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The Cauchy initial value problem in complete random normed modules
Author(s) -
Zhang Xia,
Zhang HaiLiang,
Wang BaoZheng,
Liu Ming
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6169
Subject(s) - mathematics , counterexample , lipschitz continuity , homomorphism , initial value problem , cauchy problem , bounded function , cauchy distribution , pure mathematics , discrete mathematics , mathematical analysis
We study, for the first time in the literature on the theory of random functional analysis, the Cauchy initial value problem in complete random normed modules. Under theL 0 ‐Lipschitz assumption on the solution, we prove that two kinds of Cauchy initial value problems with respect to almost surely boundedC 0semigroups of continuous module homomorphisms are well‐posed. Moreover, the counterexample also shows that it is necessary to require the almost sure boundedness for suchC 0semigroups.