Best proximity point theorems of cyclic Meir-Keeler-Kannan-Chatterjea contractions

Best proximity point theorems of cyclic Meir-Keeler-Kannan-Chatterjea contractions

In this study, by using the Meir-Keeler mapping, cyclic Kannan contraction and cyclic Chatterjee contraction, we establish the notions of cyclic Meir-Keeler-Kannan-Chatterjea contraction T from (A union B) to  (A union B) and cyclic Meir-Keeler-Kannan-Chatterjea contractive pair (T,S) of mappings T from A to  B  and S from B to A , and then we prove some best proximity point theorems for these various types of cyclic contractions. Our results generalize or improve many recent best proximity point theorems in the literature.

___

  • [1] M.A. Al-Thagafi, N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Anal., 70 (10) (2009)3665–3671.
  • [2] A. Anthony Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006)1001–1006.
  • [3] M.A. Al-Thagafi, N. Shahzad, Best proximity pairs and equilibrium pairs for Kakutani multimaps, Nonlinear Anal., 70(3) (2009) 1209–1216.
  • [4] M.A. Al-Thagafi, N. Shahzad, Best proximity sets and equilibrium pairs for a finite family of multimaps, Fixed PointTheory Appl., 3 (2008).
  • [5] C. Di Bari, T. Suzuki, C. Vetro, Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal., 69 (11) (2008)3790–3794.
  • [6] W.-S. Du, On coincidence point and fixed point theorems for nonlinear multivalued maps, Topology, and Applications,159(2012), 49–56.
  • [7] A. Anthony Eldred, P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006)1001–1006.
  • [8] S. Karpagam, S. Agrawal, Best proximity point theorems for p-cyclic Meir-Keeler contractions, Fixed Point Theory Appl.,(2009), ID 197308.
  • [9] A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl., 28(1969), 326–329.
  • [10] S. Sadiq Basha, P. Veeramani, Best approximations and best proximity pairs, Acta Sci. Math. (Szeged) 63(1997), 289–300.
  • [11] S. Sadiq Basha, P. Veeramani, Best proximity pair theorems for multifunctions with open fibers, 103 (2000) 119--129. J.Approx. Theory 103(2000), 119–129.
  • [12] S. Sadiq Basha, N. Shahzad, and R. Jeyaraj, Optimal approximate solutions of fixed point equations,h Abstract andApplied Analysis, (2011), Article ID 174560, 9 pages.
  • [13] S. Sadiq Basha, gBest proximity points: global optimal approximate solutions, Journal of Global Optimization, 49(2011),no. 1, pp. 15--21.
  • [14] K. Wlodarczyk, R. Plebaniak, A. Banach, Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces, Nonlinear Anal., 70 (9) (2009) 3332–3341.