Space Of Real-valued Continuous Functions Research Articles

Stability analysis through the Bielecki metric to nonlinear fractional integral equations of $ n $-product operators

<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 Chebyshev-type alternation theorem for best approximation by a sum of two algebras

AbstractLet X be a compact metric space, C(X) be the space of continuous real-valued functions on X and $A_{1},A_{2}$ be two closed subalgebras of C(X) containing constant functions. We consider the problem of approximation of a function $f\in C(X)$ by elements from $A_{1}+A_{2}$. We prove a Chebyshev-type alternation theorem for a function $u_{0} \in A_{1}+A_{2}$ to be a best approximation to f.

Best Approximation Results for Fuzzy-Number-Valued Continuous Functions

In this paper, we study the best approximation of a fixed fuzzy-number-valued continuous function to a subset of fuzzy-number-valued continuous functions. We also introduce a method to measure the distance between a fuzzy-number-valued continuous function and a real-valued one. Then, we prove the existence of the best approximation of a fuzzy-number-valued continuous function to the space of real-valued continuous functions by using the well-known Michael selection theorem.

When is a Locally Convex Space Eberlein–Grothendieck?

The weak topology of a locally convex space (lcs) E is denoted by w. In this paper we undertake a systematic study of those lcs E such that (E, w) is (linearly) Eberlein–Grothendieck (see Definitions 1.2 and 3.1). The following results obtained in our paper play a key role: for every barrelled lcs E, the space (E, w) is Eberlein–Grothendieck (linearly Eberlein–Grothendieck) if and only if E is metrizable (E is normable, respectively). The main applications concern to the space of continuous real-valued functions on a Tychonoff space X endowed with the compact-open topology \(C_k(X)\). We prove that \((C_k(X), w)\) is Eberlein–Grothendieck (linearly Eberlein-Grothen—dieck) if and only if X is hemicompact (X is compact, respectively). Besides this, we show that the class of E for which (E, w) is linearly Eberlein–Grothendieck preserves linear continuous quotients. Various illustrating examples are provided.

Stone–Weierstrass theorems for Riesz ideals of continuous functions

Notions of convergence and continuity specifically adapted to Riesz ideals I of the space of continuous real-valued functions on a Lindel\"of locally compact Hausdorff space are given, and used to prove Stone-Weierstra{\ss}-type theorems for I. As applications, sufficient conditions are discussed that guarantee that various types of positive linear maps on I are uniquely determined by their restriction to various point-separating subsets of I. A very special case of this is the characterization of the strong determinacy of moment problems, which is rederived here in a rather general setting and without making use of spectral theory.

Existence of nice resolutions in Cp(X) and its bidual often implies metrizability of Cp(X)

Let X be a Tychonoff space. We show that Cp(X), the space of real-valued continuous functions on X equipped with the pointwise topology, admits a resolution (an ordered covering indexed by NN) consisting of convex compact sets that swallows the local null sequences if and only if X is countable and discrete. Then we prove (main result) (i) the weak* bidual of Cp(X) admits a resolution consisting of bounded sets if and only if X is countable. Hence (ii) if Cp(X) admits a fundamental bounded resolution, X must be countable [8]. If X is first countable, then Cp(X) admits a resolution made up of bounded sets swallowing the Cauchy sequences if and only if X is countable. In the context of the present research, the weak* bidual of Cp(X) has received little or null attention so far. Result (i) fixes this situation.

An Evaluation Formula for Radon–Nikodym Derivatives Similar to Conditional Expectations over Paths

Let C[0, T] denote an analogue of Wiener space, the space of real-valued continuous functions on the interval [0, T]. For a partition $$0=t_0<t_1<\cdots <t_n=T$$ of [0, T], define $$X:C[0,T]\rightarrow \mathbb R^{n+1}$$ by $$X(x)=(x(t_0),x(t_1),\ldots ,$$ $$x(t_n))$$ . In this paper, we derive a simple evaluation formula for Radon–Nikodym derivatives similar to the conditional expectations of functions on C[0, T] with the conditioning function X which has a drift and an initial weight. As applications of the formula, we evaluate the Radon–Nikodym derivatives of the functions $$\int _0^T[x(t)]^m\mathrm{d}\lambda (t)(m\in \mathbb N)$$ and $$[\int _0^Tx(t)\mathrm{d}\lambda (t)]^2$$ on C[0, T], where $$\lambda $$ is a complex-valued Borel measure on [0, T].

Lusin and Suslin properties of function spaces

A topological space is Suslin (Lusin) if it is a continuous (and bijective) image of a Polish space. For a Tychonoff space X let C_p(X), C_k(X) and C_{{downarrow }{mathsf {F}}}(X) be the space of continuous real-valued functions on X, endowed with the topology of pointwise convergence, the compact-open topology, and the Fell hypograph topology, respectively. For a metrizable space X we prove the equivalence of the following statements: (1) X is sigma -compact, (2) C_p(X) is Suslin, (3) C_k(X) is Suslin, (4) C_{{downarrow }{mathsf {F}}}(X) is Suslin, (5) C_p(X) is Lusin, (6) C_k(X) is Lusin, (7) C_{{downarrow }{mathsf {F}}}(X) is Lusin, (8) C_p(X) is F_sigma -Lusin, (9) C_k(X) is F_sigma -Lusin, (10) C_{{downarrow }{mathsf {F}}}(X) is C_{delta sigma }-Lusin. Also we construct an example of a sequential aleph _0-space X with a unique non-isolated point such that the function spaces C_p(X), C_k(X) and C_{{downarrow }{mathsf {F}}}(X) are non-Suslin.

Selection games on continuous functions

In this paper we study the selection principle of closed discrete selection, first researched by Tkachuk in [12] and strengthened by Clontz, Holshouser in [3], in set-open topologies on the space of continuous real-valued functions. Adapting the techniques involving point-picking games on X and Cp(X), the current authors showed similar equivalences in [1] involving the compact subsets of X and Ck(X). By pursuing a bitopological setting, we have touched upon a unifying framework which involves three basic techniques: general game duality via reflections (Clontz), general game equivalence via topological connections, and strengthening of strategies (Pawlikowski and Tkachuk). Moreover, we develop a framework which identifies topological notions to match with generalized versions of the point-open game.

The contractivity of cone-preserving multilinear mappings

With the notion of the mode-j Birkhoff contraction ratio, we prove a multilinear version of the Birkhoff–Hopf and Perron–Fronenius theorems, which provide conditions on the existence and uniqueness of a solution to a large family of systems of nonlinear equations of the type , with xi being an element of a cone Ci in a Banach space . We then consider a family of nonlinear integral operators f i with positive kernel, acting on the product of spaces of continuous real-valued functions. In this setting we provide an explicit formula for the mode-j contraction ratio, which is particularly relevant in practice, as this type of operator plays a central role in numerous models and applications.

A Banach algebra of series of functions over paths

Let $C[0,T]$ denote the space of continuous real-valued functions on $[0,T]$. On the space $C[0,T]$, we introduce a Banach algebra of series of functions which are generalized Fourier-Stieltjes transforms of measures of finite variation on the product of simplex and Euclidean space. We evaluate analytic Feynman integrals of the functions in the Banach algebra which play significant roles in the Feynman integration theory and quantum mechanics.

A Characterization of the Existence of a Fundamental Bounded Resolution for the Space Cc(X) in Terms of X

We characterize in terms of the topology of a Tychonoff space X the existence of a bounded resolution for CcX that swallows the bounded sets, where CcX is the space of real-valued continuous functions on X equipped with the compact-open topology.

A Banach algebra with its applications over paths of bounded variation

‎Let $C[0,T]$ denote the space of continuous real-valued functions on $[0,T]$‎. ‎In this paper we introduce two Banach algebras‎: ‎one of them is defined on $C[0,T]$ and the other is a space of equivalence classes of measures over paths of bounded variation on $[0,T]$‎. ‎We establish an isometric isomorphism between them and evaluate analytic Feynman integrals of the functions in the Banach algebras‎, ‎which play significant roles in the Feynman integration theories and quantum mechanics‎.

A note on the criterion for a best approximation by superpositions of functions

Let $Q$ be a compact subset of the $d$-dimensional Euclidean space, and $C(Q)$ be the space of continuous real-valued functions on $Q$. We consider the problem of approximation of a function $f\in C(Q)$ by superpositions of the form $g\circ s+h\circ p$, w

Measurable functions similar to the itô integral and the Paley-Wiener-Zygmund integral over continuous paths

Let C[0,T] denote an analogue of generalizedWiener space, the space of continuous real-valued functions on the interval [0,T]. On the space C[0,T], we introduce a finite measure w?,?,? and investigate its properties, where ? is an arbitrary finite measure on the Borel class of R. Using the measure w?,?,?, we also introduce two measurable functions on C[0,T], one of them is similar to the It? integral and the other is similar to the Paley-Wiener-Zygmund integral. We will prove that if ?(R) = 1, then w?,?,? is a probability measure with the mean function ? and the variance function ?, and the two measurable functions are reduced to the Paley-Wiener-Zygmund integral on the analogue ofWiener space C[0,T]. As an application of the integrals, we derive a generalized Paley-Wiener-Zygmund theorem which is useful to calculate generalized Wiener integrals on C[0,T]. Throughout this paper, we will recognize that the generalized It? integral is more general than the generalized Paley-Wiener-Zygmund integral.

Chaotic Dynamics of Composition Operators on the Space of Continuous Functions

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 a generalization of the Stone–Weierstrass theorem

Assume X is a compact Hausdorff space and C(X) is the space of real-valued continuous functions on X. A version of the Stone–Weierstrass theorem states that a closed subalgebra \(A\subset C(X)\), which contains a nonzero constant function, coincides with the whole space C(X) if and only if A separates points of X. In this paper, we generalize this theorem to the case in which two subalgebras of C(X) are involved.

On the weak and pointwise topologies in function spaces II

For a compact space K we denote by Cw(K) (Cp(K)) the space of continuous real-valued functions on K endowed with the weak (pointwise) topology. In this paper we discuss the following basic question which seems to be open: Let K and L be infinite compact spaces. Can it happen thatCw(K)andCp(L)are homeomorphic?M. Krupski proved that the above problem has a negative answer when K=L and K is finite-dimensional and metrizable. We extend this result to the class of finite-dimensional Valdivia compact spaces K.

Definability aspects of the Denjoy integral

Author(s): Walsh, Sean | Abstract: 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.

The $\sigma$-property in $C(X)$

The $\sigma$-property of a Riesz space (real vector lattice) $B$ is: For each sequence $\{b_{n}\}$ of positive elements of $B$, there is a sequence $\{\lambda_{n}\}$ of positive reals, and $b\in B$, with $\lambda_{n}b_{n}\leq b$ for each $n$. This condition is involved in studies in Riesz spaces of abstract Egoroff-type theorems, and of the countable lifting property. Here, we examine when ``$\sigma$'' obtains for a Riesz space of continuous real-valued functions $C(X)$. A basic result is: For discrete $X$, $C(X)$ has $\sigma$ iff the cardinal $|X|< \mathfrak{b}$, Rothberger's bounding number. Consequences and generalizations use the Lindelöf number $L(X)$: For a $P$-space $X$, if $L(X)\leq \mathfrak{b}$, then $C(X)$ has $\sigma$. For paracompact $X$, if $C(X)$ has $\sigma$, then $L(X)\leq \mathfrak{b}$, and conversely if $X$ is also locally compact. For metrizable $X$, if $C(X)$ has $\sigma$, then $X$ \textit{is} locally compact.