Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal control

In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of φ-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main results.

Uniqueness of iterative positive solutions for the singular infinite-point p-Laplacian fractional differential system via sequential technique

By sequential techniques and mixed monotone operator, the uniqueness of positive solution for singular p-Laplacian fractional differential system with infinite-point boundary conditions is obtained. Green's function is derived, and some useful properties of Green' function are obtained. Based on these new properties, the existence of unique positive solutions is established, moreover, an iterative sequence and a convergence rate are given, which are important for practical application, and an example is given to demonstrate the validity of our main results.

Existence and Uniqueness of Positive Solutions for Singular Nonlinear Fractional Differential Equation via Mixed Monotone Operator Method

In this article, we discuss the existence and uniqueness of positive solution for a class of singular fractional differential equations, where the nonlinear term contains fractional derivative and an operator. By applying the fixed point theorem in cone, we get the existence and uniqueness of positive solutions for the fractional differential equation. Moreover, we give an example to demonstrate our main result.

Local existence and uniqueness of increasing positive solutions for non-singular and singular beam equation with a parameter

This paper is concerned with a class of beam equations with a parameter. By using the fixed point theorems of mixed monotone operator and the properties of cone, we study the non-singular and singular case, respectively, and obtain the sufficient conditions which guarantee the local existence and uniqueness of increasing positive solutions. Also, we present an iterative algorithm that converges to the solution. Moreover, we get some pleasant properties of the solutions for the boundary value problem dependent parameter. At last, two examples are given to illustrate the main results.

New results for a class of boundary value problems involving impulsive fractional differential equations

In this paper, a class of boundary value problems involving impulsive fractional differential equations is studied. By constructing Green?s function, a natural formula of solutions is derived. By applying fixed point theorems for mixed monotone operator with perturbation and sum operator, some new results on the existence and uniqueness of positive solutions are obtained, and an iterative sequence is constructed to approximate the positive solutions.

Some Operations on Mixed Monotone Operator in Banach Spaces

This paper discusses some operations on mixed monotone operator in Banach space, especially addition an multiplication operations. We will prove the sum and product of two mixed monotone operators. The proof using some relevant definitions. The result is the sum o of them is a mixed monotone operator and the product is too if both satisfy some conditions