**research item**

Updates

0 new

46

Recommendations

0 new

78

Followers

0 new

331

Reads

2 new

4935

## Project log

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.

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.

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.

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
- G. Arul Joseph
- M. Gunaseelan

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
- M. Gunaseelan
- G. Arul Joseph

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.

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
- M. Gunaseelan
- G. Arul Joseph

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-

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.

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.

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.

Existence of fixed points of asymptotically semicontractive operators defined on a separable uniformly convex Banach space, under certain condition is shown.

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

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.

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.

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.

We study existence of maximal fixed points of expansive mappings on fuzzy pre-ordered sets under suitable conditions.

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.

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

The existence of invariant random best approximations in Banach spaces is proved.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.