Best proximity point theorems in 1/2−modular metric spaces
الموضوعات :H. Hosseini 1 , M. Eshaghi Gordji 2
1 - Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran
2 - Department of Mathematics, Semnan University, P. O. Box 35195-363, Semnan,
Iran
الکلمات المفتاحية: Best proximity point, (α, Θ)−ω-contractions, 1/2−modular metric space,
ملخص المقالة :
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.
[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.
