Common Best Proximity Points Theorems for H-Contractive Non-Self Mappings

Fixed point theory and contractive mappings are popular tools in solving a variety of problems such as control theory, economic theory, nonlinear analysis and global analysis.There are many works on different types of contractions to find a fixed point in metric spaces. Improving and extending some kind of those, in this paper, we introduce a new version of H-contradiction for four mappings in a metric space (X,d). Then, we prove the existence and uniqueness of a common best proximity point for four non-self mappings. An example is also given to support our main result. The related fixed point theorem are also proved.

Keywords:

common best proximity points metric space, H-contractive condition,

PDF

___

[1] K. Fan, Extensions of two xed point theorems of F.E. Browder, Math . Z . 112 , 234-240 (1969)
[2] WA. Kirk, S. Reich, P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer.Funct.Anal.Optim. 24 , 851-862 (2003)
[3] MA. Alghamdi, N. Shahzad, F. Vetro, Best proximity points for some classes of proximal contractions, Abstr.Appl.Anal. 2013 , Article ID 713252 (2013)
[4] M. Derafshpour, S. Rezapour, N. Shahzad, Best proximity points of cyclic -contractions in ordered metric spaces, Topol . Methods Nonlinear Anal . 37(1) , 193-202 (2011)
[5] M. Gabeleh, Proximal weakly contractive and proximal nonexpansive non-self-mappings in metric and Banach spaces , J. Optim. Theory Appl. 158 , 615-625 (2013)
[6] M. Gabeleh, N. Shahzad, Existence and convergence theorems of best proximity points, J.Appl.Math. 2013, Article ID 101439 (2013) . doi:10.1155/2013/101439
[7] HK. Pathak, N. Shahzad, Convergence and existence results for best C-proximity points, Georgian Math.J. 19(2) , 301-316 (2012)
[8] S. Sadiq Basha, N. Shahzad, Best proximity point theorems for generalized proximal contractions, Fixed Point Theory Appl . 2012 , 42 (2012)
[9] T. Suzuki, M. Kikkawa, C. Vetro, The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal . 71 , 2918-2926 (2009)
[10] C. Vetro, Best proximity points : convergence and existence theorems for p-cyclic mappings, Nonlinear Anal . 73(7) , 2283-2291 (2010)
[11] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric space, Fixed Point Theory Appl, 2012. doi:10.1186/1687-1812-2012-94.
[12] M. Omidvari, S.M. Vaezpour, R. Saadati, Best proximity point theorems for F- contractive non-self mappings, Miskolc mathematical notes, Vol. 15(2014), No. 2, pp. 615-623