# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On some new inequalities related to certain inequalities in the theory of differential equations. (English) Zbl 0824.26010
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. 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. 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. 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.

##### MSC:
 26D15 Inequalities for sums, series and integrals of real functions 45J05 Integro-ordinary differential equations 39A10 Additive difference equations 26D10 Inequalities involving derivatives, differential and integral operators 39A12 Discrete version of topics in analysis 45G10 Nonsingular nonlinear integral equations
Full Text: