Qualitative and Quantitative Analysis of Nonlinear Systems by Michael Z. Zgurovsky & Pavlo O. Kasyanov

Author:Michael Z. Zgurovsky & Pavlo O. Kasyanov
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Repeating several lines from the proof of Theorem 5.1 we obtain that there exists such that for any and for each weak solution of Problem (5.24) on the inequality holds

(5.31)

for each

Let

for any . Let us provide the main convergence result for all weak solutions of Problem (5.24) in the strongest topologies.

Theorem 5.5

Let , weakly in H, for any . Then there exists a subsequence and such that

(5.32)

(5.33)

as for all .

Proof

The inequality (5.31), Kasyanov et al. [29, Theorem 3], Banach-Alaoglu theorem, and Cantor diagonal arguments (alternatively we may repeat several lines from the proof of Theorem 5.2) yield that there exist a subsequence and such that the following statements hold:

(a) the restrictions of and on belong to and ;

(b) the following convergence hold:

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