Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
nonlinear differential equation; curvatore operator; boundary value problem on the half line; fixed point theorem; unbounded solution
nonlinear differential equation; curvatore operator; boundary value problem on the half line; fixed point theorem; unbounded solution
Summary:
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.
References:
[1] Agarwal, R. P., Grace, S. R., O'Regan, D.: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht (2002). DOI 10.1007/978-94-017-2515-6 | MR 2091751 | Zbl 1073.34002
[2] Armellini, G.: Sopra un'equazione differenziale della dinamica. Atti Accad. Naz. Lincei, Rend., VI. Ser. 21 (1935), 111-116 Italian. MR 0035358 | Zbl 0011.20902
[3] Azzollini, A.: Ground state solutions for the Hénon prescribed mean curvature equation. Adv. Nonlinear Anal. 8 (2019), 1227-1234. DOI 10.1515/anona-2017-0233 | MR 3918427 | Zbl 1419.35047
[4] Bereanu, C., Jebelean, P., Mawhin, J.: Radial solutions for Neumann problems involving mean curvature operators in Euclidean and Minkowski spaces. Math. Nachr. 283 (2010), 379-391. DOI 10.1002/mana.200910083 | MR 2643019 | Zbl 1185.35113
[5] Bereanu, C., Mawhin, J.: Periodic solutions of nonlinear perturbations of $\phi$-Laplacians with possibly bounded $\phi$. Nonlinear Anal., Theory Methods Appl., Ser. A 68 (2008), 1668-1681. DOI 10.1016/j.na.2006.12.049 | MR 2388840 | Zbl 1147.34032
[6] Bonanno, G., Livrea, R., Mawhin, J.: Existence results for parametric boundary value problems involving the mean curvature operator. NoDEA, Nonlinear Differ. Equ. Appl. 22 (2015), 411-426. DOI 10.1007/s00030-014-0289-7 | MR 3349800 | Zbl 1410.34073
[7] Cecchi, M., Došlá, Z., Marini, M.: Oscillation of a class of differential equations with generalized phi-Laplacian. Proc. R. Soc. Edinb., Sect. A, Math. 143 (2013), 493-506. DOI 10.1017/S0308210511001156 | MR 3063571 | Zbl 1302.34054
[8] Cecchi, M., Furi, M., Marini, M.: On continuity and compactness of some nonlinear operators associated with differential equations in noncompact intervals. Nonlinear Anal., Theory Methods Appl. 9 (1985), 171-180. DOI 10.1016/0362-546X(85)90070-7 | MR 0777986 | Zbl 0563.34018
[9] Cecchi, M., Marini, M., Villari, G.: Integral criteria for a classification of solutions of linear differential equations. J. Differ. Equations 99 (1992), 381-397. DOI 10.1016/0022-0396(92)90027-K | MR 1184060 | Zbl 0761.34009
[10] Došlá, Z., Cecchi, M., Marini, M.: Asymptotic problems for differential equation with bounded $\Phi$-Laplacian. Electron. J. Qual. Theory Differ. Equ. 2009 (2009), Article ID 9, 18 pages. DOI 10.14232/ejqtde.2009.4.9 | MR 2558834 | Zbl 1203.34076
[11] Došlá, Z., Kiguradze, I.: On vanishing at infinity solutions of second order linear differential equations with advanced arguments. Funkc. Ekvacioj, Ser. Int. 41 (1998), 189-205. MR 1662336 | Zbl 1140.34433
[12] Došlá, Z., Marini, M.: On super-linear Emden-Fowler type differential equations. J. Math. Anal. Appl. 416 (2014), 497-510. DOI 10.1016/j.jmaa.2014.02.052 | MR 3188719 | Zbl 1375.34050
[13] Došlá, Z., Marini, M., Matucci, S.: Positive decaying solutions to BVPs with mean curvature operator. Rend. Ist. Mat. Univ. Trieste 49 (2017), 147-164. DOI 10.13137/2464-8728/16210 | MR 3748508 | Zbl 1448.34058
[14] Hartman, P.: Ordinary Differential Equations. Birkhäuser, Boston (1982). MR 0658490 | Zbl 0476.34002
[15] Kurzweil, J.: Sur l'équation $\ddot{x}+ f(t) x= 0$. Čas. Pěst. Mat. 82 (1957), 218-226 French. DOI 10.21136/CPM.1957.117256 | MR 0094506 | Zbl 0102.07702
[16] Kurzweil, J.: Generalized Ordinary Differential Equations. Not Absolutely Continuous Solutions. Series in Real Analysis 11. World Scientific, Hackensack (2012). DOI 10.1142/7907 | MR 2906899 | Zbl 1248.34001
[17] López-Gómez, J., Omari, P.: 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. DOI 10.1016/j.jmaa.2022.126719 | MR 4492129 | Zbl 1501.35240
[18] Moore, R. A., Nehari, Z.: Nonoscillation theorems for a class of nonlinear differential equations. Trans. Am. Math. Soc. 93 (1959), 30-52. DOI 10.1090/S0002-9947-1959-0111897-8 | MR 0111897 | Zbl 0089.06902
[19] Swanson, C. A.: Comparison and Oscillation Theory of Linear Differential Equations. Mathematics in Science and Engineering 48. Academic Press, New York (1968). DOI 10.1016/s0076-5392(08)x6131-4 | MR 0463570 | Zbl 0191.09904
[20] Takaŝi, K., Manojlović, J. V., Milošević, J.: Intermediate solutions of second order quasilinear ordinary differential equations in the framework of regular variation. Appl. Math. Comput. 219 (2013), 8178-8191. DOI 10.1016/j.amc.2013.02.007 | MR 3037526 | Zbl 1294.34057
Search
Partner of
