and Applied Analysis 3 Definition 2.1 see 13 . A time scale T is an arbitrary nonempty closed subset of the real setR with the topology and ordering inherited from R. The forward and backward jump operators σ, ρ : T → T, and the graininess μ, ν : T → R are defined, respectively, by σ t : inf{s ∈ T : s > t}, ρ t : sup{s ∈ T : s < t}, μ t : σ t − t, ν t : t − ρ t . 2.1 The point t ∈ T is called l...

Keywords: Differential equations Difference equations Stability Bohl–Perron theorem Linear delay dynamic equations Time scales Uniform exponential stability a b s t r a c t This paper is concerned with the uniform exponential stability of ordinary and delay dynamic equations. After revealing the equivalence between various types of uniform exponential stability definitions on time scales with b...

The theory of dynamic equations appears in the literature in 1988 in the Ph. D. of S. Hilger [18]. The aim of this theory consists in to study differential and difference equations under the same formulation. Thus, the concepts of ∆– derivative and ∆−integral have been defined. These two concepts cover both the classical derivative and the Riemann integral together with the forward and sum oper...

Making use of the generalized time scales exponential function, we give a new definition for the exponential stability of solutions for dynamic equations on time scales. Employing Lyapunov-type functions on time scales, we investigate the boundedness and the exponential stability of solutions to first-order dynamic equations on time scales, and some sufficient conditions are obtained. Some exam...

In this paper we introduce the notion of an exponential dichotomy for not necessarily invertible linear dynamic equations in Banach spaces within the framework of the “Calculus on Measure Chains.” Particularly this unifies the corresponding theories for difference and differential equations. We apply our approach to obtain results on perturbed systems. 2000 Mathematics Subject Classification. 3...

The asymptotic behavior of x = px is explored, with specific reference given to how the graininess of the time scale affects stability. In addition we prove a Perron type theorem for dynamic equations on time scales. The theorem gives sufficient conditions for exponential asymptotic stability of a critical point of an almost linear dynamic equation. Application to the dynamic logistic equation ...