A New Lyapunov-Krasovskii Methodology for Coupled Delay Differential Difference Equations

2006 ◽  
Author(s):  
P. Pepe ◽  
Z.-P. Jiang ◽  
E. Fridman
Keyword(s):  
Difference Equations ◽  
Delay Differential

scholarly journals Nonoscillation of Second-Order Dynamic Equations with Several Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-34 ◽  
Author(s):  
Elena Braverman ◽  
Başak Karpuz
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Comparison Theorems ◽  
Nonoscillatory Solutions ◽  
Several Delays ◽  
Delay Differential ◽  
Delay Difference Equations ◽  
Delay Difference

Existence of nonoscillatory solutions for the second-order dynamic equation(A0xΔ)Δ(t)+∑i∈[1,n]ℕAi(t)x(αi(t))=0fort∈[t0,∞)Tis investigated in this paper. The results involve nonoscillation criteria in terms of relevant dynamic and generalized characteristic inequalities, comparison theorems, and explicit nonoscillation and oscillation conditions. This allows to obtain most known nonoscillation results for second-order delay differential equations in the caseA0(t)≡1fort∈[t0,∞)Rand for second-order nondelay difference equations (αi(t)=t+1fort∈[t0,∞)N). Moreover, the general results imply new nonoscillation tests for delay differential equations with arbitraryA0and for second-order delay difference equations. Known nonoscillation results for quantum scales can also be deduced.

scholarly journals General Solution of Linear Fractional Neutral Differential Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Hai Zhang ◽  
Jinde Cao ◽  
Wei Jiang
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Difference Equations ◽  
Laplace Transform Method ◽  
Transform Method ◽  
Linear Ordinary Differential Equations ◽  
Exponential Estimates ◽  
The Laplace Transform ◽  
Gronwall Integral Inequality ◽  
Delay Differential

This paper is concerned with the general solution of linear fractional neutral differential difference equations. The exponential estimates of the solution and the variation of constant formula for linear fractional neutral differential difference equations are derived by using the Gronwall integral inequality and the Laplace transform method, respectively. The obtained results extend the corresponding ones of integer order linear ordinary differential equations and delay differential equations.

Oscillation criteria for nonlinear neutral functional dynamic equations on time scales

2013 ◽  
Vol 63 (2) ◽  
Author(s):  
I. Kubiaczyk ◽  
S. Saker ◽  
A. Sikorska-Nowak
Keyword(s):  
Difference Equations ◽  
Dynamic Equation ◽  
Time Scale ◽  
Sufficient Conditions ◽  
Dynamic Equations ◽  
Neutral Delay ◽  
Dynamic Inequality ◽  
Delay Dynamic Equations ◽  
Delay Differential ◽  
Positive Functions

AbstractIn this paper, we establish some new sufficient conditions for oscillation of the second-order neutral functional dynamic equation $$\left[ {r\left( t \right)\left[ {m\left( t \right)y\left( t \right) + p\left( t \right)y\left( {\tau \left( t \right)} \right)} \right]^\Delta } \right]^\Delta + q\left( t \right)f\left( {y\left( {\delta \left( t \right)} \right)} \right) = 0$$ on a time scale $$\mathbb{T}$$ which is unbounded above, where m, p, q, r, T and δ are real valued rd-continuous positive functions defined on $$\mathbb{T}$$. The main investigation of the results depends on the Riccati substitutions and the analysis of the associated Riccati dynamic inequality. The results complement the oscillation results for neutral delay dynamic equations and improve some oscillation results for neutral delay differential and difference equations. Some examples illustrating our main results are given.

scholarly journals Solution behaviour in a class of difference–differential equations

1998 ◽  
Vol 57 (1) ◽  
pp. 37-48 ◽  
Author(s):  
A.D. Fedorenko ◽  
V.V. Fedorenko ◽  
A.F. Ivanov ◽  
A.N. Sharkovsky
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Delay Differential Equations ◽  
Neutral Type ◽  
Time Intervals ◽  
Small Parameters ◽  
Asymptotically Periodic ◽  
The Difference ◽  
Delay Differential ◽  
Solution Behaviour

Difference equations with piecewise continuous nonlinearities and their singular perturbations, first order neutral type delay differential equations with small parameters, are considered. Solutions of the difference equations are shown to be asymptotically periodic with period-adding bifurcations and bifurcations determined by Farey's rule taking place for periods and types of solutions. Solutions of the singularly perturbed delay differential equations are considered and compared with solutions of the difference equations within finite time intervals. The comparison is based on a continuous dependence of solutions on the singular parameter.

QUANTITATIVE PROPERTIES OF MEROMORPHIC SOLUTIONS TO SOME DIFFERENTIAL-DIFFERENCE EQUATIONS

2018 ◽  
Vol 99 (2) ◽  
pp. 250-261
Author(s):  
QIONG WANG ◽  
QI HAN ◽  
PEICHU HU
Keyword(s):  
Difference Equations ◽  
Meromorphic Solutions ◽  
Quantitative Properties ◽  
Delay Differential

We investigate several quantitative properties of entire and meromorphic solutions to some differential-difference equations and generalised delay differential-difference equations. Our results are sharp in a certain sense as illustrated by several examples.

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