Best proximity points for generalized \(\alpha-\psi\)-proximal contractive type mappings. (English) Zbl 1266.47077

Jleli, Mohamed; Karapınar, Erdal; Samet, Bessem

J. Appl. Math. 2013, Article ID 534127, 10 p. (2013).

Summary: We introduce a new class of non-self-contractive mappings. For such mappings, we study the existence and uniqueness of best proximity points. Several applications and interesting consequences of our obtained results are derived.

MSC:

47H10	Fixed-point theorems
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

References:

[1]	A. Abkar and M. Gabeleh, “Best proximity points for cyclic mappings in ordered metric spaces,” Journal of Optimization Theory and Applications, vol. 151, no. 2, pp. 418-424, 2011. · Zbl 1246.54034
[2]	M. A. Al-Thagafi and N. Shahzad, “Best proximity pairs and equilibrium pairs for Kakutani multimaps,” Nonlinear Analysis, vol. 70, no. 3, pp. 1209-1216, 2009. · Zbl 1225.47056
[3]	M. A. Al-Thagafi and N. Shahzad, “Convergence and existence results for best proximity points,” Nonlinear Analysis, vol. 70, no. 10, pp. 3665-3671, 2009. · Zbl 1197.47067
[4]	C. Di Bari, T. Suzuki, and C. Vetro, “Best proximity points for cyclic Meir-Keeler contractions,” Nonlinear Analysis, vol. 69, no. 11, pp. 3790-3794, 2008. · Zbl 1169.54021
[5]	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. · Zbl 1105.54021
[6]	E. Karapınar, “Best proximity points of cyclic mappings,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1761-1766, 2012. · Zbl 1269.47040
[7]	E. Karapınar, “Best proximity points of Kannan type cylic weak phi-contractions in ordered metric spaces,” Analele Stiintifice ale Universitatii Ovidius Constanta, vol. 20, no. 3, pp. 51-64, 2012.
[8]	S. Sadiq Basha, “Extensions of Banach’s contraction principle,” Numerical Functional Analysis and Optimization, vol. 31, no. 4-6, pp. 569-576, 2010. · Zbl 1200.54021
[9]	S. Sadiq Basha, “Best proximity point theorems generalizing the contraction principle,” Nonlinear Analysis, vol. 74, no. 17, pp. 5844-5850, 2011. · Zbl 1238.54021
[10]	S. Sadiq Basha, “Best proximity point theorems: an exploration of a common solution to approximation and optimization problems,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9773-9780, 2012. · Zbl 1245.90120
[11]	V. S. Raj, “A best proximity point theorem for weakly contractive non-self-mappings,” Nonlinear Analysis, vol. 74, no. 14, pp. 4804-4808, 2011. · Zbl 1228.54046
[12]	C. Vetro, “Best proximity points: convergence and existence theorems for P-cyclic mappings,” Nonlinear Analysis, vol. 73, no. 7, pp. 2283-2291, 2010. · Zbl 1229.54066
[13]	B. Samet, C. Vetro, and P. Vetro, “Fixed point theorems for \alpha -\psi -contractive type mappings,” Nonlinear Analysis, vol. 75, no. 4, pp. 2154-2165, 2012. · Zbl 1242.54027
[14]	E. Karapınar and B. Samet, “Generalized \alpha -\psi -contractive type mappings and related fixed point theorems with applications,” Abstract and Applied Analysis, vol. 2012, Article ID 793486, 17 pages, 2012. · Zbl 1252.54037
[15]	J. H. Asl, S. Rezapour, and N. Shahzad, “On fixed points of \alpha -\psi -contractive multifunctions,” Fixed Point Theory and Applications, vol. 2012, p. 212, 2012. · Zbl 1293.54017
[16]	B. Mohammadi, S. Rezapour, and N. Shahzad, “Some results on fixed points of \alpha -\psi -Ciric generalized multifunctions,” Fixed Point Theory and Applications, vol. 2013, p. 24, 2013. · Zbl 1423.54090
[17]	M. Jleli and B. Samet, “Best proximity points for \alpha -\psi -proximal contractive type mappings and applications,” Bulletin des Sciences Mathématiques, 2013. · Zbl 1290.41024
[18]	B. Samet and M. Turinici, “Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications,” Communications in Mathematical Analysis, vol. 13, no. 2, pp. 82-97, 2012. · Zbl 1259.54024