Resolvents and solutions of weakly singular linear Volterra integral equations

The structure of the resolvent R(t,s)R(t,s) for a weakly singular matrix function B(t,s)B(t,s) is determined, where B(t,s)B(t,s) is the kernel of the linear Volterra vector integral equation equation(E )x(t)=a(t)+∫0tB(t,s)x(s)ds and a(t)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)B(t,s) satisfies certain integral conditions, R(t,s)R(t,s) has the form R(t,s)=B(t,s)+R1(t,s),R(t,s)=B(t,s)+R1(t,s), where R1(t,s)R1(t,s) is the unique continuous solution of the integral equation R1(t,s)=B1(t,s)+∫stB(t,u)R1(u,s)du and B1(t,s)B1(t,s) is defined by B1(t,s):=∫stB(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)x(t) and show that it can be expressed in terms of R(t,s)R(t,s) by x(t)=a(t)+∫0tR(t,s)a(s)ds. Finally, we show that there are parallel results for an alternative resolvent R˜(t,s) and examine when it and R(t,s)R(t,s) are equivalent.