Geodesic
On some boundary value problems for second order nonlinear differential equations
Mathematica Bohemica, Tome 137 (2012) no. 2, pp. 113-122

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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: \[ {\rm i)}\ x(0)=c>0, \ \lim _{t\rightarrow \infty }x(t)=0; \quad {\rm ii)}\ x'(0)=d0, \ \lim _{t\rightarrow \infty }x(t)=0. \]
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: \[ {\rm i)}\ x(0)=c>0, \ \lim _{t\rightarrow \infty }x(t)=0; \quad {\rm ii)}\ x'(0)=d0, \ \lim _{t\rightarrow \infty }x(t)=0. \]
DOI : 10.21136/MB.2012.142856
Classification : 34B18, 34B40, 34C10, 34D05
Keywords: boundary value problem; $p$-Laplacian; half-linear equation; positive solution; uniqueness; decaying solution; principal solution 
Došlá, Zuzana; Marini, Mauro; Matucci, Serena. On some boundary value problems for second order nonlinear differential equations. Mathematica Bohemica, Tome 137 (2012) no. 2, pp. 113-122. doi: 10.21136/MB.2012.142856
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[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). | MR

[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 | Zbl

[3] Cecchi, M., Došlá, Z., Marini, M.: On the dynamics of the generalized Emden-Fowler equations. Georgian Math. J. 7 (2000), 269-282. | DOI | MR

[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 | Zbl

[5] Cecchi, M., Došlá, Z., Marini, M.: Principal solutions and minimal sets of quasilinear differential equations. Dynam. Systems Appl. 13 (2004), 221-232. | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[8] Chanturia, T. A.: On singular solutions of nonlinear systems of ordinary differential equations. Colloq. Math. Soc. Janos Bolyai 15 (1975), 107-119. | MR

[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

[11] Došlý, O., Řehák, P.: Half-Linear Differential Equations. North-Holland Mathematics Studies 202, Elsevier, Amsterdam (2005). | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[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 | Zbl

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