Fixed Point Theory in Graph Metric Spaces

Multi-valued nonlinear contraction mappings

The flourishing field of fixed point theory started in the early days of topology with seminal contributions by Poincare, Lefschetz-Hopf, and Leray-Schauder at the turn of the 19th and early 20th centuries. The theory vigorously developed into a dense and multifaceted body of principles, results, and methods from topology and analysis to algebra and geometry as well as discrete and computational mathematics. This interdisciplinary theory par excellence provides insight and powerful tools for the solvability aspects of central problems in many areas of current interest in mathematics where topological considerations play a crucial role. Indeed, existence for linear and nonlinear problems is commonly translated into fixed point problems; for example, the existence of solutions to elliptic partial differential equations, the existence of closed periodic orbits in dynamical systems, and more recently the existence of answer sets in logic programming. The classical fixed point theorems of Banach and Brouwer marked the development of the two most prominent and complementary facets of the theory, namely, the metric fixed point theory and the topological fixed point theory.Themetric theory encompasses results and methods that involve properties of an essentially isometric nature. It originates with the concept of Picard successive approximations for establishing existence and uniqueness of solutions to nonlinear initial value problems of the 1st order and goes back as far as Cauchy, Liouville, Lipschitz, Peano, Fredholm, and most particularly, Emile Picard. However, the Polish mathematician Stefan Banach is credited with placing the underlying ideas into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Metric fixed point theory for important classes of mapping gained respectability and prominence to become a vast field of specialization partly and not only because many results have constructive proofs, but also because it sheds a revealing light on the geometry of normed spaces, not to mention its many applications in industrial fields such as image processing engineering, physics, computer science, economics, and telecommunications. A particular interest in fixed points for set-valued operators developed towards the mid-20th century with the celebrated extensions of the Brouwer and Lefschetz theorems by Kakutani and Eilenberg-Montgomery, respectively. The Banach contraction principle was later on extended to multivalued contractions by Nadler. The fixed point theory for multivalued maps found numerous applications in control theory, convex and nonsmooth optimization, differential inclusions, and economics. The theory is also used prominently in denotational semantics (e.g., to give meaning to recursive programs). In fact, it is still too early to truly estimate the importance and impact of set-valued fixed point theorems in mathematics in general as the theory is still growing and finding renewed outlets. This special issue adds to the development of fixed point theory by focusing on most recent contributions. It includes works on nonexpansive mappings in Banach and metric spaces, multivalued mappings in Banach and metric spaces, monotone mappings in ordered spaces, multivalued mappings in ordered spaces, and applications to such nonmetric spaces as modular spaces, as well as applications to logic programming and directed graphs.

Aronszajn and Panitchpakdi developed hyperconvex metric spaces to expand Hahn-theorem Banach's beyond the real line to more generic spaces. The aim of this short article is to collect and combine basic notions and results in the fixed point theory in the context of hyperconvex metric spaces. In this paper, we first introduce the definitions of hyperconvex metric spaces, nonexpansive retract, externally hyperconvex and bounded subsets, and admissible subsets. We shall review and explore some fundamental characteristics of hyperconvexity. Next, we introduce the Knaster–Kuratowski and Mazurkiewicz (KKM) theory in hyperconvex metric spaces and related results. Furthermore, we find the relationship between extremal points and hyperconvexity and related properties. Furthermore, we have highlighted some known consequences of our main results. Fixed point theorems are fundamental tools in mathematical analysis, and have been used for diverse purposes including optimization, differential equations and dynamical systems. Fixed point theory was initiated in the case of standard metric spaces and subsequently expanded to be-metrical, convex be-metallic etc., as we needed more generalised conditions for broader range of maps from a wider variety of intuitive settings. This paper investigates fixed point theorems in convex b-metric spaces, which studies by Chen et al. This study aimed to synthesize and interpret the results in terms of practical implications as well as theoretical contributions that emerge from these findings, which are discussed later on this discussion section. In situations where traditional metric constraints are restrictive, fixed point theorems can be implemented in a class of spaces using convex b-metric structure by each relaxing and combining part of triangle inequality with the condition involved from standard assumption about Banach contraction. The results in this paper are existence and uniqueness theorems, common fixed point theorem of two mappings under some contractive conditions, a common coupled coincidence point result for four self maps, thereby indicating that convex b-metric spaces can be used but desired to solve boundary value problems (BVP), stabilization of dynamic systems over bounded closed sets with Lyapunov functions. Applications show how these abstract sittings could play a more elegant role in nonlinear optimization too by identifying optimal solutions. We also compare our result of convex b-metric spaces with other generalized metrics and we discuss their advantages for future research. In particular the above review sheds light on unlimitedness of these strikingly handy class, convex b-metric spaces alongside its implication for rich maths art instead.

In the present research, we consider a fixed point problem pertaining to a hybrid multivalued contractive mapping constructed by putting together the ideas behind the two well known families of contractions known as Geraghty and Kannan type contractions respectively. The problem is formulated with respect to w-distances which are additional structures on metric spaces and are known to be instrumental in proving important results in fixed point theory. There are several corollaries dealing with the corresponding single-valued cases of the main result. An illustrative example is described. There are also some discussions, on comparisons of certain existing results, with the results derived here. A Hyers-Ulam-Rassias stability result, of the present fixed point problem, with respect to the w-distance, is established.

Functional specifications have been used to specify and verify designs of a number of reactive, discrete systems. In this paper we extend this specification style to deal with real-time and hybrid systems. As mathematical foundation we employ Banach's fixed point theory in metric spaces. The goal is to show that the theory used for discrete functional specifications smoothly carries over to real-time and hybrid systems. An example of a thermostat specification illustrates the method.

This paper explores the applications of Fixed-point theory in metric spaces, focusing on the role of real numbers in matrix spaces and iterative methods. Using Banach's Contraction Principle, the study provides mathematical proofs to establish the existence and uniqueness of fixed points, while also analyzing the convergence and stability of iterative methods. In a complete metric space (𝑋, 𝑑), a contraction mapping 𝑓: 𝑋 → 𝑋 has a unique fixed point 𝑥∗ such that 𝑓 (𝑥∗) = 𝑥∗ and 𝑑 (𝑓 (𝑥), 𝑓 (𝑦)) ≤ 𝛼 𝑑 (𝑥, 𝑦), where 0 ≤ 𝛼 < 1. The Intermediate Value Theorem ensures that for a continuous function 𝑓on [𝑎, 𝑏] with 𝑓(𝑎) ≤ 𝐿 ≤ 𝑓(𝑏), there is a point 𝑐 ∈ [𝑎, 𝑏] such that 𝑓(𝑐) = 𝐿. The Mean Value Theorem states that for a differentiable function 𝑓 on (𝑎,), there exists a point 𝑐 ∈ (𝑎, 𝑏) such that 𝑓′(𝑐)=𝑓(𝑏) – 𝑓(𝑎) / 𝑏−𝑎. The Cauchy-Schwarz inequality for vectors 𝑢and 𝑣 gives ∣⟨𝑢, 𝑣⟩∣ ≤ ∥𝑢∥ ∥𝑣∥. For the function (𝑥) = ½ (𝑥+3), the fixed point is 𝑥∗ = 3. In matrix space, the operator (𝐴) = ½ 𝐴 has a fixed point 𝐴 = 0. For 𝑥 = cos (𝑥), the successive approximations method converges to 𝑥∗ ≈ 0.7391, and the Newton-Raphson method achieves faster convergence to the same fixed point.This research highlights the practical applications of fixed-point theory, supported by real numbers, as a foundation for accurate mathematical models. It enhances outcome prediction in optimization, stability analysis, and system modeling, emphasizing the critical role of real numbers in both theoretical and applied mathematics.

There have been many attempts to generalize the definition of a metric space in order to obtain possibilities for more general fixed point results. In this paper, we give a survey of recent results on reducing fixed point theorems on generalized metric spaces to fixed point theorems on metric spaces and then investigate this fact in other generalized metric spaces. We show that many generalized metric spaces are topologically equivalent to certain metric spaces or to previously generalized metric spaces. Also, the fixed point theory in these generalized metric spaces may be a consequence of the fixed point theory in certain metric spaces or in previously generalized metric spaces.

Fixed point theory in fuzzy metric spaces has grown to become an intensive field of research.However, due to the complexity involved in the nature of fuzzy metrics, the authors need to develop innovative machinery to establish new fixed point theorems in such kind of spaces.In this paper, we propose the concepts of asymptotic fuzzy -contractive and asymptotic fuzzy Meir-Keeler mappings, and describe some new machinery by which the corresponding fixed point theorems are proved.In this sense, the techniques used for the proofs in Section 5 are completely new.

This paper presents new principles of transpose in the fixed point theory as for example: Let X be a nonempty set and let ζ be an arbitrary formula which contains terms x, y Є X, ≤, + < Ø, T : X → X, and p. Then, as assertion of the form: For every T and for every p(x, y) Є R0 + := [0,+ ∞) the following fact (A) ς(x, y Є X,≤, +, T, p implies T has a fixed point is a theorem if and only if the assertion of the form: For every T and for every p(x, y) Є C, where C is a cone of the set G of all cones, the following fact in the form (TA) ς(x, y Є X, <, Ø, T, p implies T has a fixed point is a theorem. Applications of the principles of transpose in nonlinear functional analysis and fixed point theory are numerous.

The aim of this paper is to provide some sufficient conditions under which a self-mapping T defined on a non-empty set X endowed with some convergence property is a Picard operator. A relevant example showing that such a mapping T on a non-metrizable space is a Picard operator is given. Our results can be used to obtain some known fixed point theorems on generalized metric spaces.

The iterates of the Cesaro operator were studied recently (see Fontes, F. G. and Sol ` ´ıs, F. J., Iterating the Ces´aro operators, Proc. Amer. Math. Soc., 136 (2008), No. 6, 2147–2153) on some subsets of s(C), on (C[0, 1], C) and on C([0, ∞[, C). They proved that under suitable conditions the sequence of iterates converges to a constant function. In [Andras, Sz. and Rus, I. A., ´ Iterates of Ces`aro operators, Fixed Point Theory, 11 (2010), No. 2, 171–178] the authors gave some more general results regarding the convergence of the iterates by proving that the Cesaro operator is a contraction on a ` dense subset of (C[0, 1], B), equipped with a well chosen norm, where B is a Banach space. The convergence of iterates for some general averaging operators involving one variable functions was also investigated by Sz. Andras and I. A. Rus in [ ´ Iterates of Ces`aro operators, Fixed Point Theory, 11 (2010), No. 2, 171–178]. The aim of this paper is to prove similar results involving Cesaro operators and general averaging operators for several ` variable functions. The proofs are suggested by the characterization theorem of weakly Picard operators on an L-space (see Rus, I. A., Picard operators and applications, Sci. Math. Jpn., 58 (2003), 191–219) and the method can be applied also in the study of some singular integral equations.

We introduce the new notion of generalized α − ψ rational type contractions of type I and type II in controlled metric spaces. By making use of these new notions, some fixed point theorems are also proved in the mentioned spaces for the α − admissible self maps. Some examples on the finite sets are also provided to prove the validity of our results. Some results which are direct consequences of the proved results are mentioned in the form of corollaries. As an application, an integral equation is solved with the aid of our results. Finally, a series electric circuit problem in the form of second‐order differential equation is solved by converting it into fixed point problem with the aid of the proved result.

Fixed point theory in cone metric spaces obtained via the scalarization method

We develop the theory of topological vector space valued cone metric spaces with nonnormal cones. We prove three general fixed point results in these spaces and deduce as corollaries several extensions of theorems about fixed points and common fixed points, known from the theory of (normed-valued) cone metric spaces. Examples are given to distinguish our results from the known ones.

Basic theory and applications of probabilistic metric spaces (II)