) ≤ φ(ξ (t, ζ t)

) ∀t ∈ Xζ , (4.2)

where φ is nondecreasing and {φn(t)}n∈N → 0 ∀t > 0.

Definition 4.3 The mapping ζ : F −→ F is called orbitally G-continuous if for all τ ,ρ ∈ Xand {sn}n∈N a positive sequence,

ζ snτ −→ ρ,(ζ snτ , ζ sn+1τ

) ∈ E(G) �⇒ ζ(ζ snτ

) −→ ζρ as n → ∞.

Lemma 4.4 Let (F ,℘) be a complete (CMLS) equipped by a graph G. Suppose that ζ : F →F be a Gφ-contraction. If t ∈ Xζ then there is q(t) ≥ 0 so that

℘(ζ nt, ζ n+1t

) ≤ φn(q(t)) ∀n ∈N, (4.3)

where q(t) = ℘(t, ζ t).

Proof Let t ∈ Fζ , then (t, ζ t) ∈ E(G) or (ζ t, t) ∈ E(G). Assume that w.l.o.g. (t, ζ t) ∈ E(G).Hence,

(ζ n, ζ n+1t

) ∈ E(G) ∀n ∈N. (4.4)


Hence, we deduce

℘(ζ nt, ζ n+1t

) ≤ φ(℘

(ζ n–1t, ζ nt

))

≤ φ2(℘(ζ n–2t, ζ n–1t

))

≤ ...

≤ φn(℘(t, ζ t))

= φn(q(t)). �

Theorem 4.5 Let (F ,℘, G) be a complete (CMLS) equipped by a graph G. Suppose thatζ : F → F be a Gφ-contraction, which is orbitally G-continuous. Consider the property (P)as follows: ∀{tn}n∈N in F , if tn −→ t and (tn, tn+1) ∈ E(G), then there is {tkn}n∈N where (tkn , t) ∈E(G), holds. Further, suppose that, for each s ∈ F ,

limn→∞η

(ζ is, ζ ns

)(∀i ≥ 1) and lim

n→∞η(ζ ns, ζ n+1s

)exist and are finite (4.5)

and

if limn→∞ ζ ns → u ∈ F , then ℘(ζu, u) > lim sup

n→∞η(ζu, ζ ns

)℘(ζu, ζu). (4.6)

Thus, the restriction of ζ|[s]Gto [s]G possesses a fixed point. Moreover, if for every two fixed

points v1, v2 we have η(v1, v2) > 1, then we have uniqueness of the fixed point.

Proof Consider s ∈ Fζ . By Lemma 4.4, there is q(s) ≥ 0 so that

℘(ζ ns, ζ n+1s

) ≤ φn(q(s)).

First, we show that (ζ ns)n∈N converges to some u ∈ F . It should be enough to ensure that{ζ nt}n∈N is Cauchy. Take the integers n, m. Using again Lemma 4.4, we get

℘(ζ ns, ζ n+ms

) ≤ η(ζ ns, ζ n+1s

)℘

(ζ ns, ζ n+1s

)+ η

(ζ n+1s, ζ n+ms

)℘

(ζ n+1s, ζ n+ms

)

≤ η(ζ ns, ζ n+1s

)℘

(ζ ns, ζ n+1s

)+ η

(ζ n+1s, ζ n+ms

)η(ζ n+1s, ζ n+2s

)

× ℘(ζ n+1s, ζ n+2s

)

+ η(ζ n+1s, ζ n+ms

)η(ζ n+2s, ζ n+ms

)℘

(ζ n+2s, ζ n+ms

)

≤ ...


)℘

(ζ ns, ζ n+1s

)+

n+m–2∑

i=n+1

( i∏

j=n+1

η(ζ js, ζ n+ms

))

η(ζ is, ζ i+1s

)

× ℘(ζ is, ζ i+1s

)+

n+m–1∏

k=n+1

η(ζ ks, ζ n+ms

)℘

(ζ n+m–1s, ζ n+ms

)


)φn(q(s)

)+

n+m–2∑

i=n+1

( i∏

j=n+1

η(ζ js, ζ n+ms

))

η(ζ is, ζ i+1s

)φi(q(s)

)

+n+m–1∏

k=n+1

η(ζ ks, ζ n+ms

)φn+m–1(q(s)

)η(ζ n+m–1s, ζ n+ms

)



)φn(q(s)

)+

n+m–1∑

i=n+1

( i∏

j=0

η(ζ js, ζ n+ms

))

η(ζ it, ζ i+1s

)φi(q(s)

)

≤n∏

j=0

η(ζ jt, ζ n+ms

))η

(ζ ns, ζ n+1s

)φn(q(s)

)

+n+m–1∑

i=n+1

( i∏

j=0

η(ζ js, ζ n+ms

))

η(ζ is, ζ i+1s

)φi(q(s)

)

=n+m–1∑

i=n

( i∏

j=0

η(ζ js, ζ n+ms

))

η(ζ is, ζ i+1s

)φi(q(s)

)

=n+m–1∑

i=n

�iφi(q(s)

), where �i =

( i∏

j=0

η(ζ js, ζ n+ms

))

η(ζ is, ζ i+1s

)

=m∑

i=1

�n+i–1i φn+i–1(q(s)

).

In view of the property of φ and using (4.5), we deduce that∑m

i=1 �n+i–1i φn+i–1(q(s)) is con-

vergent to 0 as n, m −→ ∞. Thus,

limn,m→∞℘

(ζ ns, ζ ms

)= 0, (4.7)

that is, the sequence {ζ ns} is Cauchy in (F ,℘). Its completeness entails that {ζ ns} convergesto some u ∈ F , so that

limn→∞℘

(ζ ns, u

)= ℘(u, u) = lim

n,m→∞℘(ζ ns, ζ ms

)= 0. (4.8)

We have ℘(u, u) = 0.Since s ∈ Fζ , one writes that ζ ns ∈ Fζ for every n ∈ N. We suppose that (s, ζ s) ∈ E(G), by

(P), we deduce that there is {ζ kn s}n∈N of {ζ ns}n∈N so that (ζ kn s, u) ∈ E(G) for any n ∈ N. Weconsider the path in G as follows: s, ζ s, . . . , ζ k1 s, u and so u ∈ [s]G. The orbital G-continuityof ζ yields

limn→∞℘

(ζ(ζ kn s

), ζu

)= ℘(ζu, ζu). (4.9)

Suppose that ℘(ζu, u) > 0. Applying (CL3), one writes

℘(ζu, u) ≤ η(ζu, ζ kn℘(ζu, ζ kn s

)+ η

(ζ ns, u

)℘

(ζ kn s, u

).

Letting n → ∞ and using (4.5), (4.8) and (4.9), we get

℘(ζu, u) ≤ lim supn→∞

η(ζu, ζ kn s

)℘(ζu, ζu).

It contradicts (4.6). Hence, ℘(ζu, u) = 0, so ζu = u, that is, u is a fixed point of ζ |[s]G.


For its uniqueness, suppose there are two fixed points v1 and v2, i.e., ζv1 = v1 and ζv2 =v2. We have

℘(v1, v2) ≤ η(v1, v1)℘(v1, v1) + η(v1, v2)℘(v1, v2)

≤ η(v1, v1)1 – η(v1, v2)

℘(v1, v1).

Since η > 1, we have ℘(v1, v2) ≤ 0, and so v1 = v2. �

5 ConclusionThe (CMLS) endowed with a graph introduced in this paper can be successfully used indifferent branches of scientific knowledge, for example when studying the kinetics of bio-chemical reaction networks, where we have the serious problem of the choice of “appro-priate distance” in the analysis of a “K-angle general kinetic reaction system”; see [34].

AcknowledgementsThe first and third authors would like to thank Prince Sultan University for funding this work through research groupNonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17. The fourth authorthank Sefako Makgatho Health Sciences University.

FundingThis research received no external funding.

Availability of data and materialsThe data used to support the findings of this study are available from the corresponding author upon request.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally in writing this article. All authors read and approved the final manuscript.

Author details1Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia.2Department of Natural Sciences, Community College Al-Riyadh, King Saud University, Riyadh, Saudi Arabia.3Department of Medical Research, China Medical University, Taichung 40402, Taiwan. 4Department of ComputerSciences and Information Engineering, Asia University, Taichung, Taiwan. 5Institut Supérieur d’Informatique et desTechniques de Communication, Université de Sousse, H. Sousse 4000, Tunisia. 6Department of Mathematics and AppliedMathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 7 May 2020 Accepted: 21 January 2021

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