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

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.

### MSC:

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

### Citations:

Zbl 0759.26012; Zbl 0458.26006
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