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On approximation of the perturbed inclusion. (English) Zbl 1130.45007

For a linear integral operator \(V:L^1([a,b],{\mathbb R}^n)\to C([a,b],{\mathbb R}^n)\) consider the inclusion \[ x\in\Psi(x)+V\Phi(x) \] where \(\Psi,\Phi\) are continuous with respect to the Hausdorff distance, \(\Psi\) assumes compact values, and \(\Phi\) is decomposable, i.e. \[ \Phi(x)=\{y:\text{\(y(t)\in\Delta(t,x)\) for a.a.\;\(t\)}\} \] for some function \(\Delta\) with values in the compact subsets of \({\mathbb R}^n\). Let the values \(\Psi(x)\) and \(\Phi(x)\) be enlarged by certain radii \(\varepsilon(x,\delta)\) and \(\eta(x,\delta)\), respectively (e.g.due to numerical perturbations). Necessary and sufficient conditions are discussed under which the solutions of the perturbed inclusion must tend to solutions of the original inclusion as \(\delta\to0\).

MSC:

45G15 Systems of nonlinear integral equations
45L05 Theoretical approximation of solutions to integral equations
47H04 Set-valued operators
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