International Journal of Nonlinear Analysis and Applications

Relation theoretic results via simulation function with applications

Document Type : Research Paper

Authors

1 Department of Mathematics H. N. B. Garhwal University, BGR Campus Pauri Garhwal-246001 Uttarakhand, India

2 Goverment Degree College Thatyur (Tehri Garhwal) Uttarakhand, India

3 Department of Mathematics, H.N.B. Garhwal university, Pauri Campus, Uttarakhand

Abstract

We introduce Ciric type $ \mathcal{Z}_\mathcal{R} $-contraction to investigate the existence of a single fixed point under a binary relation. In the sequel, we demonstrate that a variety of contractions are obtained as consequences of our contraction. Also, we provide illustrative examples to demonstrate the significance of  Ciric type $\mathcal{Z}_\mathcal{R} $-contraction in the existence of fixed point for discontinuous map via binary relation. The paper is concluded by applications to solve an integral equation and a nonlinear matrix equation.

Keywords

[1] A. Alam and M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl. 17(4) (2015) 693–702.
[2] A. Alam, and M. Imdad, Relation-theoretic metrical coincidence theorems, Filomat 31(14) (2017) 4421–4439.
[3] A. Alam and M. Imdad, Nonlinear contractions in metric spaces under locally T-transitive binary relations, Fixed Point Theory 19 (2018) 13–24.
[4] L.A. Alnasera, D. Lateefa, H. A. Fouada and J. Ahmadc, Relation theoretic contraction results in F-metric spaces. J. Nonlinear Sci. Appl. 12 (2019) 337–344.
[5] S. Antal and U.C. Gairola, Generalized Suzuki type α-Z-contraction in b-metric space, J. Nonlinear Sci. Appl. 13 (2020) 212–222.
[6] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fund. Math. 3(1) (1922) 133–181.
[7] D.W. Boyd and J.S.W. Wong, Nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464.
[8] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Sci. 29 (9)(2002) 531–536.
[9] Lj B. Ciri´c, A generalization of Banach’s contraction principle spaces, Proc. Amer. Math. Soc. 45(2) (1974) 267–273.
[10] M.A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973) 604–608.
[11] W.L. De Koning, Infinite horizon optimal control of linear discrete-time systems with stochastic parameters, Automatica 18(4) (1982) 443–453.
[12] D. Khantwal and U.C. Gairola, An extension of Matkowski’s and Wardowski’s fixed point theorems with applications to functional equations, Aequationes Math. 93(2) (2019) 433–443.
[13] F. Khojasteh, S. Shukla and S. Radenovic, A new approach to the study of fixed point theorems via simulation functions, Filomat 29(6) (2015) 1189–1194.
[14] B. Kolman, R.C. Busby and S. Ross, Discrete Mathematical Structures, 3rd ed., PHI Pvt. Ltd., New Delhi, 2000.
[15] S. Lipschutz, Schaum’s Outlines of Theory and Problems of Set Theory and Related Topics, McGraw-Hill, New York, 1964.
[16] J.H. Long, X.Y. Hu and L. Zhang, On the Hermitian positive definite solution of the nonlinear matrix equation X + A∗x −1A + B∗X−1B = I, Bull. Braz. Math. Soc. 39(3) (2015) 317–386.
[17] R.D. Maddux, Relation algebras: Studies in Logic and the Foundations of Mathematics, Elsevier B. V., Amsterdam, 2006.
[18] J. Matkowski, Integrable solutions of functional equations, Dissertations Math. 127(68) (1975).
[19] M. Olgun, O. Bicer and T. Alyildiz, A new aspect to Picard operators with simulation functions, Turkish J. Math. 40(4) (2016) 832–837.
[20] S. Radenovic, F. Vetro and J. Vujakovic, An alternative and easy approach to fixed point results via simulation functions, Demonstr. Math. 50 (1) (2017), 223–230.
[21] A. Ran and M. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132(5) (2003) 1435–1443.
[22] B.E. Rhoades, Contractive definitions and continuity, Contemp. Math. 72 (1988) 233–245.
[23] B.E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Anal. (TMA) 47 (2001) 2683–2693.
[24] A.F. Rold´an-L´opez-de-Hierro, E. Karapinar, C. Rold´an-L´opez-de-Hierro and J. Mart´ınez-Moreno, Coincidence point theorems on metric spaces via simulation functions, J. Comput. Appl. Math. 275 (2015) 345–355.
[25] B. Samet and M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal. 13 (2012) 82–97.
[26] K. Sawangsup and W. Sintunavarat, On modified Z-contractions and an iterative scheme for solving nonlinear matrix equations, J. Fixed Point Theory Appl. 20(2) (2018).
[27] A. Tomar, M. Joshi, S.K. Padaliya, B. Joshi and A. Diwedi, Fixed point under set-valued relation-theoretic nonlinear contractions and application, Filomat 33(14) (2019) 4655–4664.
[28] A. Tomar and M. Joshi, Relation-theoretic nonlinear contractions in an F-metric space and applications, Rend. Circ. Mat. Palermo 2 (2020) 1–18.
[29] M. Turinici, Abstract comparison principles and multivariable Cornwall-Bellman inequalities, J. Math. Anal. Appl. 117(1) (1986) 100–127.
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 1
March 2022
Pages 1769-1783
  • Receive Date: 03 August 2020
  • Accept Date: 08 November 2021