Best proximity points of contractive mappings on a metric space with a graph and applications

Asrifa Sultana

India

Indian Institute of Technology Madras

Mathematics

V. Vetrivel

India

Indian Institute of Technology Madras

Mathematics

Submitted: 2014-11-25

|

Accepted: 2016-11-11

|

Published: 2017-04-03

DOI: https://doi.org/10.4995/agt.2017.3424
Funding Data

Downloads

Keywords:

Fixed point, best proximity point, contraction, graph, metric space, P-property.

Supporting agencies:

University Grants Commission

New Delhi

Abstract:

We establish an existence and uniqueness theorem on best proximity point for contractive mappings on a metric space endowed with a graph. As an application of this theorem, we obtain a result on the existence of unique best proximity point for uniformly locally contractive mappings. Moreover, our theorem subsumes and generalizes many recent  fixed point and best proximity point results.
Show more Show less

References:

T. Dinevari and M. Frigon, Fixed point results for multivalued contractions on a metric space with a graph, J. Math. Anal. Appl. 405 (2013), 507-517.

https://doi.org/10.1016/j.jmaa.2013.04.014

M. Edelstein, An extension of Banach's contraction principle, Proc. Amer. Math. Soc. 12 (1961), 7-10.

K. Fan, Extensions of two fixed point theorems of F. E. Browder, Math. Z. 122 (1969), 234-240.

https://doi.org/10.1007/BF01110225

J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (2008), 1359-1373.

https://doi.org/10.1090/S0002-9939-07-09110-1

W. K. Kim and K. H. Lee, Existence of best proximity pairs and equilibrium pairs, J. Math. Anal. Appl. 316 (2006), 433-446.

https://doi.org/10.1016/j.jmaa.2005.04.053

W. K. Kim, S. Kum and K. H. Lee, On general best proximity pairs and equilibrium pairs in free abstract economies, Nonlinear Anal. TMA 68 (2008), 2216-2227.

https://doi.org/10.1016/j.na.2007.01.057

W. A. Kirk, S. Reich and P. Veeramani, Proximinal retracts and best proximity pair theorems, Numer. Funct. Anal. Optim. 24 (2003), 851-862.

https://doi.org/10.1081/NFA-120026380

L. Máté, The Hutchinson-Barnsley theory for certain non-contraction mappings, Period. Math. Hungar. 27 (1993), 21-33.

https://doi.org/10.1007/BF01877158

J. J. Nieto and R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223-239.

https://doi.org/10.1007/s11083-005-9018-5

V. Pragadeeswarar and M. Marudai, Best proximity points: approximation and optimization in partially ordered metric spaces, Optim. Lett. 7 (2013), 1883-1892.

https://doi.org/10.1007/s11590-012-0529-x

V. Sankar Raj, Best proximity point theorems for non-self mappings, Fixed Point Theory 14 (2013), 447-454.

A. C. M. Ran and M. C. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.

https://doi.org/10.1090/S0002-9939-03-07220-4

A. Sultana and V. Vetrivel, Fixed points of Mizoguchi-Takahashi contraction on a metric space with a graph and applications, J. Math. Anal. Appl. 417 (2014), 336-344.

https://doi.org/10.1016/j.jmaa.2014.03.015

A. Sultana and V. Vetrivel, On the existence of best proximity points for generalized contractions, Appl. Gen. Topol. 15 (2014), 55-63.

https://doi.org/10.4995/agt.2014.2221

Show more Show less