# Coupled Coincidence Point and Fixed Point Results for Mixed Monotone Mappings and an Application to Integro-differential Equations

• Yifan Fan
• Youqing Shen
• Chuanxi Zhu
• Zhaoqi Wu
Article

## Abstract

In this paper, we propose some new coupled coincidence point and fixed point theorems for the mappings F and g. Our results are obtained by exploring the corresponding initial value problems for the mapping F and weakening the involved contractive conditions. As an application, we study the existence and uniqueness of solution to integro-differential equations.

## Keywords

Mixed-monotone mapping coincidence point fixed point existence–uniqueness integro-differential equation

## Notes

### Acknowledgements

The authors thank the editor and the referees for their constructive comments and suggestions. The research was supported by the National Natural Science Foundation of China (11361042, 11071108, 11461045, 11701259, 11771198), the Natural Science Foundation of Jiangxi Province of China (20132BAB201001, 20142BAB211016) and partly supported by the Innovation Program of the Graduate Student of Nanchang University (cx2016147).

## References

1. 1.
Guo, D., Lakshmikantham, V.: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11(5), 623–632 (1987)
2. 2.
Bhaskar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65(7), 1379–1393 (2006)
3. 3.
Zhai, C.B., Hao, M.R.: Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems. Nonlinear Anal. 75, 2542–2551 (2012)
4. 4.
Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially orderded sets and some applications to matrix equations. Pro. Am. Math. Soc. 132, 1435–1443 (2003)
5. 5.
Berzig, M., Samet, B.: An extension of coupled fixed point’s concept in higher dimension and applications. Comput. Math. Appl. 63, 1319–1334 (2012)
6. 6.
Berinde, V.: Coupled fixed point theorems for $$\phi$$-contractive mixed monotone mappings in partially ordered metric spaces. Nonlinear Anal. 75, 3218–3228 (2012)
7. 7.
Fan, Y.F., Zhu, C.X., Wu, Z.Q.: Some $$\varphi$$-coupled fixed point results via modified $$F$$-control function’s concept in metric spaces and its applications. J. Comput. Appl. Math. 349, 70–81 (2019)
8. 8.
Berinde, V.: Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl. 64, 1770–1777 (2012)
9. 9.
Zhu, C.X.: Research on some problems for nonlinear operators. Nonlinear Anal. 71, 4568–4571 (2009)
10. 10.
Zhu, C.X.: Generalizations of Krasnoselskii’s theorem and Petryshyn’s theorem. Appl. Math. Lett. 19, 628–632 (2006)
11. 11.
Lakshmikantham, V., Ćirić, L.: Coupled fixed point theorems in partially ordered metric spaces. Nonlinear Anal. 70, 4341–4349 (2009)
12. 12.
Radenović, S.: Coupled fixed point theorems for monotone mappings in partially ordered metric spaces. Kragujev. J. Math. 38(2), 249–257 (2014)
13. 13.
Radenović, S.: Some coupled coincidence points results of monotone mappings in partially ordered metric spaces. Int. J. Nonlinear Anal. Appl. 5(2), 174–184 (2014)
14. 14.
Đorić, D., Kadelburg, Z., Radenović, S.: Coupled fixed point results for mappings without mixed monotone property. Appl. Math. Lett. 25, 1803–1808 (2012)
15. 15.
Bisht, R.K., Pant, R.P.: A remark on discontinuity at fixed point. J. Math. Anal. Appl. 445, 1239–1242 (2017)
16. 16.
Đorić, D., Kadelburg, Z., Radenović, S., Kumam, P.: A note on fixed point results without monotone property in partially ordered metric spaces. RACSAM 108, 503–510 (2014)
17. 17.
Zhai, C.B., Zhang, L.L.: New fixed point theorems for mixed monotone operators and local existence-uniqueness of positive solutions for nonlinear boundary value problems. J. Math. Anal. Appl. 382, 594–614 (2011)
18. 18.
Radenović, S., Došenović, T.: A note on some recent fixed point results for cyclic contractions in b-metric spaces and an application to integral equations. Appl. Math. Comput. 273, 155–164 (2016)
19. 19.
Nashine, H., Altun, I.: New fixed point results for maps satisfying implicit relations on ordered metric spaces and application. Appl. Math. Comput. 240, 259–272 (2014)

© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Yifan Fan
• 1
Email author
• Youqing Shen
• 1
• Chuanxi Zhu
• 1
• Zhaoqi Wu
• 1
1. 1.Department of MathematicsNanchang UniversityNanchangPeople’s Republic of China