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6 Annales Polonici Mathematici

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Eventually positive solutions for nonlinear impulsive differential equations with delays

100%

Huang S. , Cheng S.

2012

tom 104

nr 1

43-70

Several recent oscillation criteria are obtained for nonlinear delay impulsive differential equations by relating them to linear delay impulsive differential equations or inequalities, and then comparison and oscillation criteria for the latter are applied. However, not all nonlinear delay impulsive differential equations can be directly related to linear delay impulsive differential equations or inequalities. Moreover, standard oscillation criteria for linear equations cannot be applied directly since continuous coefficient functions and initial functions are required. Therefore we establish oscillation criteria for linear or nonlinear impulsive equations with piecewise continuous coefficients and initial functions. Our technique is based on transforming our problem into a fixed point problem in Banach spaces, and then establishing comparison theorems. Our results extend, improve and correct some well known results in the literature.

Mean stability of a stochastic difference equation

100%

Ungureanu V. , Cheng S.

2008

tom 93

nr 1

33-52

A simple personal saving model with interest rate based on random fluctuation of national growth rate is considered. We establish connections between the mean stochastic stability of our model and the deterministic stability of related partial difference equations. Then the asymptotic behavior of our stochastic model is studied. Although the model is simple, the techniques for obtaining its properties are not, and we make use of the theory of abstract Banach algebras and weighted spaces. It is hoped that our study will lead to more realistic random models.

Linearized comparison criteria for a nonlinear neutral differential equation

100%

Guan X. , Cheng S.

nr 2

161-173

A class of nonlinear neutral differential equations with variable coefficients and delays is considered. Conditions for the existence of eventually positive solutions are obtained which extend some of the criteria existing in the literature. In particular, a linearized comparison theorem is obtained which establishes a connection between our nonlinear equations and a class of linear neutral equations with constant coefficients.

Oscillation of a logistic equation with delay and diffusion

100%

Xie S. , Cheng S.

1995

tom 62

nr 3

219-230

This paper establishes oscillation theorems for a class of functional parabolic equations which arises from logistic population models with delays and diffusion.

Periodic solutions of nth order delay Rayleigh equations

100%

Wang G.-Q. , Cheng S.

2002

tom 78

nr 3

261-266

A priori bounds are established for periodic solutions of an nth order Rayleigh equation with delay. From these bounds, existence theorems for periodic solutions are established by means of Mawhin's continuation theorem.

Existence and uniqueness of positive periodic solutions for a class of integral equations with parameters

80%

Kang S.-G. , Shi B. , Cheng S.

2009

tom 96

nr 3

227-238

Existence of periodic solutions of functional differential equations with parameters such as Nicholson's blowflies model call for the investigation of integral equations with parameters defined over spaces with periodic structures. In this paper, we study one such equation $ϕ(x) = λ∫_{[x,x+ω]∩Ω} K(x,y)h(y)f(y,ϕ(y-τ(y)))dy$, x ∈ Ω, by means of the proper value theory of operators in Banach spaces with cones. Existence, uniqueness and continuous dependence of proper solutions are established.