## Comparison theorems related to a certain inequality used in the theory of differential equations.(English)Zbl 0867.26015

Let $$c\geq 0$$ 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 $y^2(t)\leq c^2+2\int^t_0y(s)Q(\cdot)ds,\quad t\in R_+,\tag{1}$ implies $y(t)\leq K(t)r(t),\quad t\in R_+,\tag{2}$ with $$r(t)$$ being the maximal solution of the initial problem $r'(t)=w(t,K(t)r(t)),\quad t\in R_+;\quad r(0)=c,\tag{3}$ 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.

### MSC:

 26D15 Inequalities for sums, series and integrals 39A12 Discrete version of topics in analysis 39A10 Additive difference equations