A Characterization of Semilinear Dense Range Operators and Applications

H. Leiva, N. Merentes and J. Sanchez

Abstract and Applied Analysis, 2013, vol. 2013, issue 1

Abstract: 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 : 0 0, 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.

Date: 2013
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https://doi.org/10.1155/2013/729093

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:729093

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