The unique solution of a class of sum mixed monotone operator equations and its application to fractional boundary value problems

Volume 9, Issue 5, pp 2943--2958 http://dx.doi.org/10.22436/jnsa.009.05.87

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Authors

Lishan Liu - School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, P. R. China. - Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia. Xinqiu Zhang - School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, P. R. China. Juan Jiang - School of Mathematical Sciences, Qufu Normal University, 273165, Qufu, P. R. China. Yonghong Wu - Department of Mathematics and Statistics, Curtin University, WA6845, Perth, Australia.

Abstract

In this paper we study a class of operator equations \(A(x; x) + B(x; x) = x\) in ordered Banach spaces, where A;B are two mixed monotone operators. Various theorems are established to guarantee the existence of a unique solution to the problem. In addition, associated iterative schemes have been established for finding the approximate solution converging to the fixed point of the problem. We also study the solution of the nonlinear eigenvalue equation \(A(x; x) + B(x; x) = \lambda x\) and discuss its dependency to the parameter. Our results extend and improve many known results in this field of study. We have also successfully demonstrated the application of our results to the study of nonlinear fractional differential equations with two-point boundary conditions.

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Keywords

• Mixed monotone operator
• hypo-homogeneous mixed monotone operator
• existence and uniqueness
• fractional differential equation.

MSC

•  47H07
•  47H14
•  26A33
•  34A08

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