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- Hyers-Ulam and Hyers-Ulam-Rassias stability of a class of integral equations on finite intervalsPublication . Simões, A. M.; Castro, L. P.The purpose of this work is to study different kinds of stability for a class of integral equations defined on a finite interval. Sufficient conditions are derived in view to obtain Hyers-Ulam stability and Hyers-Ulam-Rassias stability by using fixed point techniques and the Bielecki metric.
- Different Types of Hyers-Ulam-Rassias Stabilities for a Class of Integro-Differential EquationsPublication . Simões, A. M.; Castro, L. P.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 {\it 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.
- Hyers-Ulam-Rassias Stability of Nonlinear Integral Equations Through the Bielecki MetricPublication . Simões, A. M.; Castro, L. P.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.