Abstract
The existence of unbounded solutions with different asymptotic behavior for a second order nonlinear differential equation with p-Laplacian is considered. The oscillation of all solutions is investigated. Some discrepancies and similarities between equations of Emden–Fowler-type and equations with mixed nonlinearities are pointed out.
Dedicated to Professor Ivan Kiguradze on the occasion of his 80th birthday
Funding statement: The research of the third author is supported by GNAMPA, National Institute for Advanced Mathematics, Italy.
References
[1] Agarwal R. P., Grace S. R. and O’Regan D., Oscillation Theory for Second Order Dynamic Equations, Ser. Math. Anal. Appl. 5, Taylor & Francis, London, 2003. 10.4324/9780203222898Search in Google Scholar
[2] Bartušek M., Singular solutions for the differential equation with p-Laplacian, Arch. Math. (Brno) 41 (2005), no. 1, 123–128. Search in Google Scholar
[3] Bartušek M., Cecchi M., Došlá Z. and Marini M., On oscillation and nonoscillation for differential equations with p-Laplacian, Georgian Math. J. 14 (2007), no. 2, 239–252. 10.1515/GMJ.2007.239Search in Google Scholar
[4] Cecchi M., Došlá Z. and Marini M., Monotone solutions of two-dimensional nonlinear functional differential systems, Dynam. Systems Appl. 17 (2008), no. 3–4, 595–608. Search in Google Scholar
[5] Cecchi M., Došlá Z., Marini M. and Vrkoč I., Integral conditions for nonoscillation of second order nonlinear differential equations, Nonlinear Anal. 64 (2006), no. 6, 1278–1289. 10.1016/j.na.2005.06.035Search in Google Scholar
[6] Došlá Z. and Marini M., On super-linear Emden–Fowler type differential equations, J. Math. Anal. Appl. 416 (2014), no. 2, 497–510. 10.1016/j.jmaa.2014.02.052Search in Google Scholar
[7] Došlá Z. and Marini M., Positive decaying solutions for differential equations with phi-Laplacian, Bound. Value Probl. 2015 (2015), Paper No. 95. 10.1186/s13661-015-0355-zSearch in Google Scholar
[8] Došlá Z. and Marini M., Monotonicity conditions in oscillation to superlinear differential equations, Electron. J. Qual. Theory Differ. Equ. 2016 (2016), Paper No. 54. 10.14232/ejqtde.2016.1.54Search in Google Scholar
[9] Elbert Á. and Kusano T., Oscillation and nonoscillation theorems for a class of second order quasilinear differential equations, Acta Math. Hungar. 56 (1990), no. 3–4, 325–336. 10.1007/BF01903849Search in Google Scholar
[10] Kamo K. and Usami H., Asymptotic forms of weakly increasing positive solutions for quasilinear ordinary differential equations, Electron. J. Differential Equations 2007 (2007), Paper No. 126. Search in Google Scholar
[11]
Kiguradze I. T.,
On the conditions for oscillation of solutions of the differential equation
[12] Kiguradze I. T. and Chanturia T. A., Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Translated from the 1985 Russian original, Math. Appl. (Soviet Series) 89, Kluwer Academic, Dordrecht, 1993. 10.1007/978-94-011-1808-8Search in Google Scholar
[13] Kusano T., Manojlović J. V. and Milošević J., Intermediate solutions of second order quasilinear ordinary differential equations in the framework of regular variation, Appl. Math. Comput. 219 (2013), no. 15, 8178–8191. 10.1016/j.amc.2013.02.007Search in Google Scholar
[14] Li W. T. and Cheng S. S., Limiting behaviours of non-oscillatory solutions of a pair of coupled nonlinear differential equations, Proc. Edinb. Math. Soc. (2) 43 (2000), no. 3, 457–473. 10.1017/S0013091500021131Search in Google Scholar
[15] Mirzov J. D., Asymptotic Properties of Solutions of Systems of Nonlinear Nonautonomous Ordinary Differential Equations, Folia Fac. Sci. Natur. Univ. Masaryk. Brunensis. Math. 14, Masaryk University, Brno, 2004. Search in Google Scholar
[16] Wang J., Oscillation and nonoscillation theorems for a class of second order quasilinear functional-differential equations, Hiroshima Math. J. 27 (1997), no. 3, 449–466. 10.32917/hmj/1206126963Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions
- Unbounded solutions for differential equations with p-Laplacian and mixed nonlinearities
- On positive periodic solutions of linear second order functional differential equations
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- An existence result for multiple solutions to a Dirichlet problem
- Asymptotic analysis of positive solutions of first-order cyclic functional differential systems
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Articles in the same Issue
- Frontmatter
- Strict stability with respect to initial time difference for Caputo fractional differential equations by Lyapunov functions
- Unbounded solutions for differential equations with p-Laplacian and mixed nonlinearities
- On positive periodic solutions of linear second order functional differential equations
- Existence results for nonlinear fractional Dirichlet problems on the right side of the first eigenvalue
- An existence result for multiple solutions to a Dirichlet problem
- Asymptotic analysis of positive solutions of first-order cyclic functional differential systems
- Periodic solutions of Liénard–Mathieu differential equation with a small parameter
- On a class of variational problems with nonlocal integrant
- Morse theory and multiple periodic solutions of some quasilinear difference systems with periodic nonlinearities
- The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space
- Piecewise synergetic systems and applications in biochemical systems theory
- Unique solvability of the Darboux problem for linear hyperbolic functional differential equations