Existence results for nonlinear integral equations
The author gives sufficient conditions for the existence of a solution of the nonlinear Volterra integral equation \[ y(t)= h(t)+ \int^t_0 k(t, s) f(s, y(s)) ds, \qquad 0\leq t<T, \] and for the Hammerstein integral equation \[ y(t)= h(t)+ \int^1_0 k(t, s) f(s, y(s)) ds, \qquad 0\leq t\leq 1, \] using the Schauder-Tychonoff fixed-point theorem and a nonlinear alternative of Leray-Schauder type. For the Volterra equation both the case of a continuous solution and the case where the solution belongs locally to an \(L^p\)-space are considered and the solutions are not only local ones but guaranteed to exist on the whole interval \([0, T)\). The existence of solutions of the Hammerstein equation are obtained under assumptions (that in some cases are quite complicated) involving monotonicity of \(k\) and \(f\).
scientific article; zbMATH DE number 811158