Halanay Inequalities

Ravi Agarwal, Donal O’Regan and Samir Saker
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Ravi Agarwal: Texas A&M University–Kingsville, Department of Mathematics
Donal O’Regan: National University of Ireland, School of Mathematics, Statistics, and Applied Mathematics
Samir Saker: Mansoura University, Department of Mathematics

Chapter Chapter 5 in Dynamic Inequalities On Time Scales, 2014, pp 215-228 from Springer

Abstract: Abstract In 1966 Halanay [71] studied the stability of the delay differential equation x ′ ( t ) = − p x ( t ) + q x ( t − τ ) , τ > 0 , $$\displaystyle{ x^{{\prime}}(t) = -px(t) + qx(t-\tau ),\ \ \tau> 0, }$$ and proved that if f ′ ( t ) ≤ − α f ( t ) + β sup s ∈ t − τ , t f ( s ) f o r t ≥ t 0 $$\displaystyle{ f^{{\prime}}(t) \leq -\alpha f(t) +\beta \sup _{ s\in \left [t-\tau,t\right ]}f(s)\ \ for\ \ t \geq t_{0} }$$ and α > β > 0, then there exist γ > 0 and K > 0 such that f ( t ) ≤ K e − γ ( t − t 0 ) f o r t ≥ t 0 . $$\displaystyle{ f(t) \leq Ke^{-\gamma (t-t_{0})}\ \ for\ \ t \geq t_{ 0}. }$$

Keywords: Halanay-type Inequality; Delay Function; Shift Operator; Delay Construct; Inequality Dynamics (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/978-3-319-11002-8_5

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