Singular integral equations, Liapunov functionals, and resolvents

This paper, together with a recent paper by the second author on convex singular kernels, establishes a base for further investigation of mildly singular equations with Liapunov theory. We study the two nonlinear scalar integral equations x(t)=a(t)−∫0tD(t,s)[x(s)+G(s,x(s))]ds and z(t)=a(t)−∫0tD(t,s)g(s,z(s))ds where DD has a singularity at t=st=s. The first equation is decomposed into three other simpler equations. We then construct a Liapunov functional for each of the equations which will yield LpLp properties of the solutions.