Minimal conditions for implications of Gronwall-Bellman type

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• 作者：Martin Herdegen^a ; ¹ ; ^{M.Herdegen@warwick.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Sebastian Herrmann^b ; ² ; ^{sherrma@umich.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Gronwall&ndash ; Bellman inequalities ; Integral inequalities ; Semi-finite measures
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1654-1665
• 全文大小：369 K

文摘

Gronwall–Bellman type inequalities entail the following implication: if a sufficiently integrable function satisfies a certain homogeneous linear integral inequality, then it is nonpositive. We present a minimal (necessary and sufficient) condition on the Borel measure underlying the integrals for this implication to hold. The condition is also a necessary prerequisite for any nontrivial bound on solutions to inhomogeneous linear integral inequalities of Gronwall–Bellman type.