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JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS
 

Summary: JOURNAL OF INTEGRAL EQUATIONS
AND APPLICATIONS
Volume 14, Number 2, Summer 2002
ON THE NON-EXPONENTIAL CONVERGENCE
OF ASYMPTOTICALLY STABLE SOLUTIONS
OF LINEAR SCALAR VOLTERRA
INTEGRO-DIFFERENTIAL EQUATIONS
JOHN A.D. APPLEBY AND DAVID W. REYNOLDS
ABSTRACT. We study the stability of the scalar linear
Volterra equation
x (t) = -ax(t) +
t
0
k(t - s)x(s) ds, x(0) = x0
under the assumption that all solutions satisfy x(t) 0 as
t . It is shown that if k is a continuously differentiable,
positive, integrable function which is subexponential in the
sense that k (t)/k(t) 0 as t , then x(t) cannot converge
to 0 as t faster than k(t).
1. Introduction. In this note we consider the asymptotic stability

  

Source: Appleby, John - School of Mathematical Sciences, Dublin City University

 

Collections: Mathematics