Inequalities useful for certain partial differential and difference equations (Q2757050)

The author obtains some estimation theorems for solutions of the following two-variable nonlinear integral inequality and its discrete analogue: NEWLINE\[NEWLINE\begin{multlined} u^p(x, y)\leq a(x,y)+\\ +b(x,y) \int^x_0 \int^y_0 [g(s,t) u^p(s,t)+ h(s,t) u(s,t) dt ds],\;x,y\in R_+:= [0,\infty),\end{multlined}\tag{\(*\)}NEWLINE\]NEWLINE where \(p> 1\) is a real constant and \(u(x,y)\), \(a(x,y)\), \(b(x,y)\), \(g(x,y)\), \(h(x,y)\in C(R_+\times R_+, R_+)\). We note here that, by using the simple substitution \(z(x,y)= u^p(x,y)\), the last inequality can be reformulated as a special case (when \(G(\xi):= \xi^{1/p}\)) of the next inequality NEWLINE\[NEWLINE\begin{multlined} z(x,y)\leq a(x,y)+\\ +b(x,y) \int^x_0 \int^y_0 [g(s,t) z(s,t)+ h(s,t) G(z(s,t)) dt ds],\;x,y\in R_+:= [0,\infty),\end{multlined}\tag{\(**\)}NEWLINE\]NEWLINE where \(G(\xi)\in C(R_+, R_+)\) is non-decreasing.NEWLINENEWLINENEWLINEAbove inequality \((**)\) had been discussed by many authors in the last two decades; for the main results related we refer to \textit{D. S. Mitrinović}, \textit{J. E. Pečarić} and \textit{A. M. Fink} [``Inequalities involving functions and their integrals and derivatives'' (1991; Zbl 0744.26011)]. For the corresponding discrete inequalities, we refer to \textit{R. P. Agarwal} [``Difference equations and inequalities'' (2000; Zbl 0952.39001)].