Fanwei Meng

Oscillation criteria for second order quasi-linear neutral delay differential equations

Abstract In this paper, some new oscillation criteria for second order neutral nonlinear differential equation [ r ( t ) | ( x ( t ) + p ( t ) x ( ( σ ( t ) ) ) ) ′ | m - 1 ( x ( t ) + p ( t ) x ( ( σ ( t ) ) ) ) ′ ] ′ + q ( t ) f ( x ( τ ( t ) ) ) = 0 are established. New oscillation criteria are different from most known ones in the sense that they are based on a class of new functions Φ ( t , s , l ) defined in the sequel.

Some new delay integral inequalities and their applications

The main objective of this paper is to establish some new delay integral inequalities, which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain delay differential equations and delay integral equations.

Oscillation and asymptotic analysis on a new generalized Emden–Fowler equation

Abstract In this work, we analyze the new generalized Emden–Fowler equation with neutral type delays: r ( t ) | Z ′ ( t ) | α - 1 Z ′ ( t ) ′ + q ( t ) | x ( σ ( t ) ) | β - 1 x ( σ ( t ) ) = 0 , where Z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) , α ⩾ β > 0 . By use of averaging technique and specific analytical skills, some easily-accessible oscillation and asymptotic criteria are established, which have extended the results in the cited literature.

Interval oscillation criteria for a forced second-order nonlinear differential equations with damping

Abstract In this paper, we are concerned with the oscillations in a class of forced second-order differential equations with nonlinear damping terms. By using classical variational principle and averaging technique, several new interval oscillation criteria for the equations are established, which improve and extend some known results. Example is also given to illustrate the results.

Some new Volterra-Fredholm type dynamic integral inequalities on time scales

In this paper, some new explicit bounds on solutions to a class of new Volterra-Fredholm type dynamic integral inequalities on time scales are established, which can be used as effective tools in the study of certain dynamic equations. Some applications for dynamic equations are also indicated.

Integral inequalities in two independent variables for retarded Volterra equations

In this paper, we obtain some retarded Volterra integral inequalities in two independent variables which can be used as tools in the theory of partial differential and integral equations with time delays.

Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments☆

Abstract Oscillation criteria for nth order neutral differential equations with deviating arguments of the form r ( t ) | ( x ( t ) + p ( t ) x ( t - τ ) ) ( n - 1 ) | α - 1 ( x ( t ) + p ( t ) x ( t - τ ) ) ( n - 1 ) ′ + q ( t ) f ( x ( σ ( t ) ) ) = 0 , t ⩾ t 0 , n even are established, where p , q ∈ C ( [ t 0 , ∞ ) , R ) , f ∈ C ( R , R ) and α > 0 , τ ⩾ 0 are constants. The results obtained essentially improve and extend those of Dzurina and Stavroulakis [J. Dzurina, I.P. Stavroulakis, Oscillation criteria for second-order delay differential equations, Appl. Math. Comp. 140 (2003) 445–453], Sun and Meng [Y.G. Sun, F.W. Meng, Note on the paper of Dzurina and Stavroulakis, Appl. Math. Comput. 174 (2006) 1634–1641], Xu and Meng [R. Xu, F. Meng, Some new oscillation criteria for second order quasi-linear neutral delay differential equations, Appl. Math. Comput. 182 (2006) 797–803].

Some new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales

In this paper, some new explicit bounds on solutions to a class of new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales are established, which can be used as effective tools in the study of certain dynamic equations. Some applications for dynamic equations are also indicated.

Oscillation results for linear Hamiltonian systems

For linear Hamiltonian systems this paper established some new oscillation criteria which improve previous results even for self-adjoint second order differential systems. The main tool used is a generalized Riccati transformation.

Integral average method for oscillation of second order partial differential equations with delays

Using a generalized Riccati transformation, some new oscillation criteria for second order partial differential equations with delays are found through the method of integral average. These results can be considered as generalizations and improvements of the results due to Kamenev in ordinary differential equations cases.