Abstract
We examine the conditions of asymptotic stability of second-order linear dynamic equations on time scales. To establish asymptotic stability we prove the stability estimates by using integral representations of the solutions via asymptotic solutions, error estimates, and calculus on time scales.
Highlights
We examine the conditions of asymptotic stability of second-order linear dynamic equations on time scales
To establish asymptotic stability we prove the stability estimates by using integral representations of the solutions via asymptotic solutions, error estimates, and calculus on time scales
We examine asymptotic stability of second-order dynamic equation on a time scale T, L y(t) = y∇∇ + p(t)y∇(t) + q(t)y(t) = 0, t ∈ T, (1.1)
Summary
We examine asymptotic stability of second-order dynamic equation on a time scale T,. Exponential decay and stability of solutions of dynamic equations on time scales were investigated in recent papers [1, 5,6,7, 11, 12] using Lyapunov’s method. We use different approaches based on integral representations of solutions via asymptotic solutions and error estimates developed in [2, 8,9,10]. For t ∈ T we define the backward jump operator ρ : T → T by ρ(t) = sup{s ∈ T : s < t} ∀t ∈ T. The backward graininess function ν : T → [0,∞] is defined by ν(t) = t − ρ(t).
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