## Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations.(English)Zbl 0586.26008

The author gives four theorems for integral inequalities of the Gronwall- Bellman-Bihari type. The most general iequality discussed here is of the form $x(t)\leq x_ 0(t)+h(t)[\int^{t}_{0}f(s)w(x(s))ds+\int^{t}_{0}g(s)(\int^{s}_ {0}f(m)w(x(m))dm)ds],\quad t\in [0,\infty),$ where x(t), $$x_ 0(t)$$, f(t), g(t), h(t), and w(u) are nonnegative and continuous functions on [0,$$\infty)$$. Throughout this paper it is assumed that, either the function w belongs to a certain class H or it satisfies a Lipschitz condition. Here the function class H is defined as follows: Definition. A function $$w: [0,\infty)\to [0,\infty)$$ is said to belong to the class H if $$(H_ 1)$$ w(u) is nondecreasing and continuous for $$u\geq 0$$ and positive for $$u>0$$. $$(H_ 2)$$ There exists a function $$\phi$$, continuous on [0,$$\infty)$$ with $$w(au)\leq \phi (a)w(u)$$ for $$a>0$$, $$u\geq 0.$$
As examples of application of the results obtained, two theorems are also derived on the asymptotic behavior of solutions of the second order differential equation $$u''+f(t,u,u')=0.$$
{Reviewer’s remark: We note that the condition (ii) of Lemma 2 is meaningless, since G(u) is assumed to be nonnegative and continuous for $$u\geq 0$$ with $$G(0)=0$$. Further, there are some obvious misprints in this paper - in (10), (14), (24), the first line of page 157, and the 7th line of page 160.}
Reviewer: En Hao Yang

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 34C11 Growth and boundedness of solutions to ordinary differential equations
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### References:

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