Yuan Gong Sun

Note on the paper of Džurina and Stavroulakis

Abstract In this paper we will establish some new oscillation criteria for the second-order retarded differential equation of the form ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + p ( t ) | u [ τ ( t ) ] | α - 1 u [ τ ( t ) ] = 0 . The results obtained essentially improve and extend those of Džurina and Stavroulakis [Oscillation criteria for second-order delay differential equations, Appl. Math. Comput., 140 (2003) 445–453]. An open problem is proposed at the end of this paper.

Positive periodic solutions of discrete three-level food-chain model of Holling type II

With the help of differential equations with piecewise constant arguments, we first derive a discrete analogy of continuous three level food-chain model of Holling type II, which is governed by difference equations with periodic coefficients. A set of sufficient conditions is derived for the existence of positive periodic solutions with strictly positive components by using the continuation theorem in coincidence degree theory. Particularly, the upper and lower bounds of the periodic solutions are also established.

On retarded integral inequalities in two independent variables and their applications

The object of this paper is to establish some nonlinear retarded integral inequalities in two independent variables which can be used as handy tools in the theory of partial differential and integral equations with time delays.

Oscillation of second-order delay differential equations with mixed nonlinearities

Abstract In this paper, we study the following second-order delay differential equation with mixed nonlinearities ( r ( t ) | u ′ ( t ) | α - 1 u ′ ( t ) ) ′ + q 0 ( t ) | u [ τ 0 ( t ) ] | α - 1 u [ τ 0 ( t ) ] + q 1 ( t ) | u [ τ 1 ( t ) ] | β - 1 u [ τ 1 ( t ) ] + q 2 ( t ) | u [ τ 2 ( t ) ] | γ - 1 u [ τ 2 ( t ) ] = 0 , where γ > α > β > 0 . Oscillation criteria for the equation are established which generalize the results by Džurina and Stavroulakis [J.D. Džurina, I.P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comput. 140 (2003) 44–453] and by Sun and Meng [Y.G. Sun, F.W. Meng, Note on the paper of Džurina and Stavroulakis, Appl. Math. Comput. 174 (2006) 1634–1641].

Oscillation of higher-order forced nonlinear differential equations

Abstract Some new criteria are established for the oscillation of higher-order forced nonlinear differential equations by introducing a class of new auxiliary functions. No restriction is imposed on the forcing term as is generally assumed. With the help of the new auxiliary functions, the main results in this paper are different from those in the paper [Y.G. Sun, S.H. Saker, Forced oscillation of higher-order nonlinear differential equations, Appl. Math. Comput. 173 (2006) 1219–1226] and are more effective than many existing results.

Nonoscillation and oscillation of second order half-linear difference equations

This paper considers the oscillation problem for the second order half-linear difference equation Δ(∣Δxn−1∣r−1Δxn−1) + qn∣xn∣r−1xn = 0, where {qn}n=0∞ is a nonnegative sequence and r > 0. we establish new criteria for this equation to be nonoscillatory and oscillatory, respectively. When r = 1, our results are complete extensions of work by Zhang and Zhou [Oscillation and nonoscillation for second-order linear difference equations, Comp. Math. Appl. 39 (2000) 1–7], and reduce to the results in [Y.G. Sun, Oscillation and nonoscillation for second-order linear difference equations, Appl. Math. Comput. 170 (2005) 1095–1103]. When r > 0, our results generalize and complement Sun’s results in [Y.G. Sun, Oscillation and nonoscillation for half-linear second order difference equations, Indian J. Pure Appl. Math. 35 (2004) 133–142].

LMI approach to stability for a competitive Lotka–Volterra system with time-varying delays

Abstract This paper considers the problem of local asymptotic stability for a competitive Lotka–Volterra system with time-varying delays. By employing a linear matrix inequality (LMI) approach, we not only prove that the local asymptotic stability of the positive equilibrium for the Lotka–Volterra type competitive system will be preserved for suitable delays under a well known condition, but also obtain the maximal allowable length of delays by using Matlab’s Control Systems Toolbox to solve a feasible LMI. Compared with some known results, our estimate on the length of delays is less conservative.