Ulam-Hyers stability of singular integral equations, via weakly Picard operators

摘要

In this paper we investigate the Ulam-Hyers stability of several integral equations withsingularity. First we give some results concerning the Ulam-Hyers stability of integral equations withweak singularities. Our approach is also suitable for studying some fractional differential equations.In order to emphasize this aspect we prove that some conditions (5) in S. Abbas, M. Benchohra, UlamHyers stability for the Darboux problem for partial fractional differential and integro-differentialequations via Picard operators published in Results Math. 65(2014), 67-79 (respectively condition(3.1) from S. Abbas, M. Benchohra, A. Petru?sel, Ulam stability for partial fractional differentialinclusions via Picard operators theory, Electron. J. Qual. Theory Differ. Equ., 2014, No. 51,1-13) can be omitted without losing the validity of the obtained results. In the second part weestablish some generalized Ulam-Hyers-Rassias stability results for the Bessel equation and relatedequations. Our approach is based on fixed point methods and the obtained results are more generalthan those established by Byungbae Kim and Soon-Mo Jung in Bessel’s differential equation and itsHyers-Ulam stability appeared in J. Inequal. Appl., Volume 2007.