Since F is upper semicontinuous with compact values, Proposition 3.7 in [19] implies that F is closed, and so u ¯ ∈ F ( x ¯ ).
Since ϑ is compact (see Proposition 3.6), we can use the Leray Schauder alternative theorem (see Theorem 2.9) and find (widehat {u}in C) such that widehat {u}=vartheta (widehat {u}), so (widehat {u}in[0,eta]cap D_) is a solution of (1.1).
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The relative compactness of ({u}) in (L^{2}(0,T; L^{2}(Omega ))) follows from the result that (C^{(2)}) has a compact resolvent in Proposition 2.1.
(VIII) In 2006, Kum [23] applied the compact case of Proposition 5.1 to further develop the previous study [22] for a more general pseudomonotone operator.
so that, it follows from the fact that the immersion from into is compact, see [9, Proposition 3.7], Wirtinger's inequality [10, Corollary 3.2] and relation that is bounded in and, hence, is bounded in.
As R ( x, ⋅, ⋅ ), Q ( ⋅, y, ⋅ ) are closed for any ( x, y ) ∈ X × Y, and P is continuous with nonempty compact values, by Propositions 3.1, 3.3 of [14], A, B have open fibers.
Therefore, we have J ′, Ψ ′ are compact operators by [[31], Proposition 26.2].
Thus, (Psi^{prime}) is strongly continuous on X, which implies that (Psi^{prime}) is a compact operator by [44], Proposition 26.2.
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Then, is continuous in with respect to the operator norm ([15]) and also, defined by the relation (3.24) are compact operators on (see [6, Proposition 1]).
