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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 $z\left(t\right):=u{\left(t\right)}^{p}$, $p>0$, then the integral inequalities (2.1), (2.5), (2.19), (2.22) and (2.25) can be reformulated, respectively, as follows

$z\left(t\right)\le a\left(t\right)+b\left(t\right){\int }_{0}^{t}g\left(s\right)z\left(s\right)ds+b\left(t\right){\int }_{0}^{t}h\left(s\right)u{\left(s\right)}^{1/p}ds,\phantom{\rule{1.em}{0ex}}0<\frac{1}{p}<1,\phantom{\rule{2.em}{0ex}}\left(2·{1}^{\text{'}}\right)$
$z\left(t\right)\le a\left(t\right)+b\left(t\right){\int }_{0}^{t}h\left(s\right)z\left(s\right)ds+b\left(t\right){\int }_{0}^{t}k\left(t,s\right)u{\left(s\right)}^{1/p}ds,\phantom{\rule{2.em}{0ex}}\left(2·{5}^{\text{'}}\right)$
$z\left(t\right)\le a\left(t\right)+b\left(t\right){\int }_{0}^{t}\stackrel{˜}{f}\left(s,z\left(s\right)\right)ds,\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\stackrel{˜}{f}\left(t,\xi \right):=f\left(t,{\xi }^{1/p}\right),\phantom{\rule{2.em}{0ex}}\left(2·{19}^{\text{'}}\right)$
$z\left(t\right)\le a\left(t\right)+b\left(t\right)\phi \left({\int }_{0}^{t}\stackrel{˜}{f}\left(s,z\left(s\right)\right)ds\right),\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\stackrel{˜}{f}\left(t,\xi \right):=f\left(t,{\xi }^{1/p}\right),\phantom{\rule{2.em}{0ex}}\left(2·{22}^{\text{'}}\right)$

and

$z\left(t\right)\le a\left(t\right)+b\left(t\right){\int }_{0}^{t}g\left(s\right)\stackrel{˜}{W}\left[z\left(s\right)\right]ds,\phantom{\rule{4.pt}{0ex}}\text{with}\phantom{\rule{4.pt}{0ex}}\stackrel{˜}{W}\left(\xi \right):=W\left({\xi }^{1/p}\right),\phantom{\rule{2.em}{0ex}}\left(2·{25}^{\text{'}}\right)$

where $\stackrel{˜}{f}$, $\stackrel{˜}{W}$ satisfy all the conditions assumed on the functions $f$, $W$, respectively. For example, when $0 the condition (2.18) holds for $\stackrel{˜}{f}$ when $m\left(t,y\right)$ is replaced by $\stackrel{˜}{m}\left(t,y\right):=\frac{1}{p}{y}^{-1/p}m\left(t,{y}^{1/p}\right)$ and condition (2.21) holds also when $m\left(t,y\right)$ is replaced by $\stackrel{˜}{m}\left(t,y\right):={\phi }^{-1}\left[\frac{1}{p}{y}^{-1/p}\right]m\left(t,{y}^{1/p}\right)$.

##### MSC:
 26D10 Inequalities involving derivatives, differential and integral operators 39A12 Discrete version of topics in analysis 45D05 Volterra integral equations