Mutually nearest solution of a system of differential and integro-differential equations via proximal Branciari condensing operators

A new class of non-self mappings, called Branciari proximal contractions, is introduced and used to investigate the existence of best proximity points for sum and multiplication of two operators in strictly convex Banach spaces and strictly convex Banach algebras by using the proximal projection operator. In special case, we conclude a new fixed point theorem for multiplication of two self maps and then, as an application, we survey the existence of a solution for a nonlinear functional integral equation. In addition, we apply a concept of measure of noncompactness to present another class of non-self maps which is called Branciari proximal condensing operators and prove the existence of a best proximity point for these maps. Particularly, we obtain a best proximity version of Ky Fan’s best approximation theorem and apply it to study the existence of a mutually nearest solution for a system of differential and integro-differential equations. Several examples are given to support the main corollaries.