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)].


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
Full Text: DOI


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