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Comparison theorems related to a certain inequality used in the theory of differential equations. (English) Zbl 0867.26015
Let $c\ge 0$ be a constant, $y, f, g\in C(R_+,R_+)$, and $w(t,r)\in C(R_+\times R_+,R)$ be nondecreasing in $r$ for every $t$ fixed. The main result of the paper can be restated as follows: Theorem 1. The nonlinear integral inequality of Ou-Iang type $$y^2(t)\le c^2+2\int^t_0y(s)Q(\cdot)ds,\quad t\in R_+,\tag1$$ implies $$y(t)\le K(t)r(t),\quad t\in R_+,\tag2$$ with $r(t)$ being the maximal solution of the initial problem $$r'(t)=w(t,K(t)r(t)),\quad t\in R_+;\quad r(0)=c,\tag3$$ provided that $Q(\cdot)$ is given by $$Q(\cdot)=w(s,y(s)); \quad f(s)y(s)+w(s,y(s)); \quad f(s)\Bigl(y(s)+\int^s_0g(m)y(m)dm\Bigr);$$ and $$f(s)\Bigl(w(s,y(s))+\int^s_0g(m)y(m)dm\Bigr),$$ and $K(t)$ is defined by $$K(t)=1; \quad \exp\Bigl(\int^t_0f(s)ds\Bigr); \quad 1+\int^t_0f(s)\exp\int^s_0(f(m)+g(m))dm ds;$$ and $$\exp\int^t_0f(s)\int^s_0g(m)dm ds,$$ respectively. Discrete analogues of above results are also considered. Remark: Since (3) may not be a linear problem thus its maximal solution $r(t)$ cannot be continuable in general on whole $R_-$. Hence these estimates given herein hold only on a certain subinterval of $R_-$ containing the origin.

26D15Inequalities for sums, series and integrals of real functions
39A12Discrete version of topics in analysis
39A10Additive difference equations