## Integrable solutions of a mixed type operator equation.(English)Zbl 1195.45017

Let $$\Omega$$ be a measurable subset of $$\mathbb{R}^N$$ and $$X$$ be a finite dimensional Banach space. The authors deal with the existence of solutions for the operator equation
$\psi(t)=g(t,\psi(t))+(BN_fUA\psi)(t),$
where $$A$$ and $$B$$ are bounded linear operators on $$L^1(\Omega,X)$$, $$N_f$$ is the Nemytskii operator and $$U$$ is the nonlinear Urysohn operator $$(U\psi)(t)=\int_{\Omega}u(t,s,\psi(s))\,ds$$. The proof of the existence result is based on the Krasnoselskii type theorem for the sum of two operators due to K. Latrach and M. A. Taoudi [Nonlinear Anal., Theory Methods Appl. 66, 2325–2333 (2007; Zbl 1128.45006)].

### MSC:

 45G10 Other nonlinear integral equations 45N05 Abstract integral equations, integral equations in abstract spaces 47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) 47H08 Measures of noncompactness and condensing mappings, $$K$$-set contractions, etc.

Zbl 1128.45006
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### References:

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