Abstract
Let A,B be nonempty subsets of a Banach space X and let T:A→B be a non-self mapping. Under appropriate conditions, we study the existence of solutions for the minimization problem min x∈A ∥x−Tx∥.
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References
Abkar A, Gabeleh M (2010) Results on the existence and convergence of best proximity points. Fixed Point Theory Appl 386037:1–10
Abkar A, Gabeleh M (2011a) Best proximity points for cyclic mappings in ordered metric spaces. J Optim Theory Appl 150:188–193. doi:10.1007/s10957-011-9810-x
Abkar A, Gabeleh M (2011b) Global optimal solutions of noncyclic mappings in metric spaces. J Optim Theory Appl. doi:10.1007/s10957-011-9966-4. In Press
Al-Thagafi MA, Shahzad Naseer (2009) Convergence and existence results for best proximity points. Nonlinear Anal 70:3665–3671
Cho YJ, Petrot N (2010) An optimization problem related to generalized equilibrium and fixed point problems with applications. Fixed Point Theory 11:237–250
Cho YJ, Kang SM, Zhou H (2008) Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces. J Inequal Appl 2008:598191. doi:10.1155/2008/598191. 10 pp
Cho YJ, Hussain N, Pathak HK (2011) Approximation of nearest common fixed points of asymptotically I-nonexpansive mappings in Banach spaces. Commun Korean Math Soc 26:483–498
Derafshpour M, Rezapour Sh, Shahzad N (2011) Best proximity points of cyclic φ-contractions in ordered metric spaces. Topol Methods Nonlinear Anal 37:193–202
Di Bari C, Suzuki T, Vetro C (2008) Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Anal 69:3790–3794
Eldred A, Veeramani P (2006) Existence and convergence of best proximity points. J Math Anal Appl 323:1001–1006
Fan K (1969) Extensions of two fixed point theorems of FE Browder. Math Z 122:234–240
Qin X, Kang SM, Cho YJ (2010) Approximating zeros of monotone operators by proximal point algorithms. J Glob Optim 46:75–87
Sadiq Basha S (2011a) Common best proximity points: global minimal solutions. Top. doi:10.1007/s11750-011-0171-2
Sadiq Basha S (2011b) Best proximity points: global optimal approximate solutions. J Glob Optim. doi:10.1007/s10898-009-9521-0
Sankar Raj V (2011) A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal 74:4804–4808
Suzuki T, Kikkawa M, Vetro C (2009) The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal 71:2918–2926
Wang SZ, Li BY, Gao ZM (1982) Some fixed point theorems on expansion mappings. Adv Math (China) 11(2):149–153
Wlodarczyk K, Plebaniak R, Obczynski C (2010) 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
Yao Y, Cho YJ, Liou Y (2011a) Iterative algorithms for variational inclusions, mixed equilibrium problems and fixed point problems approach to optimization problems. Cent Eur J Math 9:640–656
Yao Y, Cho YJ, Liou Y (2011b) Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur J Oper Res 212:242–250
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Abkar, A., Gabeleh, M. Best proximity points of non-self mappings. TOP 21, 287–295 (2013). https://doi.org/10.1007/s11750-012-0255-7
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DOI: https://doi.org/10.1007/s11750-012-0255-7