International Journal of Nonlinear Analysis and Applications

Further investigation into the contractive condition in $\mathcal{D}$-generalized metric spaces with the transitivity

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna, Chiang Mai 50300, Thailand

2 Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University Rangsit Center, Pathum Thani 12120, Thailand

Abstract

In this paper, we present fixed point results in $\mathcal{D}$-generalized metric spaces endowed with a transitive relation that is not necessarily a partial order. We also give two examples with numerical results to support our main results while fixed-point results in the literature are not applicable. Moreover, we introduce some new notions for consideration of the multidimensional fixed point results in the $\mathcal{D}$-generalized metric spaces.

Keywords

[1] A. Alam and M. Imdad, Relation-theoretic contraction principle, Fixed point Theory Appl. 17 (2015), no. 4, 693–702.
[2] H. Ben-El-Mechaiekh, The Ran-Reurings fixed point theorem without partial order: A simple proof, J. Fixed Point Theory Appl. 16 (2014), 373–383.
[3] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65 (2006), 1379–1393.
[4] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. Theory Methods Appl. 11 (1987), 623–632.
[5] M. Jleli and B. Samet, A generalized metric space and related fixed point theorems, Fixed Point Theory Appl. 2015 (2015), no. 1.
[6] B. Kolman, R. C. Busby and S. Ross, Discrete Mathematical Structures, 3rd, Prentice-Hall, Inc., 1995.
[7] S. Lipschutz, Schaum’s Outlines of Theory and Problems of Set Theory and Related Topics, McGraw-Hill, New York, 1964.
[8] J.J. Nieto and R. Rodryguez-Lopez, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005), 223–239.
[9] J.J. Nieto and R. Rodryguez-Lopez, Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007), 2205–2212.
[10] A.C.M. Ran and 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.
[11] A.F. Roldan Lopez de Hierro, A unified version of Ran and Reuring’s theorem and Nieto and Rodrıguez-Lopez’s theorem and low-dimensional generalizations, Appl. Math. Inf. Sci. 10 (2016), no. 2, 383–393.
[12] B. Samet and C. Vetro, Coupled fixed point, f-invariant set and fixed point of N-order, Ann. Funct. Anal. 1 (2010), no. 2, 46–56.
[13] T. Senapati, L. Kanta Dey and D. Dolicanin-Dekic, Extensions of Circ and Wardowski type fixed point theorems in D-D-generalized metric spaces, Fixed point Theory Appl. 2016 (2016), no. 1, 1–14.
[14] M. Turinici, Ran-Reurings fixed point results in ordered metric spaces, Libertas Math. 31 (2011), 49–55.
[15] M. Turinici, Nieto-Lopez theorems in ordered metric spaces, Math. Student 81 (2012), 219–229.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 7
July 2023
Pages 331-344
  • Receive Date: 27 March 2021
  • Accept Date: 29 May 2021