Coupled fixed point results on orthogonal metric spaces with application to nonlinear integral equations

In this article, we prove some well-known coupled fixed point theorems in 0-complete metric spaces. Also, we present some corollaries related to our study. In addition to this, we give an example showing that our results successfully obtain the existence and uniqueness of the coupled fixed point for 0-complete metric spaces, but the results are not valid for complete metric spaces. Finally, we apply our results to examine the existence and uniqueness of a solution of the system of nonlinear integral equations.

Keywords:

orthogonal set, coupled fixed point integral equations,

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[1] M. Abbas, M. Ali Khan and S. Radenovic, Common coupled fixed point theorems in cone metric spaces for w-compatible mappings, Appl. Math. Comput. 217, 195–202, 2010.
[2] Z. Ahmadi, R. Lashkaripour and H. Baghani, A fixed point problem with constraint inequalities via a contraction in incomplete metric spaces, Filomat 32(9), 3365–3379, 2018.
[3] I. Altun and H. Simsek, Some fixed point theorems on ordered metric spaces and application, Fixed Point Theory Appl. 2010, Article ID 621492, 2010.
[4] H. Baghani, R.P. Agarwal and E. Karapınar, On coincidence point and fixed point theorems for a general class of multivalued mappings in incomplete metric spaces with an application, Filomat 33 (14), 4493–4508, 2019.
[5] H. Baghani, M.E. Gordji and M. Ramezani, Orthogonal sets: The axiom of choice and proof of a fixed point theorem, J. Fixed Point Theory Appl. 18 (3), 465–477, 2016.
[6] H. Baghani and M. Ramezani, Coincidence and fixed points for multivalued mappings in incomplete metric spaces with application, Filomat 33, 13–26, 2019.
[7] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3, 133–181, 1922.
[8] T.G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Anal. 65, 1379–1393, 2006.
[9] Y.J. Cho, B.E. Rhoades, R. Saadati, B. Samet and W. Shatanawi, Nonlinear coupled fixed point theorems in ordered generalized metric spaces with integral type, Fixed Point Theory Appl. 2012 (8), 1–14, 2012.
[10] L.j. Círíc and V. Lakshmikantham, Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces, Stoch. Anal. Appl. 27, 1246–1259, 2009.
[11] M.E. Gordji, M. Rameani, M. De La Sen and Y.J. Cho, On orthogonal sets and Banach fixed point theorem, Fixed Point Theory 18, 569–578, 2017.
[12] D. Guo and V. Lakshmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11, 623–632, 1987.
[13] E. Karapınar, Coupled fixed point theorems for nonlinear contractions in cone metric spaces, Comput. Math. Appl. 59, 3656–3668, 2010.
[14] V. Lakshmikantham and L.j. Círíc, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal. 70, 4341–4349, 2009.
[15] N.V. Luong and N.X. Thuan, Coupled fixed points in partially ordered metric spaces and application, Nonlinear Anal. 74, 983–992, 2011.
[16] G. Mani, A.J. Gnanaprakasam, J.R. Lee and C. Park, Solution of integral equations via coupled fixed point theorems in F-complete metric spaces, Open Math. 19 (1), 1223–1230, 2021.
[17] A. Mutlu, N. Yolcu and B. Mutlu, Coupled fixed point theorem for mixed monotone mappings on partially ordered dislocated quasi metric spaces, Glob. J. Math. Anal. 1 (1), 12–17, 2015.
[18] A. Mutlu, K. Özkan and U. Gürdal, Coupled Fixed Point Theorems on Bipolar Metric Spaces, Eur. J. Pure Appl. Math. 10 (4), 655–667, 2017.
[19] A. Mutlu, K. Özkan and U. Gürdal, Coupled fixed point theorem in partially ordered modular metric spaces and its an application, J. Comput. Anal. Appl. 25 (2), 1–10, 2018.
[20] A. Petruşel, G. Petruşel, B. Samet and J.C. Yao, Coupled fixed point theorems for symmetric contractions in b-metric spaces with applications to operator equation systems, Fixed Point Theory 17 (2), 457–476, 2016.
[21] M. Ramezani and H. Baghani, The Meir-Keeler fixed point theorem in incomplete modular spaces with application, J. Fixed Point Theory Appl. 19 (4), 2369–2382, 2017.
[22] M. Ramezani, O. Ege and M. De la Sen, A New fixed point theorem and a new generalized Hyers-Ulam-Rassias stability in incomplete normed spaces, Mathematics 7 (11), 1117, 2019, doi:10.3390/math7111117.
[23] F. Sabetghadam, H.P. Masiha and A.H. Sanatpour, Some coupled fixed point theorems in cone metric spaces, Fixed Point Theory Appl. 2009, Article ID 125426, 2009, doi:10.1155/2009/125426.
[24] B. Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72, 4508–4517, 2010.
[25] K. Sawangsup, W. Sintunavarat and Y.J. Cho, Fixed point theorems for orthogonal F-contraction mappings on O-complete metric spaces, J. Fixed Point Theory Appl. 22 (1), Article number: 10, 2020.
[26] W. Shatanawi, E. Karapınar and H. Aydi, Coupled coincidence points in partially ordered cone metric spaces with a c-distance, J. Appl. Math. 2012, Article ID 312078, 2012.
[27] K. Özkan, Some coupled fixed point theorems for F-contraction mappings, J. Sci. Tech. 13 (13), 97–105, 2020.