A Characterization of Semilinear Dense Range Operators and Applications

We characterize a broad class of semilinear dense range operators GH:W→Z given by the following formula, GHw=Gw+H(w), w∈W, where Z, W are Hilbert spaces, G∈L(W,Z), and H:W→Z is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator G to have dense...

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Bibliographic Details
Published in:Abstract and Applied Analysis 2013-01, Vol.2013 (2013), p.630-640-963
Main Authors: Leiva, Hugo, Merentes, Nelson, Sanchez, Jose Luis
Format: Article
Language:English
Subjects:
Approximation
Controllability
Evolution
Heat equations
Hilbert space
Linear systems
Mathematical analysis
Mathematical research
Nonlinearity
Operator theory
Operators
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Summary:We characterize a broad class of semilinear dense range operators GH:W→Z given by the following formula, GHw=Gw+H(w), w∈W, where Z, W are Hilbert spaces, G∈L(W,Z), and H:W→Z is a suitable nonlinear operator. First, we give a necessary and sufficient condition for the linear operator G to have dense range. Second, under some condition on the nonlinear term H, we prove the following statement: If Rang(G)¯=Z, then Rang(GH)¯=Z and for all z∈Z there exists a sequence {wα∈Z:00, where Z, U are Hilbert spaces, A:D(A)⊂Z→Z is the infinitesimal generator of strongly continuous compact semigroup {T(t)}t≥0 in Z,B∈L(U,Z), the control function u belongs to L2(0,τ;U), and F:[0,τ]×Z×U→Z is a suitable function. As a particular case we consider the controlled semilinear heat equation.
ISSN:1085-3375
1687-0409
DOI:10.1155/2013/729093