Various generalizations of metric spaces and fixed point theorems

The fixed point theory is the most important topic in mathematics anaysis. This topic has many applications in the different fields. It demonstrates the uniqueness of the existence of a mapping and how it can be found. The Banach contraction mapping principle is most important theorem in the fixed point theory. Most authors have introduced new contractions and they proved some fixed point theorems in metric spaces. On the other hand, some authors introduced new metric spaces as a generalization of metric spaces. Later, they proved fixed point theorems which generalize the Banach contraction principle for various mappings in several metric spaces. The purpose of this study is to introduce the existence and uniqueness of ϕ-fixed point for some new contractions in complete b-metric spaces. Firstly, we presented new definitions called (F,ϕ,α,θ)_{s} and (F,ϕ,α,θ)_{s}-weak contractions in complete b-metric spaces as a generalization of metric spaces. Later, we proved ϕ-fixed point theorems for (F,ϕ,α,θ)_{s} and (F,ϕ,α,θ)_{s}-weak contractions in complete b-metric spaces. The presented theorems extend and generalize some ϕ-fixed point results which are known in the literature. As applications, we derived some fixed point results in complete partial b-metric spaces as a generalization of partial metric spaces. The results in this paper generalizes may existing results in the literature.

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.

A new generalization of the metric space notion, named $${\mathcal {F}}$$ -metric space, was given in [M. Jleli, B. Samet, On a new generalization of metric spaces, J. Fixed Point Theory Appl. 20 (2018), no. 3, Art. 128, 20 pp.]. In this paper, we investigate some properties of $${\mathcal {F}}$$ -metric spaces. A simple proof is given to show that the natural topology induced by an $${\mathcal {F}}$$ -metric is metrizable. We present a method to construct s- $$\hbox {relaxed}_{{p}}$$ spaces and, therefore, $${\mathcal {F}}$$ -metric spaces from bounded metric spaces. We give some results that reveal differences between metric and $${\mathcal {F}}$$ -metric spaces. In particular, we show that the ordinary open and closed balls in $${\mathcal {F}}$$ -metric spaces are not necessarily topological open and closed, respectively. This answers a question posed implicitly in the quoted paper. We also show that $${\mathcal {F}}$$ -metrics are not necessarily jointly continuous functions. Despite some topological differences between metrics and $${\mathcal {F}}$$ -metrics, we show that the Nadler fixed point theorem and, therefore, the Banach contraction principle in the frame of $${\mathcal {F}}$$ -metric spaces can be reduced to their original metric versions. This reduction even happens when the Schauder fixed point theorem is investigated in $${\mathcal {F}}$$ -normed spaces structure. By applying the given technique in this paper, it turns out that some nonlinear $${\mathcal {F}}$$ -metric contractions and, therefore, the related $${\mathcal {F}}$$ -metric fixed point results can naturally be reduced to their metric versions. In addition, the same happens for some topological fixed point results.

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.

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.

In this paper, we investigate the existence of fixed points that are not necessarily unique in the setting of extended b-metric space. We state some examples to illustrate our results.

In this paper, we introduce a new space which is a generalization of function weighted metric space introduced by Jleli and Samet [On a new generalization of metric spaces, J. Fixed Point Theory Appl. 2018, 20:128] where namely function weighted $mathcal{G}$-metric space. Also, a Hausdorff $Delta$-distance is introduced in these spaces. Then several fixed point results for both single-valued and multi-valued mappings in such spaces are proved. We also construct some examples for the validity of the given results and present an application to the existence of a solution of the Volterra-type integral equation.

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)

Let (X, d) be a metric space, G be a graph associated with X and f : X → X be an operator which satisfies two main assumptions: (1) f is generalized G-monotone; (2) f is a G-contraction with respect to d. In the above framework, we will present sufficient conditions under which: (i) f is a Picard operator; (ii) the fixed point problem x = f(x), x ∈ X is well-posed in the sense of Reich and Zaslavski; (iii) the fixed point problem x = f(x), x ∈ X has the Ulam-Hyers stability property; (iv) f has the Ostrowski stability property; (v) f satisfies to some Gronwall type inequalities. Some open questions are presented.KeywordsMetric spaceDirected graph G-contractionGeneralized contractionIncreasing operatorDecreasing operatorGeneralized G-monotone operatorFixed point(Weakly) Picard operatorStabilityGronwall lemmaOpen problem2010 Mathematics Subject Classification47H1034G2045N0506A0647H0947H0754E3554H25

In this study, we introduce a new generalization of metric spaces, called Perturbed Parametric Metric Spaces (PPMS). This framework extends the classical metric space by incorporating perturbation functions and the presence of a non-negative parameter τ in its distance function, rather than a standard two-variable metric $d(\mu ,\nu )$, providing a more flexible and robust treatment of distance measurement that reflects underlying variations and imperfections in complex systems. Specifically, the measurement of distance between two points is subject to errors, often arising from factors such as instrumental inaccuracies or environmental influences. We develop the foundational properties and initiate some topological notions of PPMS, provide illustrative examples, and establish several fixed-point theorems within this setting with application to fixed-circle problem. These results demonstrate the applicability of PPMS in nonlinear analysis and show how it unifies and generalizes various existing metric-type spaces. Our approach opens new perspectives for future research in functional analysis.

Many generalizations of the traditional metric space have been introduced in the literature, such as 2−, D−, G−, S− and b−metric spaces. When the studies on these generalized metric spaces are examined, it is seen that the main motivation of the researchers is to develop and generalize the famous Banach fixed point theorem. Although introduced with a similar motivation, its ability to measure the distance between n points simultaneously distinguishes the n th order G−metric space from other generalized metric spaces. In this study, we will give new and original fixed point theorems that reveal the importance of G−metric techniques since they cannot be reduced to the framework of quasi and conventional metric spaces.