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Applications of measures of weak noncompactness and some classes of operators in the theory of functional equations in the Lebesgue space. (English) Zbl 0894.47040

The author shows that, for a subset \(X\) of \(L_1\) which is compact in measure, the continuity, demicontinuity, and weak sequential continuity of \(T: X\to X\) are equivalent. This makes it possible to enlarge the applicability of fixed point theorems involving weakly condensing operators. The author also illustrates this by means of applications to functional equations and Uryson integral equations.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J05 Equations involving nonlinear operators (general)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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