Condensing Operators of Integral Type in Busemann Reflexive Convex Spaces

Let (A, B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space X, and (E,F)⊆(A,B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(E,F)\subseteq (A,B)$$\end{document} be a nonempty and proximinal pair in X such that dist(E,F)=dist(A,B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{dist}(E,F)=\mathrm{dist}(A,B)$$\end{document}. We prove that the pair (con¯(E),con¯(F))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\overline{\mathrm{con}}(E),\overline{\mathrm{con}}(F))$$\end{document} is also proximinal, where con¯(E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathrm{con}}(E)$$\end{document} denotes the closed convex hull of the set E. Moreover, we introduce a new notion of cyclic (noncyclic) mappings involving measure of noncompactness and obtain some new existence results of best proximity points (pairs). As an application of our main conclusions, we study the existence of an optimal solution for a system of integrodifferential equations under new sufficient conditions.