## Abstract and Applied Analysis

### Fourth-Order Differential Equation with Deviating Argument

#### Abstract

We consider the fourth-order differential equation with middle-term and deviating argument ${x}^{(4)}(t)+q(t){x}^{(2)}(t)+r(t)f(x(\phi (t)))=0$, in case when the corresponding second-order equation ${h}^{″}+q(t)h=0$ is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 185242, 17 pages.

Dates
First available in Project Euclid: 14 December 2012

https://projecteuclid.org/euclid.aaa/1355495664

Digital Object Identifier
doi:10.1155/2012/185242

Mathematical Reviews number (MathSciNet)
MR2898056

Zentralblatt MATH identifier
1244.34089

#### Citation

Bartušek, M.; Cecchi, M.; Došlá, Z.; Marini, M. Fourth-Order Differential Equation with Deviating Argument. Abstr. Appl. Anal. 2012 (2012), Article ID 185242, 17 pages. doi:10.1155/2012/185242. https://projecteuclid.org/euclid.aaa/1355495664

#### References

• M. Marini, “Criteri di limitatezza per le soluzioni dell'equazione lineare del secondo ordine,” Bollettino della Unione Matematica Italiana, vol. 11, no. 1, pp. 154–165, 1975.
• J. R. Graef and J. Henderson, “Double solutions of boundary value problems for $2{m}^{\text{th}}$-order differential equations and difference equations,” Computers & Mathematics with Applications, vol. 45, no. 6–9, pp. 873–885, 2003.
• J. R. Graef, C. Qian, and B. Yang, “A three point boundary value problem for nonlinear fourth order differential equations,” Journal of Mathematical Analysis and Applications, vol. 287, no. 1, pp. 217–233, 2003.
• K. Kamo and H. Usami, “Nonlinear oscillations of fourth order quasilinear ordinary differential equations,” Acta Mathematica Hungarica, vol. 132, no. 3, pp. 207–222, 2011.
• T. Kusano and M. Švec, “On unbounded positive solutions of nonlinear differential equations with oscillating coefficients,” Czechoslovak Mathematical Journal, vol. 39, no. 1, pp. 133–141, 1989.
• K. Taka\^si and T. Tanigawa, “On the structure of positive solutions of a class of fourth order nonlinear differential equations,” Annali di Matematica Pura ed Applicata, vol. 185, no. 4, pp. 521–536, 2006.
• W. T. Li and X. Yang, “Classifications and existence criteria for positive solutions of systems of nonlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 446–462, 2004.
• M. Marini, S. Matucci, and P. Řehák, “On decay of coupled nonlinear differential systems,” Advances in Mathematical Sciences and Applications, vol. 12, no. 2, pp. 521–533, 2002.
• M. Naito and F. Wu, “On the existence of eventually positive solutions of fourth-order quasilinear differential equations,” Nonlinear Analysis, vol. 57, no. 2, pp. 253–263, 2004.
• M. Naito and F. Wu, “A note on the existence and asymptotic behavior of nonoscillatory solutions of fourth order quasilinear differential equations,” Acta Mathematica Hungarica, vol. 102, no. 3, pp. 177–202, 2004.
• R. P. Agarwal and D. O'Regan, Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
• I. T. Kiguradze and T. A. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, vol. 89 of Mathematics and its Applications (Soviet Series), Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993.
• I. T. Kiguradze, “An oscillation criterion for a class of ordinary differential equations,” Differentsial'nye Uravneniya, vol. 28, no. 2, pp. 207–219, 1992.
• M. Bartušek, M. Cecchi, Z. Došlá, and M. Marini, “Oscillation for third-order nonlinear differential equations with deviating argument,” Abstract and Applied Analysis, vol. 2010, Article ID 278962, 19 pages, 2010.
• M. Bartušek, M. Cecchi, Z. Došlá, and M. Marini, “Asymptotics for higher order differential equations with a middle term,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 1130–1140, 2012.
• W. Xiong and G. Yue, “Almost periodic solutions for a class of fourth-order nonlinear differential equations with a deviating argument,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1184–1190, 2010.