A global optimality result using Geraghty type contraction
Keywords:Contraction, proximity point, metric space, global minima, fixed point.
AbstractIn this paper we prove two proximity point results for finding the distance between two sets. Unlike the best approximation theorems they provide with globally optimal values. Here our approach is to reduce the problem to that of finding optimal approximate solutions of some fixed point equations. We use Geraghty type contractive inequalities in our theorem. Two illustrative examples are given.
Abkar, A., Gabeleh, M., Results on the existence and convergence of best proximity points, Fixed Point Theory Appl., Art. ID 386037, 10 pp (2010).
Al - Thagafi, M.A., Shahzad, N., Convergence and existence results for best proximity points, Nonlinear Anal., 70, 3665-3671 (2009). CrossRef
Amini-Harandi, A., Emani, H., A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations, Nonlinear Anal., 72, 2238-2242 (2010). CrossRef
Anthony Eldered, A., Veeramani, P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 323, 1001-1006 (2006). CrossRef
Anuradha, J., Veeramani, P., Proximal pointwise contraction, Topology and its Applications, 156, 2942-2948 (2009). CrossRef
Choudhury,B. S., Maity,P., Konar,P., A Global Optimality Result Using Nonself Mappings, Opsearch, 51(2), 312-320 (2013). CrossRef
Caballero, J., Harjani, J., Sadarangani, K., Contractive-Like mapping principles in ordered metric spaces and application to ordinary differential equations, Fixed Point Theory Appl., Art. ID 916064, 14 pp (2010).
Caballero, J., Harjani, J., Sadarangani, K., A best proximity point theorem for Geraghty-contractions, Fixed Point Theory Appl. 2012, 231 (2012). CrossRef
Di Bari, C., Suzuki, T., Vetro, C., Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal., 69, 3790-3794 (2008). CrossRef
Hobson, M.P., Efstathiou, G., Lasenby, A.N., General Reltivity, Cambridge University Press. New York. (2006).
KarapÄ±nar, E., On best proximity point of Ï†-Geraghty contractions, Fixed Point Theory Appl., 2013, 200 (2013). CrossRef
Karpagam, S., Agrawal, S., Best proximity points for cyclic orbital Meir-Keeler contractions, Nonlinear Anal., 74, 1040-1046 (2011). CrossRef
Fan, K., Extensions of two fixed point theorems of F.E. Browder, Mathematische Zeitschrift, 122, 234-240 (1969). CrossRef
Kirk, W.A., Reich, S., Veeramani, P., Proximal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim., 24, 851-862 (2003). CrossRef
Kirk, W.A., Srinivasan, P.S., Veeramani, P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, 79-89 (2003).
Rezapour, Sh., Derafshpour, M., Shahzad, N., Best proximity points of cyclic Ï• - contractions on reflexive Banach spaces, Fixed Point Theory Appl., Art. ID 946178, 2010, 7pp (2010).
Sadiq Basha, S., Global optimal approximate solutions, Optim. Lett., 5(4), 639-645 (2011). CrossRef
Sadiq Basha, S., Best proximity points: global optimal approximate solutions, J. Global Optim., 49, 15-21 (2011). CrossRef
Schutz, B.F., Geometrical Methods of Mathematical Physics, Cambridge University Press. (1980).
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