Compact Metric Space and Some Fixed-Point Theorems

On Efimov spaces and Radon measures

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.

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

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.

A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of $$C_k(X)$$ is equicontinuous, where $$C_k(X)$$ denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. The classical Ascoli theorem states that each compact space is Ascoli. We show that a pseudocompact space X is Asoli iff it is sequentially Ascoli iff it is selectively $$\omega $$ -bounded. The class of selectively $$\omega $$ -bounded spaces is studied.

Chapter 18 - Efimov's problem

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A definition of topological entropy for a free semigroup action is suggested. Suppose that a free semigroup acts on a compact metric space by continuous self-maps. To this action, we assign a skew-product transformation whose fiber entropy is taken to be the entropy of the initial action. The main result is Theorem 1, a topological analogue of the Abramov–Rokhlin formula.

Fixed point theorems due to Lal et al. (1996) and Jungck (1988) are used to derive two common fixed point theorems involving six mappings in complete and compact metric spaces, respectively.

In 1984, Khan et al. [3] established some fixed point theorems in complete and compact metric spaces by using altering distance functions. In 2017, Gordji et al. [2] described the notion of orthogonal set and orthogonal metric spaces. In their work, they held a generalization of Banach fixed point theorem in this interestingly-defined construction and in addition, applied their acquired results to demonstrate the existence of a solution of an ordinary differential equation. In this paper, some fixed point theorems on orthogonal metric spaces via altering distance functions inspired by [3] and [2] are presented.

In this paper, we introduce the concept of Eilenberg–Jachymski collection on a nonempty set. Then, we establish three results equivalent to Bourbaki–Kneser’s fixed point theorem, and, therefore, to the axiom of choice. As consequences, we present new fixed point theorems in compact topological spaces, which extend and unify those of Nemytskii–Edelstein, Liepins and Suzuki.

In this paper, we announce a beautiful and important application of the results of the paper of two Romanian mathematicians: R. Miculescu, A. Mihail, New fixed point theorems for set-valued contractions in b-metric spaces, J. Fixed Point Theory Appl., 19, 2153-2163 (2017). Namely, the method they gave in the mentioned paper for b-metric spaces can be applied without difficulty to b-metric-like spaces as well. Thus, the results on b-metric spaces established in the papers: Hassen Aydi, Monica-Felicia Bota, Erdal Karapinar and Slobodanka Mitrović, A fixed point theorem for set-valued quasi-contraction in b-metric spaces, Fixed Point Theory and Applications 2012, 2012:88, and Nicola Fabiano, Zoran Kadelburg, Stojan Radenović, and Milica Savatović, Remarks on ”On new orthogonal contractions in b-metric spaces” and ”On orthogonal partial b-metric spaces with an application”, can be extended to b-metric-like spaces as well. Also, using the significant paper at the beginning of the mentioned Romanian professors, a positive answer can be directly given to the question given at the end of the recent paper: M. Gardašević-Filipović, K. Kukić, D. Gardašević, Z. D. Mitrović, Some best proximity point results in the orthogonal 0-complete b-metric-like spaces, J. Contemp. Math. Anal. Armen. Acad. Sci., 58 (2), 105–115 (2023). In the same way, the interval [0,1s ) belonging to the coefficient of some type of contraction within the b-metric-like space can be extended to the entire interval [0,1).

<p>In this paper, we prove some common fixed point theorems for co-cyclic weak contractions in compact metric spaces.</p>

Multi-valued nonlinear contraction mappings

Let $\mathcal{I}$ be an admissible (i.e., proper and containing all finite subsets of $\mathbb{N}$) ideal of subsets of the set $\mathbb{N}^+$ of positive integers. The concept of $\mathcal{I}$-convergence of sequences in metric spaces generalizes the concept of statistical convergence and also the usual concept of convergence of sequences. In this paper we investigate some problems concerning the sets of $\mathcal{I}$-cluster points and, in particular, the sets of statistical cluster points of sequences in metric spaces which are known to be closed sets. In the first part of the paper we give a sufficient condition on a sequence $x=(x_n)_{n=1}^\infty$ in a boundedly compact metric space which ensures the connectedness of the set of all statistical cluster points of $x$. If $\Gamma_x(\mathcal{I})$ denotes the set of all $\mathcal{I}$-cluster points of the sequence $x$ and $M$ is a set of sequences in a metric space $X$ such that, for each $x\in M$, $\Gamma_x(\mathcal{I})\ne\emptyset$, then the assignment $x\mapsto\Gamma_x(\mathcal{I})$ gives a map $M\to\mathcal{F}$ where $\mathcal{F}$ is the set of all non-empty closed subsets of the space $X$ or $M\to\mathcal{C}$ where $\mathcal{C}$ is a suitable subset of $\mathcal{F}$. In the second part of the paper we study the continuity of this map with respect to the sup-metric on $M$ and some standard hypertopologies on $\mathcal{C}$ (the Vietoris topology, the Fell topology, the proximal topology and the topology given by the Hausdorff metric). We obtain some positive results in the case of locally compact and, particularly, boundedly compact metric spaces.