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

Taoudi, M. A.; Salhi, N.; Ghribi, B.

Appl. Math. Comput. 216, No. 4, 1150-1157 (2010).

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)].

Reviewer: Mirosława Zima (Rzeszow)

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.

Keywords:

fixed point theorem; integral equations; measure of weak noncompactness; strict contractions; superposition operator; integrable solutions; Banach space; bounded linear operators; Nemytskii operator; nonlinear Urysohn operator

Citations:

Zbl 1128.45006

References:

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