Document Type : Research Paper
Authors
- Mohammad Hosein Labbaf Ghasemi ^{1}
- Mohammad Reza Haddadi ^{} ^{2}
- Noha Eftekhari ^{} ^{1}
^{1} Department of pure mathematics, Faculty of mathematical sciences, Shahrekord University, Shahrekord 88186-34141, Iran.
^{2} Faculty of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran.
Abstract
In this work we investigate a class of admitting center maps on a metric space. We state and prove some fixed point and best proximity point theorems for them. We obtain some results and relevant examples. In particular, we show that if $X$ is a reflexive Banach space with the Opial condition and $T:C\rightarrow X$ is a continuous admiting center map, then $T$ has a fixed point in $X.$ Also, we show that in some conditions, the set of all best proximity points is nonempty and compact.
Keywords
Main Subjects
[1] A. Abkar and M. Gabeleh, Best proximity points of non-self mappings, Top, 21 (2013), pp. 287-295.
[2] R.P. Agarwal, E. Karapınar, D. O'Regan, and A.F. Roldán-López-de-Hierro, Fixed point theory in metric type spaces, Switzerland, Springer, 2015.
[3] R.P. Agarwal, D. O'Regan, and D.R. Sahu, Fixed point theory for Lipschitzian-type mappings with applications, New York, Springer, 2009.
[4] T.D. Benavides, J.G. Falset, E. Llorens-Fuster, and P.L. Ramírez, Fixed point properties and proximinality in Banach spaces, Nonlinear Anal., 71 (2009), pp. 1562-1571.
[5] X.P. Ding and K.K. Tan, On equilibria of non-compact generalized games, J. Math. Anal. Appl., 177 (1993), pp. 226-238.
[6] W.G. Dotson, On the Mann iterative process, Trans. Amer. Math. Soc., 149 (1970), pp. 65-73.
[7] A.A. Eldred and P. Veeramani, Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), pp. 1001-1006.
[8] J. Garcia-Falset, E. Llorens-Fuster, and S. Prus, The fixed point property for mappings admitting a center, Nonlinear Anal., 66 (2007), pp. 1257-1274.
[9] M.R. Haddadi and S.M. Moshtaghioun, Some results on the best proximity pair, Abstract and Applied Analysism, 2011 (2011).
[10] W.K. Kim and S. Kum, Best proximity pairs and Nash equilibrium pairs, J. Korean Math. Soc., 45 (2008), pp. 1297-1310.
[11] W. Kirk and N. Shahzad, Fixed point theory in distance spaces, Springer, 2016.
[12] T.D. Narang, On best coapproximation in normed linear spaces, Rocky Mountain J. Math., 1 (1992), pp. 265-287.
[13] H.K. Nashine, P. Kumam, and C. Vetro, Best proximity point theorems for rational proximal contractions, Fixed Point Theory Appl., 2013 (2013), pp. 2-11.
[14] V.S. Raj, A best proximity point theorem for weakly contractive non-self-mappings, Nonlinear Anal., 74 (2011), pp. 4804-4808.
[15] J. Zhang, Y. Su, and Q. Cheng, Best proximity point theorems for generalized contractions in partially ordered metric spaces, Fixed Point Theory Appl., 2013 (2013), pp. 1-7.