## Abstract and Applied Analysis

### New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations

Mervan Pašić

#### Abstract

We establish some new interval oscillation criteria for a general class of second-order forced quasilinear functional differential equations with ϕ-Laplacian operator and mixed nonlinearities. It especially includes the linear, the one-dimensional p-Laplacian, and the prescribed mean curvature quasilinear differential operators. It continues some recently published results on the oscillations of the second-order functional differential equations including functional arguments of delay, advanced, or delay-advanced types. The nonlinear terms are of superlinear or supersublinear (mixed) types. Consequences and examples are shown to illustrate the novelty and simplicity of our oscillation criteria.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 735360, 12 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512026

Digital Object Identifier
doi:10.1155/2013/735360

Mathematical Reviews number (MathSciNet)
MR3096826

Zentralblatt MATH identifier
1286.34054

#### Citation

Pašić, Mervan. New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 735360, 12 pages. doi:10.1155/2013/735360. https://projecteuclid.org/euclid.aaa/1393512026

#### References

• Y. Bai and L. Liu, “New oscillation criteria for second-order delay differential equations with mixed nonlinearities,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 796256, 9 pages, 2010.
• S. Murugadass, E. Thandapani, and S. Pinelas, “Oscillation criteria for forced second-order mixed type quasilinear delay differential equations,” Electronic Journal of Differential Equations, vol. 2010, article 73, 9 pages, 2010.
• T. S. Hassan, L. Erbe, and A. Peterson, “Forced oscillation of second order differential equations with mixed nonlinearities,” Acta Mathematica Scientia B, vol. 31, no. 2, pp. 613–626, 2011.
• C. Li and S. Chen, “Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 504–507, 2009.
• A. Zafer, “Interval oscillation criteria for second order super-half linear functional differential equations with delay and advanced arguments,” Mathematische Nachrichten, vol. 282, no. 9, pp. 1334–1341, 2009.
• A. F. Güvenilir and A. Zafer, “Second-order oscillation of forced functional differential equations with oscillatory potentials,” Computers & Mathematics with Applications, vol. 51, no. 9-10, pp. 1395–1404, 2006.
• A. F. Güvenilir, “Interval oscillation of second-order functional differential equations with oscillatory potentials,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 12, pp. e2849–e2854, 2009.
• Q. Kong, “Interval criteria for oscillation of second-order linear ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 229, no. 1, pp. 258–270, 1999.
• Y. G. Sun and J. S. W. Wong, “Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities,” Journal of Mathematical Analysis and Applications, vol. 334, no. 1, pp. 549–560, 2007.
• S. Jing, “A new oscillation criterion for forced second-order quasilinear differential equations,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 428976, 8 pages, 2011.
• L. H. Erbe, Q. Kong, and B. G. Zhang, Theory for Functional Differential Equations, vol. 190 of Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1994.
• M. Pašić, “Fite-Wintner-Leighton-type oscillation criteria for second-order differential equations with nonlinear damping,” Abstract and Applied Analysis, vol. 2013, Article ID 852180, 10 pages, 2013.
• M. Pašić, “New interval oscillation criteria for forced second-order differential equations with nonlinear damping,” International Journal of Mathematical Analysis, vol. 7, no. 25, pp. 1239–1255, 2013.
• D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, Germany, 2001, Reprint of the 1998 edition.
• P. L. de Nápoli and J. P. Pinasco, “A Lyapunov inequality for monotone quasilinear operators,” Differential and Integral Equations, vol. 18, no. 10, pp. 1193–1200, 2005.
• M. Feng, “Periodic solutions for prescribed mean curvature Liénard equation with a deviating argument,” Nonlinear Analysis: Real World Applications, vol. 13, no. 3, pp. 1216–1223, 2012.
• I. Rach\accent23unková and M. Tvrdý, “Second-order periodic problem with $\phi$-Laplacian and impulses,” Nonlinear Analysis: Theory, Methods and Applications, vol. 63, no. 5–7, pp. e257–e266, 2005.
• L. Ferracuti and F. Papalini, “Boundary-value problems for strongly non-linear multivalued equations involving different $\phi$-Laplacians,” Advances in Differential Equations, vol. 14, no. 5-6, pp. 541–566, 2009.
• I. Yermachenko and F. Sadyrbaev, “Quasilinearization technique for $\phi$-Laplacian type equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 975760, 11 pages, 2012.
• J. Henderson, A. Ouahab, and S. Youcefi, “Existence and topological structure of solution sets for $\phi$-Laplacian impulsive differential equations,” Electronic Journal of Differential Equations, vol. 2012, article 56, 16 pages, 2012.
• V. Kolmanovskii and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, vol. 463 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1999.
• R. Finn, Equilibrium Capillary Surface, Springer, New York, NY, USA, 1986.
• E. Giusti, Minimal Surfaces and Functions of Bounded Variations, Birkhäuser, Basel, Switzerland, 1984.
• I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics, Springer, Berlin, Germany, 2007.