Best proximity points of local contractive mappings on metric spaces endowed with binary relation
Authors: Hussain A., Arshad M., Abbas M., Dolićanin Đekić D.
Keywords: best proximity point; non-self-mapping; (α -η , ψ)-proximal contraction; closed ball
Abstract:
The aim of this paper is to present best proximity point results of (α -η;ψ)- proximal mappings satisfying local contractive conditions on a closed ball in the framework of complete metric spaces. An example is also presented to validate the result proved herein. As an application of our results, we prove existence of best proximity points of locally contractive mappings in the frame work of metric spaces endowed with binary relation. Our results extend and generalize various comparable results in the existing literature.
References:
[1] Abdeljawad, T. (2013) Meir-Keeler α-contractive fixed and common fixed point theorems. Fixed Point Theory Appl
[2] Al-Thagafi, M.A., Shahzad, N. (2009) Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Analysis: Theory, Methods & Applications, 70(3): 1209-1216
[3] Arshad, M., Shoaib, A., Beg, I. (2013) Fixed point of a pair of contractive dominated mappings on a closed ball in an ordered dislocated metric space. Fixed Point Theory and Applications, 2013(1): 115
[4] Basha, S. S. (2010) Extensions of Banach’s Contraction Principle. Numerical Functional Analysis and Optimization, 31(5): 569-576
[5] Basha, S. S. (2011) Best proximity point theorems generalizing the contraction principle. Nonlinear Analysis: Theory, Methods & Applications, 74(17): 5844-5850
[6] Basha, S. S. (2012) Best proximity point theorems: An exploration of a common solution to approximation and optimization problems. Applied Mathematics and Computation, 218(19): 9773-9780
[7] di Bari, C., Suzuki, T., Vetro, C. (2008) Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Analysis: Theory, Methods & Applications, 69(11): 3790-3794
[8] Hussain, N., Salimi, P. S. (2014) SUZUKI-WARDOWSKI TYPE FIXED POINT THEOREMS FOR $\alpha$-GF-CONTRACTIONS. Taiwanese Journal of Mathematics, 18(6)
[9] Hussain, N., Arshad, M., Shoaib, A., Fahimuddin (2014) Common fixed point results for α -ψ- contractions on a metric space endowed with graph. J. Inequal. Appl., 2014: 136
[10] Hussain, N., Karapinar, E., Salimi, P., Akbar, F. (2013) alpha-Admissible mappings and related Fixed point Theorems. Journal of Inequalities and Applications, 2013(1): 114
[11] Jleli, M., Karapınar, E., Samet, B. (2014) A short note on the equivalence between ‘best proximity’ points and ‘fixed point’ results. Journal of Inequalities and Applications, 2014(1): 246
[12] Jleli, M., Samet, B. (2013) Best proximity points for -proximal contractive type mappings and applications. Bulletin des Sciences Mathématiques, 137(8): 977-995
[13] Kadelburg, Z., Radenovic, S. (2013) A note on some recent best proximity point results for nonself mappings. Gulf Journal of Mathematics, Vol 1, 36-41
[14] Karapınar, E., Samet, B. (2012) Generalized – Contractive Type Mappings and Related Fixed Point Theorems with Applications. Abstract and Applied Analysis, 2012: 1-17
[15] Mohammadi, B., Rezapour, Sh. (2013) On Modified α -ϕ-Contractions. J. Adv. Math. Stud., 6 (2); 162-166
[16] Prolla, J.B. (1983) Fixed-point theorems for set-valued mappings and existence of best approximants. Numerical Functional Analysis and Optimization, 5(4): 449-455
[17] Raj, V. S. (2011) A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Analysis: Theory, Methods & Applications, 74(14): 4804-4808
[18] Reich, S. (1978) Approximate selections, best approximations, fixed points, and invariant sets. Journal of Mathematical Analysis and Applications, 62(1): 104-113
[19] Salimi, P., Latif, A., Hussain, N. (2013) Modified α-ψ-contractive mappings with applications. Fixed Point Theory and Applications, 2013(1): 151
[20] Samet, B., Vetro, C., Vetro, P. (2012) Fixed point theorems for -contractive type mappings. Nonlinear Analysis: Theory, Methods & Applications, 75(4): 2154-2165
[21] Sehgal, V. M., Singh, S. P. (1988) A generalization to multifunctions of Fan’s best approximation theorem. Proceedings of the American Mathematical Society, 102(3): 534-534
[22] Sehgal, V. M., Singh, S. P. (1989) A theorem on best approximations. Numerical Functional Analysis and Optimization, 10(1-2): 181-184
[23] Vetrivel, V., Veeramani, P., Bhattacharyya, P. (1992) Some extensions of fan’s best approximation theorem. Numerical Functional Analysis and Optimization, 13(3-4): 397-402
[24] Vetro, C. (2010) Best proximity points: Convergence and existence theorems for -cyclic mappings. Nonlinear Analysis: Theory, Methods & Applications, 73(7): 2283-2291