# On Generalized Caristi Type Satisfying Admissibility Mappings

Authors: Mohammed M. A. Almazah; Basel Hardan; Ahmed A. Hamoud; Faisal A. M. Ali
Journal: Journal of Mathematics
 (2023)

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## Abstract

In this article, the existence and uniqueness of a fixed point were investigated using the concept ofσ,γ-contractive in the context of Hausdorff metric space. A well-known Caristi type is primarily generalized by the new results. The result is improved by building up an example.


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## Body

## 1. Introduction

Fixed point theorems (FPTs) are essential in nonlinear functional analysis. The Banach contraction principle (BCP) [1] is one of the main findings of the FPTs that have been developed and implemented in various fields of study. BCP asserts that for any u,v∈X, where X is a complete metric space (MS), and τ is in the range of (0, 1), f has a FP.BPC has been improved and expanded, as those made by Kannan, Chatterjee, and Reich in [2–4], respectively.Newly, Isik et al. [5, 6] presented a new generalization of the BCP with an application, likewise, Biahdillah and Surjanto [7] introduced an application of the BCP in complex-valued Branciari b-MS, also Jleli et al. [8] researched further generalizations of the BCP, and on the other hand, Patil et al. [9–12] utilized the contractive, generalized contractive, Hardy-Rogers contractive, and generalized nonexpansive mappings on different spaces to get some new FPTs with applications.Caristi employed a lower semicontinuous mapping (LSC) in 1976 to obtain a discernible expansion of BCP (see [13, 14]), and LSC maps are used by Abdeljawad and Karapinar [15] to generalize Cristi FPT on cone MS. In the literature, this theorem is known as the Caristi fixed point theorem (CFPT). New publications on the CFPT type are available: Aamri et al. [16] obtained CFPTs using the Száz principle in quasi-MS, Altun et al. [17] got CFPTs and some generalizations on M-MS, Aslantas et al. [18] gave some CFPTs, and a study on CFPTs in MS with a graph was given by Chuensupantharat and Gopal [19]. Direct proof of CFP was shown by Du [20], Kuhlmann et al. [21] showed the CFPTs from the point of view of ball spaces, and Hardan et al. [22] surveyed CFPTs of contractive mapping with application. A new type of Caristi’s mapping on partial MSs shows that a partial MS is complete if and only if every Caristi mapping has an FP, given by Acar et al. [23], and on the same partial MS, Acar and Altun [24] also gave Bae and Suzuki-type generalizations of Caristi’s FPT. Karapinar [25] showed this kind of FPT in compact partial MS before generalizing it to complete partial MS.One of the helpful useful for our results is the Hausdorff space (HS), and it plays an important part in our theorem and its corollaries. The conceptα,ψ-generalized contractions in a Hausdorff partial MS and its significance in obtaining some common FPTs for a pair of self-mappings which was discussed by Nazam and Acar [26].An interesting notion is a different idea of applying the BCP, it contains the case of discontinuous functions, and this idea depends on usingσ-admissible mappings (σ-AMs) to get FPT, which was introduced by Samet et al. in [27]. For these reasons, there is more study in the literature taking this case (see [28–31]). Two various separate developments of σ-AMs were presented. Budhia et al. in [32] applied a rectangular MS, while that Ansari in [33] applied the notion of C-class maps. When we apply the researchers’ idea on the Caristi type function, we obtain new FPTs on generalized MS, and we have used these results to find the existence and uniqueness (E & U) of a solution for problems for a lot of mathematical sections. On the other hand, of the most pleasure, discussed topics in FPTs, is the investigation of the E & U coincidence fixed point (COI-FP) of different maps in the given spaces. Agarwal and Karapinar [34] mentioned some comments on coupled FPTs in G-MS. Discussion on multidimensional coincidence points introduced by Al-Mezel et al. [35] and Cho et al. [36, 37] examined multidimensional FPTs in partially ordered complete MS and partially ordered fuzzy MS. Generalization of this study on coupled and triple FPTs was obtained by Karapinar et al. [38], and also Khojasteh and Rakoćević [39] used multivalued nonself-mappings to get some new common FPTs for generalized contractive. The generalization of contractive mappings satisfying the kind of an admissibility condition with the aid of C-functions allows us to get novel common FPTs of the CFPT type in this study on HMS.


## 2. Preliminaries

In this section, we put the base for our major results.Definition 1.
(see [27]). Let f be a self-mapping on a MS X,δ and suppose σ:X×X⟶0,∞ be a function. f is called a σ-admissible function if σfu,fv≥1 whenever σu,v≥1,∀u,v∈X.Definition 2.
(see [40]). Let X,δ be a MS, and let σ,γ as in Definition 1. X is said to be σ-orderly with respect to γ if for a sequence ui in X with σui,ui+1≥γui,ui+1 for all i≥N and ui⟶u as i⟶∞; therefore, σui,u≥γui,u, for all i≥N.Definition 3.
(see [27]). Let f be a self-mapping on a MS X,δ. A map f is called a σ,η-contractive mapping if there exist two functions σ:X×X⟶0,∞ and η:0,+∞⟶0,+∞ such that(1)σu,vδfu,fv≤ηu,v∀u,v∈X,where η is a nondecreasing function.
For extra ideas ofσ-AM and σ,η-contractive mappings, see [27, 28, 30].Definition 4.
(see [31]). Suppose f be a self-mapping on a MS X,δ and let σ,γ:X×X⟶0,∞ are two mappings. A map f is called σ-AM with regard to γ if σfu,fv≥γfu,fv where σu,v≥γu,v,∀u,v∈X. Note that, if γu,v=1 for all u,v∈X, this definition leads us to Definition 1. Also, if we select σu,v=1, so we say that f is a γ-subadmissible functions.Definition 5.
(see [33]). A function ψ:ℛ+×ℛ+⟶ℛ is called C-function, if(i)
ψu1,u2≤u1(ii)
ψu1,u2=u1 implicit that either u1=0 or u2=0for allu1,u2∈0,∞.Example 1.
(see [33]). The following functions are C-functions.(i)
ψu1,u2=ηu1, where η:ℛ+⟶ℛ+ is a continuous map such that for u>0, ηu<u and η0=0(ii)
ψu1,u2=τu1, such that τ∈0,1(iii)
ψu1,u2=u1−u2Definition 6.
(see [41]). A nondecreasing continuous function η:ℛ+⟶ℛ+ is called an altering distance mapping if ηu=0 if and only if u=0Remark 1.
We refer to the class of altering distance functions byυDefinition 7.
(see [42]). Suppose f,g are self-mappings on X, then(i)
A pointz∈X is said to be a common FP of f,g if z=fz=gz.(ii)
A pointu∈X is called a COI-P of f and g if fu=gu. In case z=fu=gu, then z is said to be a COI-P of f,g.(iii)
The self-mappingsf,g are said to be weakly compatible if they commute at their COI-FP that is, fgu=gfu whenever gu=fu.
In the next section, with the help of typeC-functions, we will give a new coincidence and common from the type of CFPT by generalized of σ,η-contractive mappings satisfy σ-admissibility on HMS.


## 3. Main Results

Theorem 1.
LetX,δ be an HMS and let f,g:X⟶X are weakly compatible fulfilling σ-AMs with respect to γ. Let ψ∈C-functions and η,ξ∈υ such that for all u,v∈X.(2)σu,v≥γu,v⇒ηδfu,fv≤ψηζu,v,ξζu,v,where(3)ζu,v=maxϑfv−ϑgv,ϑfu−ϑgu.Such that for allu∈X, we have(4)ϑu=fu−gu.Suppose that(a)
There existsu0∈X such that σgu0,fu0≥γgu0,fu0(b)
σuim−1,ujm−1≥γuim−1,ujm−1, for all m⟶∞(c)
Eitherf,g are continuous or σui,z≥γui,z, for all ui⊂X,i∈N, and for some z∈XThen, there existsu∈X such that fnx=gu=u, for some n∈N, that is, u is a periodic point, but if for each periodic point u satisfying σfu,gu≥γfu,gu, then f has a fixed point. Moreover, the FP is an unique if for all u,v∈ψf=u∈X:fu=gu=u, such that σu,v≥γu,v.Proof.
Asu0∈X, given(5)σgu0,fu0≥γgu0,fu0.
Consider the iteration(6)fiu0=fui−1=gui=ϑui.
Such thatui≠ui+1, for all i∈N. So, by (6) and since f satisfied Definition 4 and use (5), we have(7)σu,v=σfu0,f2u0≥γfu0,f2u0=γu,v.
By induction, we get(8)σui,ui+1≥γui,ui+1,∀i∈N.
We will follow an incremental approach to build and accomplish our proof ofE & U.
We shall show thatδϑui,ϑui+1⟶0 as i⟶∞, i.e., δϑui,ϑui+1 is nonincreasing.
By inequality (2), we get(9)ηδϑui,ϑui+1=ηδfxi−1,fxi≤ψηζui−1,ui,ξζui−1,ui,where(10)ζui−1,ui=maxϑfxi−ϑgxi,ϑfxi−1−ϑgxi−1=maxϑui+1−ϑui,ϑui−ϑui−1=maxδϑui+1,ϑui,δϑui,ϑui−1.
We have two proposals:Proposal 1: Ifζui−1,ui=δϑui+1,ϑui, then(11)ηδϑui,ϑui+1=ηδfxi−1,fxi≤ψηδϑui+1,ϑui,ξδϑui+1,ϑui.From Definition5, we find either ηδϑui+1,ϑui=0 or ξδϑui+1,ϑui=0, i.e., δϑui+1,ϑui=0, but this is a contradiction, due to fui+1≠fuiProposal 2: Ifζui−1,ui=δϑui,ϑui−1, then(12)ηδϑui,ϑui+1=ηδfxi−1,fxi≤ψηδϑui,ϑui−1,ξδϑui,ϑui−1≤ηδϑui,ϑui−1.
Sinceη is nondecreasing map and ϑ is a continuous SLF, then(13)δϑui,ϑui+1≤δϑui,ϑui−1.
Thus,δϑui,ϑui+1 is nonincreasing sequence. So,(14)limi⟶∞δϑui,ϑui+1=χ,limi⟶∞ζui−1,ui=χ.
By the continuity property for the functionsη and ξ, we have(15)limi⟶∞ηδϑui,ϑui+1≤limi⟶∞ψηζui−1,ui,ξζui−1,ui,=ψlimi⟶∞ηζui−1,ui,limi⟶∞ξζui−1,ui,≤limi⟶∞ηδϑui,ϑui−1,or(16)ηχ≤ψηχ,ξχ≤ηχ.
Hence, by Definition5, we obtain χ=0, i.e., limi⟶∞δϑui,ϑui+1=0.
Using the abovementioned steps to proveδϑui,ϑui+2⟶0 and by inequality (2), we have(17)ηδϑui,ϑui+2=ηδfxi−1,fxi+1≤ψηζui−1,ui+1,ξζui−1,ui+1≤ηζui−1,ui+1,which could be(18)δϑui,ϑui+2≤ζui−1,ui+1,where η is changing the functions’ space. We obtain(19)ζui−1,ui+1=maxϑfxi+1−ϑgxi+1,ϑfxi−1−ϑgxi−1=maxϑui+2−ϑui+1,ϑui−ϑui−1=maxδϑui+2,ϑui+1,δϑui,ϑui−1.
Again, we have two proposals:Proposal 1: Ifζui−1,ui=δϑui+2,ϑui+1, then(20)ηδϑui+2,ϑui+1=ηδfxi+1,fxi≤ψηδϑui+2,ϑui+1,ξδϑui+2,ϑui+1.From the definition ofC-functions, we find either ηδϑui+2,ϑui+1=0 or ξδϑui+2,ϑui+1=0, i.e., δϑui+2,ϑui+1=0. But this is a contradiction, due to fui+1≠fui∀i∈N.Proposal 2: Ifζui−1,ui=δϑui,ϑui−1, this case has been discussed before.
Obviously,ϑui is not necessity to be subsequent in arranging for all i∈N of the convergence in X.
By contradiction, we shall prove thatϑui is a Cauchy sequence. For that, the following lemma is helpful for the remainder of the theorem’s proof. Its proof is classic so we skip it.Lemma 1.
LetX,δ be a MS and let ϑui be a sequence in X such that(21)limi⟶∞ϑui,ϑui+1=limi⟶∞ϑui,ϑui+2=0,where ui≠xj, for all i≠j. If ϑui is not a Cauchy sequence, then there exist ϵ>0 and two subsequences ϑuim,ϑujm⊂ϑui, where m<jm<im,m∈N. Furthermore,(22)δuim,ujm≥ϵ,δuim,ujm−1<ϵ.Thus, for the following sequences,(23)δϑuim,ϑuim−1,δϑuim,ϑuim−1,δϑujm,ϑujm−1,it satisfies(24)limi⟶∞infδϑuim,ϑujm≤limi⟶∞maxδϑuim,ϑujm≤ϵ,limi⟶∞infδϑuim,ϑuim−1≤limi⟶∞maxδϑuim,ϑuim−1≤ϵ,limi⟶∞infδϑujm,ϑujm−1≤limi⟶∞maxδϑujm,ϑujm−1≤ϵ.Now,(25)limm⟶∞δϑuim,ϑujm=limm⟶∞δfuim−1,fujm−1,for all i,j,m∈N, by inequality (2), we have(26)ηδϑuim,ϑujm=ηδfuim−1,fujm−1≤ψηζuim−1,ujm−1,ξζuim−1,ujm−1,where(27)ζuim−1,ujm−1=maxϑfxjm−1−ϑgxjm−1,ϑfxim−1−ϑgxim−1=maxϑujm−ϑujm−1,ϑuim−ϑuim−1=maxδϑujm,ϑujm−1,δϑuim,ϑuim−1.Then, by Lemma1 and since,(28)σϑujm,ϑujm−1≥γϑujm,ϑujm−1,also,(29)σϑuim,ϑuim−1≥γϑuim,ϑuim−1.Wheni,m⟶∞, the inequality (26) will come like this.(30)ηϵ≤ψηϵ,ξϵ≤ηϵ,where η,ξ,ψ,andϑ are continuous functions. Therefore, we have ηϵ=0 or ξϵ=0. Thus, ϵ=0, which is a contradiction. We conclude that ϑui is a Cauchy sequence. Since X is a complete MS, then ϑui⟶u∗ as i⟶∞, for some u∗∈X.Iff and g are continuous mappings and by iteration relation (6), we find(31)ϑui+1=gui+1=fui=fu∗,asi⟶∞.Then,f and g have a periodical point, since X is a HS, i.e., fu∗=gu∗=u∗. Otherwise, if X is an σ-regular with regard to γ, then by condition b, we obtain(32)ηδϑfui,ϑfu∗≤ψηζui,u∗,ξζui,u∗,where(33)ζui,u∗=maxϑfu∗−gϑu∗,ϑfui−gϑui=maxϑfu∗−ϑu∗,ϑui+1−ϑui=maxδϑfu∗,ϑu∗,δϑui+1,ϑui.Sinceδϑui+1,ϑui=0 as i⟶∞ (from the first step), then(34)limi⟶∞ζui,u∗=δϑfu∗,ϑu∗.By substituting equation (34) into inequality (32), we get(35)ηϑfu∗,ϑu∗≤ψηδϑfu∗,ϑu∗,ξδϑfu∗,ϑu∗,where i⟶∞. Thus, ηδϑfu∗,ϑu∗=0 or ξδϑfu∗,ϑu∗=0; therefore, δ=ϑfu∗,ϑu∗=0. Hence, ϑfu∗=ϑu∗; based on the definition of ϑ, we conclude that f and g have a periodical point fu∗=gu∗=u∗. Thus, we have shown the validity of the c part of our theory.In the remainder of the proof, we will show thatf and g have a COI-FP of the periodical point. This is what we will discuss in the next step.Supposeϑfkz=ϑgkz=z,z∈X. Obviously, z is a FP of f where k=1. We will prove ϑfk−1z=gk−1z=z∗, where k>1,∀k>1,k∈N. We have σfz,gz≥γfz,gz for a periodical point z. If potential, let ϑfk−1z≠ϑfkz and ϑgk−1z≠ϑgkz for all k>1,k∈N. Therefore, by inequalities (2),(36)ηδfk−1z,fkz≤ψηζfk−2z,fk−1z,ξζfk−2z,fk−1z,where(37)ζfk−2z,fk−1z=maxϑffk−1z−ϑgfk−1z,ϑffk−2z−ϑgfk−2z=maxϑfkz−ϑfk−1z,ϑfk−1z−ϑfk−2z=δϑfkz,ϑfk−1z,δϑfk−1z,ϑfk−2z.We have two proposals again.Proposal 1: Ifζfk−2z,fk−1z=δϑfkz,ϑfk−1z for some k, then by (36), we have(38)ηδϑfk−1z,ϑfkz≤ψηδϑfkz,ϑfk−1z,ξδϑfkz,ϑfk−1z≤ηδϑfkz,ϑfk−1z.Hence,δϑfkz,ϑfk−1z=0, by Definition 6. Thus, ϑfkz=ϑfk−1z, which is contradiction.Proposal 2: Ifζfk−2z,fk−1z=δϑfk−1z,ϑfk−2z for all k. Here, f is an σ-admissible with regard to γ and σfz,gz≥γfz,gz, and we obtain(39)ηδϑfk−1z,ϑfkz≤ψηδϑfk−1z,ϑfk−2z,ξδϑfk−1z,ϑfk−2z≤ηδϑfk−1z,ϑfk−2z.Consequently,δϑfk−1z,ϑfk−2z is a nonincreasing sequence of ℛ+, and we get(40)ηδz,ϑfz=ηδϑgz,ϑfz=ηδϑgfkz,ϑfk+1z≤ηδϑgfk−1z,ϑfkz≤ψηδϑfk−1z,ϑfk−2z,ξδϑfk−1z,ϑfk−2z≤ηδϑfk−1z,ϑfk−2z⋮≤ψηδz,ϑfz,ξδϑz,ϑfz≤ηδz,ϑfzηδϑfk−1z,ϑfkz≤ψηδϑfkz,ϑfk−1z,ξδϑfkz,ϑfk−1z.Thus,ϑfkz=ϑfk−1z. And implicitly indicates that ϑgkz=ϑgk−1z which is contradiction. Accordingly, the claim that fk−1z=gk−1z=z∗ is not true. Hence, f,g have a COI-FP z. Sincefandg are weakly compatible, and by Definition 7 (iii), then,zis a common FP offandgonX.The uniqueness of the fixed point increases its strength and renders the solution is not subject to other possibilities. Consequently, we finish our argument by explaining that the FP we have established is unique. This is what we will clarify in the final step of the proof.Suppose thatz1,z2∈X are two common FPs of f and g such that z1≠z2. Use (2) and since σz1,z2≥γz1,z2, we obtain(41)ηδz1,z2=ηδϑfz1,ϑfz2≤ψηζz1,z2,ξζz1,z2,where(42)ζz1,z2=maxϑfz2−ϑgz2,ϑfz1−ϑgz1.Then,ζz1,z2=0. Therefore, ηδz1,z2=0. Consequently, z1=z2.Thus, we have proven the uniqueness of the FP that we have found, and by this, we have completed proving our result.Now, we will present the following corollaries, which are derived from our main result.Corollary 1.
LetX,δ be a HMS let f,g:X⟶X are weakly compatible fulfilling σ-AMs with respect to γ. Let ψ∈C and η,ξ∈υ such that for all u,v∈X.(43)σu,v≥γu,v⇒ηδfu,fv≤ηζu,v−ξζu,v,where ζu,v and the conditions a, b, and c are the same as in Theorem 1. Then, there exists u∈X such that fnu=gnu=u, for some n∈N; that is, u is a periodic point, but if for each periodic point u satisfying σfu,gu≥γfu,gu, then f and g have a common FP. Moreover, the FP is an unique if for all u,v∈ψf=u∈V:fu=gu=u, such that σu,v≥γu,v.Proof.
Takeψu1−u2=u1−u2 and follow the same method proving Theorem 1.
To exemplify our findings, we will introduce the following example.Example 2.
LetX=0,1. Suppose f and g are weakly compatible self-mappings on X such that(44)fu=2u+12,u∈0,12,12,u∈12,1,gu=u,∀u∈0,1.
Letσ,γ:X×X⟶0,∞, such that σu,v=3,γu,v=2,∀u,v∈X. Suppose δ:X×X⟶0,1 be a MS. Consider ψ:0,∞×0,∞⟶ℛ and η,ξ:0,∞⟶0,∞ defined as ψu1,u2=u2−u1,3ηu=u and 5ξu=4u. Then, all the terms of Theorem 1 are satisfied. Hence, u=1/2 be a unique common FP of f and g on X.



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