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On Nemytskij operator in the space of absolutely continuous set-valued functions

Ludew J. J.

Journal of Applied Analysis

2011

Vol. 17, nr 2

277-290

We consider the Nemytskij operator, defined by (Nφ)(x) ? G(x, φ(x)), where G is a given set-valued function. It is shown that if N maps AC(I, C), the space of all absolutely continuous functions on the interval I ? [0, 1] with values in a cone C in a reflexive Banach space, into AC(I, K), the space of all absolutely continuous set-valued functions on I with values in the set K, consisting of all compact intervals (including degenerate ones) on the real line R, and N is uniformly continuous, then the generator G is of the form G(x, y) = A(x)(y) + B(x), where the function A(x) is additive and uniformly continuous for every x ∈ I and, moreover, the functions x ? A(x)(y) and B are absolutely continuous. Moreover, a condition, under which the Nemytskij operator maps the space AC(I, C) into AC(I, K) and is Lipschitzian, is given.

On Lipschitzian operators of substitution generated by set-valued functions

Ludew J. J.

Opuscula Mathematica

2007

Vol. 27, no. 1

13-24

We consider the Nemytskii operator, i.e., the operator of substitution, defined by (Nφ)(x) := G(x,φ(x)), where G is a given multifunction. It is shown that if N maps a Hölder space Hα into Hβ and N fulfils the Lipschitz condition then G(x,y) = A(x,y) + B(x), where A(x,·) is linear and A(·,y), B ∈ Hβ. Moreover, some conditions are given under which the Nemytskii operator generated by (1) maps Hα into Hβ and is Lipschitzian.