×

Best proximity points for relatively \(u\)-continuous mappings in Banach and hyperconvex spaces. (English) Zbl 1470.47044

Summary: We prove some best proximity point results for relatively \(u\)-continuous mappings in Banach and hyperconvex metric spaces. Our results generalize and extend some recent results to relatively \(u\)-continuous mappings and to general spaces.

MSC:

47H10 Fixed-point theorems
46B20 Geometry and structure of normed linear spaces
54H25 Fixed-point and coincidence theorems (topological aspects)
54E40 Special maps on metric spaces
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Eldred, A. A.; Kirk, W. A.; Veeramani, P., Proximal normal structure and relatively nonexpansive mappings, Studia Mathematica, 171, 3, 283-293, (2005) · Zbl 1078.47013
[2] Eldred, A. A.; Raj, V. S.; Veeramani, P., On best proximity pair theorems for relatively u-continuous mappings, Nonlinear Analysis: Theory, Methods & Applications, 74, 12, 3870-3875, (2011) · Zbl 1217.54044
[3] Al-Thagafi, M. A.; Shahzad, N., Convergence and existence results for best proximity points, Nonlinear Analysis: Theory, Methods & Applications, 70, 10, 3665-3671, (2009) · Zbl 1197.47067
[4] Basha, S. S.; Shahzad, N., Best proximity point theorems for generalized proximal contractions, Fixed Point Theory and Applications, 2012, (2012) · Zbl 1282.41012
[5] Basha, S. S.; Shahzad, N.; Jeyaraj, R., Best proximity points: approximation and optimization, Optimization Letters, 7, 1, 145-155, (2013) · Zbl 1283.90046
[6] Basha, S. S.; Shahzad, N.; Jeyaraj, R., Best proximity point theorems for reckoning optimal approximate solutions, Fixed Point Theory and Applications, 2012, (2012) · Zbl 1277.90097
[7] Basha, S. S.; Shahzad, N.; Jeyaraj, R., Common best proximity points: global optimal solutions, Journal of Optimization Theory and Applications, 148, 1, 69-78, (2011) · Zbl 1207.90096
[8] Gabeleh, M.; Shahzad, N., Existence and convergence theorems of best proximity points, Journal of Applied Mathematics, 2013, (2013) · Zbl 1266.47076
[9] Kirk, W. A.; Srinivasan, P. S.; Veeramani, P., Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory, 4, 1, 79-89, (2003) · Zbl 1052.54032
[10] Karpagam, S.; Agrawal, S., Existence of best proximity points of \(p\)-cyclic contractions, Fixed Point Theory, 13, 1, 99-105, (2012) · Zbl 1271.54082
[11] Sine, R., Hyperconvexity and approximate fixed points, Nonlinear Analysis: Theory, Methods & Applications, 13, 7, 863-869, (1989) · Zbl 0694.54033
[12] Espínola, R.; Lorenzo, P., Metric fixed point theory on hyperconvex spaces: recent progress, Arabian Journal of Mathematics, 1, 4, 439-463, (2012) · Zbl 1273.54052
[13] Deutsch, F.; Indumathi, V.; Schnatz, K., Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings, Journal of Approximation Theory, 53, 3, 266-294, (1988) · Zbl 0653.41029
[14] Markin, J. T., A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces, Journal of Mathematical Analysis and Applications, 321, 2, 862-866, (2006) · Zbl 1101.54025
[15] Khamsi, M. A., KKM and Ky Fan theorems in hyperconvex metric spaces, Journal of Mathematical Analysis and Applications, 204, 1, 298-306, (1996) · Zbl 0869.54045
[16] Kirk, W. A.; Shin, S. S., Fixed point theorems in hyperconvex spaces, Houston Journal of Mathematics, 23, 1, 175-188, (1997) · Zbl 0957.46033
[17] Raj, V. S.; Veeramani, P., Best proximity pair theorems for relatively nonexpansive mappings, Applied General Topology, 10, 1, 21-28, (2009) · Zbl 1213.47062
[18] Kirk, W. A.; Reich, S.; Veeramani, P., Proximinal retracts and best proximity pair theorems, Numerical Functional Analysis and Optimization, 24, 7-8, 851-862, (2003) · Zbl 1054.47040
[19] Markin, J.; Shahzad, N., Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces, Nonlinear Analysis: Theory, Methods & Applications, 70, 6, 2435-2441, (2009) · Zbl 1163.41005
[20] Abkar, A.; Gabeleh, M., Results on the existence and convergence of best proximity points, Fixed Point Theory and Applications, 2010, (2010) · Zbl 1205.47052
[21] Amini-Harandi, A.; Farajzadeh, A. P.; O’Regan, D.; Agarwal, R. P., Coincidence point, best approximation, and best proximity theorems for condensing set-valued maps in hyperconvex metric spaces, Fixed Point Theory and Applications, 2008, (2008) · Zbl 1158.54317
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.