Qualitative properties of solutions of integral equations

In this paper we study a linear integral equation x(t)=a(t)−∫0tC(t,s)x(s)ds in which the kernel fails to satisfy standard conditions yielding qualitative properties of solutions. Thus, we begin by following the standard idea of differentiation to obtain x′(t)=a′(t)−C(t,t)x(t)−∫0tCt(t,s)x(s)ds. The investigation frequently depends on x′(t)+C(t,t)x(t)=0x′(t)+C(t,t)x(t)=0 being uniformly asymptotically stable. When that property fails to hold, the investigator must turn to ad hoc methods. We show that there is a way out of this dilemma. We note that if C(t,t)C(t,t) is bounded, then for k>0k>0 the equation resulting from x′+kxx′+kx will have a uniformly asymptotically stable ODE part and the remainder can often be shown to be a harmless perturbation. The study is also continued to the pair x″+kx′x″+kx′.