Nonlinear neutral integrodifferential equations on unbounded intervals

Abstract

In this paper we prove the existence of solutions for a boundary value nonlinear neutral integrodifferential problem in R n defined on an unbounded interval. The result is obtained by using the Schaefer fixed point theorem and by using a recent result [4] on compactness of a continuous operator K : BC(I, R n ) → BC(I, R n ); here BC(I, R n ) is the Banach space of continuous functions from the (possibly) unbounded interval I ⊂ R into R n . As corollary we obtain a result of existence of solutions for a Cauchy problem for nonlinear neutral integrodifferential equations.

Local attractivity for integro-differential equations with noncompact semigroups

“…Evolution equations are used in many elds of applied mathematics [9,33].In recent years, this type of equation has gained a lot of attention [39].The theory of integro-di erential equations on in nite intervals plays a key role as it addresses certain real world problems; it has gained considerable attention recently and now represents an important nonlinear analysis branch. Broad applications for physics, mechanics, engineering, electric, chemistry, economy and other areas [2,6,35,36] have, in recent years, been found in integrodi erential equations; while numerous research papers and monographs have been written, devoting themselves to functional and integro-di erential equations [3,4,7,21,31,34,37]. Such documents contain speci c qualitative properties like existence, uniqueness, stability and asymptotic behavior of rst-order equations.…”

Second-Order Impulsive Differential Equations with Functional Initial Conditions on Unbounded Intervals

“…Differential equations with deviating argument are investigated, for example, in [6,7,9,12,14,16,17,19,20].…”

“…Various techniques are utilized in the above papers: Schauder's fixed point Theorem [6,9,16,17,19,20], Leray-Schauder's nonlinear alternative [14,18], Contractions Principle [8,13], fixed point index theory [12], and Sadovskii's fixed point Theorem [7]. Moreover, in order to prove the compactness of involved operator, the Ascoli-Arzelà Theorem is often used (see [6,9,12,14,18]).…”

“…In order to prove that is completely continuous, we make use of a variant of the compactness result in the Banach space ( , R ) of continuous bounded functions from a topological space into R due to De Pascale, Lewicki and G. Marino. We have used similar techniques and tools in [16,17,20] to study first-order, impulsive, or nonimpulsive, differential Journal of Function Spaces and Applications or integrodifferential equations on unbounded intervals. In this paper we extend these ideas to second-order impulsive equations.…”