Abstract
Let A and B be non-empty subsets of a metric space. As a non-self mapping \({T:A\longrightarrow B}\) does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.
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., Art. ID 457069, 10 pp (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)
Di Bari C., Suzuki T., Vetro C.: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 69(11), 3790–3794 (2008)
Edelstein M.: On fixed and periodic points under contractive mappings. J. London Math. Soc. 37, 74–79 (1962)
Fan K.: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 112, 234–240 (1969)
Karpagam, S., Agrawal, S.: Best proximity point theorems for p-cyclic Meir-Keeler contractions, Fixed Point Theory Appl., Art. ID 197308, 9 pp (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), 2216–2222 (2008)
Kirk W.A., Reich S., Veeramani P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851–862 (2003)
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)
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)
Sehgal V.M., Singh S.P.: A generalization to multifunctions of Fan’s best approximation theorem. Proc. Am. 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)
Vetrivel V., Veeramani P., Bhattacharyya P.: Some extensions of Fan’s best approximation theorem. Numer. Funct. Anal. Optim. 13, 397–402 (1992)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sadiq Basha, S. Best proximity points: global optimal approximate solutions. J Glob Optim 49, 15–21 (2011). https://doi.org/10.1007/s10898-009-9521-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-009-9521-0