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.


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