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Recent Advances in Fixed Point Theory in Abstract Spaces

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Volume 2021 |Article ID 6612448 | https://doi.org/10.1155/2021/6612448

Amjad Ali, Monairah Alansari, Fahim Uddin, Muhammad Arshad, Awais Asif, Ghada Ali Basendwah, "Set-Valued SU-Type Fixed Point Theorems via Gauge Function with Applications", Journal of Mathematics, vol. 2021, Article ID 6612448, 13 pages, 2021. https://doi.org/10.1155/2021/6612448

Set-Valued SU-Type Fixed Point Theorems via Gauge Function with Applications

Academic Editor: Naeem Saleem
Received26 Dec 2020
Revised21 Mar 2021
Accepted25 Mar 2021
Published19 May 2021

Abstract

In this article, we have designed two existence of fixed point theorems which are regarding to set-valued SU-type -contraction and -contraction via gauge function in the setting of metric spaces. An extensive set of nontrivial example will be given to justify our claim. At the end, we will give an application to prove the existence behavior for the system of functional equation in dynamical system and integral inclusion.

1. Introduction and Preliminaries

The most publicized famous result in nonlinear analysis is Banach contraction principle, which made clear a systematic rule to find the fixed point of a given mapping on a metric space. So far, numerous authors have studied this classical result to examine the existence and uniqueness of a solution for different forms of contractive structure.

In 2014, Jleli and Samet [1] introduced the concept of a new contraction known as the -contraction, which generalizes the Banach contraction principle in a beautiful way.

In 2015, Khojasteh et al. [2] introduced simulation function. Recently, many researchers have proved fixed point theorems for Suzuki-type (SU) mappings in metric space (see [3, 4]).

Let be a metric space. For and , let and denote the family of all nonempty closed subsets and the family of all nonempty closed bounded subsets of . Design the Pompeiu–Hausdorff metric induced by on asfor all and . A point is said to be a fixed point of , if . If, for , there exists a sequence in such that , then is said to be an orbit of . Mapping is said to be -orbitally lower semicontinuous (o.l.s.c), if a sequence in and .

A multivalued mapping is called a Nadler-contraction, if there exists such that

Nadler [5] obtained the multivalued version of the Banach contraction principle. Let be a complete metric space and be a Nadler-contraction. Then, has a fixed point. Recently, Vetro [6] proved the following result.to .

Theorem 1. Let be a complete metric space and be a multivalued mapping. Suppose that there exist and such thatwhere is the set of mapping satisfying :(i) is nondecreasing and right-continuous.(ii) For each in , .(iii) There exist and such that . Then, has at least one fixed point.

Remark 1. Let be a metric space. If is a multivalued mapping satisfying the above theorem, thenSince is nondecreasing, we obtain

Example 1. The functions , defined by and , are in .

Lemma 1. (see [6]). Let be a metric space and with . Then, for all and , there exists such that

Lemma 2. (see [6]). Let be a metric space, , and . Then, for each , there exists such that

Definition 1. (see [2]). Mapping is called a simulation function such that. for all . If such that , thenDue to , we have for all . Here, the set satisfies the conditions ()–().

Example 2. (see [2]). For , let be continuous functions such that . Functions () are in :

(i) for all , where .(ii) for all, where is a function such that

Definition 2. (see [1]). Let be a metric space and be a nonempty subset of , and is known as -admissible, if there exists a mapping such thatfor all and .

Lemma 3. (see [7]). Suppose there is a point ( is a closed subset of ) that satisfiesand for some . Then, where denotes an interval on containing 0.

Definition 3. (see [7]). (inclusion ball) Suppose and . Then, for all iterates which belongs to , we define the closed-ball with center and radius , where is defined by (13).

Lemma 4. (see [7]). Suppose there is a point that satisfies and for some ; then, and .

Definition 4. (see [7]). Let , and is known as a gauge function of order on , if it satisfies the following conditions:(i) for all and .(ii) for all .Note that the first condition of Definition 4 is equivalent to and is nondecreasing on .

Definition 5. (see [7]). A gauge function is said to be a Bianchini–Grandolfi gauge function on ifNote that a Bianchini–Grandolfi gauge function also satisfies the following functional equation:

2. Set-Valued SU -Contraction

The first main definition of this exposition is as follows.

Definition 6. Let be a metric space, be a closed subset of , and be a Bianchini–Grandolfi gauge function on . A mapping is known as set-valued SU -contraction, if there exists such that for ,which implies thatwherefor all , with , where .

Theorem 2. Let be a complete metric space and be a multivalued SU -contraction. Suppose such that for some . Then, there exists an orbit of in and such that . In addition, is a fixed point of if and only if the function is -o.l.s.c at .

Proof. Choose . In the case that , is a fixed point of . Thus, we assume that . On the other hand, we haveDefine . From (13), we have . Hence, , and so . Since , from (15) and (17), it follows thatBy right continuity of , there exists a real number such thatFromby Lemma 1, there exists such that . Since is nondecreasing, by (19), this inequality gives thatwhereWe claim thatLet . If , we have , so we obtainwhich is a contradiction. Thus, we conclude that . We assume that ; otherwise, is a fixed point of . From Remark 1, we have , and so . Next, becauseAlso, sincefrom (15), we getSince is right-continuous, there exists a real number such thatNext, fromby Lemma 1, there exists such that . By (28), this inequality gives thatwhereWe claim thatLet . If , we have , so we obtainwhich is a contradiction. Thus, we conclude that . We assume that ; otherwise, is a fixed point of . From Remark 1, we have , and so . Also, we have , sinceContinuing this setup, we have two sequences and such that , with andfor all . Then,which gives thatand by , we haveNext, we prove that is a Cauchy sequence in . Setting , from (), there exist and such thatTake . From the definition of limit, there exists such thatUsing (36) and the above inequality,This implies thatHence, there exists such thatLet . Then, using the triangular inequality and (43), we getOwing to the convergence of the series , is a Cauchy sequence in . Since is closed in , there exists such that . Note that because . Now, we claim thatorfor all . Assume, on the contrary, that there exists such thatandBy (47), we havewhich implies thatwhich together with (38) givesSincefrom contractive condition (15), we havewhereWe claim thatLet . If . Since , we havewhich is a contradiction. Thus, we conclude that . From Remark 1, we haveFrom (38), (43), and (47), we obtainwhich is a contradiction. Hence, (45) holds true:Also, we know that for all n. Thus, from (15), we havewhereWe claim thatLet . If , we have , so we obtainwhich is a contradiction. Thus, we obtain . From Remark 1, we deduceTaking limit in (64),Since is -o.l.s.c at , thenSince is closed, we have . Conversely, if is a fixed point of , then , since .

Corollary 1. Let be a complete metric space, be a Bianchini–Grandolfi gauge function on an interval , and be a given set-valued mapping. If and exist,which implies thatfor all , with . Suppose such that for some . Then, there exists an orbit of in and such that . In addition, is a fixed point of if and only if the function is -o.l.s.c at point .

Corollary 2. Let be a complete metric space, be a Bianchini–Grandolfi gauge function on an interval , and be a given set-valued mapping. If and for exist,implies thatfor all , , and . In addition, suppose such that for some . Then, there exists an orbit of in , such that and is a fixed point of if and only if the function is -o.l.s.c at .

Corollary 3. Let be a complete metric space, be a Bianchini–Grandolfi gauge function on , and be a given set-valued mapping. If and exist, thenfor all , , and . Suppose that such that for some . Then, there exists an orbit of in that converges to the fixed point