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Recent Advances in Fixed Point Theory in Abstract Spaces

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Volume 2021 |Article ID 7915448 | https://doi.org/10.1155/2021/7915448

Fahad Sameer Alshammari, K. P. Reshma, Rajagopalan R., Reny George, "Generalised Presic Type Operators in Modular Metric Space and an Application to Integral Equations of Caratheodory Type Functions", Journal of Mathematics, vol. 2021, Article ID 7915448, 20 pages, 2021. https://doi.org/10.1155/2021/7915448

Generalised Presic Type Operators in Modular Metric Space and an Application to Integral Equations of Caratheodory Type Functions

Academic Editor: Naeem Saleem
Received07 Jun 2021
Accepted07 Sep 2021
Published07 Oct 2021

Abstract

Extending the Presic type operators to modular spaces, we introduce generalised Presic type -contractive mappings and strongly -contractive mappings in a modular metric space and establish fixed-point theorems for such contractions in modular spaces. Ulam–Hyers stability of the fixed-point equation involving Presic type operators is also discussed. Our results extend and generalise some known results in the literature. The results are supported by appropriate example and an application to Caratheodory type integral equation.

1. Introduction

Maurice Ren Fréchet [1] introduced the general and axiomatic form of distance as “—space.” Felix Hausdorff [2] reexamined it as a metric space in the setting of points which has been refined, discussed, and generalised in numerous ways. Bakhtin [3], Branciari [4], George et al. [5], and Mitrović and Radenović [6] introduced the notions of a -metric, a rectangular metric, a rectangular -metric, and -metric, respectively. The basic concepts and theory of modular space was formulated in [7]. Later, in [8], the authors established fixed-point theorems in a modular function space. In 2008, Chistyakov [9] introduced the notion of a modular metric and the corresponding modular space. Chistyakov, in [10], established the existence of fixed point for contractive maps and strongly contractive maps in modular metric spaces. Umit et al. [11] introduced Bogin type -contractions and proved fixed-point theorems for such contractions in a -complete modular metric space and provided application to antiperiodic boundary value problems. Furthermore, we see that Chaipunya et al. [12] introduced Geraghty type theorems and Turkoglu and Kilinc [13] introduced Caristi type theorems in a modular metric space and gave its applications in integral equations. For more results of fixed-point theorems in modular metric space and some applications of fixed-point theorems, readers may refer to [1420]. On the other hand, Presic [21, 22] extended the Banach contraction principle to the product of a finite number of metric spaces and later Ciric and Presic [23] generalised the result of Presic [21] and proved the following.

Theorem 1. Let be a metric space, be a positive integer, and be a mapping satisfying the following condition:where are arbitrary elements in and . Then, there exists some such that . Moreover, if are arbitrary points in and for , then the sequence is convergent and if , then . If, in addition, satisfies , for all . Then, is the unique point satisfying .

Later, the above results were further extended and generalised to a -metric space, cone metric space, and cone -metric space by many authors, and fruitful applications were found for such results (see [2429]). The aim of this work is to introduce generalised Presic type contractions (which includes Ciric–Presic type contraction and Presic type contractions) in modular metric spaces and establish fixed-point theorems for such contraction mappings in a -complete modular metric space. Our results show symmetric transformation of the well-known Ciric–Presic theorem [23] and Presic theorem [21] from ordinary metric space to a modular metric space. We have introduced Ulam–Hyers stability of fixed-point equations involving Presic type operators in a modular metric space. We have also provided an application of our result to prove the existence of solution of an integral equation of Caratheodory type functions. Our results extend and generalise many known results in the literature.

2. Preliminaries

The following basic concepts of modular spaces are from [9, 10]. Let and be a given function. For any , we write as .

Definition 1. (see [10]). We call a modular metric on if for all and , the following relations hold:(mm1) (mm2) (mm3) In this case, is said to be a metric modular on . Instead of (mm1), if satisfies onlyfor all and , then is said to be pseudomodular on .
Also, is strict modular on if instead of (mm1), it satisfies (2) and for , there exists a number (possibly depending on and ), such thatClearly, if is strict modular, then it is modular, which in turn implies is pseudomodular on .
A metric modular on is convex if it satisfies the (stronger) inequality, in lieu of (mm3):for all .
The main property of a modular on a set is that (see Section 2.3 of [9]), given , the function is nonincreasing on . If , then (mm1) and (mm3) imply

Definition 2. (see [10]). Given a pseudomodular on , the two setsare said to be modular spaces (around ).

Clearly, . Note that if is metric modular on , then modular space can be equipped with a metric generated by given byfor any . Moreover, if is a convex modular on , then by Section 3.5 and Theorem 3.6 of [9], ; i.e., the two modular spaces coincide, and this common set can be endowed with a metric :

Henceforth, and represent modular metric spaces induced by .

Definition 3. (see [10]). Let be a metric modular on X. We have the following:(i)A sequence in is -convergent (or modular convergent) to some , if there exists , such that (ii)A sequence in is -Cauchy if , possibly dependent on , such that as (iii) is -complete if every -Cauchy sequence is -convergent

From [10], we have that and are closed with respect to convergence, if is pseudomodular on . Also, if is strict, then the modular limit, if exists, is unique. If is (not necessarily convex) modular on , then is always convex modular on .

3. Main Results

In this section, we introduce Presic type operators in modular metric space and prove existence of unique fixed points for such operators.

Let be the set of real numbers, and consider a function such that(a) is increasing in all variables(b), for all

Definition 4. Let be metric modular on and (where ) for some positive integer . Then, we have the following:(i) is a generalised Presic type modular contraction (or a generalised Presic w-contraction), iffor some and possibly depending on and for all and .(ii) is a generalised Presic type strongly modular contraction (or a generalised Presic strong w-contraction), iffor some and possibly depending on and for all and .(iii) is a Ciric–Presic type modular contraction (or a Ciric–Presic w-contraction), iffor some and possibly depending on and for all and .(iv) is a Ciric–Presic type strongly modular contraction (or a Ciric–Presic strong w-contraction), iffor some and possibly depending on and for all and .(v) is a Presic type modular contraction (or a Presic w-contraction), if there exists , , such thatfor some and possibly depending on and for all and .

Theorem 2. Let be a strict metric modular on and be -complete. For any positive integer , let be a generalised Presic type strongly w-contractive mapping. If there exists in , such that , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.

Proof. For arbitrary elements in with , define the sequence in given by , . By the -contractivity of , there exist two numbers and such that condition (10) holds. Let , where . Then, for any positive integer . Using the method of induction, we will prove that for any nonnegative integer ,where and . Clearly, by the definition of , (15) is true for and . Let the inequalities hold true. Then,Thus, inductive proof of (15) is complete. Set ; then, . Let integers and be such that . We set . Then, we haveTaking into account that for all , we getTherefore, is -Cauchy in . By -completeness of , there exists in such that , and by strictness of , the limit is unique. Let be arbitrary. Then, we can find a natural number such that and for all . Then, for any integer , we haveAs , we get . By the strictness of , . For uniqueness, suppose there exists , where , such that . Then, using (14), we geta contradiction (as ). Thus, the uniqueness of is established.
Now, if , thena contradiction as . Hence, is unique in .

In the next result, using a convex modular metric, we prove the existence of a fixed point for a generalised Presic type modular contractive mapping.

Theorem 3. Let be a strict convex modular on and be -complete. For any positive integer , let be a generalised Presic w-contractive mapping. If there exists in , such that , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.

Proof. We take for all and . Then, for all and , we haveSo, is strict modular on . Also, by (9), for all and , we haveThus, is a generalised Presic strong w-contraction mapping for the modular . Clearly, implies . By Theorem 2, there exists such that . The remaining part of the proof follows the same line as in Theorem 2.

Taking in Theorems 1 and 2, we get the following.

Theorem 4. Let be strict metric modular on and be -complete. For any positive integer , let be a Ciric–Presic strongly w-contractive mapping. If there exists in , such that , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.

Theorem 5. Let be strict convex modular on and be -complete. For any positive integer , let be a Ciric–Presic w-contractive mapping. If there exists in , such that , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.

Taking , with in Theorem 3, we get the following.

Theorem 6. Let be strict convex modular on and be -complete. For any positive integer , let be a Presic type w-contractive mapping. If there exists in , with , then has a fixed point; that is, there exists an such that . Moreover, for any in , with , the sequence given by converges to a fixed point of . Furthermore, if satisfiesfor all or if , then the fixed point is unique.

Remark 1. For , Theorems 4 and 5 reduce to Theorem 10 and Theorem 5.4 of [10].

Remark 2. Theorems 4 and 5 extend the results of Ciric and Presic [23] to a modular space.

Remark 3. Theorem 6 extends the result of Presic [21] to a modular space.

Example 1. Let . Define the mapping by . Note that for all , then and is a complete modular metric space. Let be given by . Let given by . Then,If , then for all . Thus, is a generalised Presic type strongly modular contraction and Ciric–Presic type strongly modular contraction with constant . Theorems 2 and 4 are applicable, and is the unique fixed point of .

Example 2. Let . Define the mapping by . Note that for all <