Smoothed integral equations (Q626178)

The authors consider a perturbed Volterra integral equation \[ z(t)=a(t)-\int_{0}^{t}B(t,s)\Bigl[z(s)+G(s,z(s))\Bigr]ds, \] where \(G(s,z(s))\) defines the perturbation function. For the unperturbed linear equation \(x(t)=a(t)-\int_{0}^{t}B(t,s)x(s)ds\), there is a resolvent equation \(R(t,s)=B(t,s)-\int_{s}^{t}B(t,u)R(u,s)du\) and a variation of parameters formula \(x(t)=a(t)-\int_{0}^{t}R(t,s)a(s)ds\). It is assumed that \(B\) is a perturbed convex and that \(a(t)\) may be badly behaved in several ways. When these equations are treated separately by means of a Lyapunov functional, restrictive conditions are required separately on \(a(t)\) and \(B(t,s)\). Here, the authors treat them as a single equation \(f(t)=S(t)-\int_{0}^{t}B(t,u)f(u)du\) where \(S\) is an integral combination of \(a(t)\) and \(B(t,s)\). There are two distinct advantages. First, the possibly bad behavior of \(a(t)\) is smoothed. Second, properties of \(S\) needed in the Lyapunov functional can be obtained from an array of properties of \(a(t)\) and \(B(t,s)\) yielding considerable flexibility not seen in standard treatment. The results are used to treat nonlinear perturbation problems. Moreover, the function \(y(t)=a(t)-\int_{0}^{t}B(t,s)a(s)ds\) is shown to converge pointwise in \(L^2[0;\infty)\) to \(x(t)\).