Non-Decreasing Solutions for (k,Υ)-Fractional Quadratic Integral Equations of Urysohn–Volterra Type

Spaces of continuous functions over a Ψ-space

c-14 - Spaces of Functions in Pointwise Topology

The Denjoy integral is an integral that extends the Lebesgue integral and can integrate any derivative. In this paper, it is shown that the graph of the indefinite Denjoy integral $f\mapsto \int_a^x f$ is a coanalytic non-Borel relation on the product space $M[a,b]\times C[a,b]$, where $M[a,b]$ is the Polish space of real-valued measurable functions on $[a,b]$ and where $C[a,b]$ is the Polish space of real-valued continuous functions on $[a,b]$. Using the same methods, it is also shown that the class of indefinite Denjoy integrals, called $ACG_{\ast}[a,b]$, is a coanalytic but not Borel subclass of the space $C[a,b]$, thus answering a question posed by Dougherty and Kechris. Some basic model theory of the associated spaces of integrable functions is also studied. Here the main result is that, when viewed as an $\mathbb{R}[X]$-module with the indeterminate $X$ being interpreted as the indefinite integral, the space of continuous functions on the interval $[a,b]$ is elementarily equivalent to the Lebesgue-integrable and Denjoy-integrable functions on this interval, and each is stable but not superstable, and that they all have a common decidable theory when viewed as $\mathbb{Q}[X]$-modules.

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator [Formula: see text] exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol [Formula: see text] admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

On continuous selections for metric projections in spaces of continuous functions

This paper is concerned with periodic traveling wave solutions of the forced generalized nearly concentric Korteweg‐de Vries equation in the form of . The authors first convert this equation into a forced generalized Kadomtsev‐Petviashvili equation, , and then to a nonlinear ordinary differential equation with periodic boundary conditions. An equivalent relationship between the ordinary differential equation and nonlinear integral equations with symmetric kernels is established by using the Green′s function method. The integral representations generate compact operators in a Banach space of real‐valued continuous functions. The Schauder′s fixed point theorem is then used to prove the existence of nonconstant solutions to the integral equations. Therefore, the existence of periodic traveling wave solutions to the forced generalized KP equation, and hence the nearly concentric KdV equation, is proved.

Let T be a compact Hausdor topological space and let M denote an n-dimensional subspace of the space C(T ), the space of real-valued continuous functions on T and let the space be equipped with the uniform norm. Zukhovitskii (7) attributes the Basic Theorem to E.Ya.Remez and gives a proof by duality. He also gives a proof due to Shnirel'man, which uses Helly's Theorem, now the paper obtains a new proof of the Basic Theorem. The significance of the Basic Theorem for us is that it reduces the characterization of a best approximation to f 2 C(T ) from M to the case of finite T , that is to the case of approximation in l 1 (r). If one solves the problem for the finite case of T then one can deduce the solution to the general case. An immediate consequence of the Basic Theorem is that for a finite dimensional subspace M of C0(T ) there exists a separating measure for M and f 2 C0(T )\M, the cardinality of whose support is not greater than dimM+1. This result is a special case of a more general abstract result due to Singer (5). Then the Basic Theorem is used to obtain a general characterization theorem of a best approximation from M to f 2 C(T ). We also use the Basic Theorem to establish the suciency of Haar's condition for a subspace M of C(T ) to be Chebyshev.

Several equivalent conditions are given for the existence of real-valued Baire functions of all classes on a type of K-analytic spaces, called disjoint analytic spaces, and on all pseudocompact spaces. The sequential stability index for the Banach space of bounded continuous real-valued functions on these spaces is shown to be either 0,1, or Ω (the first uncountable ordinal). In contrast, the space of bounded real-valued Baire functions of class 1 is shown to contain closed linear subspaces with index α for each countable ordinal α. The sequential stability index for linear subspaces of continuous real-valued functions on a compact space is shown to be invariant under isomorphic embeddings in the space of continuous real-valued functions on any compact space.

Distance to spaces of continuous functions

Let S be an arbitrary topological space, and let C(S) be the space of continuous real-valued functions on S. A certain class of topologies on C(S) is studied. Some cases are indicated in which topologies of a given class on C(S) are topologies of uniform convergence on compact sets of S.

A classical theorem of Hurewicz characterizes spaces with the Hurewicz covering property as those having bounded continuous images in the Baire space. We give a similar characterization for spaces X which have the Hurewicz property hereditarily. We proceed to consider the class of Arhangel'skiĭ \alpha_1 spaces, for which every sheaf at a point can be amalgamated in a natural way. Let C_p(X) denote the space of continuous real-valued functions on X with the topology of pointwise convergence. Our main result is that C_p(X) is an \alpha_1 space if, and only if, each Borel image of X in the Baire space is bounded. Using this characterization, we solve a variety of problems posed in the literature concerning spaces of continuous functions.

We prove a Korovkin type approximation theorem for positive linear operators on weighted spaces of continuous real-valued functions on a compact Hausdorff space X. These spaces comprise a variety of subspaces of C (X) with suitable locally convex topologies and were introduced by Nachbin 1967 and Prolla 1977. Some early Korovkin type results on the weighted approximation of real-valued functions in one and several variables with a single weight function are due to Gadzhiev 1976 and 1980.

Let C(K) be the Banach space of real-valued continuous functions on a compact Hausdorff space with the supremum norm and let Xbea closed subspace of C(K) which separates points of K. Necessary and sufficient conditions are given for X to be the range of a projection of norm one in C(K).It is shown that the form of a projection of norm one is determined by a real-valued continuous function which is defined on a subset of K and which satisfies conditions imposed by X.When there is a projection of norm one onto X, it is shown that there is a one-to-one correspondence between the continuous functions which satisfy the conditions imposed by X and the projections of norm one onto X.Using well-known results of Nachbin, Goodner and Kelley (see [2, p. 2]), it is easily demonstrated that if Sis an extremely disconnected compact Hausdorff space, then X, a closed subspace of C(S), is the range of a norm one projection in C(S) iff X itself is isometric to the continuous functions on an extremely disconnected compact Hausdorff space.The main purpose of this research is to give necessary and sufficient conditions for a closed separating subspace X of C(K), K a compact Hausdorff space, to be the range of a norm one projection in C(K).Theorem 3.1 supplies these conditions.Proposition 1.2 shows that the form of each contractive projection onto X can be given in terms of a real-valued continuous function, determined by the projection, which is defined on a subset of K.This continuous function satisfies certain conditions which depend only on the subspace X. Theorem 3.1 also shows that when there is a contractive projection onto X there is a one-to-one correspondence between contractive projections onto X and the continuous functions which satisfy these conditions imposed by X.We give another proof to the fact, proved in [3], that the Banach space Y is isometric to the range of a contractive

<abstract><p>This work is devoted to the analysis of Hyers, Ulam, and Rassias types of stabilities for nonlinear fractional integral equations with $ n $-product operators. In some special cases, our considered integral equation is related to an integral equation which arises in the study of the spread of an infectious disease that does not induce permanent immunity. $ n $-product operators are described here in the sense of Riemann-Liouville fractional integrals of order $ \sigma_i \in (0, 1] $ for $ i\in \{1, 2, \dots, n\} $. Sufficient conditions are provided to ensure Hyers-Ulam, $ \lambda $-semi-Hyers-Ulam, and Hyers-Ulam-Rassias stabilities in the space of continuous real-valued functions defined on the interval $ [0, a] $, where $ 0 < a < \infty $. Those conditions are established by applying the concept of fixed-point arguments within the framework of the Bielecki metric and its generalizations. Two examples are discussed to illustrate the established results.</p></abstract>

A theorem due to Milutin [12] (see also [13]) asserts that for any two uncountable compact metric spaces Ω1 and Ω2 the spaces of continuous real-valued functions C(Ω1) and C(Ω2) are linearly isomorphic. It immediately follows from consideration of tensor products that if X is any Banach space then C(Ω1;X) and C(Ω2;X) are isomorphic.