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Positive $$L^p$$ solutions of Hammerstein integral equations. (English) Zbl 0981.45001
The authors establish the existence of positive $$L^p$$-solution of the Hammerstein integral equation $y(t)=h(t)+\int_0^T k(t,s)f(s,y(s)) ds \quad\text{for a.e. } t\in[0,T)$ where $$[0,T)$$ is either bounded or unbounded interval contained in $$R$$. First, for simplicity, the case when $$h\equiv 0$$ is considered. The function $$f$$ is a.e. positive, nondecreasing in view of the second variable, satisfies the classical Caratheódory’s assumptions and some kind of growth condition. The kernel $$k$$ is measurable and satisfies some regularity conditions.
The main tool in the proofs is Krasnoselskii’s fixed point theorem for completely continuous operators acting on a cone contained in a Banach space.
To illustrate the obtained results the authors consider the case $$f(t,y)=y^n$$, $$n\geq 0$$.

##### MSC:
 45G10 Other nonlinear integral equations 45M20 Positive solutions of integral equations 45G05 Singular nonlinear integral equations
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