Let A and B be two nonempty subsets of a Banach space X. A mapping T : is said to be cyclic relatively nonexpansive if T(A) and T(B) and for all ( ) . In this paper, we introduce a geometric notion of seminormal structure on a nonempty, bounded, closed, and convex pair of subsets of a Banach space X. It is shown that if (A, B) is a nonempty, weakly compact, and convex pair and (A, B) has seminormal structure, then a cyclic relatively nonexpansive mapping T : has a fixed point. We also discuss stability of fixed points by using the geometric notion of seminormal structure. In the last section, we discuss sufficient conditions which ensure the existence of best proximity points for cyclic contractive type mappings. 1. Introduction Let be a Banach space and . Recall that a mapping is nonexpansive provided that for all . A closed convex subset of a Banach space has normal structure in the sense of Brodskii and Milman [1] if for each bounded, closed, and convex subset of which contains more than one point, there exists a point which is not a diametral point; that is, where is the diameter of . The set is said to have fixed point property (FPP) if every nonexpansive mapping has a fixed point. In 1965, Kirk proved the following famous fixed theorem. Theorem 1 (see [2]). Let be a nonempty, weakly compact, and convex subset of a Banach space . If has normal structure, then has the FPP. We mention that every compact and convex subset of a Banach space has normal structure (see [3]) and so has the FPP. Moreover, every bounded, closed, and convex subset of a uniformly convex Banach space has normal structure (see [4]) and then by Theorem 1 has the FPP. It is interesting to note that there exists a weakly compact and convex subset of which does not have the fixed point property (see [5] for more information). In particular, cannot have normal structure. In the current paper, we introduce a geometric notion of seminormal structure on a nonempty, closed, and convex pair of subsets of a Banach space and present a new fixed point theorem which is an extension of Kirk’s fixed point theorem. We also study the stability of fixed points by using this geometric property. Finally, we establish a best proximity point theorem for a new class of mappings. 2. Preliminaries In [6], Kirk et al. obtained an interesting extension of Banach contraction principle as follows. Theorem 2 (see [6]). Let and be nonempty closed subsets of a complete metric space . Suppose that is a cyclic mapping, that is, and . If for some and for all , , has a unique fixed point in . An interesting feature
K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
M. A. Khamsi and W. A. Kirk, “Pure and applied mathematics,” in An Introduction to Metric Spaces and Fixed Point Theory, pp. 303–304, Wiley-Interscience, New York, NY, USA, 2001.
A. Amini-Harandi, “Best proximity points theorems for cyclic strongly quasi-contraction mappings,” Journal of Global Optimization, vol. 56, no. 4, pp. 1667–1674, 2012.
W. A. Kirk, P. S. Srinivasan, and P. Veeramani, “Fixed points for mappings satisfying cyclical contractive conditions,” Fixed Point Theory, vol. 4, no. 1, pp. 79–89, 2003.
A. A. Eldred, W. A. Kirk, and P. Veeramani, “Proximal normal structure and relatively nonexpansive mappings,” Studia Mathematica, vol. 171, no. 3, pp. 283–293, 2005.
A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001–1006, 2006.
M. A. Al-Thagafi and N. Shahzad, “Convergence and existence results for best proximity points,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3665–3671, 2009.
C. Di Bari, T. Suzuki, and C. Vetro, “Best proximity points for cyclic Meir-Keeler contractions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3790–3794, 2008.
R. Espínola, “A new approach to relatively nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 136, no. 6, pp. 1987–1995, 2008.
A. Fernández-León, “Existence and uniqueness of best proximity points in geodesic metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 4, pp. 915–921, 2010.
M. Gabeleh, “Proximal weakly contractive and proximal nonexpansive non-self-mappings in metric and Banach spaces,” Journal of Optimization Theory and Applications, vol. 158, no. 2, pp. 615–625, 2013.
C. Mongkolkeha and P. Kumam, “Best proximity point theorems for generalized cyclic contractions in ordered metric spaces,” Journal of Optimization Theory and Applications, vol. 155, no. 1, pp. 215–226, 2012.
S. Rezapour, M. Derafshpour, and N. Shahzad, “Best proximity points of cyclic -contractions on reflexive Banach spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 946178, 7 pages, 2010.
S. S. Basha and N. Shahzad, “Best proximity point theorems for generalized proximal contractions,” Fixed Point Theory and Applications, vol. 2012, article 42, 2012.
S. Sadiq Basha, N. Shahzad, and R. Jeyaraj, “Best proximity points: approximation and optimization,” Optimization Letters, vol. 7, no. 1, pp. 145–155, 2013.
S. S. Basha, N. Shahzad, and R. Jeyaraj, “Best proximity point theorems for reckoning optimal approximate solutions,” Fixed Point Theory and Applications, vol. 2012, article 202, 2012.
M. A. Al-Thagafi and N. Shahzad, “Best proximity pairs and equilibrium pairs for Kakutani multimaps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 3, pp. 1209–1216, 2009.
M. A. Al-Thagafi and N. Shahzad, “Best proximity sets and equilibrium pairs for a finite family of multimaps,” Fixed Point Theory and Applications, vol. 2008, Article ID 457069, 10 pages, 2008.
A. Abkar and M. Gabeleh, “Proximal quasi-normal structure and a best proximity point theorem,” Journal of Nonlinear and Convex Analysis, vol. 14, pp. 653–659, 2013.
G. S. R. Kosuru and P. Veeramani, “A note on existence and convergence of best proximity points for pointwise cyclic contractions,” Numerical Functional Analysis and Optimization, vol. 32, no. 7, pp. 821–830, 2011.