Topological Structure of the Solution Set for Evolution Inclusions by Yong Zhou Rong-Nian Wang & Li Peng
Author:Yong Zhou, Rong-Nian Wang & Li Peng
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore
(4.28)
Thus, in accordance with Definition 4.8, is the solution to (4.23) on [0, T] with equal to some Bochner integrable function such that for a.e. . Since has closed values, is also a solution to (4.5) on [0, T]. Moreover, taking into account Remark 4.8 (ii), we have that
for a.e. . By means of Theorem 4.6, we obtain that
(4.29)
for all .
Let be the maximal solution to the Cauchy problem , . Since , are bounded on [0, T], we can assume (without loss of generality) that is globally integrally bounded, so any solution of the equation can be continued up to a global one, i.e., defined on the whole interval. So, by (4.29) and Lemma 4.1, we obtain that for any . By Lemma 4.8, converges uniformly to 0 on [0, T], when . This completes the proof.
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