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Fixed Point Theory for Set Valued Maps

Goal: Fixed point theory is an active area of research with wide range of applications in various directions. It is concerned with the results which state that under certain conditions a self map f on a set X admit one or more fixed point. Fixed point theory started almost immediately after the classical analysis began its rapid development. The further growth was motivated mainly by the need to prove existence theorems for differential and integral equations. Thus the fixed point theory started as purely analytical theory. Fixed point theory can be divided into three major areas: Metric fixed point theory, Topological fixed point theory and Discrete fixed
point theory. Classical and major results in these areas are: Banach’s fixed point theorem, Brouwer’s fixed point theorem and Tarski’s fixed point theorem.
In 1922, Polish mathematician Stefan Banach formulated and proved a theorem which concerns under appropriate conditions the existence and uniqueness of a fixed point in a complete metric space. His result is known as Banach’s fixed point theorem or the Banach contraction principle. Due to its simplicity and generality, the contraction principle has drawn attention of a very large number of mathematicians.
After enormous development of linear functional analysis the
time was ripe to focus on nonlinear problems. Then the role of the analytical fixed point theory became even more important.
The study of fixed points for set valued contractions and non expansive maps using the Hausdorff metric was initiated by Markin. Later, an interesting and rich fixed point theory for such maps has been developed. The theory of set valued maps has applications in control theory, convex optimization, differential inclusions and economics. Following the Banach contraction principle Nadler introduced the concept of set valued contractions and established that a set valued contraction possesses a fixed point in a complete metric space. Subsequently many authors generalized Nadler’s fixed point theorem in different way.
A constructive proof of a fixed point theorem makes the theorem twice as worthy because it yields an algorithm for computing a fixed point. Indeed, many fixed point theorems have constructive proofs, of which we might mention the geometric fixed point results due to Banach and Nadler, for single valued and set valued mappings. These results are of particular importance and play a fundamental role in nonlinear analysis. They are used prominently in denotational semantics, for example to give meaning to recursive programs. In fact, it is hard to overestimate their applicability and importance in mathematics. Among other applications, they are used to show the existence of solutions to differential equations, as well as the
existence of equilibria in game theory. Tarski’s fixed point theorem guarantees the existence of a fixed point of an order preserving function defined on a nonempty complete lattice. In theoretical computer
science, least fixed points of monotone functions are used to define program semantics. Tarski’s fixed point theorem has important applications in formal semantics of programming languages. Although Tarski’s proof is beautiful and elegant, but non constructive.
Recently there have been so many exciting developments in the field of existence of fixed point in partially ordered sets. This trend was started by Ran and Reurings; they extended the Banach contraction principle in partially ordered sets with some application to matrix equation. Their results are hybrid of the two classical theorems; Banach’s fixed point theorem and Tarski’s fixed point theorem. The results are applicable in some cases where neither Tarski’s theorem nor Knaster-Tarski or
Amman theorem which requires existence of supremum for every chain in X, are useful. Following the trend several researchers have recently obtained many significant results and shown there applications in different areas. This project will continue to further explore this area with the intension to look for new results with applications.

Methods: Iterative Methods

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Ismat Beg
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In this paper we introduce the concept of ordered uniform convexity in ordered convex metric spaces and study some properties of order uniform convexity. Finally as application we connect our results with existence of fixed points for monotone non-expansive mappings defined on these spaces.
Ismat Beg
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The aim of this article is to introduce a new notion of ordered convex metric spaces and study some basic properties of these spaces. Several characterizations of these spaces are proven that allow making geometric interpretations of the new concepts. 1. Introduction Menger [1] initiated the study of convexity in metric spaces which was further developed by many authors [2–4]. The terms “metrically convex” and “convex metric space” are due to [2]. Subsequently, Takahashi [5] introduced the notion of convex metric spaces and studied their geometric properties. Takahashi also proved that all normed spaces and their convex subsets are convex metric spaces and gave an example of a convex metric space which is not embedded in any normed/Banach space. Kirk [6] showed that a metric space of hyperbolic type is a convex metric space. Afterward, Shimizu and Takahashi [7] gave the concept of uniformly convex metric space, studied its properties, and constructed examples of a uniformly convex metric space. Beg [8] established some inequalities in uniformly convex complete metric spaces analogous to the parallelogram law in Hilbert spaces and their applications. Beg [9] proved that a closed convex subset of uniformly convex complete metric spaces is a Chebyshev set. Recently, Abdelhakim [10] studied convex functions on these spaces. The aim of this note is to further continue the research in this direction by introducing the concept of ordered convex metric spaces and study their structure. We conclude with the plan of the paper. In Section 2, we recall some basic notations and definitions from the existing literature on convex metric spaces, order structure, and general topology. In Section 3, we introduce the new concept of ordered convex metric spaces and study some basic properties. Several characterizations of these spaces are also proven that allow making geometric interpretations of the new concepts Finally, Section 4 concludes with a summary statement. 2. Preliminaries In this section, basic results about convex metric spaces and order structure are given. Definition 1 (see [5]). Let be a metric space and . A mapping is said to be a convex structure on if for each and , Metric space together with the convex structure is called a convex metric space. A nonempty subset is said to be convex if whenever . Remark 2 (see [5, 10]). The convex metric space has the following properties: (i), , (ii)Open spheres and closed spheres are convex(iii)If is a family of convex subsets of , then is convex Any normed space and a convex subset of a normed space is a convex metric space. There are several examples in the existing literature [5, 7, 8, 10] of convex metric spaces which are not embedded in any normed space. Definition 3 (see [11]). A binary relation defined for some pairs of elements of a set is called an order relation in if is reflexive, transitive, and antisymmetric. A reflexive and transitive relation is called a preorder. Remark 4 (see [11]). Let be a binary relation on a set . By we mean and Relation is defined as if and The inverse of is defined as if . Incomparable elements and (i.e., and ) are denoted by Transitivity of order relation implies for all . Definition 5 (see [11]). An ordered set is called totally ordered if it has no incomparable elements. Proposition 6 (see Proposition 4.1 of [12]). A topological space is disconnected if and only if it has a nonempty subset that is both open and closed. Proposition 7 (see [13]). Let be a connected topological space. If is a connected subset of such that is the union of nonempty, pairwise disjoint open (in ) sets , then is connected for all . 3. Ordered Convex Metric Spaces In this section, first, we introduce the property on a convex metric space. Next, we present some notations and definitions related to an order relation on a convex metric space. Finally, we define ordered convex metric space and prove several interesting results related to ordered convex metric spaces. Definition 8. A convex metric space is said to have property if for all in and in , we have Each normed space has property , if we define In Definition 8, taking and using Remark 2, we obtain Let be a convex metric space and be an ordered relation on . First, we define some notation for subsequent use. For any in and Definition 9. (i) A relation on a convex metric space is said to be continuous if for all in , the sets and are closed. (ii) A relation on a convex metric space is said to be Archimedean if for all in with , there exists such that and When relation is total, the space is called Archimedean. (iii) A relation on a convex metric space is said to have betweenness property if for all in , all and all if and only if Definition 10. A relation on a convex metric space , is (i)(o)-convex if and implies (ii)(o)-concave if and implies (iii)(o)-linear if implies Definition 11. A convex metric space with order relation is called ordered convex metric space if is continuous. Proposition 12. Let be an ordered convex metric space with property Then, is Archimedean and are open for any in Proof. Let be an Archimedean relation on and be closed. Without loss of generality, we can assume that is nonempty. Choose in Now, continuity of and imply that there exists such that is contained in the complement of . Assume there exists Continuity of implies that is an open set in Thus, is union of at most countably many mutually disjoint open intervals. Axiom of choice further implies that there exists among these intervals an open interval such that If , then set ; otherwise, Then, By Definition 8 (ii), we have Obviously, ; thus, It contradicts that is Archimedean. Hence, is open. Similarly, we can prove that is open. Assume that and are open. Choose in such that Remark 2 (i) implies that Since is open, thus there exists such that Also, and is open; therefore, there exists such that Now obviously, and Next, we give Example 13 to show that we cannot drop any condition from Proposition 12. Example 13. Consider the convex metric space with usual Euclidean distance and convex structure defined by Assume is a reflexive relation on such that for all and no other element are comparable. Then, each of and either contains at most two elements or is equal Therefore, and are closed. Also is not open. Moreover, but for all and all Thus, is continuous, and are not open, and is not Archimedean. Now, the following proposition is obvious. Proposition 14. Any totally ordered convex metric space with property is Archimedean. Theorem 15. A nontrivial continuous Archimedean order on a convex metric space with property is totally ordered. Proof. Let be not totally ordered on the convex metric space . Then, there exists such that and with Let Then, using Remark 4 or Since , therefore Thus, and Now, we prove Obviously, To prove other inclusions, choose If then it follows from transitivity and that , which is a contradiction to Therefore, , i.e., In a similar way, if then which contradicts Thus, Now, and Remark 2 (i) imply that and Continuity of further implies that is closed. Using Equality 3, we obtain that is a closed set. On the other hand, Proposition 12 implies that is an open set. Thus, we have a nonempty closed-open proper subset of Since is connected, therefore it is a contradiction to Proposition 6 Similarly, we can show a contradiction for the case Hence, is a totally ordered relation. Corollary 16. Let be an ordered convex metric space with property If is an Archimedean relation, then the space is also Archimedean. Proposition 17. Let be an ordered convex metric space with property ; then, is (o)-linear is convex. Proof. Let be (o)-linear. Choose and Define and Then, and Transitivity of implies It follows from (o)-linearity of that By Definition 8 (ii), we obtain Transitivity of further implies that Therefore, Hence, is convex. Assume is convex. Choose such that and . From reflexivity of Remark 2 implies Since by assumption is convex. Therefore, Thus, Now transitivity of and imply that and Property (see Definition 8 (i)) of implies . Hence, is (o)-linear. Proposition 18. Let be an ordered convex metric space with property , then is (o)-convex if and only if is convex. Proof. Suppose that is (o)-convex. Choose and Define and Then, and (o)-convexity of implies that Using Definition 8, we have Thus, Hence, is convex. Assume that is convex. Choose such that and Remark 2 implies As is convex, therefore Thus, Hence, is (o)-convex. Proposition 19. Let be an ordered convex metric space with property ; then, is (o)-concave if and only if is convex. Proof. Similar to Proposition 18. Theorem 20. Let be an Archimedean-ordered convex metric space with property The relation is (o)-linear if and only if is (o)-convex and (o)-concave. Proof. Assume is (o)-linear. Proposition 17 implies is convex, thus a connected subset of Obviously, Clearly, , and are mutually disjoint sets. Proposition 12 and continuity of imply that these all three sets are open sets. Since Proposition 18 and Proposition 19 further imply that (o)-concavity of is equivalent to the convexity of and (o)-convexity of is equivalent to the convexity of Now, Proposition 7 implies that is (o)-convex and (o)-concave. Assume that is convex and is concave for all in . As Therefore, is convex. Proposition 17 further implies that is (o)-linear. 4. Concluding Remarks Order, convexity, and metric are three fundamental concepts in mathematics. These ideas have beautiful geometric properties with significant applications in approximation and optimization (see [14, 15]). In this work, we tried to combine these three indispensable notions of order, convexity, and metric. We introduced the new concept of ordered convex metric spaces and studied some of their properties. Several characterizations (Propositions 12, 17, 18, and 19 and Theorem 20) of these spaces are proven that allow to make geometric interpretations of the new concepts. This author’s recommendation is to study other applications of ordered convex metric spaces to economics, preference modelling, control theory, functional analysis, etc. Data Availability No data were used to support this study. Conflicts of Interest Author declares that he has no conflict of interest.
Ismat Beg
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This paper introduces the concept of k-ordered proximal contractions and then study best proximity point results for these mappings. An example is given to show accuracy and significance of our claims.
Ismat Beg
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We prove Ekeland's variational principle in S JS-metric spaces. A generalization of Caristi fixed point theorem on S JS-metric spaces is obtained as a consequence.
Ismat Beg
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Sufficient conditions for existence of common fixed point on complex partial b-metric spaces are obtained. Our results generalize and extend several well-known results. In the end we explore applications of our key results to solve a system of Urysohn type integral equations.
Ismat Beg
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In this paper, we introduce the concept of generalized F-proximal contraction mappings and prove some best proximity point theorems for a non-self mapping in a complete metric space. Then some of the well known results in the existing literature are generalized/extended using these newly obtained results. An example is being given to demonstrate usefulness of our results.
Ismat Beg
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In this paper, we introduce the concept of a generalized convex metric space as a generalization of a convex metric space, which is due to Takahashi, and give the iterative scheme due to Xiao et al.. We also establish strong convergence of this scheme to a unique common fixed point of a finite family of asymptotically quasi-nonexpansive mappings. Our results generalize/extend several existing results
Ismat Beg
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In this paper, we introduce the concept of generalized orthogonal -Suzuki contraction mapping and prove some fixed point theorems on orthogonal -metric spaces. Our results generalize and extend some of the well-known results in the existing literature. As an application of our results, we show the existence of a unique solution of the first-order ordinary differential equation. 1. Introduction and Preliminaries Banach contraction principle is one of the famous and useful results in mathematics. In last 100 years, it is extended in many different directions. The substitution of the metric space by other generalized metric spaces is one normal way to strengthen the Banach contraction principle. The concept of a -metric space was introduced by Bakhtin [1] and Czerwik [2]. They also established the fixed point result in the setting of -metric spaces which is a generalization of the Banach contraction principle. In 2015, Alsulami et al. [3] introduced the concepts of generalized -Suzuki type contraction mappings and proved the fixed point theorems on complete -metric spaces. On the other hand, Gordji et al. [4] introduced the new concept of an orthogonality in metric spaces and proved the fixed point result for contraction mappings in metric spaces endowed with this new type of orthogonality. Furthermore, they gave the application of this results for the existence and uniqueness of the solution of a first-order ordinary differential equation, while the Banach contraction mapping principle cannot be applied in this situation. This new concept of an orthogonal set has many applications, and there are also many types of the orthogonality. Afterward Eshaghi Gordji and Habibi [5] proved fixed point in generalized orthogonal metric spaces. Recently, Sawangsup et al. [6] introduced the new concept of an orthogonal -contraction mappings and proved the fixed point theorems on orthogonal-complete metric spaces. Subsequently, many other researchers [7–11] studied the orthogonal contractive type mappings and obtained significant results. This paper is in continuation of these studies; first, we introduced the new concepts of generalized orthogonal -Suzuki contraction mappings on an orthogonal -metric space and then prove the fixed point theorems on orthogonal -complete metric space with examples and applications to differential equations. Over results generalize/extend several results from the existing literature. In this paper, we denote by , , and the set of positive integers, the set of positive real numbers, and the nonempty set, respectively. Next, we state the concept of a control function which was introduced by Wardowski [12]. Let denote the family of all functions satisfying the following properties: (F1) is strictly increasing (F2) for each sequence of positive numbers, we have Bakhtin [1] and Czerwik [2] gave the concept of a -metric space as follows. Definition 1 (see [2]). Let be a nonempty set and . Suppose that the mapping satisfies the following conditions for all : (i) if and only if (ii)(iii) Then, is called a -metric space with the coefficient . Example 2 (see [2]). Define a mapping by for all . Then, is a -metric space with the coefficient . The idea of generalized -Suzuki type contraction and -Suzuki type contraction in complete -metric spaces is due to Alsulami et al. [3]. Gordji et al. [4] introduced the notion of an orthogonal set (or -set). Definition 3 (see [4]). Let and be a binary relation. If satisfies the following condition: then it is called an orthogonal set (briefly -set). We denote this -set by . Example 4 (see [4]). Let be the set all people in the world. Define the binary relation on by if can give blood to . According to the Table 1, if is a person such that his (her) blood type is -, then we have for all . This means that is an -set. In this -set, (in Definition 3) is not unique. Note that, in this example, may be a person with blood type . In this case, we have for all . Type You can give blood to You can receive blood from A+ A+AB+ A+A-O+O- O+ O-A+B+AB+ O+O- B+ B+AB+ B+B-O+O- AB+ AB+ Everyone A- A+A-AB+AB- A-O- O- Everyone O- B- B+B-AB+AB- B-O- AB- AB+AB- AB-B-O-A-
Ismat Beg
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In the paper we obtain sufficient conditions for the existence of common fixed point for a pair of contractive type mappings in bicomplex valued metric spaces.
Ismat Beg
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We introduce the idea of S^{JS}-metric spaces which is a generalization of S-metric spaces. Next we study the properties of S^{JS}-metric spaces and prove several theorems. We also deal with abstract S^{JS}-topological spaces induced by S^{JS}-metric and obtain several classical results including Cantor's intersection theorem in this setting.
Ismat Beg
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In 1969, Meir and Keeler obtained a remarkable generalization of Banach's results. In this paper, a Meir-Keller type fixed point theorem for a pair of maps on locally convex topological spaces is proved.
A result regarding invariant random approximation is proved.
Ismat Beg
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Existence of fixed points of asymptotically semicontractive operators defined on a separable uniformly convex Banach space, under certain condition is shown.
Ismat Beg
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A general random fixed point theorem, that incorporate several known random fixed point theorems, involving continuous random operators is proved.
By using measurable selection method a random approximation theorem is obtained. As an application of our theorem, a random fixed point theorem is derived. A random coincidence theorem for a pair of continuous random operators is also proved
Ismat Beg
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In this paper we prove random fixed point theorems in reflexive Banach spaces for nonexpansive random operators satisfying inward or Leray-Schauder condition and establish a random approximation theorem.
Ismat Beg
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A general random fixed point theorem, that incorporates several known random fixed point theorems, involving continuous random operators, is proved.
The existence of best approximation in random Banach space under suitable conditions is proved.
Existence of fixed points of fuzzy multivalued mappings with values in fuzzy ordered sets under suitable conditions is proved.
Ismat Beg
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We prove the existence of fixed points and study the structure of the set of fixed points for nonexpansive mappings on uniformly convex complete metric spaces.
Ismat Beg
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We study existence of maximal fixed points of expansive mappings on fuzzy pre-ordered sets under suitable conditions.
Ismat Beg
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Common random fixed points for a R-weakly commuting random operators are obtained under suitable conditions. A result regarding invariant random approximation is also derived.
Ismat Beg
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By using Hahn-Banach theorem, a characterization of random approximations is obtained.
The aim of this paper is to study random fixed points in connection with random approximations. We prove random approximation theorems for more general random operators, i.e., 1-set-contractive random operators or continuous random operators. As applications of our theorems, we derive results regarding random coincidence points and random fixed points
Ismat Beg
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The existence of invariant random best approximations in Banach spaces is proved.
Ismat Beg
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We prove the existence of a fixed point for asymptotically nonexpansive mappings defined on a uniformly convex metric space. A Mann type iteration scheme is constructed which converges to fixed point.
Ismat Beg
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We prove the parallelogram inequalities in metric spaces and use them to obtain the fixed points of involutions.
Let (Ω,Σ) be a measurable space, (E,P) be an ordered separable Banach space and let [a, b] be a nonempty order interval in E.It is shown that if f:Ω×[a, b]→E is an increasing compact random map such that a≤f(ω, a) and f(ω, b)≤b for each ω∈Ω then f possesses a minimal random fixed point α and a maximal random fixed point β.
Ismat Beg
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The existence of minimal and maximal fixed points for monotone operators defined on probabilistic Banach spaces is proved. We obtained sufficient conditions for the existence of coupled fixed point for mixed monotone condensing multivalued operators
We proved random analogue of Edelstein fixed point theorem for contractive random multivalued maps. We also proved existence of unique common random fixed point for a hybrid of two compatible contractive random maps.
Ismat Beg
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Existence of fixed point for monotone maps on a pre-ordered set under suitable condition is proved.
It is shown that every contractive mapping on a probabilistic Banach space has a unique fixed point.
Probabilistic version of the invariance of domain for contractive field and Schauder invertibility theorem are proved. As an application, the stability of probabilistic open embedding is established.
Ismat Beg
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The aim of this paper is to study the behaviour of the iterate of a nonexpansive random map on an arbitrary Banach space. We proved the existence of unique random fixed points of nonexpansive random mappings in Banach spaces. Our results are noteworthy in the sense that no geometric assumption is required on the underlying space.
We proved a fuzzy variational principle and as an application obtained a characterization of complete fuzzy metric spaces in terms of existence of fixed points in fuzzy metric spaces.
Ismat Beg
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We obtain necessary conditions for convergence of the Cauchy Picard sequence of iterations for Tricomi mappings defined on a uniformly convex linear complete metric space.
We obtained sufficient conditions for the existence of fixed point, almost fixed point and best approximation of *- non-expansive multi valued map defined on a non-convex un-bounded subset of a Banach spaces (as as a hyper convex space). Random version of several results is also proved. Our results generalize and extend several known results, both in deterministic and random cases.
Ismat Beg
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We establish the existence and approximation of solutions to the operator inclusion y Ty for deterministic and random cases for a nonexpansive and *-nonexpansive multivalued mapping T defined on a closed bounded (not necessarily convex) subset C of a Banach space. We also prover random fixed points and approximation results for*-nonexpansive random operators defined on an unbounded subject C of a uniformly convex Banach space.
Ismat Beg
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We proved the existence of common fixed points of two expansive mappings defined on fuzzy ordered set.
An important property of metric spaces is the existence and uniqueness of completion. In this paper, we prove existence and uniqueness of completion of a complex valued strong b-metric space.
We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and quasi nonexpansive maps defined on a compact convex subset of a uniformly convex complete metric space. We obtain results on best approximation as a fixed point in a strictly convex metric space.
Ismat Beg
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The purpose of this article is to prove existence of random fixed point for asymptotically nonexpansive random maps. We also construct a Mann type iteration process for asymptotically nonexpansive random maps which converges to the random fixed point.
We obtain necessary and sufficient conditions for the existence of essential random fixed point of a random operator defined on a compact metric space. The structure of the set of essential random fixed points is also studied.
Ismat Beg
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We prove the existence of unique best approximation for an element of a uniformly convex complete metric space from a closed convex subset. As application a fixed point theorem is proved.
The purpose of this paper is to study the convergence problem of Mann and Ishikawa type iterative schemes of weakly contractive mapping in a complete convex metric space. We establish the results on invariant ap-proximation for the mapping defined on a class of nonconvex sets in a convex metric space. Finally, we obtain the existence of common fixed points of two asymptotically nonexpansive mappings through the convergence of iteratively defined sequence in a uniformly convex metric space.
We prove the existence of coincidence point and common fixed point for mappings satisfying generalized weak contractive condition. As an application, related results on invariant approximation are derived. Our results generalize various known results in the literature.
Ismat Beg
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We prove the existence of fixed point for weakly contractive multivalued maps satisfying the inwardness condition in the framework of a convex metric space. Fixed point theorems for multivalued contraction mapping taking the closed values are also obtained. These theorems extend several known results.
Ismat Beg
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We generate a sequence of measurable mappings iteratively and study necessary conditions for its strong convergence to a random fixed point of strongly pseudocontractive random operator. We establish the weak convergence of an implicit random iterative procedure to common random fixed point of a finite family of nonexpansive random operators in Hilbert spaces. We prove the equivalence between the convergence of random Ishikawa and random Mann iterative schemes for contraction random operator and strongly pseudocontractive random operator. We also examine the stability of random fixed point iterative procedures for the random operators satisfying certain contractive conditions in the context of metric spaces.
We construct random iterative processes for weakly contractive and asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. It is shown that they converge to the random fixed points of these operators in the setting of Banach spaces. We also proved that an implicit random iterative process converges to the common random fixed point of a finite family of asymptotically quasi-nonexpansive random operators in uniformly convex Banach spaces.
Ismat Beg
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We construct a random iteration scheme and study necessary conditions for its convergence to a common random fixed point of two pairs of compatible random operators satisfying Meir-Keeler type conditions in Polish spaces. Some random fixed point theorems for weakly compatible random operators under generalized contractive conditions in the framework of symmetric spaces are also proved.
We prove the existence of a common random fixed point of two asymptotically nonexpansive random operators through strong and weak convergences of an iterative process. The necessary and sufficient condition for the convergence of sequence of measurable functions to a random fixed point of asymptotically quasi-nonexpansive random operators in uniformly convex Banach spaces is also established.
Ismat Beg
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In the present paper, we introduce new types of convergence of a sequence in left dislocated and right dislocated metric spaces. Also, we generalize Banach contraction principle in these newly defined generalized metric spaces.
Ismat Beg
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In this paper we first introduce a new approach to the classical fixed point theorems for H⁺-type nonexpansive multivalued mappings in Banach spaces by reformulating the arguments in an ultrapower context which helps to illuminate many underlying ideas and obtained a generalization of classical Nadler's fixed point theorem. Secondly using this generalization of Nadler's fixed point theorem, we study invariant approximation and obtain several new results by replacing multivalued nonexpansive mapping with H⁺-type multivalued nonexpansive mapping. In doing so we apply weakly compact and starshaped conditions on certain nonempty subset of P_{K}(x₀). Several examples are constructed to further illustrate our results.
Ismat Beg
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In this paper we prove existence of fixed point theorems for Z-contractive map, Geraghty type contractive map and interpolative Hardy-Rogers type contrac-tive mapping in the setting of S JS metric spaces with two metrics. Examples are constructed to high light the significance of newly obtained results.
Ismat Beg
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We introduce the notion of sequentially compactness on S JS-metric spaces and study the properties of sequentially compact S JS-metric spaces. As an application, we obtain the results on fixed points of mappings defined on sequentially compact S JS-metric spaces.
Ismat Beg
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We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and quasi nonexpansive maps defined on a compact convex subset of a uniformly convex complete metric space. We obtain results on best approximation as a fixed point in a strictly convex metric space.
We introduce a new class of uniformly R-subweakly commuting mappings and then using this class study the problem of approximation of common fixed points of asymptotically S-nonexpansive mappings in a Banach space with uniformly Gateaux differentiable norm.
Ismat Beg
added 7 research items
We prove the parallelogram inequalities in Menger convex metric spaces and use them to obtain fixed points of involutions and asymptotically non-expansive mappings. We construct Mann type iterative sequences in a Menger convex metric space and study its convergence.
We iteratively generate a sequence of measurable mappings and study necessary conditions for its convergence to a random fixed point of random nonexpansive operator. A random fixed point theorem for random nonexpansive operator, relaxing the convexity condition on the underlying space, is also proved. As an application, we obtained random fixed point theorems for Caristi type random operators.
A common fixed point result for C q -commuting and compatible maps in Menger convex metric space is obtained. As application, various invariant approximation theorems are derived.
Ismat Beg
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A new contraction condition for multivalued maps in metric spaces is introduced and then, based on this new condition, we prove two fixed point theorems for such contractions. The new condition uses the altering distance technique and a Pompeiu type metric on the family of nonempty and closed subsets of a metric space. Our results essentially compliments and generalizes some well known results. As application, we model a nonconvex Hammerstein type integral inclusion and prove an existence theorem for this problem.
Ismat Beg
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We obtain sufficient conditions for existence of fixed points of integral type contractive mappings on S^{JS}- metric spaces. We also study common fixed point and couple fixed point of integral type mappings and construct examples to support our results.
Ismat Beg
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A new general composite implicit random iteration scheme with perturbed mapping is proposed and obtain necessary and sufficient conditions for strong convergence of proposed iteration scheme to random fixed point of a finite family of random nonexpansive mappings are obtained.
Ismat Beg
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Results regarding the existence of random fixed points of a nonexpansive random operator defined on an unbounded subset of a Banach space are proved.
The aim of this paper is to prove some random fixed point theorems for asymptotically nonexpansive random operator defined on an unbounded closed and starshaped subset of a Banach space.
Ismat Beg
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We introduced a notion of topological vector space valued cone metric space and obtained some common fixed point results. Our results generalize some recent results in the literature.
The existence of random fixed points of random multivalued operators satisfying certain contractive conditions is established. The results regarding common random fixed point of a pair of random R-multivalued operators are also obtained. Our work generalizes, refines and improve several earlier known results.
Ismat Beg
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A modified general composite implicit random iteration process is proposed and necessary and sufficient conditions for strong convergence of this iteration process to a common random fixed point of a finite family of random asymptotically nonexpansive mappings, are obtained.
Ismat Beg
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We obtain necessary and sufficient conditions for existence of a common fixed point of three maps f, g and h in a complete Menger space under a general contractive condition. A common fixed point theorem for a pair of weakly biased mappings, which is more general than weakly compatible mappings, has also been proved.
Ismat Beg
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Let (X,) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Let F : X × X X be a mixed monotone set valued mapping. We obtain sufficient conditions for the existence of a coupled fixed point of F.
Aim of this paper is to provide necessary conditions for the existence of fixed points of nonexpansive mappings on a nonconvex domain of a random normed space. As an application, related results on invariant approximation are also derived.
Ismat Beg
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Fixed point theory is an active area of research with wide range of applications in various directions. It is concerned with the results which state that under certain conditions a self map f on a set X admit one or more fixed point. Fixed point theory started almost immediately after the classical analysis began its rapid development. The further growth was motivated mainly by the need to prove existence theorems for differential and integral equations. Thus the fixed point theory started as purely analytical theory. Fixed point theory can be divided into three major areas: Metric fixed point theory, Topological fixed point theory and Discrete fixed point theory. Classical and major results in these areas are: Banach’s fixed point theorem, Brouwer’s fixed point theorem and Tarski’s fixed point theorem. In 1922, the Polish mathematician Stefan Banach formulated and proved a theorem which concerns under appropriate conditions the existence and uniqueness of a fixed point in a complete metric space. His result is known as Banach’s fixed point theorem or the Banach contraction principle. Due to its simplicity and generality, the contraction principle has drawn attention of a very large number of mathematicians. After the period of enormous development of linear functional analysis the time was ripe to focus on nonlinear problems. Then the role of the analytical fixed point theory became even more important. The study of fixed points for set valued contractions and nonexpansive maps using the Hausdorff metric was initiated by Markin. Later, an interesting and rich fixed point theory for such maps has been developed. The theory of set valued maps has applications in control theory, convex optimization, differential inclusions and economics. Following the Banach contraction principle Nadler introduced the concept of set i valued contractions and established that a set valued contraction possesses a fixed point in a complete metric space. Subsequently many authors generalized Nadler’s fixed point theorem in different way. A constructive proof of a fixed point theorem makes the theorem twice as worthy because it yields an algorithm for computing a fixed point. Indeed, many fixedpoint theorems have constructive proofs, of which we might mention the geometric fixed point results due to Banach and Nadler, for single valued and set valued mappings. These results are of particular importance and play a fundamental role in nonlinear analysis. They are used prominently in denotational semantics, for example to give meaning to recursive programs. In fact, it is hard to overestimate their applicability and importance in mathematics. Among other applications, they are used to show the existence of solutions to differential equations, as well as the existence of equilibria in game theory. Tarski’s fixed point theorem guarantees the existence of a fixed point of an orderpreserving function defined on a nonempty complete lattice. In theoretical computer science, least fixed points of monotone functions are used to define program semantics. Tarski’s fixed point theorem has important applications in formal semantics of programming languages. Although Tarski’s proof is beautiful and elegant, but non constructive. Recently there have been so many exciting developments in the field of existence of fixed point in partially ordered sets. This trend was started by Ran and Reurings; they extended the Banach contraction principle in partially ordered sets with some application to matrix equation. Their results are hybrid of the two classical theorems; Banach’s fixed point theorem and Tarski’s fixed point theorem. The results are applicable in some cases where neither Tarski’s theorem nor Knaster-Tarski or Amman theorem which requires existence of supremum for every chain in X, are useful. Following the trend this thesis focuses on the existence of fixed point in partially ordered metric spaces for set valued mappings. The thesis is divided into three chapters. Chapter 1, is essentially an introduction, where we fix notations, terminology to be used. It is a survey aimed at recalling some basic definitions and facts. While some of the classical and recent results about fixed point existence are also presented in ii this chapter Chapter 2, concerned with the study of fixed point of hybrid contractions on ordered metric spaces. The first half of the chapter deals with the existence of fixed points for set valued mappings where the contraction condition is assumed for comparable elements of ordered set. A coupled fixed point theorem is also established. In the second half G-contractions and G-contractive mappings are considered to establish an existence of fixed point in metric spaces endowed with a graph. Chapter 3, is devoted for the study of common fixed points and coincidence point for a pair of set valued mappings in ordered metric spaces. It has been shown that an implicit relation cover many definitions of contraction mappings in one go. We explicit some implicit relation for a pair of set valued mapping to obtain their fixed points. A pre order relations along with an implicit relation is also used to prove existence of fixed point. Fixed point of Kannan type set valued mappings is also proved. iii
A notion of generalized cone metric space is introduced, and some convergence properties of sequences are proved. Also some fixed point results for mappings satisfying certain contractive conditions are obtained. Our results complement, extend and unify several well known results in the literature.
Ismat Beg
added 2 research items
A notion of generalized cone metric space is introduced, and some convergence properties of sequences are proved. Also some fixed point results for mappings satisfying certain contractive conditions are obtained. Our results complement, extend and unify several well known results in the literature.
Ismat Beg
added 3 research items
Sufficient conditions for existence of random fixed point of a nonexpansive rotative random operator are obtained and existence of random periodic points of a random operator is proved. We also derive random periodic point theorem for ǫ-expansive random operator.
In common fixed point results for two maps defined on a G−cone metric space, satisfying generalized contractive conditions are obtained. These results generalize, improve and unify several well known comparable results in the literature.
Sufficient conditions for a random solution of random operator inclusions involving random multivalued operators satisfying a general contractive condition are obtained. A simultaneous random solution of the system of two random operator inclusions is also derived. Keywordsrandom operator inclusion-random operators-random iterative process-metric space-measurable space 2000 Mathematics Subject ClassificationPrimary 47H09, 47H10, 47H40-Secondary 54H25, 60H25
Ismat Beg
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In this paper we introduce the concept of R-continuity and R-closedness for a pair of multivalued mappings. A new class of (R, α)-generalized rational multivalued contraction mappings is defined. After that we establish the existence of common fixed point of such mappings in the framework of b-metric spaces endowed with an arbitrary binary relation R. These results unify, generalize and complement several results in the existing literature. Several examples are constructed in support of our new results. Existence of fixed and common fixed points of single valued mappings is also shown in the setting of R-complete b-metric spaces.
Let (X,d) be a metric space and F:X↝X be a set valued mapping. We obtain sufficient conditions for the existence of a fixed point of the mapping F in the metric space X endowed with a graph G such that the set V(G) of vertices of G coincides with X and the set of edges of G is E(G)={(x,y):(x,y)∈X×X}.
Let (X, d,) be a partially ordered metric space. Let F, G be two set valued mappings on X. We obtained sufficient conditions for the existence of a common fixed point of F , G satisfying an implicit relation in X.
Ismat Beg
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We obtain results concerning strong convergence to common fixed points of asymptotically I-nonexpansive maps T for which (T,I) is a Banach operator pair in a Banach space with uniformly Gâteaux differentiable norm. Several common fixed point and best approximation results for this newly defined class of maps are proved.
Fixed point theorems for two hybrid pairs of single valued and multivalued noncompatible maps under strict contractive condition are proved, without appeal to continuity of any map involved therein and completeness of underlying space. These results extend, unify and improve the earlier comparable known results.
Ismat Beg
added 11 research items
In this monograph those aspects of random solution of random operator equations and random operator inclusion, which fall within the scope of investigation of random fixed point are discussed. This book is divided into three chapters. Chapter 1 is essentially an introductory in nature. Here we fix notations, recall some basic definitions and summarize some of the familiar classical and recent results about random fixed points, which are essential for study in the sequel. In chapter 2, different random iterative algorithms helpful in solving various random operator equations are investigated. For the purpose of obtaining the solution of random operator equations involving single valued random operators, sequence of measurable mappings is generated through iterative methods and necessary conditions for the convergence of these procedures are worked out. This chapter includes the study of convergence of three step random iterative process for asymptotically non expansive random operator to obtain the random solution of the random fixed point equation The question of equivalence between the convergence of random iterative schemes and their stability has also been taken into account. The latter part of this chapter deals with the existence of common random fixed points of two asymptotically non expansive random operators. Common random fixed points of two pairs of compatible random operators and finite family of non expansive random operators is also studied. In chapter 3, the existence of solution to the nonlinear random multivalued inclusion is established. The existence theorems concerning random periodic point of random multivalued contractive operators are proved in the framework of separable metric spaces. Some random fixed point theorems for random multivalued operators satisfying certain contractive conditions are obtained. The results regarding common random fixed point of a pair of random multivalued operators are established. Finally, sufficient conditions are developed for the existence of coupled random fixed point of multivalued random operator in the context of an ordered Banach space.
We establish some fixed point theorems in convex metric spaces for (k, L)− Lipschitzian mappings. Our results generalize and extend corresponding results in the existing literature.
We establish results on invariant approximation for fuzzy nonexpansive mappings defined on fuzzy metric spaces. As an application a result on the best approximation as a fixed point in a fuzzy normed space is obtained. We also define the strictly convex fuzzy normed space and obtain a necessary condition for the set of all t-best approximations to contain a fixed point of arbitrary mappings. A result regarding the existence of an invariant point for a pair of commuting mappings on a fuzzy metric space is proved. Our results extend, generalize and unify various known results in the existing literature.
Ismat Beg
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et (X,�...) be a partially ordered set and d be a metric on X such that (X, d) is a complete metric space. Assume that X satisfies; if a non-decreasing sequence xn → x in X, then xn � x, for all n. Let F be a set valued mapping from X into X with nonempty closed bounded values satisfying; (i) there exists κ ∈ (0, 1) with D(F(x), F(y)) ≤ κd(x, y), for all x � y, (ii) if d(x, y) < ε < 1 for some y ∈ F(x) then x � y, (iii) there exists x0 ∈ X, and some x1 ∈ F(x0) with x0 � x1 such that d(x0, x1) < 1. It is shown that F has a fixed point. Several consequences are also obtained.
Let ( X , ⪯ ) be a partially ordered set and d be a complete metric on X. The notion of f-contractive for a set-valued mapping due to Latif and Beg is extended through an implicit relation. Coincidence and fixed point results are obtained for mappings satisfying generalized contractions in a partially ordered metric space X. Our results improve and extend several known results in the existing literature. MSC: 47H10, 47H04, 47H07.
Ismat Beg
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We present a common fixed point theorem for two pairs of self-mappings by using the notions of compatibility and subsequential continuity (alternatively subcompatibility and reciprocal continuity) in Menger space and give some examples. As an application to our main result, we also obtain the corresponding common xed point theorem in metric spaces. Our results improve several well-known results in the literature.
Ismat Beg
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You can see my talk on "Fixed Point" during RAMMMA2019 0n http://121.52.153.178:8080/xmlui/handle/123456789/16660
 
Ismat Beg
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The purpose of this paper is to extend the notion of (φ,ψ)−weak contraction to intuitionistic fuzzy metric spaces, by using an altering distance function. We obtain common fixed point results in intuitionistic fuzzy metric spaces, which generalize several known results from the literature.
In this paper, existence of common fixed points of two multi-valued mappings satisfying generalized ϕ-contractive condition in the setting of ordered G-metric spaces, is established. An example to support our results is also presented.
Ismat Beg