scholarly journals Positive Solutions for Neutral Difference Equations with Continuous Arguments

2007 ◽  
Vol 14 (4) ◽  
pp. 699-710
Author(s):  
Xianyi Li
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Comparison Principle ◽  
Difference Equations ◽  
Linear Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Neutral Difference Equations ◽  
Necessary And Sufficient

Abstract Some “sharp” conditions are established for a kind of linear neutral difference equations with continuous arguments not to possess eventually positive solutions. The existence and asymptotic behavior are obtained for positive solutions of the kind of equations. The results for linear cases are further extended to nonlinear ones. A comparison principle, which is a necessary and sufficient condition, for linear equations not to possess eventually positive solutions is also presented.

The uniform exponential stability of a class of linear differential–difference equations in a Hilbert space

1981 ◽  
Vol 89 (3-4) ◽  
pp. 201-215 ◽  
Author(s):  
Richard Datko
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Exponential Stability ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Exponential Stability ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

scholarly journals A blow-up dichotomy for semilinear fractional heat equations

2020 ◽  
Author(s):  
Robert Laister ◽  
Mikołaj Sierżęga
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Positive Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Heat Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Whole Space ◽  
Necessary And Sufficient

Abstract We derive a blow-up dichotomy for positive solutions of fractional semilinear heat equations on the whole space. That is, within a certain class of convex source terms, we establish a necessary and sufficient condition on the source for all positive solutions to become unbounded in finite time. Moreover, we show that this condition is equivalent to blow-up of all positive solutions of a closely-related scalar ordinary differential equation.

5.—The Uniform Asymptotic Stability of Certain Linear Differential-difference Equations

1976 ◽  
Vol 74 ◽  
pp. 71-79
Author(s):  
R. Datko
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Asymptotic Stability ◽  
Linear Ordinary Differential Equations ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is developed for determination of the uniform stability of a class of non-autonomous linear differential-difference equations. This condition is the analogue of the Liapunov criterion for linear ordinary differential equations.

scholarly journals Existence of Ψ Bounded Solutions for a Lyapunov Matrix Dierential Equation

2014 ◽  
Vol 52 (2) ◽  
Author(s):  
Aurel Diamandescu
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Bounded Solution ◽  
Sufficient Condition ◽  
Matrix Differential Equation ◽  
Bounded Solutions ◽  
Necessary And Sufficient Condition ◽  
Lyapunov Matrix ◽  
Matrix Differential ◽  
Necessary And Sufficient

AbstractIt is proved a necessary and sufficient condition for the existence of at least one Ψ- bounded solution of a linear non- homogeneous Lyapunov matrix differential equation. In addition, it is given a result in connection with the asymptotic behavior of the Ψ- bounded solutions of this equation.

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