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2014, 4(2): 133-140. doi: 10.3934/naco.2014.4.133

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  May 2013 Revised  April 2014 Published  May 2014

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: normal cone, Cone metric, generalized cone proximal $\varphi$-cyclic contraction pairs, non-self mappings., best proximity points.
Mathematics Subject Classification: 47H10, 54H2.
Citation: Byung-Soo Lee. Existence and convergence results for best proximity points in cone metric spaces. Numerical Algebra, Control & Optimization, 2014, 4 (2) : 133-140. doi: 10.3934/naco.2014.4.133
References:
[1]

M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points,, Nonlinear Analysis, 70 (2009), 3665.  doi: 10.1016/j.na.2008.07.022.  Google Scholar

[2]

M. A. Al-Thagafi and N. Shahzad, Best proximity sets and equilibrium pairs for a finite family of mulimaps,, Fixed Point Theory Appl., 10 (2008).   Google Scholar

[3]

M. A. Al-Thagafi and N. Shahzad, Best proximity pairs and equilibrium pairs for Kakutani multimaps,, Nonlinear Analysis, 70 (2009), 1209.  doi: 10.1016/j.na.2008.02.004.  Google Scholar

[4]

S. S. Basha, Best proximity point theorems: resolution of an important non-linear programming problem,, Optim. Lett., 7 (2013), 1167.  doi: 10.1007/s11590-012-0493-5.  Google Scholar

[5]

A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points,, J. Math. Anal. Appl., 232 (2006), 1001.  doi: 10.1016/j.jmaa.2005.10.081.  Google Scholar

[6]

K. Fan, Extensions of two fixed point theorems of F. E. Browder,, Math. Z., 122 (1969), 234.   Google Scholar

[7]

M. Gabeleh and A. Abkar, Best proximity points for semi-cyclic contractive pairs in Banach spaces,, Int. Math. Forum, 6 (2011), 2179.   Google Scholar

[8]

L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings,, J. Math. Anal. Appl., 332 (2007), 1468.  doi: 10.1016/j.jmaa.2005.03.087.  Google Scholar

[9]

S. Karpagam and S. Agrawal, Best proximity point theorems for p-cyclic Meir-Keeler contraction,, Fixed Point Theory Appl., 9 (2009).   Google Scholar

[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.  doi: 10.1016/j.na.2007.01.057.  Google Scholar

[11]

W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorems,, Numer. Func. Anal. Optim., 24 (2003), 851.  doi: 10.1081/NFA-120026380.  Google Scholar

[12]

B. S. Lee, Cone metirc version of existence and convergence for best proximity points,, Universal J. Appl. Math., 2 (2014), 104.   Google Scholar

[13]

C. Mongkalkeha and P. Kumam, Some common best proximity points for proximity commuting mappings,, Optim. Lett., 7 (2013), 1825.  doi: 10.1007/s11590-012-0525-1.  Google Scholar

[14]

D. Turkoglu, M. Abuloha and T. Abdeljawad, KKM mappings in cone metric spaces and some fixed point theorems,, Nonlinear Analysis, 72 (2010), 348.  doi: 10.1016/j.na.2009.06.058.  Google Scholar

[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.   Google Scholar

show all references

References:
[1]

M. A. Al-Thagafi and N. Shahzad, Convergence and existence results for best proximity points,, Nonlinear Analysis, 70 (2009), 3665.  doi: 10.1016/j.na.2008.07.022.  Google Scholar

[2]

M. A. Al-Thagafi and N. Shahzad, Best proximity sets and equilibrium pairs for a finite family of mulimaps,, Fixed Point Theory Appl., 10 (2008).   Google Scholar

[3]

M. A. Al-Thagafi and N. Shahzad, Best proximity pairs and equilibrium pairs for Kakutani multimaps,, Nonlinear Analysis, 70 (2009), 1209.  doi: 10.1016/j.na.2008.02.004.  Google Scholar

[4]

S. S. Basha, Best proximity point theorems: resolution of an important non-linear programming problem,, Optim. Lett., 7 (2013), 1167.  doi: 10.1007/s11590-012-0493-5.  Google Scholar

[5]

A. A. Eldred and P. Veeramani, Existence and convergence of best proximity points,, J. Math. Anal. Appl., 232 (2006), 1001.  doi: 10.1016/j.jmaa.2005.10.081.  Google Scholar

[6]

K. Fan, Extensions of two fixed point theorems of F. E. Browder,, Math. Z., 122 (1969), 234.   Google Scholar

[7]

M. Gabeleh and A. Abkar, Best proximity points for semi-cyclic contractive pairs in Banach spaces,, Int. Math. Forum, 6 (2011), 2179.   Google Scholar

[8]

L. G. Huang and X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings,, J. Math. Anal. Appl., 332 (2007), 1468.  doi: 10.1016/j.jmaa.2005.03.087.  Google Scholar

[9]

S. Karpagam and S. Agrawal, Best proximity point theorems for p-cyclic Meir-Keeler contraction,, Fixed Point Theory Appl., 9 (2009).   Google Scholar

[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.  doi: 10.1016/j.na.2007.01.057.  Google Scholar

[11]

W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorems,, Numer. Func. Anal. Optim., 24 (2003), 851.  doi: 10.1081/NFA-120026380.  Google Scholar

[12]

B. S. Lee, Cone metirc version of existence and convergence for best proximity points,, Universal J. Appl. Math., 2 (2014), 104.   Google Scholar

[13]

C. Mongkalkeha and P. Kumam, Some common best proximity points for proximity commuting mappings,, Optim. Lett., 7 (2013), 1825.  doi: 10.1007/s11590-012-0525-1.  Google Scholar

[14]

D. Turkoglu, M. Abuloha and T. Abdeljawad, KKM mappings in cone metric spaces and some fixed point theorems,, Nonlinear Analysis, 72 (2010), 348.  doi: 10.1016/j.na.2009.06.058.  Google Scholar

[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.   Google Scholar

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