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**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.}

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 |

### Keywords:

integral inequalities of the Gronwall-Bellman-Bihari type; asymptotic behavior; second order differential equation
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\textit{F. M. Dannan}, J. Math. Anal. Appl. 108, 151--164 (1985; Zbl 0586.26008)

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