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In this article, we study the convergence behavior of G -nonexpansive mappings for three-step iterations in convex metric space along with the directed graph. We demonstrate the Δ -convergence as well as the strong convergence results for these mappings. For supporting our main results we presented a numerical example and convergence comparison of modified Thianwan iteration with SP iteration scheme and Noor iteration scheme. [ABSTRACT FROM AUTHOR]

In this article, the probabilistic metric distance between two disjoint sets is utilised to define the essential criteria for the existence and uniqueness of the best proximity point, which takes into account the global optimization problem. In order to solve this problem, we pretend that we are trying to obtain the optimal approximation to the solution of a fixed point equation. Here, we introduce two types of probabilistic proximal contraction mappings and use a geometric property called Ω-property in the context of probabilistic metric spaces. We also obtain some consequences for self-mappings, which give the fixed point results. Some examples are provided to validate the findings. As an application, we obtain the solution to a second-order boundary value problem using a minimum t-norm in the context of probabilistic metric spaces. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to find sufficient conditions for the existence and uniqueness of common best proximity points for discontinuous non-self mappings in the setting of a complete metric space. We introduce and analyze new concepts such as proximally reciprocal continuous, proximally weak reciprocal continuous, R-proximally weak commuting of types MΛ and MΓ for non-self mappings. Furthermore, we obtain a common best proximity point theorem for such mappings. In addition, we provide an example to support our main result. [ABSTRACT FROM AUTHOR]

In this manuscript, we prove the existence and uniqueness of a common best proximity point for a pair of non-self mappings satisfying the iterative mappings in a complete fuzzy multiplicative metric space. We consider the pair of non-self mappings X: P → G and Z: P → G and the mappings do not necessarily have a common fixed-point. In a complete fuzzy multiplicative metric space, if φ satisfy the condition φ (b, Zb, ς) = φ (P,G, ς) = φ (b, Xb, ς), then b is a common best proximity point. Further, we obtain the common best proximity point for the real valued functions L,M: (0, 1] → R by using a generalized fuzzy multiplicative metric space in the setting of (L,M)- iterative mappings. Furthermore, we utilize fuzzy multiplicative versions of the (L,M)-proximal contraction, (L,M)-interpolative Reich-Rus-Ciric type proximal contractions, (L,M)-Kannan type proximal contraction and (L,M)-interpolative Hardy-Rogers type proximal contraction to examine the common best proximity points in fuzzy multiplicative metric space. Moreover, we provide differential non-trivial examples to support our results. [ABSTRACT FROM AUTHOR]

In this work, we prove the existence of the best proximity point results for ⊥-contraction (orthogonal-contraction) mappings on an O-complete metric space (orthogonal-complete metric space). Subsequently, these existence results are employed to establish the common best proximity point result. Finally, we provide suitable examples to demonstrate the validity of our results. [ABSTRACT FROM AUTHOR]

In this paper, we present some generalized multi-valued contraction results on cone metric spaces. We use some maximum and sum types of contractions for a pair of multi-valued mappings to prove some common fixed point theorems on cone metric spaces without the condition of normality. We present an illustrative example for multi-valued contraction mappings to support our work. Moreover, we present a supportive application of nonlinear integral equations to validate our work. This new theory, can be modified in different directions for multi-valued mappings to prove fixed point, common fixed point and coincidence point results in the context of different types of metric spaces with the application of different types of integral equations. [ABSTRACT FROM AUTHOR]

In a recent paper, Khojasteh et al. presented a new collection of simulation functions, said Z-contraction. This form of contraction generalizes the Banach contraction and makes different types of nonlinear contractions. In this article, we discuss a pair of nonlinear operators that applies to a nonlinear contraction including a simulation function in a partially ordered metric space. For this pair of operators with and without continuity, we derive some results about the coincidence and unique common fixed point. In the following, many known and dependent consequences in fixed point theory in a partially ordered metric space are deduced. As well, we furnish two interesting examples to explain our main consequences, so that one of them does not apply to the principle of Banach contraction. Finally, we use our consequences to create a solution for a particular type of nonlinear integral equation. [ABSTRACT FROM AUTHOR]

The goal of this paper is to present a new class of contraction mappings, so-called ηϑℓ- contractions. Also, in the context of partially ordered metric spaces, some coupled fixed-point results for ηϑℓ-contraction mappings are introduced. Furthermore, to support our results, two examples are provided. Finally, the theoretical results are applied to obtain the existence of solutions to coupled fractional differential equations with a Mittag-Leffler kernel. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of fuzzy-controlled bipolar metric space and prove some fixed-point theorems in this space. Our results generalize and expand some of the literature's well-known results. We also provide some applications of our main results to integral equations. [ABSTRACT FROM AUTHOR]

In this manuscript, a novel general class of contractions, called Jaggi–Suzuki-type hybrid (G-α-ϕ)-contraction, is introduced and some fixed point theorems that cannot be deduced from their akin in metric spaces are proved. The dominance of this family of contractions is that its contractive inequality can be specialized in various manners, depending on multiple parameters. Nontrivial comparative examples are constructed to validate the assumptions of our obtained theorems. Consequently, a number of corollaries that reduce our result to some prominent results in the literature are highlighted and analyzed. Additionally, we examine Ulam-type stability and well-posedness for the new contraction proposed herein. Finally, one of our obtained corollaries is applied to set up unprecedented existence conditions for solution to a class of integral equations. For future aspects of our results, an open problem is noted concerning the discretized population balance model, whose solution may be analyzed using the techniques established herein. [ABSTRACT FROM AUTHOR]

This paper aims to study the convergence of Picard's iteration to a best proximity pair for a class of noncyclic mappings with the help of projections in hyperbolic uniformly convex metric spaces. Some sufficient conditions are provided to guarantee the existence of a common best proximity pair for a pair of noncyclic mappings. Moreover, the existence and convergence of best proximity pairs for asymptotic pointwise noncyclic orbital contractions is studied. The main conclusions are supported with illustrative examples. [ABSTRACT FROM AUTHOR]

The aim of this article is to obtain common fixed point results on complex-valued extended b-metric spaces for rational contractions involving control functions of two variables. Our theorems generalize some famous results in the literature. We supply an example to show the originality of our main result. As an application, we develop common fixed point results for rational contractions involving control functions of one variable in the class of complex-valued extended b-metric spaces. [ABSTRACT FROM AUTHOR]

We introduce a new type of interpolative proximal contractive condition that ensures the existence of the best proximity points of fuzzy mappings in the complete non-archimedean fuzzy metric spaces. We establish certain best proximity point theorems for such proximal contractions. We improve and generalize the fuzzy proximal contractions by introducing fuzzy proximal interpolative contractions. The obtained results improve and generalize the best proximity point theorems published in Fuzzy Information and Engineering, 5 (2013), 417--429. Moreover, we provide many nontrivial examples to validate our best proximity point theorem. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of an optimum solution of a system of ordinary differential equations via the best proximity points. In order to obtain the optimum solution, we have developed the best proximity point results for generalized multivalued contractions of b-metric spaces. Examples are given to illustrate the main results and to show that the new results are the proper generalization of some existing results in the literature. [ABSTRACT FROM AUTHOR]

We investigate the existence of fixed point problems on a partial metric space. The results obtained are for set contractions in the domain of sets and the pattern for the partial metric space is constructed on a directed graph. Essentially, our main strategy is to employ generalized ϕ-contractions in order to prove our results, where the fixed points are investigated with a graph structure. Moreover, we state and prove the well-posedness of fixed point based problems of the generalized ϕ-contractive operator in the framework of a partial metric space. We illustrate the main results in this manuscript by providing several examples. [ABSTRACT FROM AUTHOR]

This paper studies a general p -contractive condition of a self-mapping T on X , where X   , d is either a metric space or a dislocated metric space, which combines the contribution to the upper-bound of d T x   ,   T y , where x and y are arbitrary elements in X of a weighted combination of the distances d x , y   ,   d x , T x , d y , T y , d x , T y , d y , T x ,   d x , T x − d y , T y and d x , T y − d y , T x . The asymptotic regularity of the self-mapping T on X and the convergence of Cauchy sequences to a unique fixed point are also discussed if X , d is complete. Subsequently, T ,   S generalized cyclic p -contraction pairs are discussed on a pair of non-empty, in general, disjoint subsets of X. The proposed contraction involves a combination of several distances associated with the T ,   S -pair. Some properties demonstrated are: (a) the asymptotic convergence of the relevant sequences to best proximity points of both sets is proved; (b) the best proximity points are unique if the involved subsets are closed and convex, the metric is norm induced, or the metric space is a uniformly convex Banach space. It can be pointed out that both metric and a metric-like (or dislocated metric) possess the symmetry property since their respective distance values for any given pair of elements of the corresponding space are identical after exchanging the roles of both elements. [ABSTRACT FROM AUTHOR]

A concept of fuzzy projection operator is introduced and use to investigate the nonemptiness of the fuzzy proximal pairs. We then consider the classes of noncyclic contractions and noncyclic relatively nonexpansive mappings and survey the existence of best proximity pairs for such mappings. In the case that the considered mapping is noncyclic relatively nonexpansive, we need a geometric notion of fuzzy proximal normal structure defined on a nonempty and convex pair in a convex fuzzy metric space. We also prove that every nonempty, compact and convex pair of subsets of a strictly convex fuzzy metric space has the fuzzy proximal normal structure. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of non-empty fixed point sets of newly introduced generalized set-valued F-contractions of b-metric spaces. Some illustrative examples show that the new results in this paper generalize properly, unify and extend some related results in the existing literature. Moreover, we extract some important consequences of the results in b-metric spaces. Particularly, by setting b-metric constant equal to one, we obtain some specific cases showing notable enhancement of existing results yet in metric spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove some new fixed point results for a pair of multivalued dominated locally contractive mappings in b-multiplicative metric space. Further, fixed point theorems for multigraph-dominated mappings are also established. Some new fixed point results on a closed ball are obtained for a pair of multigraph-dominated mappings endowed with graphic structure in b-multiplicative metric space. An illustrative example is given to show the validity of the hypothesis of our obtained result. Moreover, applications for a coupled system of nonlinear Volterra-type integral equations and functional equations in dynamic programming are presented to show the novelty of our results. [ABSTRACT FROM AUTHOR]

This manuscript was built to generalize Ekeland variational principle for mixed monotone functions in the setting of partially ordered complete metric spaces. The results obtained are applied to give different proofs for tripled fixed points of mixed monotone mappings in the mentioned space by using a variational technique. The results presented in our manuscripts generalize and expand many of the findings presented in the earlier period. For the sobriety and enhancement of our paper, two examples are given and the existence and uniqueness of the solution to a periodic boundary value problem are studied as applications. [ABSTRACT FROM AUTHOR]

The objective of this manuscript is to present new tripled fixed point results for mixed-monotone mappings by a pivotal lemma in the setting of partially ordered complete metric spaces. Our outcomes sum up, enrich, and generalize several results in the current writing. Moreover, some examples have been discussed to strengthen and support our theoretical results. Finally, the theoretical results are applied to study the existence and uniqueness of the solution to an integro-differential equation. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to present some fixed-point results for a pair of fuzzy dominated mappings which are generalized V-contractions in modular-like metric spaces. Some theorems using a partial order are discussed and also some useful results to graphic contractions for fuzzy-graph dominated mappings are developed. To explain the validity of our results, 2D and 3D graphs have been constructed. Also, applications are provided to show the novelty of our obtained results and their usage in engineering and computer science. [ABSTRACT FROM AUTHOR]

In this manuscript, the existence theorem for a unique solution to a coupled system of impulsive fractional differential equations in complex-valued fuzzy metric spaces is studied and the fuzzy version of some fixed point results by using the definition and properties of a complex-valued fuzzy metric space is presented. Ultimately, some appropriate examples are constructed to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper involves extended b − metric versions of a fractional differential equation, a system of fractional differential equations and two-dimensional (2D) linear Fredholm integral equations. By various given hypotheses, exciting results are established in the setting of an extended b − metric space. Thereafter, by making consequent use of the fixed point technique, short and simple proofs are obtained for solutions of a fractional differential equation, a system of fractional differential equations and a two-dimensional linear Fredholm integral equation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a notion of the Górnicki-Proinov type contraction. Then, we prove the uniqueness and existence of the fixed point for such mappings in the framework of the complete metric spaces. Some illustrative examples are also expressed to strengthen the observed results. [ABSTRACT FROM AUTHOR]

This paper is aimed at studying the uniqueness of coincidence best proximity point for ϑ , α + , g -proximal contractions in complete Branciari metric space. Throughout this article, discontinuity of the Branciari metric space is used and we obtained the desired results without assuming it as a continuous. Some examples are provided to validate the results proved herein. As an application, we derive the best proximity point results in the setup of complete Branciari metric space endowed with graph. Further, our results extend and generalize the existing ones in literature. [ABSTRACT FROM AUTHOR]

In the current manuscript, the notion of a cone b 2 -metric space over Banach's algebra with parameter b ≻ ¯ e is introduced. Furthermore, using α -admissible Hardy-Rogers' contractive conditions, we have proven fixed-point theorems for self-mappings, which generalize and strengthen many of the conclusions in existing literature. In order to verify our key result, a nontrivial example is given, and as an application, we proved a theorem that shows the existence of a solution of an infinite system of integral equations. [ABSTRACT FROM AUTHOR]

In this article, we introduce Reich type contractions and (α , F)-contractions in the class of controlled metric spaces and establish some new related fixed point theorems. Our results are generalizations of some known results of literature. Some examples and certain consequences are given to illustrate significance of presented results. [ABSTRACT FROM AUTHOR]

The aim of this research work is to find out some results in fixed point theory for a pair of families of multivalued mappings fulfilling a new type of U -contractions in modular-like metric spaces. Some new results in graph theory for multigraph-dominated contractions in modular-like metric spaces are developed. An application has been presented to ensure the uniqueness and existence of a solution of families of nonlinear integral equations. [ABSTRACT FROM AUTHOR]

In this research, under some appropriate conditions, we approximate stationary points of multivalued Suzuki mappings through the modified Agarwal-O'Regan-Sahu iteration process in the setting of 2-uniformly convex hyperbolic spaces. We also provide an illustrative numerical example. Our results improve and extend some recently announced results of the current literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of α L ψ -rational contractive and cyclic α L ψ - rational contractive mappings and establish the existence and uniqueness of fixed points for such mappings in complete metric-like spaces (dislocated metric spaces). The results presented here substantially generalize and extend several comparable results in the existing literature. As an application, we prove new fixed point results for ψ L -graphic and cyclic ψ L -graphic rational contractive mappings. Moreover, some examples and an application to integral equation are presented here to illustrate the usability of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the new notion of Suzuki-type (a, b, q, g)-contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak Pl-property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings T integrated in a pair (g, T) of mappings. In particular, g is a non-contractive fuzzy self-mapping, in the framework of non-Archimedean ordered fuzzy complete metric spaces and T is a p-cyclic proximal contraction. Two types of such contractions (so called of type I and of type II) are dealt with. In particular, the existence, uniqueness and limit properties for sequences to optimal fuzzy best proximity coincidence points are investigated for such pairs of mappings. [ABSTRACT FROM AUTHOR]

This paper investigates in a formal context some fundamental controllability properties "from" and "to" the origin of probabilistic discrete-time dynamic systems as well as their uniform versions and complete controllability in a class of probabilisticmetric spaces or probabilistic normed spaces, in particular, in probabilistic Menger spaces. Some related approximate probabilistic controllability properties are also investigated for the case when a nominal controllable systemis subject to either parametrical perturbations or unmodelled dynamics. In this context, the approximate controllability of a perturbed system is a robustness-type approximate controllability provided that the nominal system is controllable. Some illustrative examples are also given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce best proximal contractions in complete ordered non-Archimedean fuzzy metric space and obtain some proximal results. The obtained results unify, extend, and generalize some comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

We present fixed point results of mappings satisfying generalized contractive conditions in complex valued metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of generalized contractive-type mappings involved in cocyclic representation of a nonempty subset of a complex valued metric space are also obtained. Some examples are also presented to support the results proved herein. These results extend and generalize many results in the existing literature. [ABSTRACT FROM AUTHOR]

In this manuscript we extend very recent fixed point theorems in the setting of Menger spaces in three senses: on the one hand, we introduce the notion of Menger probabilistic metric-like space by avoiding a non-necessary constraint (at least, for our purposes) in the properties that define a Menger space; on the other hand, we consider a more general class of auxiliary functions in the contractivity condition; finally, we show that the function $t\rightarrow1/t-1$ (which appears in many fixed point theorems in the fuzzy context) can be replaced by more appropriate and general functions. We illustrate our main statements with an example in which previous results cannot be applied. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present fixed point result of mappings satisfying a generalized rational contractive condition in the setup of multiplicative metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of pair of rational contractive types mappings involved in cocyclic representation of a nonempty subset of a multiplicative metric space are also obtained. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature. [ABSTRACT FROM AUTHOR]

In this paper, using α —admissibility and the concept of simulation functions, some near-fixed point results in the setting of metric interval and normed interval spaces are established. The results have been proved using Z -contractions. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present a new notion of a tripled fixed point (TFP) findings by virtue of a control function in the framework of fuzzy cone metric spaces (FCM-spaces). This function is a continuous one-to-one self-map that is subsequentially convergent (SC) in FCM-spaces. Moreover, by using the triangular property of a FCM, some unique TFP results are shown under modified contractive-type conditions. Additionally, two examples are discussed to uplift our work. Ultimately, to examine and support the theoretical results, the existence and uniqueness solution to a system of Volterra integral equations (VIEs) are obtained. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to introduce and study some fixed point results for rational multivalued G-contraction and F-Khan-type multivalued contraction mappings on a metric space with a graph. At the end, we give an illustrative example. [ABSTRACT FROM AUTHOR]

The main result of this paper is a fixed-point theorem for multivalued contractions obtained through an inequality with rational terms. The contraction is an F-type contraction. The results are obtained in a metric space endowed with a graph. The main theorem is supported by illustrative examples. Several results as special cases are obtained by specific choices of the control functions involved in the inequality. The study is broadly in the domain of setvalued analysis. The methodology of the paper is a blending of both graph theoretic and analytic methods. [ABSTRACT FROM AUTHOR]

We define the notion of C * -algebra-valued fuzzy metric spaces and we study the topology induced by these spaces. [ABSTRACT FROM AUTHOR]

The symmetry concept is a congenital characteristic of the metric function. In this paper, our primary aim is to study the fixed points of a broad category of set-valued maps which may include discontinuous maps as well. To achieve this objective, we newly extend the notions of orbitally continuous and asymptotically regular mappings in the set-valued context. We introduce two new contractive inequalities one of which is of Geraghty-type and the other is of Boyd and Wong-type. We proved two new existence of fixed point results corresponding to those inequalities. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to state and prove some convergence theorems for the Mann and Ishikawa iteration schemes involving C * -algebra-multi-valued contractive mappings in the setting of convex C * -algebra-valued metric spaces. The convergence theorems of the proposed iterations to a common fixed point of finite and infinite family of such mappings are also established. [ABSTRACT FROM AUTHOR]

In this paper, we prove convergence theorems for viscosity approximation processes involving * −nonexpansive multi-valued mappings in complete convex metric spaces. We also consider finite and infinite families of such mappings and prove convergence of the proposed iteration schemes to common fixed points of them. Our results improve and extend some corresponding results. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is the investigation of possibilities for improvements and generalizations contractive condition of Ćirić in the fuzzy metric spaces. Various versions of fuzzy contractive conditions are studied in two directions. First, motivated by recent results, more general contractive conditions in fuzzy metric spaces are achieved and secondly, quasi-contractive type of mappings are investigated in order to obtain fixed point results with a wider class of t-norms. [ABSTRACT FROM AUTHOR]

The symmetry concept is an intrinsic property of metric spaces as the metric function generalizes the notion of distance between two points. There are several remarkable results in science in connection with symmetry principles that can be proved using fixed point arguments. Therefore, fixed point theory and symmetry principles bear significant correlation between them. In this paper, we introduce the new definition of the eventually Δ -restrictive set-valued map together with the concept of p-orbital continuity. Further, we introduce another new concept called the Δ (ϵ) -restrictive set-valued map. We establish several fixed point results related to these maps and proofs of these results also provide us with schemes to find a fixed point. In a couple of results, the stronger condition of compactness of the underlying metric space is assumed. Some results are illustrated with examples. [ABSTRACT FROM AUTHOR]

The concept of symmetry is inherent in the study of metric spaces due to the presence of the symmetric property of the metric. Significant results, such as with the Borsuk–Ulam theorem, make use of fixed-point arguments in their proofs to deal with certain symmetry principles. As such, the study of fixed-point results in metric spaces is highly correlated with the symmetry concept. In the current paper, we first define a new and modified Ćirić-Reich–Rus-type contraction in a b-metric space by incorporating the constant s in its definition and establish the corresponding fixed-point result. Next, we adopt an interpolative approach to establish some more fixed-point theorems. Existence of fixed points for ω -interpolative Ćirić-Reich–Rus-type contractions are investigated in this context. We also illustrate the validity of our results with some examples. [ABSTRACT FROM AUTHOR]