Mixed monotone method for a class of fractional q-difference boundary value problems

Abstract

In order to study the existence and uniqueness of nontrivial solutions for boundary value problems of a class of nonlinear fractional difference equations.Firstly, a new concept is defined on a new set. Then, by using the definition of normal cone, the existence of unique fixed points for two mixed monotone operators is established. Green functions for linear fractional-order boundary value problems are obtained, and the upper and lower bounds of Green functions are estimated. Thus, the expression of the particular solution can be obtained.Secondly, by using the abstract theorem, we discuss the non-linear terms which meet the conditions of the theorem, establish the existence of the unique solution of the above problem, and obtain the iteration sequence approaching the unique solution. Furthermore, the existence and uniqueness of nontrivial solutions for boundary value problems of fractional difference equations are proved. Finally, we give an example to illustrate the validity of the main theorems and results. The results show that the conditions of the theorem are proved and the nontrivial solution of the boundary value problem satisfies the existence and uniqueness. The research method has obtained good results in theoretical proof and boundary value problem, and has certain reference significance for other boundary value problems.

Keywords

• nonlinear partial differential equations
• fractional q-difference equation
• mixed monotone operator
• existence and uniqueness
• nontrivial solution