On exact rates of decay of solutions of linear systems of Volterra equations with delay

This paper considers the resolvent of a finite-dimensional linear convolution Volterra integral equation. The main results give conditions which ensure that the exact rate of decay of the resolvent can be determined using a positive weight function related to the kernel. The decay rates can be exponential or subexponential. Many other related results on exact rates of exponential and subexponential decay of solutions of Volterra integro-differential equations are given. We also present an application to a linear compartmental system with discrete and continuous lags.

[1]  J. Appleby,et al.  Subexponential solutions of linear integro-differential equations and transient renewal equations , 2002, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[2]  I. Györi.  Connections between compartmental systems with pipes and integro-differential equations , 1986 .

[3]  Olof J. Staffans,et al.  Volterra Integral and Functional Equations , 1990 .

[4]  J. Jacquez,et al.  Qualitative theory of compartmental systems with lags. , 2002, Mathematical biosciences.

[5]  István Győri.  Interaction between oscillations and global asymptotic stability in delay differential equations , 1990 .

[6]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[7]  Stephen Wainger,et al.  Variants of the Wiener-Levy Theorem, with Applications to Stability Problems for Some Volterra Integral Equations , 1975 .

[8]  I. M. Gelfand,et al.  Commutative Normed Rings. , 1967 .

[9]  Subexponential solutions of linear ItoVolterra equations with a damped perturbation , 2004 .

[10]  P. Ney,et al.  Functions of probability measures , 1973 .

[11]  István Győri.  STABILITY IN A CLASS OF INTEGRODIFFERENTIAL SYSTEMS , 1992 .

[12]  John A. D. Appleby,et al.  On the Non-Exponential Convergence of Asymptotically Stable Solutions of Linear Scalar Volterra Integro-Differential Equations , 2002 .

[13]  R. L. Wheeler,et al.  Weighted $L^1$-Remainder Theorems for Resolvents of Volterra Equations , 1980 .

[14]  R. Agarwal.  Recent Trends in Differential Equations , 1992 .

[15]  J. Eller,et al.  Compartmental systems with pipes. , 1981, Mathematical biosciences.

[16]  Subexponential solutions of scalar linear integro-differential equations with delay , 2004 .

[17]  V. Chistyakov.  A Theorem on Sums of Independent Positive Random Variables and Its Applications to Branching Random Processes , 1964 .