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On certain new integral inequalities and their applications. (English) Zbl 1026.26008

In this paper, the authors establish some integral inequalities in two independent variables. These inequalities complement the recent results obtained by B. G. Pachpatte [JIPAM, J. Inequal. Pure Appl. Math. 2, No. 2, Paper 15 (2001; Zbl 0989.26011)]. Specifically, four major results are obtained in this paper but we choose to state one of the results to convey the importance of the paper: Let $u\left(x,y\right),a\left(x,y\right),b\left(x,y\right),c\left(x,y\right),d\left(x,y\right)$ be nonnegative continuous functions defined for $x,y\in {ℝ}^{+}=\left[0,\infty \right)$ and $g:\left[0,\infty \right)\to \left[0,\infty \right)$ satisfies (i) $g\left(u\right)$ is positive, nondecreasing and continuous for $u\ge 0$, (ii) $\left(1/v\right)g\left(u\right)\le g\left(u/v\right)$, $u>0$, $v\ge 1·$ If the function $z\left(x,y\right)$ is defined by

$z\left(x,y\right)=a\left(x,y\right)+c\left(x,y\right){\int }_{0}^{x}{\int }_{y}^{\infty }d\left(s,t\right)u\left(s,t\right)dtds$

with $z\left(x,y\right)$ nondecreasing in $x$ and $z\left(x,y\right)\ge 1$ for $x,y\in {ℝ}^{+}$ and

$u\left(x,y\right)\le z\left(x,y\right)+{\int }_{\alpha }^{x}b\left(s,y\right)g\left(u\left(s,y\right)\right)ds,$

for $\alpha ,x,y,\in {ℝ}^{+}$ and $\alpha \le x$, then

$u\left(x,y\right)\le p\left(x,y\right)\left[a\left(x,y\right)+c\left(x,y\right)e\left(x,y\right)exp\left({\int }_{0}^{x}{\int }_{y}^{\infty }d\left(s,t\right)p\left(s,t\right)c\left(s,t\right)dtds\right)\right],$

where

$p\left(x,y\right)={G}^{-1}\left(G\left(1\right)+{\int }_{\alpha }^{x}b\left(s,y\right)ds\right),$
$e\left(x,y\right)={\int }_{0}^{x}{\int }_{y}^{\infty }d\left(s,t\right)p\left(s,t\right)a\left(s,t\right)dtds,$
$G\left(u\right)={\int }_{{u}_{0}}^{u}\frac{ds}{g\left(s\right)},\phantom{\rule{1.em}{0ex}}u\ge {u}_{0}>0,$

${G}^{-1}$ is the inverse function of $G$ and

$G\left(1\right)+{\int }_{\alpha }^{x}b\left(s,y\right)ds\in \text{Dom}\left({G}^{-1}\right)·$

Other inequalities obtained in this paper are similar to the above result and the methods of proof are also similar. Applications of these results in obtaining boundedness and uniqueness of solutions to some partial differential equations are also given.

##### MSC:
 26D10 Inequalities involving derivatives, differential and integral operators 26D15 Inequalities for sums, series and integrals of real functions