Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Karaj Branch‎, ‎Islamic Azad University‎, ‎Karaj‎, ‎Iran

2 Department of Mathematics‎, ‎Semnan University‎, ‎P‎. ‎O‎. ‎Box 35195-363‎, ‎Semnan‎, Iran

Abstract

‎In this paper‎, ‎first we introduce the notion of $\frac{1}{2}$-modular metric spaces and weak $(\alpha,\Theta)$-$\omega$-contractions in this spaces and we establish some results of best proximity points‎. ‎Finally‎, ‎as consequences of these theorems‎, ‎we derive best proximity point theorems in modular metric spaces endowed with a graph and in partially ordered metric spaces‎. ‎We present an example to illustrate the usability of these theorems‎.
 

Keywords

Main Subjects

[1] A. A. N. Abdou, M. A. Khamsi, On the fixed points of nonexpansive mappings in modular metric spaces, Fixed Point Theory Appl. (2013), 2013:229.
[2] V. V. Chistyakov, Modular metric spaces, I: Basic concepts, Nonlinear Anal. 72 (1) (2010), 1-14.
[3] V. V. Chistyakov, Modular metric spaces, II: Application to superposition operators, Nonlinear Anal. 72 (1) (2010), 15-30.
[4] L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. thesis, University of Freiburg, Germany, 2002.
[5] P. Harjulehto, P. Hst, M. Koskenoja, S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal. 25 (3) (2006), 205-222.
[6] J. Heinonen, T. Kilpelinen, O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, 1993.
[7] N. Hussain, M. A. Kutbi, P.Salimi, Best proximity point results for modified α-ψ-proximal rational contractions, Abstr. Appl. Anal. (2013), 2013:927457.
[8] N. Hussain, A. Latif, P. Salimi, Best proximity point results for modified Suzuki α-ψ-proximal contractions, Fixed Point Theory Appl. (2014), 2014:10.
[9] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 1 (136) (2008), 1359-1373.
[10] M. Jleli, B. Samet, A new generalization of the Banach contraction principle, J. Inequal. Appl. (2014), 2014:38.
[11] M. Jleli, B. Samet, Best proximity points for α-ψ-proximal contractive type mappings and applications, Bull. des Sciences Mathématiques., Doi:10.1016/j.bulsci.2013.02.003.
[12] M. A. Khamsi, W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley, New York, 2001.
[13] M. A. Khamsi, W. K. Kozlowski, S. Reich, Fixed point theory in modular function spaces, Nonlinear Anal. 14 (1990), 935-953.
[14] W. M. Kozlowski, Modular Function Spaces, Series of Monographs and Textbooks in Pure and Applied Mathematics, Dekker, New York/Basel, 1988.
[15] A. Latif, M. Hezarjaribi, Peyman Salimi and Nawab Hussain, Best proximity point theorems for α-ψ-proximal
contractions in intuitionistic fuzzy metric spaces, J. Inequal. Appl. (2014), 2014:352.
[16] C. Mongkolkeha, Y. J. Cho, P. Kumam, Best Proximity point for generalized proximal C-contraction mappings in metric spaces with partial orders, J. Inequal. Appl. (2013), 2013:94
[17] J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math, Springer, Berlin, 1983.
[18] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, 1950.
[19] W. Orlicz, Collected Papers, Part I, II, PWN Polish Scientific Publishers, Warsaw, 1988.
[20] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004), 1435-1443.
[21] S. Sadiq Basha, P. Veeramani, Best proximity point theorem on partially ordered sets, Optim. Lett. (2012), doi: 10.1007/s11590-012-0489-1
[22] B. Samet, C. Vetro, P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012), 2154-2165.