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Very recently, P. Saha et al. [A global optimality result in probabilistic spaces using control function. Optimization. 2021;70:2387–2400] studied the existence of best proximity points for Banach type ϕ-proximal contractions in the framework of Menger probabilistic metric spaces. In this paper, we present a simple proof of their main existence result which is based on a fixed point theorem for ϕ-contractive self-mappings in complete Menger PM-spaces due to B.S. Choudhury and K. Das [A new contraction principle in Menger spaces. Acta Math Sin. 2008;24:13791386]. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the existence of the best proximity points in binormed linear spaces. As a consequence, we obtain some fixed point results. We also provide some illustrations to support our claims. As applications, we obtain the existence of a solution to split feasible and variational inequality problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a weak MT-cyclic Reich type contractions and obtain the existence theorems for best proximity point for self-mappings defined on the complete metric spaces. Our results improve and generalize some results in literature. Also, we give some applications of our results to solving some classes of non-linear integral and differential equations. [ABSTRACT FROM AUTHOR]

In the current paper, we discuss sufficient and necessary conditions for the existence of best proximity points for cyclic f- contractive mappings in the setting of strictly convex Banach spaces. Extensions of Edelstein's theorem are considered as well as an extension of a main result in Park [Park, S. (1978). Fixed points of f-contractive maps. Rocky Mountain J. Math. 8:743–750]. Another existence result of best proximity points will be obtained for asymptotically relatively nonexpansive mappings under different conditions with respect to the recent paper of Rajesh and Veeramani [Rajesh, S., Veeramani, P. (2016). Best Proximity point theorems for asymptotically relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 37:80–91]. [ABSTRACT FROM AUTHOR]

In this paper, we present two main results about best proximity points for multivalued mappings. First, taking into account Klim and Wardowski's approach in fixed-point theory, we obtain a new result for multivalued mappings which includes the main theorem of Kamran [Kamranm, T. (2009). Mizoguchi–Takahashi's type fixed point theorem. Comput. Math. Appl. 57(3):507–511]. Second, combining this approach with cyclic contraction, we give a general best proximity point result, and so we obtain many well-known fixed-point theorems such as Mizoguchi–Takahashi, Feng–Liu and Klim–Wardowski [Mizoguchi, N., Takahashi, W. (1989). Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1):177–188; Feng, Y., Liu, S. (2006). Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 317(1):103–112; Klim, D., Wardowski, D. (2007). Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl. 334(1):132–139]. Also, we support our results by providing some nontrivial, illustrative and comparative examples. [ABSTRACT FROM AUTHOR]