Topological Vector Space-Valued Cone Metric Spaces and Fixed Point Theorems

In this paper, the author first introduces the concept of generalized algebraic cone metric spaces and some elementary results concerning generalized algebraic cone metric spaces. Next, by using these results, some new fixed point theorems on generalized (complete) algebraic cone metric spaces are proved and an example is given. As a consequence, the main results generalize the corresponding results in complete algebraic cone metric spaces and generalized complete metric spaces.

In order to verify that some fixed point theorems are valid in the the partial cone metric space. Some fixed point theorems in the cone metric space and partial metric space are extended into partial cone metric space by constructing sequences and iteration, getting some new fixed point results.

The present study aims to establish fixed point theorems for four self-mappings defined on cone metric spaces by employing the EA and CLR properties. The results not only unify and extend several well-known fixed point theorems in the existing literature but also provide a general framework for analyzing operator systems in ordered topological vector spaces.

A new concept of the c‐distance in cone metric space has been introduced recently in 2011. The aim of this paper is to extend and generalize some fixed point theorems on c‐distance in cone metric space.

The study of common fixed point theory with different types of mappings and contractive conditions takes a significant role in present research activity. At first, the great mathematician Brouwer introduced the famous fixed-point theory in 1912. According to Brouwer, “Every continuous function from the closed and bounded subset of Rn into itself has a fixed point”. After that, the great mathematician S. Banach in 1922 invented the well-known fixed point theory, namely Banach Fixed Point Theory. “Huang & Zhang [3] have introduced the concept of cone metric space, where the set of real numbers is replaced by an ordered Banach space [3]”. The present research article is about some common fixed-point theorems for “self-mappings and commuting mappings” on Cone and Cone Rectangular Metric Space.

Let \(\left( E,C,t\right) \) be a real ordered topological vector space and let (X, d) be a tvs-cone metric space over cone C. Using Proposition 19.9 of Deimling (Nonlinear functional analysis, Springer, Berlin, 1985), we show that E can be equipped with a norm such that C is a normal monotone solid cone. Hence, a tvs-cone metric space \(\left( X,d\right) \) over a solid cone C is a normal cone metric space over the same cone C. This assures that tvs-cone metric spaces are not a genuine generalization of cone metric spaces introduced by Huang and Zhang, recently. Further, if the cone C is solid then we have only cone metric spaces over normal solid cone (with coefficient of normality \(K=1\)). Here, we introduce also the notion of Sehgal–Guseman–Perov type mappings and we establish a result of existence and uniqueness of fixed points for this class of mappings.

The explicit aim of this manuscript is to obtain fixed point consequences under novel ψ-contraction mappings in a complete cone metric space over Banach algebra. We connect and relate different fixed point theorems by using the idea of ψ-contraction mappings, providing a thorough viewpoint that deepens our comprehension of this topic. Our theorems generalize and unify many results in the scientific literature. These prospective extensions offer intriguing research directions and have the potential to further advance the study of fixed point theory. The investigation of examples plays an extremely crucial role in verifying the effectiveness and validity of our theoretical results. Moreover, to support the theoretical results, some examples are investigated to emphasize these results. Ultimately, the existence and uniqueness of the solution to the Urysohn integral and nonlinear fractional differential equation are cooperated as applications to provide an authoritative basis for dealing with actual problems that include these equations.

In this paper we obtain a unique common fixed point theorem for two pairs of weakly compatible mappings in \(D^{*}\)-cone metric spaces.

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.

The notion of C-class functions in Banach algebras is introduced. By using such concept, a new fixed point theorem is established. An example to illustrate main theorem is given. Finally, applications of our main result to cyclic mappings and weak contraction type mappings are given.

Introduction. In his study of the structure of distribution kernels (centering around his celebrated kernel theorem), L. Schwartz [11; 13] has studied the subclasses of regular and very regular distribution kernels, which may be characterized roughly by the fact that they carry infinitely differentiable functions into infinitely differentiable functions. It is our object in the present paper to extend this study to kernels which are analytically regular, i.e., which, again roughly, carry analytic functions into analytic functions. The motivation for such a study is provided by the fundamental solutions of elliptic equations with analytic coefficients. One consequence of our results is that for such equations the standard theorems on the analyticity of regular solutions imply the analyticity of the fundamental solution. The principal tool in our investigation is the general theory of topological vector spaces (particularly in certain forms given to it by A. Grothendieck) and especially the theory of topological tensor products. In ?1.1, we summarize the definitions from the theory of topological vector spaces which we shall use. In ?1.2, we give a brief summary of the definitions and principal results of Grothendieck'stheory of topological tensor products. In ?I.3, we summarize the results of Schwartz's study of distribution kernels, at least insofar as they relate to the generalizations to be given to analytic kernels. ?II is devoted to an intensive consideration of the space Gt(K) of real analytic functions on a compact subset of Rn. ?III states and proves our principal results on analytic distribution kernels. The writer should like to thank Professors L. Nachbin and F. E. Browder for discussion, suggestions, and criticism. The results of the present paper constitute a portion of the writer's Doctoral Dissertation [1] at the University of Sao Paulo, prepared while in residence at the Institute for Pure and Applied Mathematics, Rio de Janeiro.

On a vector space, one can define an order by using a cone in the vector space. In this way, Huang and Zhang reviewed cone metric spaces as a generalization of metric spaces with a different view (J. Math. Anal. Appl. 2007; 332:1468–1476). Most known cones are normal with normal constant M = 1. In this paper, we shall give some results about fixed point of multifunctions on the cone metric spaces with normal constant M = 1. In this way, we shall provide a generalization for main results of Kunze et al. (J. Math. Anal. Appl. 2007; 330:159–173).

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.