Positive \(L^p\) solutions of Hammerstein integral equations. (English) Zbl 0981.45001

Meehan, Maria; O’Regan, Donal

Arch. Math. 76, No. 5, 366-376 (2001).

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 \(f(t,y)=y^n\), \(n\geq 0\).

Reviewer: Dariusz Bugajeweski (Poznań)

MSC:

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

Keywords:

positive solution; Hammerstein integral equation; Krasnoselskii’s fixed point theorem; cone; Banach space