In this paper, we introduce a new family of non-self mappings, called proximal expansive operators and survey the existence of a best proximity point for such mappings and used to obtain a best proximity version of Krasnoselskii’s fixed point problem in reflexive and strictly convex Banach spaces. We also consider De Blasi measure of weak noncompactness and present a class of proximal \(\omega\) -condensing operators to prove the other best proximity point theorems as generalizations of Schauder’s fixed point theorem. As an application we investigate the existence of an optimum solution for the following system of nonlinear mixed problems under some sufficient conditions \(\begin{aligned} {\left\{ \begin{array}{ll} y'(t)=f_1\Big (t,y\big (y(t)\big )\Big ) \ ; &\quad y(t_0)=\alpha _2,\\ z'(t)=f_2\Big (t,\int _{t_0}^{t}k\big (t,z(s)\big )ds\Big ) \ ; &\quad z(t_0)=\alpha _1, \end{array}\right. } \end{aligned}\) in the space C[0, 1], consists of all continuous real valued functions defined on [0, 1], which is renormed with an equivalent norm of the supremum norm in order to obtain the strict convexity assumption on C[0, 1].
