Space Of Continuous Functions Research Articles

Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra

We analyze properties of semigroups generated by Schrödinger operators Δ−V or polyharmonic operators −(−Δ)m, on metric graphs both on Lp-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kreĭn-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.

Non-linear modular Birkhoff-James orthogonality preservers between spaces of continuous functions

A new notion of non-linear isomorphisms between Banach modules is introduced based on modular Birkhoff-James orthogonality. It is shown that a (possibly non-linear) bijective modular Birkhoff-James orthogonality preserver in both direction between spaces of continuous functions (vanishing at infinity) induces a homeomorphism between underlying locally compact Hausdorff spaces. Moreover, characterizations of linear and additive modular Birkhoff-James orthogonality preservers between spaces of continuous functions are also given in terms of the induced homeomorphisms and continuous functions having constant absolute values.

Ascoli and sequentially Ascoli spaces

A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of Ck(X) is evenly continuous, where Ck(X) denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. Various properties of (sequentially) Ascoli spaces are studied, and we give several characterizations of sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show that a hereditary Ascoli space is Fréchet–Urysohn. A locally compact abelian group G with the Bohr topology is sequentially Ascoli iff G is compact. If X is totally countably compact or near sequentially compact then it is a sequentially Ascoli space. The product of a locally compact space and an Ascoli space is Ascoli. If additionally X is a μ-space, then X is locally compact iff the product of X with any Ascoli space is an Ascoli space. Extending one of the main results of [18] and [16] we show that Cp(X) is sequentially Ascoli iff X has the property (κ). We give a necessary condition on X for which the space Ck(X) is sequentially Ascoli. For every metrizable abelian group Y, Y-Tychonoff space X, and nonzero countable ordinal α, the space Bα(X,Y) of Baire-α functions from X to Y is κ-Fréchet–Urysohn and hence Ascoli.

Range preserving maps between the spaces of continuous functions with values in a locally convex space

Let C ( X , E ) C(X,E) be the linear space of all continuous functions on a compact Hausdorff space X X with values in a locally convex space E E . We characterize maps T : C ( X , E ) → C ( Y , E ) T:C(X,E)\to C(Y,E) which satisfy R a n ( T F − T G ) ⊂ R a n ( F − G ) \mathrm {Ran}(TF-TG)\subset \mathrm {Ran}(F-G) for all F , G ∈ C ( X , E ) F,G\in C(X,E) .

Semigroup-theoretic approach to diffusion in thin layers separated by semi-permeable membranes

Using techniques of the theory of semigroups of linear operators, we study the question of approximating solutions to equations governing diffusion in thin layers separated by a semi-permeable membrane. We show that as thickness of the layers converges to 0, the solutions, which by nature are functions of 3 variables, gradually lose dependence on the vertical variable and thus may be regarded as functions of 2 variables. The limit equation describes diffusion on the lower and upper sides of a two-dimensional surface (the membrane) with jumps from one side to the other. The latter possibility is expressed as an additional term in the generator of the limit semigroup, and this term is built from permeability coefficients of the membrane featuring in the transmission conditions of the approximating equations (i.e., in the description of the domains of the generators of the approximating semigroups). We prove this convergence result in the spaces of square integrable and continuous functions, and study the way the choice of transmission conditions influences the limit.

Projections in the convex hull of two isometries of absolutely continuous function spaces

In this paper we provide a complete description of projections in the convex hull of two surjective linear isometries (carrying a weighted composition operator form) on absolutely continuous function space AC(X, E), where X is a compact subset of $$\mathbb R$$ with at least two points and E is a strictly convex normed space. Among the consequences of the main result, it is shown that generalized bi-circular projections are the only projections on AC(X) expressed as the convex combination of two surjective linear isometries, and an affirmative answer is given to the question in Botelho and Jamison (Canad Math Bull 53:398–403, 2010) for such spaces.

Finite Element Solvers for Biot’s Poroelasticity Equations in Porous Media

We study and compare five different combinations of finite element spaces for approximating the coupled flow and solid deformation system, so-called Biot’s equations. The permeability and porosity fields are heterogeneous and depend on solid displacement and fluid pressure. We provide detailed comparisons among the continuous Galerkin, discontinuous Galerkin, enriched Galerkin, and two types of mixed finite element methods. Several advantages and disadvantages for each of the above techniques are investigated by comparing local mass conservation properties, the accuracy of the flux approximation, number of degrees of freedom (DOF), and wall and CPU times. Three-field formulation methods with fluid velocity as an additional primary variable generally require a larger number of DOF, longer wall and CPU times, and a greater number of iterations in the linear solver in order to converge. The two-field formulation, a combination of continuous and enriched Galerkin function space, requires the fewest DOF among the methods that conserve local mass. Moreover, our results illustrate that three out of the five methods conserve local mass and produce similar flux approximations when conductivity alteration is included. These comparisons of the key performance indicators of different combinations of finite element methods can be utilized to choose the preferred method based on the required accuracy and the available computational resources.

Asymptotic behaviour of fast diffusions on graphs

We study a diffusion process on a finite graph with semipermeable membranes on vertices. We prove, in L^1 and L^2-type spaces that for a large class of boundary conditions, describing communication between the edges of the graph, the process is governed by a strongly continuous semigroup of operators, and we describe asymptotic behaviour of the diffusion semigroup as the diffusions’ speed increases at the same rate as the membranes’ permeability decreases. Such a process, in which communication is based on the Fick law, was studied by Bobrowski (Ann. Henri Poincaré 13(6):1501–1510, 2012) in the space of continuous functions on the graph. His results were generalized by Banasiak et al. (Semigroup Forum 93(3):427–443, 2016). We improve, in a way that cannot be obtained using a very general tool developed recently by Engel and Kramar Fijavž (Evolut. Equ. Control Theory 8(3)3:633–661, 2019), the results of J. Banasiak et al.

BSDEs driven by cylindrical martingales with application to approximate hedging in bond markets

We consider Lipschitz-type backward stochastic differential equations (BSDEs) driven by cylindrical martingales on the space of continuous functions. We show the existence and uniqueness of the solution of such infinite-dimensional BSDEs and prove that the sequence of solutions of corresponding finite-dimensional BSDEs approximates the original solution. We also consider the hedging problem in bond markets and prove that, for an approximately attainable contingent claim, the sequence of locally risk-minimizing strategies based on small markets converges to the generalized hedging strategy.

Existence and uniqueness for the mild solution of the stochastic heat equation with non-Lipschitz drift on an unbounded spatial domain

We prove the existence and uniqueness of the mild solution for a nonlinear stochastic heat equation defined on an unbounded spatial domain. The nonlinearity is not assumed to be globally, or even locally, Lipschitz continuous. Instead the nonlinearity is assumed to satisfy a one-sided Lipschitz condition. First, a strengthened version of the Kolmogorov continuity theorem is introduced to prove that the stochastic convolutions of the fundamental solution of the heat equation and a spatially homogeneous noise grow no faster than polynomially. Second, a deterministic mapping that maps the stochastic convolution to the solution of the stochastic heat equation is proven to be Lipschitz continuous on polynomially weighted spaces of continuous functions. These two ingredients enable the formulation of a Picard iteration scheme to prove the existence and uniqueness of the mild solution.

Existence results for Riemann–Liouville fractional evolution inclusions in Banach spaces

The aim of this work is to study the existence of mild solution for semi-linear fractional evolution inclusions involving Riemann–Liouville derivative in Banach space. We prove our main result by introducing a regular measure of noncompactness in weighted space of continuous functions and using the condensing multivalued maps theory. Our result improve and complement several earlier related works. An example is given to illustrate the applications of the abstract result.

Optimal Extension of Positive Order Continuous Operators with Values in Quasi-Banach Lattices

The goal of this article is to present some method of optimal extension of positive order continuous and $ \sigma $ -order continuous operators on quasi-Banach function spaces with values in Dedekind complete quasi-Banach lattices. The optimal extension of such an operator is the smallest extension of the Bartle–Dunford–Schwartz type integral. It is also shown that if a positive operator sends order convergent sequences to quasinorm convergent sequences, then its optimal extension is the Bartle–Dunford–Schwartz type integral.

Integral Equations with Multidimensional Partial Integrals

We obtain Fredholm criteria for linear equations with multidimensional partial integrals in the space of continuous functions of three variables provided that the kernels are continuous vector-valued functions on a rectangle with the values in spaces of integrable functions. We establish the Fredholm criterion for equations with degenerate kernels and describe the scheme of studying the Fredholm properties of linear equations with partial integrals in spaces of continuous functions of at least four variables.

Fredholm Property of Linear Equations with Partial Integrals in the Space of Integrable Functions

We study the Fredholm property of linear equations with partial integrals and kernels in some classes. We show that a linear operator with partial integrals acts in the space of integrable functions and in the space of continuous functions defined in a square. We give an example of a non-Fredholm homogeneous linear equation of the second kind with partial integrals and continuous kernels. We obtain Fredholm criteria for linear operators and equations with partial integrals in the space of integrable functions. Bibliography: 6 titles.

The vanishing discount problem for monotone systems of Hamilton-Jacobi equations. Part 1: linear coupling

We establish a convergence theorem for the vanishing discount problem for a weakly coupled system of Hamilton-Jacobi equations. The crucial step is the introduction of Mather measures and their relatives for the system, which we call respectively viscosity Mather and Green-Poisson measures. This is done by the convex duality and the duality between the space of continuous functions on a compact set and the space of Borel measures on it. This is part 1 of our study of the vanishing discount problem for systems, which focuses on the linear coupling, while part 2 will be concerned with nonlinear coupling.

SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index

In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index H∈(14,1). We prove that the solution of each of the above equations is continuous in terms of the index H, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.

Invariant measures of stochastic delay lattice systems

<p style='text-indent:20px;'>This paper is concerned with the existence and uniqueness of invariant measures for infinite-dimensional stochastic delay lattice systems defined on the entire integer set. For Lipschitz drift and diffusion terms, we prove the existence of invariant measures of the systems by showing the tightness of a family of probability distributions of solutions in the space of continuous functions from a finite interval to an infinite-dimensional space, based on the idea of uniform tail-estimates, the technique of diadic division and the Arzela-Ascoli theorem. We also show the uniqueness of invariant measures when the Lipschitz coefficients of the nonlinear drift and diffusion terms are sufficiently small.

Entropic regularization of continuous optimal transport problems

We analyze continuous optimal transport problems in the so-called Kantorovich form, where we seek a transport plan between two marginals that are probability measures on compact subsets of Euclidean space. We consider the case of regularization with the negative entropy with respect to the Lebesgue measure, which has attracted attention because it can be solved by the very simple Sinkhorn algorithm. We first analyze the regularized problem in the context of classical Fenchel duality and derive a strong duality result for a predual problem in the space of continuous functions. However, this problem may not admit a minimizer, which prevents obtaining primal-dual optimality conditions. We then show that the primal problem is naturally analyzed in the Orlicz space of functions with finite entropy in the sense that the entropically regularized problem admits a minimizer if and only if the marginals have finite entropy. We then derive a dual problem in the corresponding dual space, for which existence can be shown by purely variational arguments and primal-dual optimality conditions can be derived. For marginals that do not have finite entropy, we finally show Gamma-convergence of the regularized problem with smoothed marginals to the original Kantorovich problem.

Ascoli’s theorem for pseudocompact spaces

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.

On the universal completion of pointfree function spaces

This paper approaches the construction of the universal completion of the Riesz space C(L) of continuous real functions on a completely regular frame L in two different ways. Firstly as the space of continuous real functions on the Booleanization of L. Secondly as the space of nearly finite Hausdorff continuous functions on L. The former has no counterpart in the classical theory, as the Booleanization of a spatial frame is not spatial in general, and it offers a lucid way of representing the universal completion as a space of continuous real functions. As a corollary we obtain that C(L) and C(M) have isomorphic universal completions if and only if the Booleanization of L and M are isomorphic and we characterize frames L such that C(L) is universally complete as almost Boolean frames. The application of this last result to the classical case C(X) of the space of continuous real functions on a topological space X characterizes those spaces X for which C(X) is universally complete. Finally, we present a pointfree version of the Maeda-Ogasawara-Vulikh representation theorem and use it to represent the universal completion of an Archimedean Riesz space with weak unit as a space of continuous real functions on a Boolean frame.