Solutions of Integro-Differential Equations on the Half-Axis with Rapidly Decreasing Non-Difference Kernels (original) (raw)
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.
2012
In this article, we consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. In the first part, we deal with two-dimensional integralalgebraic equations. Next, we analyze Volterra integral equations of the first kind with a degenerate matrix-kernel on the diagonal. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.
Central European Journal of Mathematics, 2014
We consider systems of integral-algebraic and integro-differential equations with weakly singular kernels. Although these problem classes are not in the focus of the main stream literature, they are interesting, not only in their own right, but also because they may arise from the analysis of certain classes of differential-algebraic systems of partial differential equations. In the first part of the paper, we deal with two-dimensional integral-algebraic equations. Next, we analyze Volterra integral equations of the first kind in which the determinant of the kernel matrix k(t, x) vanishes when t = x. Finally, the third part of the work is devoted to the analysis of degenerate integro-differential systems. The aim of the paper is to specify conditions which are sufficient for the existence of a unique continuous solution to the above problems. Theoretical findings are illustrated by a number of examples.
On the solution of the integro-differential equation with an integral boundary condition
Numerical Algorithms, 2013
In this paper, we present an existence of solution for a functional integrodifferential equation with an integral boundary condition arising in chemical engineering, underground water flow and population dynamics, and other field of physics and mathematical chemistry. By using the techniques of noncompactness measures, we employ the basic fixed point theorems such as Darbo's theorem to obtain the mentioned aim in Banach algebra. Then this paper presents a powerful numerical approach based on Sinc approximation to solve the equation. Then convergence of this technique is discussed by preparing a theorem which shows exponential type convergence rate and guarantees the applicability of that. Finally, some numerical examples are given to confirm efficiency and accuracy of the numerical scheme.