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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.

Holomorphic extension of locally holder functions

Rusev P.

Demonstratio Mathematica

2003

Vol. 36, nr 2

335--342

Let γ be a smooth Jordan curve in the extended complex plane passing trough the point at infinity. In the paper are given sufficiently conditions under which a complex function defined on γ admits a holomorphic extensions into a region complementary to γ.