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Zbl 0824.26010
Pachpatte, B.G.
On some new inequalities related to certain inequalities in the theory of differential equations.
(English)
[J] J. Math. Anal. Appl. 189, No.1, 128-144 (1995). ISSN 0022-247X

The present article is devoted to new generalizations and their discrete analogues of a very useful integral inequality due to {\it L. Ou-Iang} [Shuxue Jinzhan 3, 409-415 (1957)]. Six theorems are contained in the paper. The first half of them are concerned with the continuous case and the theorems of the other half give discrete analogues of the integral inequalities obtained.\par The fundamental results for the continuous case are embodied in Theorem 1, which yields a priori bound on solutions to three special cases $(a\sb 1)$--$(a\sb 3)$ of the following integral inequality: $$u\sp 2(t)\le c\sp 2+ 2 \int\sp t\sb 0 u(s)\Biggl\{ f\sb 1(s) u(s)+ f\sb 2(s) \int\sp s\sb 0 g(m) u(m) dm+ f\sb 3(s)\Biggr\} ds,\tag A$$ where $t\in R\sb += [0, \infty)$, all functions involved are real-valued, non-negative and continuous functions and $c\ge 0$ is a constant.\par The above-mentioned special cases $(a\sb 1)$--$(a\sb 3)$ of (A) are: $f\sb 2(s)\equiv 0$, $f\sb 1(s)\equiv f\sb 2(s)$, and $f\sb 1(s)\equiv 0$, respectively.\par In Theorem 2, the next inequality is discussed $$u\sp 2(t)\le \Biggl( c\sp 2\sb 1+ 2 \int\sp t\sb 0 f(s) u(s) ds\Biggr) \Biggl( c\sp 2\sb 2+ 2\int\sp t\sb 0 h(s) u(s) ds\Biggr),\quad t\in R\sb +,$$ where $u$, $f$ and $h$ are real-valued, nonnegative and continuous functions on $R\sb +$, and $c\sb 1$, $c\sb 2$ are nonnegative constants. A special system consists of two $(a\sb 1)$-type like integral inequalities also contained in Theorem 2. The Theorem 3 yields some more general integral inequalities. Applications of Theorem 1 and Theorem 4 to certain integrodifferential and finite difference equations, respectively, are also indicated. The author claims that, all the inequalities in the paper (except $(a\sb 4)$ and $(b\sb 4)$) can be extended to functions of several variables.
[Yang En-Hao (Guangzhou)]
MSC 2000:
*26D15 Inequalities for sums, series and integrals of real functions
45J05 Integro-ordinary differential equations
39A10 Difference equations
26D10 Inequalities involving derivatives, diff. and integral operators
39A12 Discrete version of topics in analysis
45G10 Nonsingular nonlinear integral equations

Keywords: nonlinear integral inequalities; discrete analogues; bound on solutions

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