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An existence theorem for Hammerstein integral equations. (English) Zbl 0823.45004
The author considers the following Hammerstein integral equation $x(t) = g(t) + \lambda \int_ D k(t,x) f(s,x(s)) ds, \tag{HIE}$ $$D$$ a compact subset of $$\mathbb{R}^ n$$, $$g,k,f$$ functions with values in finite- dimensional Banach spaces.
From the author’s introduction: “In a recent paper [J. Integral Equations Appl. 4, No. 1, 89-94 (1992; Zbl 0755.45005)] we have been able to dispense with all of these assumptions just assuming that $$k$$ and $$f$$ satisfy Caratheodory conditions; but, as observed by J. Banas, even if such a hypothesis is completely natural for $$f$$, it is sometimes restrictive when applied to $$k$$; for instance, if to some $$k(t,s) = p(t) q(s)$$ it implies the continuity of $$q$$, whereas requiring that $$q$$ belongs to some $$L^ r$$-space would be more natural.
Here we want to show that actually it is possible to have (HIE) under this more general hypothesis; indeed, we present a result in which we assume that $$f$$ is a Caratheodory function such that $$F$$ maps $$L^ 1 (D,X)$$ into $$L^ 2 (D,Y)$$, continuously, and $$k$$ is a measurable function such that the functions $$s \to k(t,s)$$ belong to $$L^ \infty$$ and $$K$$ is a linear, continuous operator from $$L^ 1 (D,Y)$$ into $$L^ 1 (D,X)$$, where $$X,Y$$ are finite-dimensional Banach spaces”.
Reviewer: U.Kosel (Freiberg)

##### MSC:
 45G10 Other nonlinear integral equations 45N05 Abstract integral equations, integral equations in abstract spaces
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