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Global optimal solutions of noncyclic mappings in metric spaces. (English) Zbl 1262.90134
In the case that the fixed point equation has no solution, the best approximate solution is of interest. The authors prove the existence of such a globally optimal solution for noncyclic mappings in metric spaces, and apply this result for the solution of related problems in the theory of analytic functions.

90C26Nonconvex programming, global optimization
90C48Programming in abstract spaces
Full Text: DOI
[1] Eldred, A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001--1006 (2006) · Zbl 1105.54021 · doi:10.1016/j.jmaa.2005.10.081
[2] Abkar, A., Gabeleh, M.: Results on the existence and convergence of best proximity points. Fixed Point Theory Appl. 2010, 386037 (2010), 10 pp. · Zbl 1205.47052
[3] Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70, 3665--3671 (2009) · Zbl 1197.47067 · doi:10.1016/j.na.2008.07.022
[4] Suzuki, T., Kikkawa, M., Vetro, C.: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 71, 2918--2926 (2009) · Zbl 1178.54029 · doi:10.1016/j.na.2009.01.173
[5] Derafshpour, M., Rezapour, Sh., Shahzad, N.: Best proximity points of cyclic {$\phi$}-contractions in ordered metric spaces. Topol. Methods Nonlinear Anal. 37, 193--202 (2011) · Zbl 1227.54046
[6] Di Bari, C., Suzuki, T., Vetro, C.: Best proximity points for cyclic Meir-Keeler contractions. Nonlinear Anal. 69, 3790--3794 (2008) · Zbl 1169.54021 · doi:10.1016/j.na.2007.10.014
[7] 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, 3332--3341 (2009) · Zbl 1182.54024 · doi:10.1016/j.na.2008.04.037
[8] 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. (2008). doi: 10.1016/j.na.2008.04.037 . Nonlinear Anal. 71(7--8), 3585--3586 (2009)
[9] Abkar, A., Gabeleh, M.: Best proximity points for cyclic mappings in ordered metric spaces. J. Optim. Theory Appl. 150, 188--193 (2011) · Zbl 1232.54035 · doi:10.1007/s10957-011-9810-x
[10] Vetro, C.: Best proximity points: convergence and existence theorems for p-cyclic mappings. Nonlinear Anal. 73(7), 2283--2291 (2010) · Zbl 1229.54066 · doi:10.1016/j.na.2010.06.008
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[12] Sankar Raj, V.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74, 4804--4808 (2011) · Zbl 1228.54046 · doi:10.1016/j.na.2011.04.052
[13] Khamsi, M.A., Kirk, W.A.: An Introduction to Metric Spaces and Fixed Point Theory. Wiley, New York (2001) · Zbl 1318.47001
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