Previous |  Up |  Next

Article

Keywords:
$p$-Laplacian; differential equation; asymptotic integration
Summary:
In this paper we deal with the problem of asymptotic integration of nonlinear differential equations with $p-$Laplacian, where $1 < p < 2$. We prove sufficient conditions under which all solutions of an equation from this class are converging to a linear function as $t \rightarrow \infty $.
References:
[1] Agarwal, R.P., Djebali, S., Moussaoui, T., Mustafa, O.G.: On the asymptotic integration of nonlinear differential equations. J. Comput. Appl. Math 202 (2007), 352–376. DOI 10.1016/j.cam.2005.11.038 | MR 2319962 | Zbl 1123.34038
[2] Bartušek, M., Medveď, M.: Existence of global solutions for systems of second-order functional-differential equations with $p$-Laplacian. EJDE 40 (2008), 1–8. MR 2392944 | Zbl 1171.34335
[3] Bartušek, M., Pekárková, E.: On the existence of proper solutions of quasilinear second order differential equations. EJQTDE 1 (2007), 1–14.
[4] Bihari, I.: A generalization of a lemma of Bellman and its applications to uniqueness problems of differential equations. Acta Math. Hungar. 7 (1956), 81–94. DOI 10.1007/BF02022967 | MR 0079154
[5] Caligo, D.: Comportamento asintotico degli integrali dell’equazione $y^{\prime \prime }(x)+A(x)y(x)=0,$ nell’ipotesi $\lim _{x\rightarrow +\infty }A(x)=0$. Boll. Un. Mat. Ital. (2) 3 (1941), 286–295. MR 0005219
[6] Cohen, D.S.: The asymptotic behavior of a class of nonlinear differntial equations. Proc. Amer. Math. Soc. 18 (1967), 607–609. DOI 10.1090/S0002-9939-1967-0212289-3 | MR 0212289
[7] Constantin, A.: On the asymptotic behavior of second order nonlinear differential equations. Rend. Mat. Appl. (7) 13 (4) (1993), 627–634. MR 1283990
[8] Constantin, A.: Solutions globales d’équations différentielles perturbées. C. R. Acad. Sci. Paris Sér. I Math. 320 (11) (1995), 1319–1322. MR 1338279
[9] Constantin, A.: On the existence of positive solutions of second order differential equations. Ann. Mat. Pura Appl. (4) 184 (2) (2005), 131–138. DOI 10.1007/s10231-004-0100-1 | MR 2149089
[10] Dannan, F.M.: Integral inequalities of Gronwall-Bellman-Bihari type and asymptotic behavior of certain second order nonlinear differential equations. J. Math. anal. Appl. 108 (1) (1985), 151–164. DOI 10.1016/0022-247X(85)90014-9 | MR 0791139
[11] Kusano, T., Trench, W.F.: Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations. J. London Math. Soc.(2) 31 (3) (1985), 478–486. DOI 10.1112/jlms/s2-31.3.478 | MR 0812777
[12] Kusano, T., Trench, W.F.: Existence of global solutions with prescribed asymptotic behavior for nonlinear ordinary differential equations. Ann. Mat. Pura Appl. (4) 142 (1985), 381–392. MR 0839046
[13] Lipovan, O.: On the asymptotic behaviour of the solutions to a class of second order nonlinear differential equations. Glasgow Math. J. 45 (1) (2003), 179–187. DOI 10.1017/S0017089502001143 | MR 1973349
[14] Medveď, M., Moussaoui, T.: Asymptotic integration of nonlinear $\Phi -$Laplacian differential equations. Nonlinear Anal. 72 (2010), 1–8. DOI 10.1016/j.na.2009.09.042 | MR 2577598
[15] Medveď, M., Pekárková, E.: Existence of global solutions for systems of second-order differential equations with $p$-Laplacian. EJDE 2007 (136) (2007), 1–9. MR 2349964
[16] Medveď, M., Pekárková, E.: Long time behavior of second order differential equations with $p$-Laplacian. EJDE 2008 (108) (2008), 1–12.
[17] Mustafa, O.G., Rogovchenko, Y.V.: Global existence of solutions with prescribed asymptotic behavior for second-order nonlinear differential equations. Nonlinear Anal. 51 (2002), 339–368. DOI 10.1016/S0362-546X(01)00834-3 | MR 1918348 | Zbl 1017.34005
[18] Mustafa, O.G., Rogovchenko, Y.V.: Asymptotic behavior of nonoscillatory solutions of second-order nonlinear differential equations. Dynamic Systems and Applications 4 (2004), 312–319. MR 2117799 | Zbl 1082.34042
[19] Pekárková, E.: Estimations of noncontinuable solutions of second order differential equations with $p$-Laplacian. Arch. Math.( Brno) 46 (2010), 135–144. MR 2684255 | Zbl 1240.34187
[20] Philos, Ch.G., Purnaras, I.K., Tsamatos, P.Ch.: Large time asymptotic to polynomials solutions for nonlinear differential equations. Nonlinear Anal. 59 (2004), 1157–1179. DOI 10.1016/S0362-546X(04)00323-2 | MR 2098511
[21] Rogovchenko, S.P., Rogovchenko, Y.V.: Asymptotics of solutions for a class of second order nonlinear differential equations. Portugal. Math. 57 (1) (2000), 17–32.
[22] Rogovchenko, Y.V.: On asymptotic behavior of solutions for a class of second order nonlinear differential equations. Collect. Math. 49 (1) (1998), 113–120. MR 1629766
[23] Tong, J.: The asymptotic behavior of a class of nonlinear differential equations of second order. Proc. Amer. Math. Soc. 54 (1982), 235–236. DOI 10.1090/S0002-9939-1982-0637175-4 | MR 0637175 | Zbl 0491.34036
[24] Trench, W.F.: On the asymptotic behavior of solutions of second order linear differential equations. Proc. Amer. Math. Soc. 54 (1963), 12–14. DOI 10.1090/S0002-9939-1963-0142844-7 | MR 0142844
Partner of
EuDML logo