Basic theory and applications of probabilistic metric spaces (II)

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 geometric approach to the study of the partial stability of linear systems is proposed, which is based on the application of the geometric theory of linear operators. Using the theory of conjugate spaces and conjugate linear operators, bases are constructed in which the system under study takes the canonical form. A cyclic subspace with respect to the conjugate linear operator is considered. A basis is constructed for the dual space of a linear operator, in which its matrix takes the canonical form. This basis corresponds to the dual basis of the original linear space. Then, in a pair of bases of dual spaces the system under study takes the simplest form. The geometric properties of the system are realized using a non-singular linear transformation in the space of a part of the components of the system’s phase vector. This allows us to decompose the system under study in order to obtain necessary and sufficient conditions for the partial stability of the linear system. In an equivalent system, an independent subsystem is distinguished, whose nature of stability determines the behavior of the investigated component of the original system’s phase vector. The relationship between the partial stability of the system and the existence of an invariant subspace of a linear operator characterizing the dynamics of the system is established. The canonical form of the resulting subsystem makes it easy to exclude auxiliary variables and write an equation equivalent to this system. The application of the obtained results to the solution of the problem of partial stability for linear systems with constant coefficients belonging to the classes of ordinary differential equations, discrete systems and systems with deviating argument is shown. An example of a linear system of differential equations is given to illustrate the result obtained.

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 theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

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.

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

This paper extends the known results on the equivalence between market completeness and the uniqueness of martingale measures for finite asset economies, to the infinite asset case. Our arguments employ results from the theory of linear operators between locally convex topological vector spaces. This theory of linear operators provides an operational approach to the issue of completeness and uniqueness, that is also more closely connected with the mainstream of empirical asset pricing, than was hitherto available.

CHAPTER II - DIMENSION OF METRIC SPACES

Copula Theory and Its Applications

Generally, finding a solution to a theoretical mathematical modelling problem is equivalent to finding a fixed point for a suitable operator. Accordingly, fixed point theory is therefore a very important and crucial in many areas, such as mathematics, sciences, and engineering. A very popular and important fixed point theory is those formulated by Stefan Banach in 1922. The theory is related to a complete normed space and known as the Banach fixed point theory. Recently there have been numerous generalization of the Banach fixed point theory. One of them is a fixed point theory in modular spaces. In this paper, we will formulate some fixed point theorems in modular spaces by using $C^*$-class functions. The obtained results generalize and improve some results in [Supama, On Some Common Fixed Point Theorems inModulared Spaces, International Mathematical Forum, Vol. 7, no. 52, (2012), 2571 - 2579.].