A Liapunov functional for a singular integral equation

We consider a scalar integral equation x(t)=a(t)−∫0tC(t,s)g(s,x(s))ds where a∈L2[0,∞)a∈L2[0,∞), while C(t,s)C(t,s) has a significant singularity, but is convex when t−s>0t−s>0. We construct a Liapunov functional and show that g(t,x(t))−a(t)∈L2[0,∞)g(t,x(t))−a(t)∈L2[0,∞) and that x(t)−a(t)→0x(t)−a(t)→0 pointwise as t→∞t→∞. Small perturbations are also added to the kernel. In addition, we consider both infinite and finite delay problems. This paper offers a first step toward treating discontinuous kernels with Liapunov functionals.