Existence of Solutions of Integral Equations Defined in Unbounded Domains Via Spectral Theory

Abstract

In this work we study integral equations defined on the whole real line. Using a suitable Banach space, we look for solutions which satisfy some certain kind of asymptotic behavior. We will consider spectral theory in order to find fixed points of the integral operator. MSC: 45P05; 34K08; 45J05; 34K25Keywords: asymptotic behavior; integral operators; unbounded domain; spectral theory IntroductionIn this paper we will study the existence of fixed points of the following integral operatorWhen working with integral problems defined in unbounded intervals, the main difficulty is the lack of compactness of the operator. In the recent literature (see [4,6,[12][13][14]), most of the authors use the following relatively compactness criterion (see [3,16]) to deal with this problem: 1 ([16, Theorem 1]). Let E be a Banach space and ( , E) the space of all bounded continuous functions x : → E. For a set D ⊂ ( , E) to be relatively compact, it is necessary and sufficient that:

Asymptotic properties of PDEs in compact spaces

“…These kind of criteria are reworkings of the classical 0123456789(). : V,-vol Ascoli-Arzelà Theorem and have been used in a different way in [4,5]. In these works the authors are able to apply Ascoli-Arzelà Theorem by compactifying the domain of the functions involved in the ODE, thus allowing for a study of the asymptotic properties of the solutions.…”

“…In this article we combine both the study of solutions of PDEs with the study of asymptotic properties of the solutions via compactification of the domain. Furthermore, we take the opportunity to fix some of the shortcomings in [4,5] and provide an example of application. It is worth noticing that our results are not constrained to partial differential equations of a particular type (parabolic, elliptic, hyperbolic) since the results obtained are presented for the integral form of the equations; but also that, in general, the conditions to be checked for a particular problem can become quite unwieldy, which can be a limiting factor when it comes to apply the results to more convoluted problems.…”

Solvability of Coupled Systems of Generalized Hammerstein-Type Integral Equations in the Real Line

“…On the other hand, Cabada et al [23] deal with Hammerstein-type integral equations in unbounded domains via spectral theory. More concretely they study the existence of fixed points of the integral operator…”