Singular integral equations, Liapunov functionals, and resolvents (Q765842)

The authors consider the two nonlinear scalar integral equations \[ x(t) = a(t) - \int_0^t D(t,s)(x(s) + G(s,x(s)))ds \tag{1} \] and \[ z(t) = a(t) - \int_0^t D(t,s)g(s, z(s))ds, \tag{2} \] where \(D(t,s)\) has a singularity at \(t=s\). They construct Lyapunov functionals for the equations (1) and (2). Using these functionals, properties of the solutions to the equations (1), (2) are investigated.