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- Hyers-Ulam and Hyers-Ulam-Rassias stability of a class of Hammerstein integral equationsPublication . Simões, A. M.; Castro, L. P.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.
- 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.
- Stabilities for a class of higher order integro-differential equationsPublication . Castro, L. P.; Simões, A. M.This work is devoted to analyse different kinds of stabilities for higher order integro-differential equations within appropriate metric spaces. We will consider the σ-semi-Hyers-Ulam stability which is a new kind of stability somehow between the Hyers-Ulam and the Hyers-Ulam-Rassias stabilities. Sufficient conditions are obtained in view to guarantee Hyers-Ulam, σ-semiHyers-Ulam and Hyers-Ulam-Rassias stabilities for such a class of integro-differential equations. We will be considering finite and infinite intervals as integration domains. Among the used techniques, we have fixed point arguments and generalizations of the Bielecki metric.
- Hyers-Ulam and Hyers-Ulam-Rassias Stability for a Class of Integro-Differential EquationsPublication . Castro, L. P.; Simões, A. M.The concept of stability for functional, differential, integral and integro-differential equations has been studied in a quite extensive way during the last six decades and has earned particular interest due to their great number of applications (see [1, 3, 5, 6, 8–16, 18–23, 26] and the references therein). [...]
- 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.