A generalization of Darbo's theorem with application to the‎ ‎solvability of systems of integral-differential equations in Sobolev spaces

Document Type : Research Paper


1 Department of Mathematics‎, ‎Mashhad Branch‎, ‎Islamic Azad University‎, ‎Mashhad‎, ‎Iran

2 Department of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.


‎In this article‎, ‎we introduce the notion of $(\alpha,\beta)$-generalized Meir-Keeler condensing operator in a‎ ‎Banach space‎, ‎a characterization using strictly L-functions and provide an extension of Darbo's fixed point theorem associated with measures‎ of noncompactness‎. ‎Then‎, ‎we establish some results on the existence of coupled fixed points for a‎ ‎class of condensing operators in Banach spaces‎. ‎As an application‎, ‎we study the‎ ‎problem of existence of entire solutions for a general system of nonlinear integral-differential equations in a Sobolev space‎. ‎Further‎, an example is presented to verify the effectiveness and applicability of our main results‎.