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

##### MSC:
 45G10 Other nonlinear integral equations
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