The purpose of this paper is to study different kinds of stability for a class of Hammerstein integral equations. Sufficient conditions are derived in view to obtain Hyers-Ulam stability and Hyers-Ulam-Rassias stability for such a class of Hammerstein integral equations. The consequent different cases of a finite interval and an infinite interval are considered, and some concrete examples are included to illustrate the results.
Abstract.We study different kinds of stabilities for a class of very general nonlinear integro-differential equations involving a function which depends on the solutions of the integro-differential equations and on an integral of Volterra type. In particular, we will introduce the notion of semi-Hyers-Ulam-Rassias stability, which is a type of stability somehow in-between the Hyers-Ulam and Hyers-Ulam-Rassias stabilities. This is considered in a framework of appropriate metric spaces in which sufficient conditions are obtained in view to guarantee Hyers-Ulam-Rassias, semi-Hyers-Ulam-Rassias and Hyers-Ulam stabilities for such a class of integro-differential equations. We will consider the different situations of having the integrals defined on finite and infinite intervals. Among the used techniques, we have fixed point arguments and generalizations of the Bielecki metric. Examples of the application of the proposed theory are included.
We analyse different kinds of stabilities for classes of nonlinear integral equations of Fredholm and Volterra type. Sufficient conditions are obtained in order to guarantee Hyers‐Ulam‐Rassias, σ‐semi‐Hyers‐Ulam and Hyers‐Ulam stabilities for those integral equations. Finite and infinite intervals are considered as integration domains. Those sufficient conditions are obtained based on the use of fixed point arguments within the framework of the Bielecki metric and its generalizations. The results are illustrated by concrete examples.
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