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A fixed point approach in the study of the solution sets of Lipschitzian functional-differential inclusions. (English) Zbl 0735.34016

Given topological and Banach spaces \((\Lambda,{\mathcal T})\) and \((Z,\|\cdot\|)\) a set-valued function \(F: \Lambda\to N(Z)\) is said to have a retractive representation if there exist a set \(Y\in N(Z)\) and a continuous mapping \(f:\Lambda \times Y\to Z\) such that for every \((\lambda,y)\in \Lambda\times Y\) one has \(f(\lambda,y)\in F(\lambda)\) and \(f(\lambda,y)=y\) if and only if \(y\in F(\lambda)\). A retractive representation \(f\) is said to be equi-continuous, and when \(\Lambda\) is a uniform space, uniformly continuous or equi-uniformly continuous if the mappings \(f(\cdot,y)\), \(y\in Y\), are equi-continuous and uniformly or equi-uniformly continuous, respectively. The present paper contains sufficient conditions implying that the solution set mappings of some functional-differential inclusions in Banach spaces depending on parameters \(\lambda\in\Lambda\) have retractive and equi-uniformly continuous retractive representations. Here and in the present paper \(N(Z)\) denotes the family of all nonempty subsets of \(Z\).

MSC:

34A60 Ordinary differential inclusions
34K05 General theory of functional-differential equations
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