Principal Matrix Solutions and Variation of Parameters for Volterra Integro-Dynamic Equations on Time Scales

Abstract

Abstract. We introduce the principal matrix solution Z(t, s) of the linear Volterratype vector integro-dynamic equationand prove that it is the unique matrix solution ofWe also show that the solution ofis unique and given by the variation of parameters formula2010 Mathematics Subject Classification. 34N05, 45D05, 39A13, 45J05.

Necessary and sufficient conditions for uniform stability of Volterra integro-dynamic equations using new resolvent equation

“…By a similar argument as in [1,Theorem 12], we conclude that for a given y 0 ∈ R n the unique solution of (6) satisfying y(t) = y 0 is…”

“…T is defined in [1]. The rest of the proof is identical to that of Lemma 7 and Lemma 8 of [1] and hence we omit.…”

“…In this section, we make use of the adjoint equation of (5) to show that the matrix solution Z(t, s) of (5), Adıvar proved its existence in [1], is equivalent to our resolvent R(t, s). That would put the existence of R(t, s) to rest.…”

“…a(t, s)x(s) ∆s, t ∈ [t 0 , ∞) T (1) and showed the existence of the resolvent equation equation r(t, s) of (1) that satisfies r(t, s) = −a(t, s) + t σ(t) r(t, u)a(u, s) ∆u.…”

“…In [1] the first author introduced the principal matrix solution Z(t, s) of the linear Volterra vector integro-dynamic equation …”