### New results of mixed monotone operator equations

DOI: http://dx.doi.org/10.12775/TMNA.2019.003

#### Abstract

These mixed monotone operators and monotone operators are $e$-concave-convex operators and $e$-concave operators respectively.

Without using compactness or continuity, we obtain the existence and uniqueness of fixed points by monotone iterative techniques and properties of cones. Our main results extended and improved some existing results. Also, we applied the results to some differential equations.

#### Keywords

#### References

Y. Chen, The existence of a fixed point for the sum of two monotone operators, Positivity 12 (2008), 643–652.

D. Guo and V. Lakskmikantham, Coupled fixed points of nonlinear operators with applications, Nonlinear Anal. 11 (1987), no. 5, 623–632.

D. Guo, V. Lakskmikantham and X. Liu, Nonlinear integral equations in abstract spaces. Kluwer Acad. Publ. Dordrecht, 1996.

F. Li and Z. Liang, Fixed point of φ-concave(-φ-convex) operator and application, J. Systems Sci. Math. Sci. 14 (1994), no. 4, 355–360. (in Chinese)

J. Liu, F. Li and L. Lu, Fixed point and applications of mixed monotone operators with superlinear nonlinearity, Acta Math. Sci. Ser. A 23 (2003), no. 1, 19–24. (in Chinese)

L. Liu, X. Zhang, J. Jiang and Y. Wu, The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems, J. Nonlinear Sci. Appl. 9 (2016), 2943–2958.

H. Wang and L. Zhan, The solution for a class of sum operator equation and its application to fractional differential equation boundary value problems, J. Math. Anal. Appl. 203 (2015), DOI: 10.1186/s13661-015-0467-5.

X. Wu, New fixed point theorems and applications of mixed monotone operator, J. Math. Anal. Appl. 341 (2008), 883–893.

Y. Wu and Z. Liang, Existence and uniqueness of fixed points for mixed monotone operators with applications, Nonlinear Anal. 67 (2007), 2752–2762.

C. Zhai and DR. Anderson, A sum operator equation and applications to nonlinear elastic beam equations and Lane–Emden–Fowler equations, J. Math. Anal. Appl. 375 (2015), 388–400.

C. Zhai and M. Hao, Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems, Nonlinear Anal. 75 (2012), 2542–2551.

C. Zhai, C. Yang and C. Guo, Positive solutions of operator equation on ordered Banach spaces and applications, Comput. Math. Appl. 56 (2008), 3150–3156.

C. Zhai and L. Zhang, 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 (2011), 594–614.

X. Zhang, L. Liu and Y. Wu, Fixed point theorems for the sum of three classes of mixed monotone operators and applications, Fixed Point Theory Appl. 49 (2016), DOI: 10.1186/s13663-016-0533-4.

Z. Zhao, Uniqueness and existence of fixed points on some mixed monotone mappings in order linear spaces, J. Sysems Sci. Math. Scis. 19 (1999), no. 2, 217–224. (in Chinese)

Z. Zhao, Existence and uniqueness of fixed points for some mixed monotone operators, Nonlinear Anal. 73 (2010), 1481–1490.

Z. Zhao and X. Du, Fixed points of generalized e-concave (generalized e-convex) operators and their applications, J. Math. Anal. Appl. 334 (2007), 1426–1438.

### Refbacks

- There are currently no refbacks.