Coincidence best proximity points for geraghty type proximal cyclic contractions

Volume 18, Issue 1, pp 98--114 http://dx.doi.org/10.22436/jmcs.018.01.11

Publication Date: January 10, 2018 Submission Date: January 20, 2017

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Authors

Somayya Komal - Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand Azhar Hussain - Department of Mathematics, University of Sargodha, Sargodha-40100, Pakistan Nazra Sultana - Department of Mathematics, University of Sargodha, Sargodha-40100, Pakistan Poom Kumam - Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

Abstract

In this paper, we study the notions of generalized Geraghty proximal cyclic contractions for non-self mapping and obtain coincidence best proximity point theorems in the framework of complete metric spaces. Some examples are given to show the validity of our results. Our results extended and unify many existing results in the literature.

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Keywords

• \(\alpha\)-Geraghty proximal contraction of first and second kind
• \(\alpha\)-proximal cyclic contraction
• \(\alpha\)-proximal admissible maps

MSC

•  55M20

References

• [1] A. Abkar, M. Gabeleh, Generalized cyclic contractions in partially ordered metric spaces, Optim. Lett., 6 (2012), 1819– 1830
  • View Article
  • MathSciNet
  • Google Scholar


• [2] A. Abkar, M. Gabeleh, Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl., 153 (2012), 298–305
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] M. Abbas, A. Hussain, P. Kumam , A coincidence best proximity point problem in G-metric spaces, Abstr. Appl. Anal., 2015 (2015), 12 pages
  • View Article
  • MathSciNet
  • Google Scholar


• [4] R. P. Agarwal, M. A. El-Gebeily, D. O’Regan, Generalized contractions in partially ordered metric spaces, Appl. Anal. , 87 (2008), 109–116
  • View Article
  • Google Scholar


• [5] M. A. Al-Thagafi, N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal., 70 (2009), 3665–3671.
  • View Article
  • MathSciNet
  • Google Scholar


• [6] A. Amini-Harandi, H. Emami, A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., 72 (2010), 2238–2242
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181
  • Google Scholar


• [8] J. Caballero, J. Harjani, K. Sadarangani, A best proximity point theorem for Geraghty-contractions, Fixed Point Theory Appl., 2012 (2012 ), 9 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] S.-H. Cho, J.-S. Bae, E. Karapınar, Fixed point theorems for \(\alpha\)-Geraghty contraction type maps in metric spaces, Fixed Point Theory Appl., 2013 (2013 ), 11 pages
  • View Article
  • Google Scholar


• [10] C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal., 69 (2008), 3790–3794
  • View Article
  • Google Scholar


• [11] A. A. Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001–1006
  • View Article
  • Google Scholar


• [12] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc., 40 (1973), 604–608
  • View Article
  • Google Scholar


• [13] E. Hille, R. S. Phillips, Functional analysis and semi-groups, rev. ed., American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, R. I.,/ 31 (1957)
  • MathSciNet


• [14] L.-G. Huang, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468–1476
  • View Article
  • Google Scholar


• [15] M. A. Khamsi, V. Y. Kreinovich, Fixed point theorems for dissipative mappings in complete probabilistic metric spaces, Math. Japon., 44 (1996), 513–520
  • MathSciNet
  • Google Scholar
  • MATH


• [16] W. A. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24 (2003), 851–862
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] P. Kumam, P. Salimi, C. Vetro, Best proximity point results for modified \(\alpha\)-proximal C-contraction mappings, Fixed Point Theory Appl., 2014 (2014 ), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [18] C. Mongkolkeha, Y. J. Cho, P. Kumam, Best proximity points for Geraghty’s proximal contraction mappings, Fixed Point Theory Appl., 2013 (2013 ), 17 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), 257– 290
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] S. Sadiq Basha, Extensions of Banach’s contraction principle, Numer. Funct. Anal. Optim., 31 (2010), 569–576
  • View Article
  • MathSciNet
  • Google Scholar


• [21] S. Sadiq Basha, Best proximity point theorems generalizing the contraction principle, Nonlinear Anal., 74 (2011), 5844– 5850
  • View Article
  • MathSciNet
  • Google Scholar


• [22] S. Sadiq Basha, Best proximity points: global optimal approximate solutions, J. Global Optim., 49 (2011), 15–21
  • View Article
  • MathSciNet
  • Google Scholar


• [23] S. Sadiq Basha, Common best proximity points: global minimization of multi-objective functions, J. Global Optim., 54 (2012), 367–373
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] S. Sadiq Basha, N. Shahzad, R. Jeyaraj, Best proximity points: approximation and optimization, Optim. Lett., 7 (2013), 145–155
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] S. Sadiq Basha, P. Veeramani, Best proximity pair theorems for multifunctions with open fibres, J. Approx. Theory, 103 (2000), 119–129
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] N. Shahzad, S. Sadiq Basha, R. Jeyaraj , Common best proximity points: global optimal solutions, J. Optim. Theory Appl., 148 (2011), 69–78
  • View Article
  • MathSciNet
  • Google Scholar


• [27] C. Vetro, Best proximity points: convergence and existence theorems for p-cyclic mappings, Nonlinear Anal., 73 (2010), 2283–2291
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] K. Włodarczyk, R. Plebaniak, A. Banach, Best proximity points for cyclic and noncyclic set-valued relatively quasiasymptotic contractions in uniform spaces, Nonlinear Anal., 70 (2009), 3332–3341
  • View Article
  • Google Scholar


• [29] K. Włodarczyk, R. Plebaniak, A. Banach, Erratum to: ”Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces” [Nonlinear Anal., (2008), doi: 10.1016/j.na.2008.04.037] [ MR2503079], Nonlinear Anal., 71 (2009), 3585–3586
  • View Article
  • MathSciNet
  • MATH


• [30] K. Włodarczyk, R. Plebaniak, C. Obczyński, Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces, Nonlinear Anal., 72 (2010), 794–805
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [31] S.-K. Yang, J.-S. Bae, S.-H. Cho, Coincidence and common fixed and periodic point theorems in cone metric spaces, Comput. Math. Appl., 61 (2011), 170–177
  • View Article
  • MathSciNet
  • Google Scholar