Resolvents and solutions of weakly singular linear Volterra integral equations

The structure of the resolvent R(t, s) for a weakly singular matrix function B(t, s) is determined, where B(t, s) is the kernel of the linear Volterra vector integral equation x(t) = a(t) + integral(t)(0) B(t, s)x(s) ds and a(t) is a given continuous vector function. Using contraction mappings in a Banach space of continuous vector functions with an exponentially weighted norm, we show that when B(t, s) satisfies certain integral conditions, R(t, s) has the form R(t, s) = B(t, s) + R(1)(t, s), where R(1)(t, s) is the unique continuous solution of the integral equation R(1)(t, s) = B(1)(t, s) + integral(t)(s) B(t, u) R(1)(u, s) du and B(1)(t, s) is defined by B(1)(t, s) := integral(t)(s) B(t, u)B(u, s) du. As examples, the formulas of resolvents for a couple of weakly singular kernels of practical interest are derived. We also obtain conditions under which a weakly singular integral equation (E) has a unique continuous solution x(t) and show that it can be expressed in terms of R(t, s) by x(t) = a(t) + integral(t)(0) R(t, s)a(s) ds. Finally, we show that there are parallel results for an alternative resolvent. (R) over tilde (t, s) and examine when it and R(t, s) are equivalent. (C) 2010 Elsevier Ltd. All rights reserved.