Ti-Jun Xiao

Nonlocal impulsive problems for nonlinear differential equations in Banach spaces

We study the existence and uniqueness of mild and classical solutions for a nonlinear impulsive differential equation with nonlocal conditions {u^(t)=Au(t)+f(t,u(t)),[emailxa0protected][emailxa0protected]?K,t ti,u(0)+g(u)=u0,@Du(ti)=Ii(u(ti)),i=1,2,...,p,0

Solutions to the Cauchy problem for differential equations in Banach spaces with fractional order

In this paper, we use the monotone iterative technique combined with cone theory to investigate the existence of solutions to the Cauchy problem for Caputo fractional differential equations in Banach spaces. New existence theorems are obtained for the case of a cone P being normal and fully regular respectively. Moreover, two examples are given to illustrate the abstract results.

Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations

This article is concerned with some properties of Stepanov-like almost automorphic (S p -a.a.) functions. We establish a composition theorem about S p -a.a. functions, and with its help, study the existence and uniqueness of almost automorphic solutions for semilinear evolution equations in Banach spaces. Moreover, integration and differentiation of S p -a.a. functions are discussed. Some theorems extend earlier results.

Solutions to Fractional Differential Equations with Nonlocal Initial Condition in Banach Spaces

A new existence and uniqueness theorem is given for solutions to differential equations involving the Caputo fractional derivative with nonlocal initial condition in Banach spaces. An application is also given.

Almost Automorphic Solutions to Abstract Fractional Differential Equations

A new and general existence and uniqueness theorem of almost automorphic solutions is obtained for the semilinear fractional differential equation , in complex Banach spaces, with Stepanov-like almost automorphic coefficients. Moreover, an application to a fractional relaxation-oscillation equation is given.

Weighted pseudo almost periodic functions and applications to evolution equations with delay

In this paper, we introduce a new class of vector-valued weighted pseudo almost periodic functions and present some basic properties about these new functions. Moreover, we obtain an new existence theorem of vector-valued weighted pseudo almost periodic solution to the evolution equations x^(t)=Ax(t)+f(t,xt) with delay in a Banach space X.

Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales

We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.

Positive almost automorphic solutions for a class of non-linear delay integral equations

This article is concerned with a class of non-linear delay integral equations, which unify some extensively studied delay integral equations. We establish a new existence and uniqueness theorem about positive almost automorphic solutions of the delay integral equations. Our theorem can deal with some cases to which many known results are not applicable. Two examples are given to illustrate our results.

Notes on Interpolation Inequalities

An easy proof of the John-Nirenberg inequality is provided by merely using the Calderón-Zygmund decomposition. Moreover, an interpolation inequality is presented with the help of the John-Nirenberg inequality.

Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions

We introduce and study some new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. General results for asymptotically strict pseudocontractions are established. The main result extends the previous results.