Fixed point theorems for mixed monotone operators with perturbation and applications to fractional differential equation boundary value problems

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

“…Our results in this paper will extend and improve many known results in the field and in particular those in [2,7,21,27,28,32]. The rest of the paper is organized as follows.…”

“…Thereafter, many authors have investigated various kinds of nonlinear mixed monotone operators in Banach spaces such as nonlinear operators with concave-convex (see [4]), mixed monotone operators with α-concave-convex (see [33,34]), nonlinear operators with φ-concave-convex (see [10,26]), nonlinear operators with e-concave-convex (see [39]), and also obtained a lot of important results on mixed monotone operators (see [1,2,7,8,11,13,16,17,21,22,23,24,27,28,29,31,32,38]). These studies not only have theoretical significance but also have a wide range of applications in engineering, nuclear physics, biology, chemistry, technology, etc.…”

“…In [28], using the fixed point theorem of mixed monotone operators, Zhai and Hao studied the existence and uniqueness of positive solutions to the following operator equation in Banach spaces E A(x, x) + Bx = x, x ∈ P, where P is a cone in E and A : P × P → E is an α-concave mixed monotone operator, B : P → E is an increasing hypo-homogeneous operator.…”

Positive solutions for a class of fractional boundary value problems with fractional boundary conditions

“…Some examples are also presented in order to illustrate the obtained result. Our approach is based on a mixed monotone operator method introduced in [29].…”

Uniqueness of positive solutions of fractional boundary value problems with non-homogeneous integral boundary conditions

“…The existence of positive solutions of a BVP can be proved by constructing an associated operator and finding the fixed point of the operator. This idea has been widely applied to BVPs of integer order (for example, [4,11,13,15,18,22,25,27,28,29,33,35]), as well as to the study of fractional BVPs ( [1,5,14,17,20,23,26,39,41,43]). The construction of the associated operators is a key step in this approach.…”