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**On some new inequalities related to a certain inequality arising in the theory of differential equations.**
*(English)*
Zbl 0987.26010

The author obtains bounds on solutions to some nonlinear integral inequalities and their discrete analogues. Unfortunately, these integral inequalities (and hence their discrete analogues) are not new. Because most of them can be reduced to known results studied by the authors under the same (or even weaker) conditions. Indeed, letting \(z(t):= u(t)^p\), \(p> 0\), then the integral inequalities (2.1), (2.5), (2.19), (2.22) and (2.25) can be reformulated, respectively, as follows
\[
z(t)\leq a(t)+ b(t) \int^t_0 g(s)z(s) ds+ b(t) \int^t_0 h(s)u(s)^{1/p} ds,\quad 0< {1\over p}< 1,\tag{2.1\('\)}
\]

\[ z(t)\leq a(t)+ b(t) \int^t_0 h(s)z(s) ds+ b(t) \int^t_0 k(t,s)u(s)^{1/p} ds,\tag{2.5\('\)} \]

\[ z(t)\leq a(t)+ b(t) \int^t_0\widetilde f(s, z(s)) ds,\text{ with }\widetilde f(t,\xi):= f(t, \xi^{1/p}),\tag{2.19\('\)} \]

\[ z(t)\leq a(t)+ b(t) \phi\Biggl(\int^t_0 \widetilde f(s, z(s)) ds\Biggr),\text{ with }\widetilde f(t,\xi):= f(t, \xi^{1/p}),\tag{2.22\('\)} \] and \[ z(t)\leq a(t)+ b(t) \int^t_0 g(s) \widetilde W[z(s)] ds,\text{ with }\widetilde W(\xi):= W(\xi^{1/p}),\tag{2.25\('\)} \] where \(\widetilde f\), \(\widetilde W\) satisfy all the conditions assumed on the functions \(f\), \(W\), respectively. For example, when \(0< y\leq x\) the condition (2.18) holds for \(\widetilde f\) when \(m(t,y)\) is replaced by \(\widetilde m(t, y):={1\over p} y^{-1/p} m(t, y^{1/p})\) and condition (2.21) holds also when \(m(t,y)\) is replaced by \(\widetilde m(t,y):= \phi^{-1} [{1\over p} y^{-1/p}] m(t,y^{1/p})\).

\[ z(t)\leq a(t)+ b(t) \int^t_0 h(s)z(s) ds+ b(t) \int^t_0 k(t,s)u(s)^{1/p} ds,\tag{2.5\('\)} \]

\[ z(t)\leq a(t)+ b(t) \int^t_0\widetilde f(s, z(s)) ds,\text{ with }\widetilde f(t,\xi):= f(t, \xi^{1/p}),\tag{2.19\('\)} \]

\[ z(t)\leq a(t)+ b(t) \phi\Biggl(\int^t_0 \widetilde f(s, z(s)) ds\Biggr),\text{ with }\widetilde f(t,\xi):= f(t, \xi^{1/p}),\tag{2.22\('\)} \] and \[ z(t)\leq a(t)+ b(t) \int^t_0 g(s) \widetilde W[z(s)] ds,\text{ with }\widetilde W(\xi):= W(\xi^{1/p}),\tag{2.25\('\)} \] where \(\widetilde f\), \(\widetilde W\) satisfy all the conditions assumed on the functions \(f\), \(W\), respectively. For example, when \(0< y\leq x\) the condition (2.18) holds for \(\widetilde f\) when \(m(t,y)\) is replaced by \(\widetilde m(t, y):={1\over p} y^{-1/p} m(t, y^{1/p})\) and condition (2.21) holds also when \(m(t,y)\) is replaced by \(\widetilde m(t,y):= \phi^{-1} [{1\over p} y^{-1/p}] m(t,y^{1/p})\).

Reviewer: Yang En-Hao (Guangzhou)

### MSC:

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

39A12 | Discrete version of topics in analysis |

45D05 | Volterra integral equations |

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\textit{B. G. Pachpatte}, J. Math. Anal. Appl. 251, No. 2, 736--751 (2000; Zbl 0987.26010)

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### References:

[1] | Mitrinović, D.S., Analytic inequalities, (1970), Springer-Verlag Berlin/New York · Zbl 0199.38101 |

[2] | Pachpatte, B.G., Inequalities for differential and integral equations, (1998), Academic Press New York · Zbl 1032.26008 |

[3] | Pachpatte, B.G., On the discrete generalizations of Gronwall’s inequality, J. Indian math. soc., 37, 147-156, (1973) · Zbl 0331.26017 |

[4] | Pachpatte, B.G., Some new finite difference inequalities, Comput. math. appl., 28, 227-241, (1994) · Zbl 0809.26009 |

[5] | Pachpatte, B.G., On some discrete inequalities useful in the theory of certain partial finite difference equations, Ann. differential equations, 12, 1-12, (1996) · Zbl 0864.26008 |

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