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

  • Yifan FanEmail author
  • Youqing Shen
  • Chuanxi Zhu
  • Zhaoqi Wu


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.


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



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).


  1. 1.
    Guo, D., Lakshmikantham, V.: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11(5), 623–632 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bhaskar, T.G., Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65(7), 1379–1393 (2006)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)CrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Berinde, V.: Coupled coincidence point theorems for mixed monotone nonlinear operators. Comput. Math. Appl. 64, 1770–1777 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Zhu, C.X.: Research on some problems for nonlinear operators. Nonlinear Anal. 71, 4568–4571 (2009)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Zhu, C.X.: Generalizations of Krasnoselskii’s theorem and Petryshyn’s theorem. Appl. Math. Lett. 19, 628–632 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Lakshmikantham, V., Ćirić, L.: Coupled fixed point theorems in partially ordered metric spaces. Nonlinear Anal. 70, 4341–4349 (2009)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Radenović, S.: Coupled fixed point theorems for monotone mappings in partially ordered metric spaces. Kragujev. J. Math. 38(2), 249–257 (2014)MathSciNetCrossRefGoogle Scholar
  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)zbMATHGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Bisht, R.K., Pant, R.P.: A remark on discontinuity at fixed point. J. Math. Anal. Appl. 445, 1239–1242 (2017)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetCrossRefGoogle Scholar
  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)MathSciNetzbMATHGoogle Scholar
  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)MathSciNetzbMATHGoogle Scholar

Copyright information

© 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

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