##
**Bounded and zero-convergent solutions of a class of Stieltjes integro- differential equations.**
*(English)*
Zbl 0739.45005

Consider the equation (1) \(G(p(t)x'(t))=C+\int^ t_ a f(x(s))d\sigma(s)\), \(t\geq a\), where \(p,\sigma: [a,\infty)\to R\) are locally of bounded variation with \(p>0\), \(p^{-1}\in L^ 1_{loc}\) and \(\sigma\) nondecreasing. Let \(f,G\in C(R;R)\) satisfy \(xf(x)>0\), \(xG(x)>0\), \(x\neq 0\), with \(| G(x)|\to\infty\) as \(| x|\to\infty\), \(f\) nondecreasing and \(G\) increasing. Under these conditions one has that the set \(X\) of nonconstant solutions of (1) can be written as \(X=A\cup B\) where \(A=\{\exists t_ x\) such that \(x(t)x'(t)>0\) for \(t>t_ x\}\), \(B=\{\exists t_ x\) such that \(x(t)x'(t)\leq 0\) for \(t>t_ x\}\). The authors give conditions under which solutions in \(A\) are bounded. In addition, the structure of \(B\) is analyzed in detail.

Reviewer: S.O.Londen (Espoo)

### MSC:

45J05 | Integro-ordinary differential equations |

45G10 | Other nonlinear integral equations |

45D05 | Volterra integral equations |

### Keywords:

bounded and zero-convergent solutions; Stieltjes integro-differential equations; nonlinear Volterra equations; asymptotic behavior; measure- valued kernels
PDFBibTeX
XMLCite

\textit{S. Chen} et al., Proc. Am. Math. Soc. 113, No. 4, 999--1008 (1991; Zbl 0739.45005)

Full Text:
DOI

### References:

[1] | Mariella Cecchi, Mauro Marini, and Gabriele Villari, On the monotonicity property for a certain class of second order differential equations, J. Differential Equations 82 (1989), no. 1, 15 – 27. · Zbl 0694.34035 · doi:10.1016/0022-0396(89)90165-4 |

[2] | Shao Zhu Chen, Terminal value problems for \?”=\?(\?,\?,\?’), Ann. Differential Equations 5 (1989), no. 4, 389 – 395. · Zbl 0685.34006 |

[3] | Shao Zhu Chen, Existence and uniqueness of solutions of limit boundary value problems for second-order differential systems, Acta Math. Appl. Sinica 10 (1987), no. 3, 324 – 332 (Chinese, with English summary). · Zbl 0637.34011 |

[4] | S. S. Cheng, H. J. Li and W. T. Patula, Bounded and zero convergent solutions of second order differential equations, preprint. |

[5] | Zhong Chao Liang and Shao Zhu Chen, Asymptotic behavior of solutions to second-order nonlinear differential equations, Chinese Ann. Math. Ser. B 6 (1985), no. 4, 481 – 490. A Chinese summary appears in Chinese Ann. Math. Ser. A 6 (1985), no. 6, 762. · Zbl 0593.34055 |

[6] | Mauro Marini and Pierluigi Zezza, On the asymptotic behavior of the solutions of a class of second-order linear differential equations, J. Differential Equations 28 (1978), no. 1, 1 – 17. · Zbl 0371.34032 · doi:10.1016/0022-0396(78)90075-X |

[7] | Mauro Marini, On nonoscillatory solutions of a second-order nonlinear differential equation, Boll. Un. Mat. Ital. C (6) 3 (1984), no. 1, 189 – 202. · Zbl 0574.34022 |

[8] | Mauro Marini, Monotone solutions of a class of second order nonlinear differential equations, Nonlinear Anal. 8 (1984), no. 3, 261 – 271. · Zbl 0552.34053 · doi:10.1016/0362-546X(84)90048-8 |

[9] | Bing Li You and Qing Guang Huang, Fundamental theory of a class of functional equations and its application in the study of Kneser’s theorem, Ann. Differential Equations 5 (1989), no. 1, 107 – 128. · Zbl 0676.34039 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.