# On the Power of Simulation and Admissible Functions in Metric Fixed Point Theory

Authors: Areej S. S. Alharbi; Hamed H. Alsulami; Erdal Karapinar
Journal: Journal of Function Spaces
 (2017)

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## Abstract

We investigate the existence and uniqueness of certain operators which form a new contractive condition via the combining of the notions of admissible function and simulation function contained in the context of completeb-metric spaces. The given results not only unify but also generalize a number of existing results on the topic in the corresponding literature.


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## Body

## 1. Introduction

The crucial notion of this research is the simulation function which is defined by Khojasteh et al. [1]. After that, Argoubi et al. [2] relaxed the conditions of the notion of simulation function a little bit to guarantee that the considered set is nonempty.In this manuscript, we respond to the question, how do we guarantee the existence of fixed points of the new contraction defined by the help of the admissible function and the simulation function in the frame of completeb-metric spaces? The presented main theorem of the paper covers and unifies a huge number of published results on the topic in the related literature.Definition 1 (see [2], cf. [1]).
Letσ:R0+×R0+→R be a mapping that satisfies the following inequality and the condition below: (S1)
σ(r,s)<s-r for each r,s>0.(S2)
if{rn},{sn} are sequences in (0,∞) such that limn→∞⁡rn=limn→∞⁡sn>0, then(1)limsupn→∞⁡σrn,sn<0.We shall use the letter S to indicate the class of all simulation functions σ:R0+×R0+→R. It is obvious from the axiom (S2) that(2)σr,r<0for  everyr>0.Note that the conditionσ(0,0)=0 in the original definition of the simulation function is removed in Definition 1. Indeed, this condition gives a contradiction when one takes s=r in the first condition (S1). For further detail on the discussion, see, for example, [2].Throughout the paper, we shall useR0+ to represent nonnegative real numbers.The following example [1, 3, 4] shall be helpful to illustrate the worth of the notion of simulation function.Example 2.
Suppose thatΦ denotes the set of all continuous functions ϕ:R0+→R0+ such that ϕ(r)=0 if, and only if, r=0. The following functions σ1-σ6 form a simulation function. (i)σ1(t,s)=ϕ1(s)-ϕ2(t)foreacht,s∈R0+ and ϕ1,ϕ2∈Φ, where ϕ1(t)<t≤ϕ2(t) for each t>0.(ii) LetR,T:R0+→(0,∞) be two continuous functions with respect to each variable and the inequality R(t,s)>T(t,s) holds for each t,s>0. Then,(3)σ2t,s=s-Rt,sTt,stfor  eacht,s∈R0+.(iii)σ3(t,s)=s-ϕ3(s)-t for each t,s∈[0,∞).(iv) For eachs,t∈R0+,(4)σ4t,s=sφs-t,whereφ:R0+→[0,1) with limsupt→r+φ(t)<1 for each r>0.(v) For eachs,t∈R0+(5)σ5t,s=ηs-t,whereη∈Φ and it is upper semicontinuous.(vi) For eachs,t∈[0,∞),(6)σ6t,s=s-∫0tϕudu,whereϕ is a self-mapping on R0+ with the following properties:
(1)∫0εϕ(u)du exists,(2) for everyε>0, ∫0εϕ(u)du>ε.For the further attracted simulation function examples see, for example, [1, 3, 4].In 1993 Czerwik [5] proposed a more general frame for the notion of standard metric, so called a b-metric.Definition 3.
ForM≠∅, let db:M×M→R0+ be a function satisfying the following conditions: (1)
db(p,q)=0 if and only if p=q.(2)
db(p,q)=db(q,p) for each p,q∈M.(3)
db(p,q)≤s[db(p,r)+db(r,q)] for each p,q,r∈M, where s≥1.
Here,db is called a b-metric. Further, the triple (M,db,s) is called a b-metric space.For the special case ofs=1, the notion of b-metric turns into the standard metric. Consequently, the notion of b-metric is more general than the standard metric.For the sake of completeness, we recollect standard but interesting three examples ofb-metric spaces; see, for example, [6, 7] and the related references therein.Example 4.
LetM=R. For all p,q∈M, we define a function db as(7)dbp,q=p-q2.Then, db is a b-metric on real numbers. The first two axioms are fulfilled in a straightforward way. The last axiom is satisfied for s=2: (8)p-q2≤2p-r2+r-q2.Example 5.
For a fixedp∈(0,1), consider (9)M=lpR=t=tn⊂R:∑n=1∞tnp<∞. We introduce the corresponding distance functions as (10)dbt,s=∑n=1∞tn-snp1/pfor  eacht,s∈M.Then, (M,db,s) forms a b-metric space with the constant s=21/p.Example 6.
Suppose thatE is a Banach space with the zero vector 0E of E. Take P as a cone in E such that int(P)≠∅ and further, ⪯ is partial ordering with respect to P. For a nonempty set M, we define a mapping c:M×M→E as follows: (M1)
0⪯c(p,q) for each p,q∈M.(M2)
c(p,q)=0 if and only if p=q.(M3)
c(p,q)⪯c(x,z)+c(z,y), for each p,q∈M.(M4)
c(p,q)=c(q,p) for each p,q∈M.
Then, the mappingc is called cone metric on M. Moreover, the pair (M,c) is said to be a cone metric space.
If a normal coneP in E is normal with the normality constant K, then, the mapping D:M×M→R0+, defined by Db(x,y)=c(x,y), forms a b-metric space where the function c:M×M→E is a cone metric. Moreover, the triple (M,Db,s) forms a b-metric space with the constant s≔K≥1.Suppose that(M,db,s) is a b-metric space. A self-mapping T on M is said to be a S-contraction with respect to σ [1], if the following inequality is fulfilled: (11)σdbTp,Tq,dbp,q≥0for  eachp,q∈M,σ∈S.On account of (σ2), we derive that(12)dbTp,Tq≠dbp,qfor  each  distinctp,q∈M.Taking (12) into account, we find that T cannot be an isometry whenever T is a S-contraction. Moreover, if T is a S-contraction in the setting of b-metric space with a fixed point, then the desired fixed point is necessarily unique.Theorem 7.
In a completeb-metric space, each S-contraction has a unique fixed point.This theorem can be stated also as follows: eachS-contraction yields a Picard sequence that converges to a unique fixed point.For a familyΨ:=ψ:R0+→R0+, if the following two conditions are fulfilled,(i)
each functionψ∈Ψ is nondecreasing;(ii)
there exista∈0,1 and k0∈N and a convergent series of nonnegative terms ∑k=1∞vk such that for any t∈R+ and for k≥k0 we have (13)ψk+1t≤aψkt+vk.Here, Ψ is called the class of c-comparison functions (see [8]). For a ψ∈Ψ the notation ψn indicates the nth iteration of the function ψ. The following lemma is recollected from [8].Lemma 8.
For aψ∈Ψ, we have (i)
for eacht∈R+, the sequence ψntn∈N converges to 0 as n→∞;(ii)
for anyt∈R+, the inequalities ψt<t are fulfilled;(iii)
each auxiliary functionψ is continuous at 0;(iv)
for anyt∈R+, the series ∑k=1∞ψkt is convergent.Berinde [9] characterized (c)-comparison functions to use for the contraction mappings in the setting of b-metric spaces, as follows.Definition 9.
Fix a real numbers≥1. An increasing function ϕ:R0+→R0+ is said to be (b)-comparison if there exist a convergent nonnegative series ∑k=1∞vk,  k0∈N, and a∈(0,1) such that for any t≥0 and for k≥k0,(14)sk+1ϕk+1t≤askϕkt+vk.Lemma 10 (see [10]).
Letϕ:R0+→R0+ be a (b)-comparison function. Define a self-mapping bs on R0+ as bs=∑k=0∞skϕk(t): (1)
For anyt∈R0+, the series ∑k=0∞skϕk(t) is convergent.(2)
The functionbs is increasing and continuous at t=0.Remark 11.
It is obvious that each(b)-comparison function is a comparison function. Consequently, on account of Lemma 8, we deduce that any (b)-comparison function ψ satisfies the inequality ψ(t)<t.Popescu [11] introduces the notion of the α-orbital admissible as follows.Definition 12 (see [11]).
Suppose thatT is a self-mapping over a nonempty set M and α:M×M→R0+ is a function. The mapping T is called an α-orbital admissible if the following implication is provided: (15)αx,Tx≥1⟹αTx,T2x≥1.We should mention the notion of theα-orbital admissible [11, 12] inspired from the notion of the α-orbital admissible notion defined in [13, 14].In this paper, by combining the notion of the simulation function together with the admissible functions, we shall consider a new type contractive mapping in the frame of completeb-metric spaces. Accordingly, our results improve and extend the main results in [15] in twofold: first, we investigate the existence and uniqueness of a fixed point in b-metric spaces instead of standard metric space. Secondly, we extend the condition σ(α(p,q)db(Tp,Tq),(db(p,q))≥0 for each p,q∈M by adding an auxiliary function ϕ into account. Consequently, we investigate the existence and uniqueness of a fixed point in the new extended condition σ(α(p,q)db(Tp,Tq),ϕ(db(p,q)))≥0 for each p,q∈M. We illustrate that the class of the new contractive mapping covers several well-known contractive mappings.


## 2. Main Results

We start this section by defining the(α-ϕ)-type S-contraction which is a generalization of the notion of S-contraction.Definition 13.
LetM be a nonempty set s≥1 and α:M×M→[0,∞) be function. Suppose that T is a self-mapping defined over a b-metric space (M,db,s). The self-mapping T is called an (α-ϕ)-type S-contraction with respect to σ if there are σ∈S and ϕ∈Φ such that(16)σαp,qdbTp,Tq,ϕdbp,q≥0for  eachp,q∈M.Before stating our main theorem, we shall give lemmas that have a crucial role in the proof of the main result.Lemma 14.
LetM be a nonempty set. Suppose that α:M×M→R0+ is a function and T:M→M is an α-orbital admissible mapping. If there exists p0∈M such that α(p0,Tp0)≥1 and pn=Tpn-1 for n=0,1,…, then, we have(17)αpn,pn+1≥1,for  eachn=0,1,….Proof.
On account of the assumptions of the theorem, there existsp0∈M such that α(p0,Tp0)≥1. Owing to the fact that T is α-orbital admissible, we find(18)αp0,p1=αp0,Tp0≥1⟹αTp0,Tp1=αp1,p2≥1. By iterating the above inequality, we derive that (19)αpn,pn+1=αTpn-1,Tpn≥1,for  eachn=0,1,….Theorem 15.
LetM be a nonempty set, s≥1, and α:M×M→R0+ be a function. Suppose that a continuous self-mapping T over a complete b-metric space (M,db,s) is α-orbital admissible. Suppose also the mapping T forms an (α-ϕ)-type S-contraction with respect to σ. If there exists p0∈M such that α(p0,Tp0)≥1, then there exists p∈M such that Tp=p.Proof.
Based on the assumption, there existsp0∈M such that α(p0,Tp0)≥1. We shall construct an iterative sequence {pn} in M by setting pn+1=Tpn for each n≥0. By Lemma 14, we have (17); that is, (20)αpn,pn+1≥1,for  eachn=0,1,….
Taking (16) and (17) into account, for each n≥1, we derive that(21)0≤σαpn,pn-1dbTpn,Tpn-1,ϕdbpn,pn-1=σαpn,pn-1dbpn+1,pn,ϕdbpn,pn-1<ϕdbpn,pn-1-αpn,pn-1dbpn+1,pn.Accordingly, from (16) and (21) we conclude that(22)dbpn,pn+1≤αpn,pn-1dbpn,pn+1≤ϕdbpn,pn-1<dbpn,pn-1for  eachn=1,2,….Hence, we conclude that the constructive sequence {db(pn,pn-1)} is bounded from below by zero, and moreover, it is nondecreasing. Hereby, there exists θ≥0 such that limn→∞⁡db(pn,pn-1)=θ≥0. We shall indicate that(23)limn→∞⁡dbpn,pn-1=0.Suppose, on the contrary, that θ>0. On account of inequality (22), we find that(24)limn→∞⁡αpn,pn-1dbpn,pn+1=θ,limn→∞⁡ϕdbpn,pn-1=θ.Taking sn=α(pn,pn-1)db(pn,pn+1) and tn=db(pn,pn-1) together with the condition (σ3), we derive that(25)0≤limsupn→∞⁡σαpn,pn-1dbpn+1,pn,ϕdbpn,pn-1<0,a contradiction. Consequently, we have θ=0.
In the next step, we shall show that the constructive sequence{pn} is Cauchy. By iteration on the inequality (22), we derive that(26)dbpn+1,pn≤ψndbp1,p0,for  eachn≥1.From (26) and using the triangular inequality, for each p≥1, we have(27)dbpn,pn+p≤s·dbpn,pn+1+s2·dbpn+1,pn+2+⋯+sp-2·dbpn+p-3,pn+p-2+sp-1·dbpn+p-2,pn+p-1+sp·dbpn+p-1,pn+p≤s·dbp0,p1+s2·dbp0,p1+⋯+sp-2·dbp0,p1+sp-1·dbp0,p1+sp·dbp0,p1=1sn·sn+1·dbp0,p1+⋯+sn+p-1·dbp0,p1+sn+p·dbp0,p1≤1sn·sn+1·dbp0,p1+⋯+sn+p·dbp0,p1=1sn·∑i=n+1n+psi·dbp0,p1<1sn·∑i=n+1∞si·dbp0,p1.
The precedent inequality is(28)dbpn,pn+p<1sn·∑i=n+1∞si·dbp0,p1⟶0asn⟶∞, which yields that {pn} is a Cauchy sequence in (M,db,s). Since (M,db,s) is complete, there exists p∈M such that(29)limn→∞⁡dbpn,p=0.Since T is continuous, we obtain from (29) that(30)limn→∞⁡dbpn+1,Tp=limn→∞⁡dbTpn,Tp=0.Combining the uniqueness of the limit together with (29) amd (30), we find that p forms a fixed point of T; that is, Tp=p.The continuity condition can be relaxed in Theorem15 by replacing a suitable condition like the given below.Definition 16.
Lets≥1. We say that a b-metric space (M,db,s) isregular if {pn} is a sequence in M such that α(pn,pn+1)≥1 for each n and pn→x∈M as n→∞; then there is a subsequence {pn(k)} of {pn} such that α(pn(k),x)≥1 for each k.By removing the continuity condition from the main result, Theorem15 is possible. But, in this case, we should add the “regularity” condition which is mentioned in Definition 16.Theorem 17.
LetM be a nonempty set, s≥1, and α:M×M→R0+ be a function. Suppose that (M,db,s) is regular and a self-mapping T on a complete b-metric space (M,db,s) is α-orbital admissible. Suppose also the mapping T forms an (α-ϕ)-type S-contraction with respect to σ. If there exists p0∈M such that α(p0,Tp0)≥1, then there exists p∈M such that Tp=p.Proof.
By repeating the steps in the proof of Theorem15, we guarantee that the sequence {pn} defined by pn+1=Tpn for each n≥0 converges for some p∈M. From (17) and regularity of the metric, there exists a subsequence {pn(k)} of {pn} such that α(pn(k),p)≥1 for each k. By implementing (16), for each k, we get that(31)0≤σαpnk,pdbTpnk,Tp,ϕdbpnk,p=σαpnk,pdbpnk+1,Tp,ϕdbpnk,p<ϕdbpnk,p-αpnk,pdbpnk+1,Tp,which leads to(32)dbpnk+1,Tp=dbTpnk,Tp≤αpnk,pdbTpnk,Tp≤ϕdbpnk,p<dbpnk,p.Taking k→∞ in inequality (32), we deduce that db(u,Tp)=0; that is, p=Tp.Note that, in Theorems15 and 17, we observe only the existence of the fixed point of the given operator. As a next step, we shall investigate the uniqueness of the obtained fixed point. Let Fix(T) represent the set of all fixed points of operator T. For this purpose, we need the following additional condition:(U)
α(p,q)≥1 for each p,q∈Fix(T).Theorem 18.
Under the assumption of additional condition(U), the obtained fixed point p of the operator T defined in Theorem 15 (resp., Theorem 17) turns to be unique fixed point.Proof.
LetT be an α-orbital admissible S-contraction with respect to σ. Regarding Theorem 15 or Theorem 17, we guarantee the existence of a fixed point of the mapping T; namely, p=Tp. Suppose p is not the unique fixed point of T; thus, there exists q with p≠q. So, we have db(p,q)>0. Regarding the condition (U), the definition of T yields that(33)σαp,qdbTp,Tq,ϕdbp,q≥0.Due to definition of the auxiliary function σ, the inequality above implies that(34)dbp,q=dbTp,Tq≤αp,qdbTp,Tq,ϕdbp,q<dbp,q, which is a contradiction. Thus, p is the unique fixed point of T.


## 3. Consequences

### 3.1. Consequences in the Setting ofb-Metric Space

Consider a mappingϕ1∈ϕ related with an α-orbital admissible S-contraction with respect to σ, namely T; that is, σE(α(p,q)db(Tp,Tq),ϕ1(db(p,q)). For a function ψ∈Ψ, we set(35)σEt,s=ψs-tfor  eachs,t∈R0+.It is straightforward that σBW is a simulation function. Combining the observations above, we have(36)σEαp,qdbTp,Tq,ϕ1dbp,q=ψϕ1dbp,q-αp,qdbTp,Tqfor  eachs,t∈R0+, which is equivalent to(37)αp,qdbTp,Tq≤ϕdbp,q,for  eachp,q∈M,where(38)ϕt≔ψϕ1t∈Φ.Thus, the above sample of simulation function together with Theorem18 yields the following result.Theorem 19.
Let(M,d) be a complete b-metric space and let T:M→M be defined as(39)αp,qdbTp,Tq≤ϕdbp,q,for  eachp,q∈M,where α:M×M→R0+ and ϕ∈Φ. Suppose that (i)
T is α-orbital admissible;(ii)
there existsp0∈M such that α(p0,Tp0)≥1;(iii)
eitherT is continuous or (M,d) is regular.
Then there existsp∈M such that Tp=p. Moreover, if the condition (U) is fulfilled, then we guarantee that the obtained fixed point p of T is unique.Lettingϕ(t)=kt with k∈[0,1), Theorem 19 implies the following.Theorem 20.
Let(M,d) be a complete b-metric space and let T:M→M be defined as(40)αp,qdbTp,Tq≤kdbp,q,for  eachp,q∈M, where k∈[0,1), α:M×M→R0+, and ϕ∈Φ. Suppose that (i)
T is α-orbital admissible;(ii)
there existsp0∈M such that α(p0,Tp0)≥1;(iii)
eitherT is continuous or (M,d) is regular.
Then there existsp∈M such that Tp=p. Moreover, if the condition (U) is fulfilled, then we guarantee that the obtained fixed point p of T is unique.By lettingα(p,q)=1 in Theorems 19 and 20 we get the main results of Theorems 1 and 2 of Czerwik [5]. Notice that, in this case, conditions (i)–(iii) of Theorems 19 and 20 are fulfilled trivially.


### 3.2. Consequences in the Setting of Standard Metric Space

In this section, we consider the results in the setting of standard metric. Thus, we considers=1 throughout this section. We shall show that a number of existing fixed point results in the literature are the simple consequence of our main results. In particular, by taking Example 2 into consideration, we can list many well-known results as a consequence of our main results.Ifψ∈Ψ and we define(41)σEt,s=ψs-tfor  eachs,t∈R0+, then σBW is a simulation function (cf. Example 2 (v)).First, we derive the very interesting recent results of Samet et al. [13] as a corollary of Theorem 18.Theorem 21.
Theorems2.1 and 2.2 in [13] are consequences of the following.Proof.
TakingσE(r,s)=ψ(s)-rfor  eachs,r∈[0,∞) in Theorem 18, we derive that (42)αp,qdbTp,Tq≤ψdbp,q,for  eachp,q∈M. We skip the details.As is well known, the main theorem in [13] covers several fixed point results, including the pioneer fixed point theorem of Banach. Moreover, as is shown in [13, 16], several fixed point theorems in different settings (in the sense of partially ordered set, in the sense of cyclic mapping, etc.) can be concluded from Theorems 2.1 and 2.2 in [13] by setting α(p,q) in a proper way.Notice also that one can express the main result of Khojasteh et al. [1] as a straight consequence of our main result.Theorem 22.
Theorem18 yields Theorem 7.Proof.
It is sufficient to takeα(p,q)=1 for each p,q∈M in Theorem 21.It is obvious that all presented results in [1] follow from our main result.


## 3.1. Consequences in the Setting ofb-Metric Space

Consider a mappingϕ1∈ϕ related with an α-orbital admissible S-contraction with respect to σ, namely T; that is, σE(α(p,q)db(Tp,Tq),ϕ1(db(p,q)). For a function ψ∈Ψ, we set(35)σEt,s=ψs-tfor  eachs,t∈R0+.It is straightforward that σBW is a simulation function. Combining the observations above, we have(36)σEαp,qdbTp,Tq,ϕ1dbp,q=ψϕ1dbp,q-αp,qdbTp,Tqfor  eachs,t∈R0+, which is equivalent to(37)αp,qdbTp,Tq≤ϕdbp,q,for  eachp,q∈M,where(38)ϕt≔ψϕ1t∈Φ.Thus, the above sample of simulation function together with Theorem18 yields the following result.Theorem 19.
Let(M,d) be a complete b-metric space and let T:M→M be defined as(39)αp,qdbTp,Tq≤ϕdbp,q,for  eachp,q∈M,where α:M×M→R0+ and ϕ∈Φ. Suppose that (i)
T is α-orbital admissible;(ii)
there existsp0∈M such that α(p0,Tp0)≥1;(iii)
eitherT is continuous or (M,d) is regular.
Then there existsp∈M such that Tp=p. Moreover, if the condition (U) is fulfilled, then we guarantee that the obtained fixed point p of T is unique.Lettingϕ(t)=kt with k∈[0,1), Theorem 19 implies the following.Theorem 20.
Let(M,d) be a complete b-metric space and let T:M→M be defined as(40)αp,qdbTp,Tq≤kdbp,q,for  eachp,q∈M, where k∈[0,1), α:M×M→R0+, and ϕ∈Φ. Suppose that (i)
T is α-orbital admissible;(ii)
there existsp0∈M such that α(p0,Tp0)≥1;(iii)
eitherT is continuous or (M,d) is regular.
Then there existsp∈M such that Tp=p. Moreover, if the condition (U) is fulfilled, then we guarantee that the obtained fixed point p of T is unique.By lettingα(p,q)=1 in Theorems 19 and 20 we get the main results of Theorems 1 and 2 of Czerwik [5]. Notice that, in this case, conditions (i)–(iii) of Theorems 19 and 20 are fulfilled trivially.


## 3.2. Consequences in the Setting of Standard Metric Space

In this section, we consider the results in the setting of standard metric. Thus, we considers=1 throughout this section. We shall show that a number of existing fixed point results in the literature are the simple consequence of our main results. In particular, by taking Example 2 into consideration, we can list many well-known results as a consequence of our main results.Ifψ∈Ψ and we define(41)σEt,s=ψs-tfor  eachs,t∈R0+, then σBW is a simulation function (cf. Example 2 (v)).First, we derive the very interesting recent results of Samet et al. [13] as a corollary of Theorem 18.Theorem 21.
Theorems2.1 and 2.2 in [13] are consequences of the following.Proof.
TakingσE(r,s)=ψ(s)-rfor  eachs,r∈[0,∞) in Theorem 18, we derive that (42)αp,qdbTp,Tq≤ψdbp,q,for  eachp,q∈M. We skip the details.As is well known, the main theorem in [13] covers several fixed point results, including the pioneer fixed point theorem of Banach. Moreover, as is shown in [13, 16], several fixed point theorems in different settings (in the sense of partially ordered set, in the sense of cyclic mapping, etc.) can be concluded from Theorems 2.1 and 2.2 in [13] by setting α(p,q) in a proper way.Notice also that one can express the main result of Khojasteh et al. [1] as a straight consequence of our main result.Theorem 22.
Theorem18 yields Theorem 7.Proof.
It is sufficient to takeα(p,q)=1 for each p,q∈M in Theorem 21.It is obvious that all presented results in [1] follow from our main result.


## 4. Conclusion

It is very easy to see that one can list a further outcome of our main results by letting the mappingsσ,α,ψ,ϕ in a suitable way like in Example 2. More precisely, by following the techniques in [13, 16] one can easily derive a number of well-known fixed point results in the distinct settings (such as in the frame ofcyclic contraction and in the setting ofpartially ordered set endowed with a metric). We prefer not to list all consequences due to our concerns on the length of the paper. This paper can be also considered as a continuation of the recent paper [17].



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