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Keywords:
boundary value problem; $p$-Laplacian; half-linear equation; positive solution; uniqueness; decaying solution; principal solution
Summary:
We investigate two boundary value problems for the second order differential equation with $p$-Laplacian \[ (a(t)\Phi _{p}(x'))'=b(t)F(x), \quad t\in I=[0,\infty ), \] where $a$, $b$ are continuous positive functions on $I$. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: \[ {\rm i)}\ x(0)=c>0, \ \lim _{t\rightarrow \infty }x(t)=0; \quad {\rm ii)}\ x'(0)=d<0, \ \lim _{t\rightarrow \infty }x(t)=0. \]
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 Acad., Dordrecht (2003).
[2] Cecchi, M., Došlá, Z., Kiguradze, I., Marini, M.: On nonnegative solutions of singular boundary value problems for Emden-Fowler type differential systems. Differ. Integral Equ. 20 (2007), 1081-1106. MR 2365203 | Zbl 1212.34044
[3] Cecchi, M., Došlá, Z., Marini, M.: On the dynamics of the generalized Emden-Fowler equations. Georgian Math. J. 7 (2000), 269-282. MR 1779551
[4] Cecchi, M., Došlá, Z., Marini, M.: On nonoscillatory solutions of differential equations with $p$-Laplacian. Adv. Math. Sci. Appl. 11 (2001), 419-436. MR 1842385 | Zbl 0996.34039
[5] Cecchi, M., Došlá, Z., Marini, M.: Principal solutions and minimal sets of quasilinear differential equations. Dynam. Systems Appl. 13 (2004), 221-232. MR 2140874 | Zbl 1123.34026
[6] Cecchi, M., Došlá, Z., Marini, M., Vrkoč, I.: Integral conditions for nonoscillation of second order nonlinear differential equations. Nonlinear Anal., Theory Methods Appl. 64 (2006), 1278-1289. DOI 10.1016/j.na.2005.06.035 | MR 2200492 | Zbl 1114.34031
[7] 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
[8] Chanturia, T. A.: On singular solutions of nonlinear systems of ordinary differential equations. Colloq. Math. Soc. Janos Bolyai 15 (1975), 107-119. MR 0591720
[9] Chanturia, T. A.: On monotonic solutions of systems of nonlinear differential equations. Russian Ann. Polon. Math. 37 (1980), 59-70.
[10] Došlá, Z., Marini, M., Matucci, S.: A boundary value problem on a half-line for differential equations with indefinite weight. (to appear) in Commun. Appl. Anal. MR 2867356
[11] Došlý, O., Řehák, P.: Half-Linear Differential Equations. North-Holland Mathematics Studies 202, Elsevier, Amsterdam (2005). MR 2158903 | Zbl 1090.34001
[12] Garcia, H. M., Manasevich, R., Yarur, C.: On the structure of positive radial solutions to an equation containing a $p$-Laplacian with weight. J. Differ. Equations 223 (2006), 51-95. DOI 10.1016/j.jde.2005.04.012 | MR 2210139 | Zbl 1170.35404
[13] Lian, H., Pang, H., Ge, W.: Triple positive solutions for boundary value problems on infinite intervals. Nonlinear Anal., Theory Methods Appl. 67 (2007), 2199-2207. DOI 10.1016/j.na.2006.09.016 | MR 2331870 | Zbl 1128.34011
[14] Mirzov, J. D.: Asymptotic properties of solutions of systems of nonlinear nonautonomous ordinary differential equations. Folia Fac. Sci. Nat. Univ. Masaryk. Brun. Math. 14 (2004). MR 2144761 | Zbl 1154.34300
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