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Zbl 0987.26010
Pachpatte, B.G.
On some new inequalities related to a certain inequality arising in the theory of differential equations.
(English)
[J] J. Math. Anal. Appl. 251, No.2, 736-751 (2000). ISSN 0022-247X

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 $$z(t)\le a(t)+ b(t) \int^t_0 g(s)z(s) ds+ b(t) \int^t_0 h(s)u(s)^{1/p} ds,\quad 0< {1\over p}< 1,\tag{2.1'}$$ $$z(t)\le a(t)+ b(t) \int^t_0 h(s)z(s) ds+ b(t) \int^t_0 k(t,s)u(s)^{1/p} ds,\tag{2.5'}$$ $$z(t)\le a(t)+ b(t) \int^t_0\widetilde f(s, z(s)) ds,\text{ with }\widetilde f(t,\xi):= f(t, \xi^{1/p}),\tag{2.19'}$$ $$z(t)\le a(t)+ b(t) \phi\Biggl(\int^t_0 \widetilde f(s, z(s)) ds\Biggr),\text{ with }\widetilde f(t,\xi):= f(t, \xi^{1/p}),\tag{2.22'}$$ and $$z(t)\le a(t)+ b(t) \int^t_0 g(s) \widetilde W[z(s)] ds,\text{ with }\widetilde W(\xi):= W(\xi^{1/p}),\tag{2.25'}$$ where $\widetilde f$, $\widetilde W$ satisfy all the conditions assumed on the functions $f$, $W$, respectively. For example, when $0< y\le x$ 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 $\widetilde m(t,y):= \phi^{-1} [{1\over p} y^{-1/p}] m(t,y^{1/p})$.
[Yang En-Hao (Guangzhou)]
MSC 2000:
*26D10 Inequalities involving derivatives, diff. and integral operators
39A12 Discrete version of topics in analysis
45D05 Volterra integral equations

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

Cited in: Zbl 1168.26307

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