Abstract
We present necessary and sufficient conditions for the existence of unbounded increasing solutions to ordinary differential equations with mean curvature operator. The results illustrate the asymptotic proximity of such solutions with those of an auxiliary linear equation on the threshold of oscillation. A new oscillation criterion for equations with mean curvature operator, extending Leighton criterion for linear Sturm-Liouville equation, is also derived.
Article PDF
Avoid common mistakes on your manuscript.
References
R. P. Agarwal, S. R. Grace, D. O’Regan: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht, 2002.
G. Armellini: Sopra un’equazione differenziale della dinamica. Atti Accad. Naz. Lincei, Rend., VI. Ser. 21 (1935), 111–116. (In Italian.)
A. Azzollini: Ground state solutions for the Hénon prescribed mean curvature equation. Adv. Nonlinear Anal. 8 (2019), 1227–1234.
C. Bereanu, P. Jebelean, J. Mawhin: Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces. Math. Nachr. 283 (2010), 379–391.
C. Bereanu, J. Mawhin: Periodic solutions of nonlinear perturbations of φ-Laplacians with possibly bounded φ. Nonlinear Anal., Theory Methods Appl., Ser. A 68 (2008), 1668–1681.
G. Bonanno, R. Livrea, J. Mawhin: Existence results for parametric boundary value problems involving the mean curvature operator. NoDEA, Nonlinear Differ. Equ. Appl. 22 (2015), 411–426.
M. Cecchi, Z. Došlá, M. Marini: Oscillation of a class of differential equations with generalized phi-Laplacian. Proc. R. Soc. Edinb., Sect. A, Math. 143 (2013), 493–506.
M. Cecchi, M. Furi, M. Marini: On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals. Nonlinear Anal., Theory Methods Appl. 9 (1985), 171–180.
M. Cecchi, M. Marini, G. Villari: Integral criteria for a classification of solutions of linear differential equations. J. Differ. Equations 99 (1992), 381–397.
Z. Došlá, M. Cecchi, M. Marini: Asymptotic problems for differential equation with bounded Φ-Laplacian. Electron. J. Qual. Theory Differ. Equ. 2009 (2009), Article ID 9, 18 pages.
Z. Došlá, I. Kiguradze: On vanishing at infinity solutions of second order linear differential equations with advanced arguments. Funkc. Ekvacioj, Ser. Int. 41 (1998), 189–205.
Z. Došlá, M. Marini: On super-linear Emden-Fowler type differential equations. J. Math. Anal. Appl. 416 (2014), 497–510.
Z. Došlá, M. Marini, S. Matucci: Positive decaying solutions to BVPs with mean curvature operator. Rend. Ist. Mat. Univ. Trieste 49 (2017), 147–164.
P. Hartman: Ordinary Differential Equations. Birkhäuser, Boston, 1982.
J. Kurzweil: Sur l’équation ẍ + f(t)x = 0. Čas. Pĕst. Mat. 82 (1957), 218–226. (In French.)
J. Kurzweil: Generalized Ordinary Differential Equations. Not Absolutely Continuous Solutions. Series in Real Analysis 11. World Scientific, Hackensack, 2012.
J. López-Gómez, P. Omari: Optimal regularity results for the one-dimensional prescribed curvature equation via the strong maximum principle. J. Math. Anal. Appl. 518 (2023), Article ID 126719, 22 pages.
R. A. Moore, Z. Nehari: Nonoscillation theorems for a class of nonlinear differential equations. Trans. Am. Math. Soc. 93 (1959), 30–52.
C. A. Swanson: Comparison and Oscillation Theory of Linear Differential Equations. Mathematics in Science and Engineering 48. Academic Press, New York, 1968.
K. Takaŝi, J. V. Manojlović, J. Milošević: Intermediate solutions of second order quasilinear ordinary differential equations in the framework of regular variation. Appl. Math. Comput. 219 (2013), 8178–8191.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Jaroslav Kurzweil
The second and third authors were partially supported by INdAM-GNAMPA Projects (Italy). Open access funding provided by Masaryk University.
Rights and permissions
Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Došlá, Z., Marini, M. & Matucci, S. On unbounded solutions for differential equations with mean curvature operator. Czech Math J (2023). https://doi.org/10.21136/CMJ.2023.0111-23
Received:
Published:
DOI: https://doi.org/10.21136/CMJ.2023.0111-23
Keywords
- nonlinear differential equation
- curvatore operator
- boundary value problem on the half line
- fixed point theorem
- unbounded solution