Necessary and Sufficient Conditions for Stability of Volterra Integro-dynamic Equation on Time Scales

(2) x∆(t) = G(t, x(s); 0 ≤ s ≤ t) := G(t, x(·)) on a time scale T that is unbounded above with 0 ∈ T, where x ∈ R andG : [0,∞)T× R 7→ R is a is rd-continuous function in t and x with G(t, 0) = 0. Throughout this paper, for each continuous function φ : [0, t0]T 7→ R there exists at least one continuous function x(t) = x(t, t0, φ) on an interval [t0, a]T such that it satisfies (2) for t ∈ [t0, a]T and x(t, t0, φ) = φ(t) for t ∈ [0, t0]T. For the existence and extendibility of solutions of (2) we refer the reader to [6] and to [14] for Volterra integral equations on time scales. When T = R, we refer the reader to [19], and [18] for results concerning boundedness of solutions of functional differential equations.