A Note on the Unique Solution of the Integral Equations in the Framework of Fixed Point Theorem on Partially Ordered Metric Space

Youhua Qian^1, , Juan Wu¹ and Yafei Zhang¹

¹College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang, China

Cite this paper

Youhua Qian, Juan Wu and Yafei Zhang. A Note on the Unique Solution of the Integral Equations in the Framework of Fixed Point Theorem on Partially Ordered Metric Space. American Journal of Applied Mathematics and Statistics. 2016; 4(5):154-160. doi: 10.12691/AJAMS-4-5-3

Abstract

In this paper, we obtained the unique solution of the integral and coupled integral equation in the framework of fixed point theorem on partially ordered metric space. Our results unified some methods in studying the existence of unique solution for the integral equation. Moreover, all results are much more brief. In addition, the examples are given to illustrate the usability of the obtained results.

Keywords

coupled integral equations, fixed point theorems, partially ordered metric spaces

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References

[1]	W. Y. Feng and G. Zhang, New fixed point theorems on order intervals and their applications, Fixed Point Theory Appl. (2015), 2015: 218.
[2]	M. A. Khamsi, Generalized metric spaces: A survey, J. Fixed Point Theory Appl. 17. (2015), 455-475.
[3]	S. Radenovi´ c, T. Doˇ senovi´ c,T.A. Lampert and Z. Golubov´ ı´ c, A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equa-tions, Applied Mathematics and Computation . 273 (2016), 155-164.
[4]	M. Jleli and B. Samet, Best proximity point results for MK-proximal contractions on ordered sets, J. Fixed Point Theory Appl. 17 (2015), 439-452.
[5]	S. Sadiq Basha, Best proximity point theorems on partially ordered sets, Optim Lett. (2013), 7:1035-1043.
[6]	S. Sadiq Basha, Common best proximity points: global minimal solution, Top (2013), 21: 182-188.
[7]	S. Sadiq Basha, Common best proximity points: global minimization of multi-objective func-tions, J Glob Optim. (2012), 54: 367-373.
[8]	B. Samet, Coupled point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Analysis. 72 (2010), 4508-4517.
[9]	T. Gnana Bhaskar, V.Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Analysis 65 (2006), 1379-1393.
[10]	J. J. Nieto and R. Rodr´ ıguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22 (2005), 223-239.
[11]	A. C. 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.
[12]	T. G. Bhaskar and V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis. 65 (2006), 1379-1393.
[13]	Umit Aksoy, Erdal Karapinar and ` Inci M. Erhan Fixed points of generalized admissible contractions on b-metric spaces with an application to boundary value problems, (2016), 17(6), 1095-1108.
[14]	M. F. Bota, A. Petruşel, G. Petruşel, B. Samet. Coupled fixed point theorems for single-valued operators in b-metric spaces. Fixed Point Theory & Applications, (1)(2015):1-15.
[15]	W. Sintunavarat, Nonlinear integral equations with new admissibility types in b-metric spaces, Journal of Fixed Point Theory and Applications, (2015):1-20.