On some new integrodifferential inequalities of the Gronwall and Wendroff type. (English) Zbl 0796.26007

Cheung, Wing-Sum

J. Math. Anal. Appl. 178, No. 2, 438-449 (1993).

Applying bounds for the solution of the differential inequality \(\dot f(x) \leq k(x) f(x)\) in some class of real valued continuously differentiable functions, the author considers (in three theorems) Gronwall-Wendroff inequalities of the form \[ \dot y(x) \leq \sum^ n_{i=1} g_ i (x_ i) + \delta y(x)+ \int^ x_ 0p(s) \bigl[ \gamma y(s)+ \dot y(s) \bigr] ds \] (with \(\delta = 0\), \(\gamma=1\) in Theorem 1; \(\delta=M\) (constant), \(\gamma = 1\) in Theorem 2; \(\delta = M\), \(\gamma=0\) in Theorem 3), where \(y,p\) are nonnegative real valued functions, \(\dot y\) nonnegative and \(y(x) = 0\) if \(x_ i= 0\) for some \(i \in \{1, \dots, n\}\). In these theorems bounds for \(\dot y\) are obtained. Here \(x=(x_ 1, \dots, x_ n) \in R^ n\), \(\dot f(x) = \partial^ nf/ \partial x_ 1 \dots \partial x_ n\) (similarly \(\dot y)\), \(ds = ds_ 1\dots ds_ n\) and \(\int^ x_ 0 \dots ds\) denotes multiple integral.
The inequality considered in Theorem 4 can be found, e.g., in D. Bainov and P. Simeonov [Integral inequalities and applications (1992; Zbl 0759.26012); Theorem 12.9] and C.-C. Yeh [J. Math. Anal. Appl. 78, 78-87 (1980; Zbl 0458.26006)]. Two applications on boundedness and continuous dependence of some partial integrodifferential equations are added.

Reviewer: J.Popenda (Poznań)

MSC:

26D10	Inequalities involving derivatives and differential and integral operators
45K05	Integro-partial differential equations

Keywords:

integrodifferential inequalities; Gronwall-Wendroff inequalities; partial integrodifferential equations

Citations:

Zbl 0759.26012; Zbl 0458.26006