On some new inequalities related to certain inequalities in the theory of differential equations.

*(English)*Zbl 0824.26010The present article is devoted to new generalizations and their discrete analogues of a very useful integral inequality due to 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 \((a_ 1)\)–\((a_ 3)\) of the following integral inequality: \[ u^ 2(t)\leq c^ 2+ 2 \int^ t_ 0 u(s)\Biggl\{ f_ 1(s) u(s)+ f_ 2(s) \int^ s_ 0 g(m) u(m) dm+ f_ 3(s)\Biggr\} ds,\tag{A} \] where \(t\in R_ += [0, \infty)\), all functions involved are real-valued, non-negative and continuous functions and \(c\geq 0\) is a constant.

The above-mentioned special cases \((a_ 1)\)–\((a_ 3)\) of (A) are: \(f_ 2(s)\equiv 0\), \(f_ 1(s)\equiv f_ 2(s)\), and \(f_ 1(s)\equiv 0\), respectively.

In Theorem 2, the next inequality is discussed \[ u^ 2(t)\leq \Biggl( c^ 2_ 1+ 2 \int^ t_ 0 f(s) u(s) ds\Biggr) \Biggl( c^ 2_ 2+ 2\int^ t_ 0 h(s) u(s) ds\Biggr),\quad t\in R_ +, \] where \(u\), \(f\) and \(h\) are real-valued, nonnegative and continuous functions on \(R_ +\), and \(c_ 1\), \(c_ 2\) are nonnegative constants. A special system consists of two \((a_ 1)\)-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 \((a_ 4)\) and \((b_ 4)\)) can be extended to functions of several variables.

The fundamental results for the continuous case are embodied in Theorem 1, which yields a priori bound on solutions to three special cases \((a_ 1)\)–\((a_ 3)\) of the following integral inequality: \[ u^ 2(t)\leq c^ 2+ 2 \int^ t_ 0 u(s)\Biggl\{ f_ 1(s) u(s)+ f_ 2(s) \int^ s_ 0 g(m) u(m) dm+ f_ 3(s)\Biggr\} ds,\tag{A} \] where \(t\in R_ += [0, \infty)\), all functions involved are real-valued, non-negative and continuous functions and \(c\geq 0\) is a constant.

The above-mentioned special cases \((a_ 1)\)–\((a_ 3)\) of (A) are: \(f_ 2(s)\equiv 0\), \(f_ 1(s)\equiv f_ 2(s)\), and \(f_ 1(s)\equiv 0\), respectively.

In Theorem 2, the next inequality is discussed \[ u^ 2(t)\leq \Biggl( c^ 2_ 1+ 2 \int^ t_ 0 f(s) u(s) ds\Biggr) \Biggl( c^ 2_ 2+ 2\int^ t_ 0 h(s) u(s) ds\Biggr),\quad t\in R_ +, \] where \(u\), \(f\) and \(h\) are real-valued, nonnegative and continuous functions on \(R_ +\), and \(c_ 1\), \(c_ 2\) are nonnegative constants. A special system consists of two \((a_ 1)\)-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 \((a_ 4)\) and \((b_ 4)\)) can be extended to functions of several variables.

Reviewer: Yang En-Hao (Guangzhou)

##### MSC:

26D15 | Inequalities for sums, series and integrals |

45J05 | Integro-ordinary differential equations |

39A10 | Additive difference equations |

26D10 | Inequalities involving derivatives and differential and integral operators |

39A12 | Discrete version of topics in analysis |

45G10 | Other nonlinear integral equations |