Existence results for nonlinear integral equations. (English) Zbl 0851.45003

O’Regan, Donal

J. Math. Anal. Appl. 192, No. 3, 705-726 (1995).

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 \(f\).

Reviewer: G.Gripenberg (Helsinki)

MSC:

45G10

Other nonlinear integral equations

Keywords:

Lp-solution; nonlinear Volterra integral equation; Hammerstein integral equation; continuous solution