Resolvents and solutions of singular Volterra integral equations with separable kernels in honor of Professor T. A. Burton for his seminal contributions to Liapunov theory for integral and fractional differential equations

摘要

The Volterra integral equation x(t)=a(t)+∫0tB(t,s)x(s)dswith a kernel of the form B(t,s)=p(t)q(s) is investigated, where a, p, and q are functions that are defined a.e. on an interval [0,T] and are measurable. The main result of this paper states that if qa is Lebesgue integrable on [0,T], the sign of B(t,s) does not change for almost all (t,s), and if there is a function f that is continuous on [0,T], except possibly at countably many points, with B(t,t)=f(t) a.e. on [0,T], then the function x defined by x(t):= a(t)+∫_0 ~tR(t,s)a(s)ds,where R(t,s):=B(t,s) e∫_s ~(tB(u,u)du),solves (1) a.e. on [0,T]. Three diverse examples illustrate the efficacy of using (2) and (3) to calculate solutions of (1). Two of the examples involve singular kernels: the solution of one of them is nowhere continuous on (0,T).