## 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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### References:

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