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We obtain a new maximum principle of the periodic solutions when the corresponding impulsive equation is linear. If the nonlinear is quasi-monotonicity, we study the existence of the minimal and maximal periodic mild solutions for impulsive partial differential equations by using the perturbation method, the monotone iterative technique and the method of upper and lower solution. We give an example in last part to illustrate the main theorem.

In this paper, a generalized Gronwall inequality is demonstrated, playing an important role in the study of fractional differential equations. In addition, with the fixed-point theorem and the properties of Mittag–Leffler functions, some results of the existence as well as asymptotic stability of square-mean S-asymptotically periodic solutions to a fractional stochastic diffusion equation with fractional Brownian motion are obtained. In the end, an example of numerical simulation is given to illustrate the effectiveness of our theory results.

In this paper, we study the initial boundary value problem for a class of fourth order damped wave equations with arbitrary positive initial energy. In the framework of the energy method, we further exploit the properties of the Nehari functional. Finally, the global existence and finite time blow-up of solutions are obtained.

We investigate some initialboundary value problems for time-fractional diffusion equations of order (0,1). Such equations model anomalous diffusion on fractals. The existence of solution irrelevant to is established only if the external force function f is weighted Hlder continuous, which is weaker than Hlder continuous. Some interesting versions of maximal and spatial regularity criteria depending on the fractional exponent are also discussed.

We discuss the existence and regularity of solutions to some fractional evolution equations in the q-norm. The linear part generates a noncompact semigroup, and the nonlinear part satisfies some conditions with respect to the fractional power norm of the linear part. In the end, we apply the obtained results to a control system.

This paper deals with the existence and uniqueness of periodic solutions,S-asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.

We investigate the initial value problem for a class of fractional evolution equations in a Banach space. Under some monotone conditions and noncompactness measure conditions of the nonlinearity, the well-known monotone iterative technique is then extended for fractional evolution equations which provides computable monotone sequences that converge to the extremal solutions in a sector generated by upper and lower solutions. An example to illustrate the applications of the main results is given.

This paper deals with the existence of periodic solutions for the 2nth-order ordinary differential equation u(2n)(t)=f(t,u(t),u″(t),…,u(2n−2)(t)), where the nonlinear term f:R×Rn→R is a continuous odd function and f(t,x0,x1,…,xn−1) is 2π-periodic in t. Some existence results for odd 2π-periodic solutions are obtained under the condition that f satisfies some linear, superlinear or sublinear growth conditions on x0,x1,…,xn−1.

In this article, the theory of positive semigroup of operators and the monotone iterative technique are extended for the impulsive fractional evolution equations with nonlocal initial conditions. The existence results of extremal mild solutions are obtained. As an application that illustrates the abstract results, an example is given.

The periodic boundary value problem is discussed for a class of fractional evolution equations. The existence and uniqueness results of mild solutions for the associated linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. With the aid of the estimation, the existence and uniqueness results of positive mild solutions are obtained by using the monotone iterative technique. As an application that illustrates the abstract results, an example is given.

We study the periodic boundary value problem for semilinear fractional differential equations in an ordered Banach space. The method of upper and lower solutions is then extended. The results on the existence of minimal and maximal mild solutions are obtained by using the characteristics of positive operators semigroup and the monotone iterative scheme. The results are illustrated by means of a fractional parabolic partial differential equations.

In this article, the well-known monotone iterative technique is extended for impulsive fractional evolution equations. Under some monotone conditions and noncompactness measure conditions of the nonlinearity, some existence and uniqueness results are obtained. A generalized Gronwall inequality for fractional differential equation is also used. As an application that illustrates the abstract results, an example is given. 2000 MSC: 26A33; 34K30; 34K45.

This article is concerned with the existence of mild solutions to the initial value problem for a class of semilinear evolution equations with fractional order. New existence theorems are obtained by means of fixed point theorem for condensing maps. The results extend some related results in this direction.