Abstract
We study the existence and uniqueness of best proximity points for two classes of non-self-contractive mappings: almost (φ,θ)-contractive mappings and Meir–Keeler-type contractive mappings.
Similar content being viewed by others
References
Fan, K.: Extensions of two fixed point theorems of F.E. Browder. Math. Z. 112(3), 234–240 (1969)
Kirk, W.A., Reich, S., Veeramani, P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851–862 (2003)
Eldred, A.A., Veeramani, P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001–1006 (2006)
Di Bari, C., Suzuki, T., Vetro, C.: Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Anal. 69(11), 3790–3794 (2008)
Al-Thagafi, M.A., Shahzad, N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70(10), 3665–3671 (2009)
Anuradha, J., Veeramani, P.: Proximal pointwise contraction. Topol. Appl. 156, 2942–2948 (2009)
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)
Sankar Raj, V., Veeramani, P.: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 10(1), 21–28 (2009)
Vetro, C.: Best proximity points: convergence and existence theorems for p-cyclic mappings. Nonlinear Anal. 73, 2283–2291 (2010)
Fernàndez-León, A.: Existence and uniqueness of best proximity points in geodesic metric spaces. Nonlinear Anal. 73, 915–921 (2010)
Sadiq Basha, S.: Extensions of Banach’s contraction principle. Numer. Funct. Anal. Optim. 31(5), 569–576 (2010)
Suzuki, T., Vetro, C.: Three existence theorems for weak contractions of Matkowski type. Int. J. Math. Stat. 6, 110–120 (2010)
Shahzad, N., Sadiq Basha, S., Jeyaraj, R.: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 148, 69–78 (2011)
Sankar Raj, V.: A best proximity point theorem for weakly contractive non-self-mappings. Nonlinear Anal. 74(14), 4804–4808 (2011)
Akbar, A., Gabeleh, M.: Global optimal solutions of noncyclic mappings in metric spaces. J. Optim. Theory Appl. 153, 298–305 (2012)
Sadiq Basha, S., Shahzad, N., Jeyaraj, R.: Common best proximity points: global optimization of multi-objective functions. Appl. Math. Lett. 24, 883–886 (2011)
Sadiq Basha, S.: Best proximity points: global optimal approximate solutions. J. Glob. Optim. 49, 15–21 (2011)
Babu, G.V.R., Sandhya, M.L., Kameswari, M.V.R.: A note on a fixed point theorem of Berinde on weak contractions. Carpath. J. Math. 24(1), 8–12 (2008)
Banach, S.: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 3, 133–181 (1922)
Suzuki, T.: Meir–keeler contractions of integral type are still Meir–Keeler contractions. Int. J. Math. Math. Sci. 2007, 39281 (2007)
Meir, A., Keeler, E.: A theorem on contraction mappings. J. Math. Anal. Appl. 28, 326–329 (1969)
Branciari, A.: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 29(9), 531–536 (2002)
Acknowledgement
This work is supported by the Research Center, College of Science, King Saud University.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Samet, B. Some Results on Best Proximity Points. J Optim Theory Appl 159, 281–291 (2013). https://doi.org/10.1007/s10957-013-0269-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-013-0269-9