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

In this paper we investigate the Ulam-Hyers stability of several integral equations with singularity. First we give some results concerning the Ulam-Hyers stability of integral equations with weak singularities. Our approach is also suitable for studying some fractional di erential 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 di erential and integro-di erential equations via Picard operators published in Results Math. 65(2014), 67-79 (respectively condition (3.1) from S. Abbas, M. Benchohra, A. Petrusel, Ulam stability for partial fractional di erential inclusions via Picard operators theory, Electron. J. Qual. Theory Di er. Equ., 2014, No. 51, 1-13) can be omitted without losing the validity of the obtained results. In the second part we establish some generalized Ulam-Hyers- Rassias stability results for the Bessel equation and related equations. Our approach is based on xed point methods and the obtained results are more general than those established by Byungbae Kim and Soon-Mo Jung in Bessel's di erential equation and its Hyers-Ulam stability appeared in J. Inequal. Appl., Volume 2007.

Ulam-Hyers stability ; Picard operators ; Bessel equation ; Integral equations with singularities ; Fractional di erential equations

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