JinRong Wang

Nonlinear impulsive problems for fractional differential equations and Ulam stability

In this paper, the first purpose is treating Cauchy problems and boundary value problems for nonlinear impulsive differential equations with Caputo fractional derivative. We introduce the concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively. By using a new fixed point theorem, we obtain many new existence, uniqueness and data dependence results of solutions via some generalized singular Gronwall inequalities. The second purpose is discussing Ulam stability for impulsive fractional differential equations. Some new concepts in stability of impulsive fractional differential equations are offered from different perspectives. Some applications of our results are also provided.

On recent developments in the theory of boundary value problems for impulsive fractional differential equations

This paper is motivated from some recent papers treating the boundary value problems for impulsive fractional differential equations. We first make a counterexample to show that the formula of solutions in cited papers are incorrect. Second, we establish a general framework to find the solutions for impulsive fractional boundary value problems, which will provide an effective way to deal with such problems. Third, some sufficient conditions for the existence of the solutions are established by applying fixed point methods. Meanwhile, data dependence is obtained by using a new generalized singular Gronwall inequality. Finally, three examples are given to illustrate the results.

Nonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls

This paper investigates nonlocal problems for a class of fractional integrodifferential equations via fractional operators and optimal controls in Banach spaces. By using the fractional calculus, Holder inequality, p-mean continuity and fixed point theorems, some existence results of mild solutions are obtained under the two cases of the semigroup T(t), the nonlinear terms f and h, and the nonlocal item g. Then, the existence conditions of optimal pairs of systems governed by a fractional integrodifferential equation with nonlocal conditions are presented. Finally, an example is given to illustrate the effectiveness of the results obtained.

A survey on impulsive fractional differential equations

Abstract Recently, in series of papers we have proposed different concepts of solutions of impulsive fractional differential equations (IFDE). This paper is a survey of our main results about IFDE. We present several types of such equations with various boundary value conditions as well. Concept of solutions, existence results and examples are presented. Proofs are only sketched.

Controllability of Fractional Functional Evolution Equations of Sobolev Type via Characteristic Solution Operators

The paper is concerned with the controllability of fractional functional evolution equations of Sobolev type in Banach spaces. With the help of two new characteristic solution operators and their properties, such as boundedness and compactness, we present the controllability results corresponding to two admissible control sets via the well-known Schauder fixed point theorem. Finally, an example is given to illustrate our theoretical results.

Nonlocal impulsive fractional differential inclusions with fractional sectorial operators on Banach spaces

This paper investigates existence of PC-mild solutions of impulsive fractional differential inclusions with nonlocal conditions when the linear part is a fractional sectorial operators like in Bajlekova (2001) 1 on Banach spaces. We derive two existence results of PC-mild solutions when the values of the semilinear term F is convex as well as another existence result when its values are nonconvex. Further, the compactness of the set of solutions is characterized.

On the Solvability and Optimal Controls of Fractional Integrodifferential Evolution Systems with Infinite Delay

In this paper, we study the solvability and optimal controls of a class of fractional integrodifferential evolution systems with infinite delay in Banach spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and continuous dependence of mild solutions are investigated by utilizing the techniques of a priori estimation and extension of step by steps. Finally, existence of optimal controls for system governed by fractional integrodifferential evolution systems with infinite delay is proved.

Hermite–Hadamard-type inequalities for Riemann–Liouville fractional integrals via two kinds of convexity

In this article, two fundamental integral identities including the second-order derivatives of a given function via Riemann–Liouville fractional integrals are established. With the help of these two fractional-type integral identities, all kinds of Hermite–Hadamard-type inequalities involving left-sided and right-sided Riemann–Liouville fractional integrals for m-convex and (s, m)-convex functions, respectively. Our methods considered here may be a stimulant for further investigations concerning Hermite–Hadamard-type inequalities involving Hadamard fractional integrals.

Nonlocal Controllability of Semilinear Dynamic Systems with Fractional Derivative in Banach Spaces

In this paper, we establish two sufficient conditions for nonlocal controllability for fractional evolution systems. Since there is no compactness of characteristic solution operators, our theorems guarantee the effectiveness of controllability results under some weakly compactness conditions.

Optimal feedback control for semilinear fractional evolution equations in Banach spaces

Abstract In this paper, we study optimal feedback controls of a system governed by semilinear fractional evolution equations via a compact semigroup in Banach spaces. By using the Cesari property, the Fillippove theorem and extending the earlier work on fractional evolution equations, we prove the existence of feasible pairs. An existence result of optimal control pairs for the Lagrange problem is presented.