Existence and convergence results for best proximity points in cone metric spaces

Byung-Soo  Lee

doi:10.3934/naco.2014.4.133

Article Contents

2014, Volume 4, Issue 2: 133-140. Doi: 10.3934/naco.2014.4.133

This issue Previous Article Mixed integer programming model for scheduling in unrelated parallel processor system with priority consideration Next Article A weighted-path-following method for symmetric cone linear complementarity problems

Existence and convergence results for best proximity points in cone metric spaces

Byung-Soo Lee^1,

1.
Department of Mathematics, Kyungsung University, Busan 608-736

Received: April 30, 2013

Revised: March 31, 2014

Published: April 2014

Abstract Related Papers Cited by

Abstract

In this paper, the author introduces generalized cone proximal $\varphi$-cyclic contraction pairs in cone metric spaces and considers the existence and convergence of best proximity point for a pair in cone metric spaces. His results generalize the corresponding results in [1, 4, 5, 7, 8, 12, 13, 15].

Keywords:

Cone metric,
generalized cone proximal $\varphi$-cyclic contraction pairs,
normal cone,
best proximity points,
non-self mappings.

Mathematics Subject Classification: 47H10, 54H25.

Citation:

Related Papers

Cited by

References

[1]	M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points, Nonlinear Analysis, 70 (2009), 3665-3671.doi: 10.1016/j.na.2008.07.022.
[2]	M. A. Al-Thagafi and N. Shahzad, Best proximity sets and equilibrium pairs for a finite family of mulimaps, Fixed Point Theory Appl., Article ID 457069, 10 pages, 10 (2008).
[3]	M. A. Al-Thagafi and N. Shahzad, Best proximity pairs and equilibrium pairs for Kakutani multimaps, Nonlinear Analysis, 70 (2009), 1209-1216.doi: 10.1016/j.na.2008.02.004.
[4]	S. S. Basha, Best proximity point theorems: resolution of an important non-linear programming problem, Optim. Lett., 7 (2013), 1167-1177.doi: 10.1007/s11590-012-0493-5.
[5]	A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 232 (2006), 1001-1006.doi: 10.1016/j.jmaa.2005.10.081.
[6]	K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z., 122 (1969), 234-240.
[7]	M. Gabeleh and A. Abkar, Best proximity points for semi-cyclic contractive pairs in Banach spaces, Int. Math. Forum, 6 (2011), 2179-2186.
[8]	L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 332 (2007), 1468-1476.doi: 10.1016/j.jmaa.2005.03.087.
[9]	S. Karpagam and S. Agrawal, Best proximity point theorems for p-cyclic Meir-Keeler contraction, Fixed Point Theory Appl., Art. ID 197308, 9 (2009).
[10]	W. K. Kim, S. Kum and K. H. Lee, On general best proximity pairs and equilibrium pairs in free abstract economies, Nonlinear Analysis, 68 (2008), 2216-2227.doi: 10.1016/j.na.2007.01.057.
[11]	W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Func. Anal. Optim., 24 (2003), 851-862.doi: 10.1081/NFA-120026380.
[12]	B. S. Lee, Cone metirc version of existence and convergence for best proximity points, Universal J. Appl. Math., 2 (2014), 104-108.
[13]	C. Mongkalkeha and P. Kumam, Some common best proximity points for proximity commuting mappings, Optim. Lett., 7 (2013), 1825-1826.doi: 10.1007/s11590-012-0525-1.
[14]	D. Turkoglu, M. Abuloha and T. Abdeljawad, KKM mappings in cone metric spaces and some fixed point theorems, Nonlinear Analysis, 72 (2010), 348-353.doi: 10.1016/j.na.2009.06.058.
[15]	D. Xu and L. Deng, Cone semi-metric spaces and fixed point theorems for generalized weak contractive mappings, Nonlinear Analysis Forum, 18 (2013), 57-64.