Zhenlai Han

Remarks on the paper [Appl. Math. Comput. 207 (2009) 388-396]

In this paper, some sufficient conditions are established for the oscillation of second-order neutral differential equations(r(t)@j(x(t))|Z^(t)|^@a^-^1Z^(t))^+q(t)f(x(@s(t)))=0,t>=t0>0, where Z(t)=x(t)+p(t)x(t-@t) and @a>0,0==t0>0, where @!t0^~dtr(t)<~,0=

Positive solutions for eigenvalue problems of fractional differential equation with generalized p-Laplacian

In this paper, we investigate the existence of positive solutions for the eigenvalue problem of nonlinear fractional differential equation with generalized p-Laplacian operator D 0 + β ( ? ( D 0 + α u ( t ) ) ) = λ f ( u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 0 ) = u ( 1 ) = 0 , ? ( D 0 + α u ( 0 ) ) = ( ? ( D 0 + α u ( 1 ) ) ) = 0 , where 2 < α ≤ 3 , 1 < β ≤ 2 are real numbers, D 0 + α , D 0 + β are the standard Riemann-Liouville fractional derivatives, ? is a generalized p-Laplacian operator, λ 0 is a parameter, and f : ( 0 , + ∞ ) ? ( 0 , + ∞ ) is continuous. By using the properties of Green function and Guo-Krasnoselskii fixed-point theorem on cones, several new existence results of at least one or two positive solutions in terms of different eigenvalue interval are obtained. Moreover, the nonexistence of positive solution in term of the parameter λ is also considered.

Three-point boundary value problems of fractional functional differential equations with delay

AbstractIn this paper, we study three-point boundary value problems of the following fractional functional differential equations involving the Caputo fractional derivative:n n where DαC, DβC denote Caputo fractional derivatives, 2<α<3, 0<β<1, η∈(0,1), 1<λ<12η. We use the Green function to reformulate boundary value problems into an abstract operator equation. By means of the Schauder fixed point theorem and the Banach contraction principle, some existence results of solutions are obtained, respectively. As an application, some examples are presented to illustrate the main results.MSC:34A08, 34K37.

On forced oscillation of higher order functional differential equations

Abstract In this paper, we establish some new oscillation criteria for the following forced functional differential equation x ( n ) ( t ) + q ( t ) | x ( τ ( t ) ) | λ - 1 x ( τ ( t ) ) = e ( t ) , t ∈ [ t 0 , ∞ ) . No restriction is imposed on the forcing term e ( t ) to be the n th derivative of an oscillatory function. Two cases have been taken into consideration: (i) q ( t ) 0 , λ > 1 and τ ( t ) ⩽ t ( ⩾ t ) ; (ii) q ( t ) changes its sign, 0xa0 λ τ ( t ) ⩽ t ( ⩾ t ) . The main results generalize some existing results and can be applied to the oscillation problem that are not covered in the literature.

The Globally Exponentially Stability of Switched Impulsive Systems with Mixed Delays

This paper studies the exponential stability of a class of switched impulsive systems with mixed delays. Based on some impulsive delay differential inequalities, a sufficient condition for globally exponential stability of switched impulsive systems with mixed time delays is estabilished.

Order of Convergence Based on Discrepancy Principle for the Dynamical Systems Method

This paper investigates the order of convergence based on discrepancy principle for the dynamical systems method. Firstly, proposing the dynamical systems method, we estimate the order of convergence of solution based on discrepancy principle. Then, we prove order of convergence.

Existence of Solutions for Fractional Differential Equations Boundary Value Problem with Delay

In this paper, we study a class of fractional functional differential equation with delay. We use Green function to reformulate the boundary value problem into an abstract operator equation. The existence of positive solutions of the boundary value problem for this class of fractional functional differential equation is obtained by using fixed point theorems on cone.

Oscillation of Even Order Damped Neutral Delay Differential Equations with Mixed Nonlinearities

By means of Riccati transformation technique and inequalities, we study the even order damped neutral delay differential equations with mixed nonlinearities. Three sufficient conditions are obtained for oscillatory behavior of the equation which extend and improve some known results in literature.

⌽ 0 -stability for discrete hybrid system with initial time difference

In this paper, we consider the discrete hybrid system with different initial times. By employing variational cone-valued Lyapunov-like functions, we get some comparison theorems and several ⌽0-stability criteria for discrete hybrid system with different initial values.

Criteria on boundedness in terms of two measures for dynamic systems

In this paper, we consider the following dynamic systems on time scales x(σ(t)) = f(t, x<inf>t</inf>), xt<inf>0</inf> = x<inf>0</inf> y(σ(t)) = g(t, y<inf>t</inf>), yt<inf>0</inf> = y<inf>0</inf> where σ(t) > t. By employing the concept of upper monotone increasing function and Lyapunov method, we obtain a new comparison theorem first, on this basis, some criteria on the boundedness in terms of two measures for these dynamic systems are obtained.