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Existence results for a generalized nonlinear Hammerstein equation on \(L_{1}\) spaces. (English) Zbl 1128.45006

The authors show the existence of at least one solution for the following nonlinear Hammerstein integral equation \[ \psi(t)=g(t,\psi(t))+\lambda\int_\Omega\zeta(t,s)f(s,\psi(s))\,ds \tag{1} \] in \(L^1(\Omega,X)\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) and \(X\) is a finite dimensional Banach space. Their assumptions are the following:
(i) \(g:\Omega\times X\rightarrow X\), is a contraction with respect to its second variable.
(ii) \(f:\Omega\times X\rightarrow Y\), is Caratheodory and its corresponding Nemitskii operator \(N_f\) maps \(L^1(\Omega,X)\) into \(L^1(\Omega,Y)\), where \(Y\) is another finite dimensional Banach space.
(iii) \(\zeta:\Omega\times\Omega\rightarrow L(Y,X)\), (where \(L(Y,X)\) is the space of bounded linear operators from \(Y\) into \(X\)), is strongly measurable and the linear operator \(B\) defined by \[ (B\psi)(t)=\int_\Omega\zeta(t,s)\psi(s)\,ds \] maps \(L^1(\Omega,X)\) into \(L^1(\Omega,Y)\).
(iv) \(\rho(t):\Omega\rightarrow L(Y,X)\), defined by \(\rho(t)(s)=\zeta(t,s)\) belongs to \(L^\infty(\Omega, L(Y,X))\).
(v) \(\alpha+\eta| \lambda| \,\|B\|<1\), where \(\alpha\) is the contraction constant of \(g\) and \(\eta\) is a constant related to the function \(f\).
The authors solve problem (1) using a variant of the Krasnosel’skii fixed-point theorem, concerning the sum of two operators \(A\) and \(B\). In particular, they replace the compactness of the operator \(A\) (assumed in the original version of the theorem) with a weak type of contractiveness for \(A+B\), involving the De Blasi measure of weak noncompactness.

MSC:

45G10 Other nonlinear integral equations
47H10 Fixed-point theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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