An existence theorem for Hammerstein integral equations.

*(English)*Zbl 0823.45004The 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”.

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 |