Best proximity point (pair) results via MNC in Busemann convex metric spaces




Coupled best proximity point (pair), Cyclic (noncyclic) condensing operator, Optimum solution, Busemann convex space


In this paper, we present a new class of cyclic (noncyclic) α-ψ and β-ψ condensing operators and survey the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in the setting of reflexive Busemann convex spaces. Then an application of the main existence result to study the existence of an optimal solution for a system of differential equations is demonstrated.


Download data is not yet available.

Author Biographies

Moosa Gabeleh, Ayatollah Boroujerdi University

Department of Mathematics

Pradip Ramesh Patle, Amity University Madhya Pradesh

Department of Mathematics, Amity School of Engineering and Technology


R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, vol. 55, Birkhäuser, Basel, 1992.

M. Ayerbe Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of noncompactness in metric fixed point theory, Operator Theory: Advances and Applications, vol. 99, Birkhäuser, Basel (1997).

G. C. Ahuja, T. D. Narang and S. Trehan, Best approximation on convex sets in a metric space, J. Approx. Theory 12 (1974), 94-97.

A. Berdellima, Complete sets and closure of their convex hulls in CAT(0) spaces, arXiv:2109.06002v1.

M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Springer-Verlag, Berlin Heidelberg, 1999.

H. Busemann, Geometry of geodesics, Academic Press, (1955) New York.

A. A. Eldred, W. A. Kirk and P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171 (2005), 283-293.

R. Espínola and A. Nicolae, Mutually nearest and farthest points of sets and the drop theorem in geodesic spaces, Monatsh. Math. 165 (2012), 173-197.

R. Espínola and B. Piatek, Fixed point property and unbounded sets in CAT(0) spaces, J. Math. Anal. Appl. 408 (2013), 638-654.

R. Espínola, O. Madiedo and A. Nicolae, Borsuk-Dugundji type extensions theorems with Busemann convex target spaces, Annales Academiae Scientiarum Fennicae Mathematica 43 (2018), 225-238.

A. Fernández León and A. Nicolae, Best proximity pair results relatively nonexpansive mappings in geodesic spaces, Numer. Funct. Anal. Optim. 35 (2014), 1399-1418.

M. Gabeleh and J. Markin, Optimum solutions for a system of differential equations via measure of noncompactness, Indagationes Mathematicae 29 (2018), 895-906.

M. Gabeleh and H. P. Künzi, Condensing operators of integral type in Busemann reflexive convex spaces, Bull. Malays. Math. Sci. Soc. 43 (2020), 1971-1988.

M. Gabeleh and H. P. Künzi, Mappings of generalized condensing type in metric spaces with Busemann convex structure, Bull. Iran. Math. Soc. 46 (2020), 1465-1483.

M. Gabeleh, Best proximity points for cyclic mappings, Ph.D Thesis (in Persian) (2012).

M. Gabeleh and J. Markin, Global optimal solutions of a system of differential equations via measure of noncompactness, Filomat 35 (2021), 5059-5071.

A. Latif, N. Saleem and M. Abbas, α-optimal best proximity point result involving proximal contraction mappings in fuzzy metric spaces, J. Nonlin. Sci. Appl. 10 (2017), 92-103.

P. R. Patle, D. K. Patel and R. Arab, Darbo type best proximity point results via simulation function with application, Afrika Mathematika 31 (2020), 833-845.

H. Rehman, D. Gopal and P. Kumam, Generalizations of Darbo's fixed point theorem for a new condensing operators with application to a functional integral equation, Demonstr. Mat. 52 (2019), 166-182.

N. Saleem, H. Ahmad, H. Aydi and Y. U. Gaba, On some coincidence best proximity point results, J. Math. 2021 (2021), 1-19.

N. Saleem, M. Abbas, B. Bin-Mohsin and S. Radenovic, Pata type best proximity point results in metric spaces, Miskolc Math. Notes 21 (2020), 367-386.

N. Saleem, M. Abbas and K. Sohail, Approximate fixed point results for (α-η)-type and (β-ψ)-type fuzzy contractive mappings in b-fuzzy metric spaces, Malaysian J. Math. Sci. 15 (2019), 367-386.

B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 19 (2012), 2154-2165.




How to Cite

M. Gabeleh and P. R. Patle, “Best proximity point (pair) results via MNC in Busemann convex metric spaces”, Appl. Gen. Topol., vol. 23, no. 2, pp. 405–424, Oct. 2022.



Regular Articles