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Thus by (H4), we obtain the assertion of Lemma 4.3.

Applying Theorem 4.1, we obtain the assertion of Theorem 4.2.

Now, by Theorem 3.1, we obtain the assertion of this corollary.

Combining Theorems 2.1 and 2.2, and Lemma 2.3, we obtain the assertion of the theorem.

By using Lemma 1, we obtain the assertion of Theorem 1. □.

Summing up over ν, we obtain the assertion of the theorem.

Further, using the spectral decomposition of the operator A, from equalities (3.6) we obtain the assertions of the theorem.

From estimates (3.27 - 3.28) we obtain the assertion for problem (3.24).

We can assume that uλ is a strictly local minimizer of J ̃ λ in the X-topology, otherwise we have obtained the assertion of Theorem 1.2.

end{aligned} (2.6) Comparing the coefficients of (frac{z^{n}}{n!}) on both sides of (2.6), we obtain the assertion (2.3) of Theorem 2.3.

Inserting this into the right-hand side of (A.2), we obtain the assertion (4.5) of Theorem 4.3.