Zhenhai Liu

New exact solutions to the perturbed nonlinear Schrödinger’s equation with Kerr law nonlinearity

In this paper, by using the modified mapping method and the extended mapping method, we derive some new exact solutions of the perturbed nonlinear Schrodinger’s equation with Kerr law nonlinearity, which are the linear combination of two different Jacobi elliptic functions and we also consider the solutions in the limit cases.

On the Controllability of Impulsive Fractional Evolution Inclusions in Banach Spaces

In this paper, we deal with the controllability of a class of impulsive fractional evolution inclusions in Banach spaces. We establish some sufficient conditions of controllability by use of the well-known fixed point theorem for multivalued maps due to Dhage associated with an evolution system. At the end of the paper, a concrete application is given to illustrate our main results.

APPROXIMATE CONTROLLABILITY OF FRACTIONAL EVOLUTION SYSTEMS WITH RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVES ∗

In this paper, we deal with the control systems governed by fractional evolution differential equations involving Riemann--Liouville fractional derivatives in Banach spaces. Our main purpose in this article is to establish suitable assumptions to guarantee the existence and uniqueness results of mild solutions. Under these conditions, the approximate controllability of the associated fractional evolution systems involving Riemann--Liouville fractional derivatives is formulated and proved.

A class of variational–hemivariational inequalities of elliptic type

This paper is devoted to the existence of solutions for variational–hemivariational inequalities of elliptic type, with a higher order quasilinear principal part, at resonance as well as at nonresonance. The approach relies on the use of pseudomonotone operators. By means of the notion of Clarkes generalized gradient and the properties of the first eigenfunction of the quasilinear principal part, we also build a Landesman–Lazer theory in the nonsmooth framework of variational–hemivariational inequalities of elliptic type.

Nonlinear boundary value problems of fractional differential systems

In this paper, we consider the existence of generalized solutions for fractional differential systems with nonlinear boundary value conditions. We first establish a new comparison theorem. By applying the monotone iterative technique and the method of lower and upper generalized solutions, we obtain sufficient conditions for existence of extremal generalized solutions.

An inverse coefficient problem for a nonlinear parabolic variational inequality

This study is related to inverse coefficient problems for a nonlinear parabolic variational inequality with an unknown leading coefficient in the equation for the gradient of the solution. An inverse method, involving minimization of a least-squares cost functional, is developed to identify the unknown coefficient. It is proved that the solution of the corresponding direct problem depends continuously on the coefficient. On the basis of this, the existence of a quasisolution of the inverse problem is obtained in the appropriate class of admissible coefficients.

Iteration methods on sideways parabolic equations

We consider an inverse heat conduction problem (IHCP) in the quarter plane, where data are given at x = 1. The problem is called a sideways parabolic equation and is severely ill-posed. Numerical methods such as Tikhonov and Fourier regularization methods have been developed. However, they contain an a priori bound of the solution in their parameter choice. A large estimate bound may cause bad numerical results. In this paper, we introduce a new class of iteration methods to solve the IHCP and prove that our methods are of order optimal under both a priori and a posteriori stopping rules. An appropriate selection of a parameter in the iteration scheme helps to reduce the iterative steps and get a satisfactory approximate solution. Furthermore, if we use the discrepancy principle, we can avoid the selection of the a priori bound.

Generalized quasi-variational hemi-variational inequalities

This paper is devoted to the existence of solutions for generalized quasi-variational hemi-variational inequalities with multivalued, discontinuous pseudomonotone operators. We obtain results which generalize and extend previously known theorems.

EXISTENCE AND GLOBAL BIFURCATION OF PERIODIC SOLUTIONS TO A CLASS OF DIFFERENTIAL VARIATIONAL INEQUALITIES

In this paper, by using the topological degree theory for multivalued maps and the method of guiding functions, the existence and global bifurcation for periodic solutions of a class of differential variational inequalities are studied.

Anti-periodic fractional boundary value problems for nonlinear differential equations of fractional order

By using Schauder’s fixed point theorem and the contraction mapping principle, we discuss the existence of solutions for nonlinear fractional differential equations with fractional anti-periodic boundary conditions. Some examples are given to illustrate the main results.