Gronwall's Inequality - Integral Form With Locally Finite Measures

Integral Form With Locally Finite Measures

Let I denote an interval of the real line of the form or ) < ∞ for all t ∈ I (this is certainly satisfied when μ is a locally finite measure). Assume that u is integrable with respect to μ in the sense that

and that u satisfies the integral inequality

If, in addition,

  • the function α is non-negative or
  • the function t → μ is continuous for t ∈ I and the function α is integrable with respect to μ in the sense that

then u satisfies Grönwall's inequality

for all t ∈ I, where Is,t denotes to open interval (s, t).

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