Abstract
In this paper, we prove new common best proximity point theorems for a proximity commuting mapping in a complete metric space. Our results generalized a recent result of Sadiq Basha [Common best proximity points: global minimization of multi-objective functions, J. Glob. Optim., (2011)] and some results in the literature.
Similar content being viewed by others
References
Al-Thagafi M.A., Shahzad N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70(10), 3665–3671 (2009)
Al-Thagafi M.A., Shahzad N.: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70(3), 1209–1216 (2009)
Al-Thagafi, M.A., Shahzad, N.: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl. 457069, 10 (2008)
Anthony Eldred A., Veeramani P.: Existence and Convergence of best proximity points. J. Math. Anal. Appl. 323, 1001–1006 (2006)
Anthony Eldred A., Kirk W.A., Veeramani P.: Proximinal normal structure and relatively nonexpanisve mappings. Studia Math. 171(3), 283–293 (2005)
Alber, Ya.I., Guerre-Delabriere, S.: Principle of Weakly Contractive Maps in Hilbert Spaces, New Results in Operator Theory and its Applications. Birkhauser, Basel 98, pp. 7–22 (1997)
Di Bari C., Suzuki T., Vetro C.: Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Anal. 69(11), 3790–3794 (2008)
Fan K.: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 112, 234–240 (1969)
Harjani J., Sadarangani K.: Fixed point theorems for weakly contractive mappings in partially odered sets. Nonlinear Anal. 71, 3403–3410 (2009)
Karpagam, S., Agrawal, S.: Best proximity point theorems for p-cyclic Meir- Keeler contractions, Fixed Point Theory Appl. 197308, 9 (2009)
Kim W.K., Kum S., Lee K.H.: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68(8), 222–2216 (2008)
Kirk W.A., Reich S., Veeramani P.: Proximinal retracts and best proximity pair theorems. Numer. Func. Anal. Optim. 24, 851–862 (2003)
Mongkolkeha, C., Kumam, P.: Best proximity point theorems for generalized cyclic contractions in ordered metric spaces. J. Optim. Theory Appl. (2012). doi:10.1007/s10957-012-9991-y
Prolla, J.B.: Fixed point theorems for set valued mappings and existence of best approximations. Numer. Funct. Anal. Optim. 5, 449–455 (1982–1983)
Reich S.: Approximate selections, best approximations, fixed points and invariant sets. J. Math. Anal. Appl. 62, 104–113 (1978)
Rhoades B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. 47, 2683–2693 (2001)
Sadiq Basha S., Veeramani P.: Best approximations and best proximity pairs. Acta. Sci. Math. (Szeged) 63, 289–300 (1997)
Sadiq Basha S., Veeramani P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119–129 (2000)
Sadiq Basha S., Veeramani P., Pai D.V.: Best proximity pair theorems. Indian J. Pure Appl. Math. 32, 1237–1246 (2001)
Sadiq Basha S.: Best proximity point theorems generalizing the contraction principle. Nonlinear Anal. 74, 5844–5850 (2011)
Sadiq Basha, S.: Common best proximity points: global minimization of multi-objective functions. J. Glob. Optim. doi:10.1007/s10898-011-9760-8 (2011)
Sankar Raj V.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74, 4804–4808 (2011)
Sehgal V.M., Singh S.P.: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Amer. Math. Soc. 102, 534–537 (1988)
Sehgal V.M., Singh S.P.: A theorem on best approximations. Numer. Funct. Anal. Optim. 10, 181–184 (1989)
Srinivasan P.S.: Best proximity pair theorems. Acta Sci. Math. (Szeged) 67, 421–429 (2001)
Wlodarczyk K., Plebaniak R., Banach A.: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 70(9), 3332–3341 (2009)
Wlodarczyk K., Plebaniak R., Banach A.: Erratum to: Best proximity points for cyclic and noncyclic set-valued relatively quasi-asymptotic contractions in uniform spaces. Nonlinear Anal. 71, 3583–3586 (2009)
Wlodarczyk K., Plebaniak R., Obczynski C.: Convergence theorems, best approximation and best proximity for set-valued dynamic systems of relatively quasi-asymptotic contractions in cone uniform spaces. Nonlinear Anal. 72, 794–805 (2010)
Sintunavarat, W., Kumam, P.: Coupled best proximity point theorem in metric spaces. Fixed Point Theory Appl. 2012, 93 (2012). doi:10.1186/1687-1812-2012-93
Zhang Q., Song Y.: Fixed point theory for generalized φ-weak contractions. Appl. Math. Lett. 22(1), 75–78 (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mongkolkeha, C., Kumam, P. Some common best proximity points for proximity commuting mappings. Optim Lett 7, 1825–1836 (2013). https://doi.org/10.1007/s11590-012-0525-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-012-0525-1