Abkar, A.; Gabeleh, M. Best proximity points for asymptotic cyclic contraction mappings. (English) Zbl 1263.47065 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 7261-7268 (2011). The authors prove existence and convergence of unique best proximity points for asymptotic cyclic contractions in metric spaces satisfying the property UC and in uniformly convex Banach spaces. They also prove existence and convergence result for an asymptotic proximal pointwise contraction mapping in uniformly convex Banach spaces. Moreover, they prove existence of best proximity points for generalized cyclic contraction mappings in Banach spaces which do not necessarily satisfy the geometric property UC. Reviewer: D. S. Diwan (Bhilai) Cited in 1 ReviewCited in 35 Documents MSC: 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects) Keywords:best proximity points; fixed points; proximal pointwise contraction; cyclic contraction; property UC PDFBibTeX XMLCite \textit{A. Abkar} and \textit{M. Gabeleh}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 7261--7268 (2011; Zbl 1263.47065) Full Text: DOI References: [1] Kirk, W. A.; Srinivasan, P. S.; Veeramani, P., Fixed points for mappings satisfying cyclic contractive conditions, Fixed Point Theory, 4, 1, 79-86 (2003) · Zbl 1052.54032 [2] Eldred, A. A.; Veeramani, P., Existence and convergence of best proximity points, J. Math. Anal. Appl., 323, 1001-1006 (2006) · Zbl 1105.54021 [3] Suzuki, T.; Kikkawa, M.; Vetro, C., The existence of best proximity points in metric spaces with the property UC, Nonlinear Anal., 71, 2918-2926 (2009) · Zbl 1178.54029 [4] Kirk, W. A., Fixed points of asymptotic contractions, J. Math. Anal. Appl., 277, 645-650 (2003) · Zbl 1022.47036 [5] Arandelovic, Ivan D., On a fixed point of Kirk, J. Math. Anal. Appl., 301, 384-385 (2005) · Zbl 1075.47031 [6] Jachymski, J., A note on a paper of I.D. Arandelovic on asymptotic contractions, J. Math. Anal. Appl., 358, 491-492 (2009) · Zbl 1165.47039 [7] W.A. Kirk, Asymptotic contractions, in: Plenary Lecture, the 8th International Conference on Fixed Point Theory and its Applications, Chiang Mai University, Thailand, July 16-22, 2007.; W.A. Kirk, Asymptotic contractions, in: Plenary Lecture, the 8th International Conference on Fixed Point Theory and its Applications, Chiang Mai University, Thailand, July 16-22, 2007. [8] Kirk, W. A.; Xu, Hong-Kun, Asymptotic pointwise contractions, Nonlinear Anal., 69, 4706-4712 (2008) · Zbl 1172.47038 [9] Anuradha, J.; Veeramani, P., Proximal pointwise contraction, Topology Appl., 156, 2942-2948 (2009) · Zbl 1180.47035 [10] Sankara Raju Kosuru, G.; Veeramani, P., A note on existence and convergence of best proximity points for pointwise cyclic contractions, Numer. Funct. Anal. Optim., 32, 7, 8213-8830 (2011) · Zbl 1237.54052 [11] Eldred, A. A.; Kirk, W. A.; Veeramani, P., Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171, 283-293 (2005) · Zbl 1078.47013 [12] Al-Thagafi, M. A.; Shahzad, Naseer, Convergence and existence results for best proximity points, Nonlinear Anal., 70, 3665-3671 (2009) · Zbl 1197.47067 [13] Abkar, A.; Gabeleh, M., Results on the existence and convergence of best proximity points, Fixed Point Theory Appl. (2010), Art. ID 386037, 10 pages · Zbl 1205.47052 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.