Abstract
Let (A, B) be a nonempty, closed and convex pair in a reflexive and Busemann convex space X, and \((E,F)\subseteq (A,B)\) be a nonempty and proximinal pair in X such that \(\mathrm{dist}(E,F)=\mathrm{dist}(A,B)\). We prove that the pair \((\overline{\mathrm{con}}(E),\overline{\mathrm{con}}(F))\) is also proximinal, where \(\overline{\mathrm{con}}(E)\) 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.
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Acknowledgements
The first author was partially supported by a grant from Ayatollah Boroujerdi University (No. 15664-164295). The second author acknowledges partial support from the National Research Foundation of South Africa under Grant 114773.
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Rosihan M. Ali.
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Gabeleh, M., Künzi, HP.A. Condensing Operators of Integral Type in Busemann Reflexive Convex Spaces. Bull. Malays. Math. Sci. Soc. 43, 1971–1988 (2020). https://doi.org/10.1007/s40840-019-00785-x
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DOI: https://doi.org/10.1007/s40840-019-00785-x