On Generalized Caristi Type Satisfying Admissibility Mappings

Abstract

In this article, the existence and uniqueness of a fixed point were investigated using the concept of -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.

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 , where is a complete metric space (MS), and is in the range of (0, 1), 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 concept-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 using -admissible mappings (-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 -AMs were presented. Budhia et al. in [32] applied a rectangular MS, while that Ansari in [33] applied the notion of -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 -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 be a self-mapping on a MS and suppose be a function. is called a -admissible function if whenever .

Definition 2. (see [40]). Let be a MS, and let as in Definition 1. is said to be -orderly with respect to if for a sequence in with for all and as ; therefore, , for all .

Definition 3. (see [27]). Let be a self-mapping on a MS . A map is called a -contractive mapping if there exist two functions and such thatwhere is a nondecreasing function.
For extra ideas of -AM and -contractive mappings, see [27, 28, 30].

Definition 4. (see [31]). Suppose be a self-mapping on a MS and let are two mappings. A map is called -AM with regard to if where . Note that, if for all , this definition leads us to Definition 1. Also, if we select , so we say that is a -subadmissible functions.

Definition 5. (see [33]). A function is called -function, if(i)(ii) implicit that either or  for all .

Example 1. (see [33]). The following functions are -functions.(i), where is a continuous map such that for , and (ii), such that (iii)

Definition 6. (see [41]). A nondecreasing continuous function is called an altering distance mapping if if and only if

Remark 1. We refer to the class of altering distance functions by

Definition 7. (see [42]). Suppose are self-mappings on X, then(i)A point is said to be a common FP of if .(ii)A point is called a COI-P of and if . In case , then is said to be a COI-P of .(iii)The self-mappings are said to be weakly compatible if they commute at their COI-FP that is, whenever .In the next section, with the help of type -functions, we will give a new coincidence and common from the type of CFPT by generalized of -contractive mappings satisfy -admissibility on HMS.

3. Main Results

Theorem 1. Let be an HMS and let are weakly compatible fulfilling -AMs with respect to . Let -functions and such that for all .where

Such that for all , we have

Suppose that(a)There exists such that (b), for all (c)Either are continuous or , for all , and for some

Then, there exists such that , for some , that is, is a periodic point, but if for each periodic point satisfying , then has a fixed point. Moreover, the FP is an unique if for all , such that .

Proof. As , givenConsider the iterationSuch that , for all . So, by (6) and since satisfied Definition 4 and use (5), we haveBy induction, we getWe will follow an incremental approach to build and accomplish our proof of E & U.
We shall show that as , i.e., is nonincreasing.
By inequality (2), we getwhereWe have two proposals: Proposal 1: If , then From Definition 5, we find either or , i.e., , but this is a contradiction, due to  Proposal 2: If , thenSince is nondecreasing map and is a continuous SLF, thenThus, is nonincreasing sequence. So,By the continuity property for the functions and , we haveorHence, by Definition 5, we obtain , i.e., .
Using the abovementioned steps to prove and by inequality (2), we havewhich could bewhere is changing the functions’ space. We obtainAgain, we have two proposals: Proposal 1: If , then From the definition of -functions, we find either or , i.e., . But this is a contradiction, due to . Proposal 2: If , this case has been discussed before.Obviously, is not necessity to be subsequent in arranging for all of the convergence in .
By contradiction, we shall prove that 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. Let be a MS and let be a sequence in such thatwhere , for all . If is not a Cauchy sequence, then there exist and two subsequences , where . Furthermore,

Thus, for the following sequences,it satisfies

Now,for all , by inequality (2), we havewhere

Then, by Lemma 1 and since,also,

When , the inequality (26) will come like this.where are continuous functions. Therefore, we have or . Thus, , which is a contradiction. We conclude that is a Cauchy sequence. Since is a complete MS, then as , for some .

If and are continuous mappings and by iteration relation (6), we find

Then, and have a periodical point, since is a HS, i.e., . Otherwise, if is an -regular with regard to , then by condition , we obtainwhere

Since as (from the first step), then

By substituting equation (34) into inequality (32), we getwhere . Thus, or ; therefore, . Hence, ; based on the definition of , we conclude that and have a periodical point . Thus, we have shown the validity of the part of our theory.

In the remainder of the proof, we will show that and have a COI-FP of the periodical point. This is what we will discuss in the next step.

Suppose . Obviously, is a FP of where . We will prove , where . We have for a periodical point . If potential, let and for all . Therefore, by inequalities (2),where

We have two proposals again. Proposal 1: If for some , then by (36), we have Hence, , by Definition 6. Thus, , which is contradiction. Proposal 2: If for all . Here, is an -admissible with regard to and , and we obtain

Consequently, is a nonincreasing sequence of , and we get

Thus, . And implicitly indicates that which is contradiction. Accordingly, the claim that is not true. Hence, have a COI-FP . Sinceand are weakly compatible, and by Definition 7 (iii), then,is a common FP ofandon.

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 that are two common FPs of and such that . Use (2) and since , we obtainwhere

Then, . Therefore, . Consequently, .

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. Let be a HMS let are weakly compatible fulfilling -AMs with respect to . Let and such that for all .where and the conditions a, b, and c are the same as in Theorem 1. Then, there exists such that , for some ; that is, is a periodic point, but if for each periodic point satisfying , then and have a common FP. Moreover, the FP is an unique if for all , such that .

Proof. Take and follow the same method proving Theorem 1.
To exemplify our findings, we will introduce the following example.

Example 2. Let . Suppose and are weakly compatible self-mappings on such thatLet , such that . Suppose be a MS. Consider and defined as and . Then, all the terms of Theorem 1 are satisfied. Hence, be a unique common FP of and on .

Data Availability

No data were used in this investigation.

Conflicts of Interest

All authors declare that they have no conflicts of interest.

Acknowledgments

Their authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through Larg Groups (project under grant number: RGP.2/4/43). This research was financially supported by the authors.