Solutions of Integro-Differential Equations on the Half-Axis with Rapidly Decreasing Non-Difference Kernels (original) (raw)

Singular perturbations of integro-differential equations

Applied Mathematics and Computation, 2006

We study the singular perturbation problem (E) 2 u (t) + u (t) = Au (t) + (K * Au)(t) + f (t), t ≥ 0, > 0, for the integrodifferential equation (E) w (t) = Aw(t) + (K * Aw)(t) + f (t), t ≥ 0, in a Banach space, when → 0 +. Under the assumption that A is the generator of a strongly continuous cosine family and under some regularity conditions on the scalar-valued kernel K we show that problem (E) has a unique solution u (t) for each small > 0. Moreover u (t) converges to u(t) as → 0 + , the unique solution of equation (E).

Characterizations of linear Volterra integral equations with nonnegative kernels

Journal of Mathematical Analysis and Applications, 2007

We first introduce the notion of positive linear Volterra integral equations. Then, we offer a criterion for positive equations in terms of the resolvent. In particular, equations with nonnegative kernels are positive. Next, we obtain a variant of the Paley-Wiener theorem for equations of this class and its extension to perturbed equations. Furthermore, we get a Perron-Frobenius type theorem for linear Volterra integral equations with nonnegative kernels. Finally, we give a criterion for positivity of the initial function semigroup of linear Volterra integral equations and provide a necessary and sufficient condition for exponential stability of the semigroups.

Integro-differential equations of Volterra type

Bulletin of the Australian Mathematical Society, 1970

The aim of this paper is concerned with studying the stability properties of an integro-differential system "by reducing it into a scalar integro-differential equation. A theorem is stated about the existence of a maximal solution of such systems and a "basic result on integro-differential inequalities. Utilizing these results we obtain sufficient conditions for uniform asymptotic stability of the trivial solution of the integro-differential system of the form x'{t) = F(t, x(t), Ax) , V=jfc

On oscillation of integro-differential equations

TURKISH JOURNAL OF MATHEMATICS, 2018

We study the oscillatory behavior of solutions for integro-differential equations of the form x ′ (t) = e(t) − ∫ t 0 (t − s) α−1 k(t, s)f (s, x(s)) ds, t ≥ 0, where 0 < α < 1. Our method is based on the use of the beta function and asymptotic behavior of nonoscillatory solutions. An example is given to illustrate the main result. Equations of this form include Caputo type fractional differential equations, so the results are applicable to some fractional type differential equations as well.