A non-oscillation theorem for nonlinear differential equations with p-Laplacian

The equation considered in this paper is tp(φp(x′))′ + g(x) = 0, where φp(x′) = |x′|p−2x′ with p > 1, and g(x) satisfies the signum condition xg(x) > 0 if x ≠ 0 but is not assumed to be monotone. Our main objective is to establish a criterion on g(x) for all non-trivial solutions to be non-oscillatory. The criterion is the best possible. The method used here is the phase-plane analysis of a system equivalent to this differential equation. The asymptotic behaviour is also examined in detail for eventually positive solutions of a certain half-linear differential equation.

[1]  A nonoscillation theorem for the first variation and applications , 1977 .

[2]  N. Yoshida,et al.  Nonoscillation Theorems for a Class of Quasilinear Differential Equations of Second-Order , 1995 .

[3]  J. Wong.  Oscillation theorems for second-order nonlinear differential equations of Euler type , 1996 .

[4]  J. Sugie,et al.  Oscillation of solutions of second-order nonlinear self-adjoint differential equations , 2004 .

[5]  Ravi P. Agarwal,et al.  Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations , 2002 .

[6]  J. Sugie,et al.  Nonlinear oscillations of second order differential equations of Euler type , 1996 .

[7]  Jitsuro Sugie,et al.  Oscillation criteria for second order nonlinear differential equations of Euler type , 2001 .

[8]  J. D Mirzov,et al.  On some analogs of Sturm's and Kneser's theorems for nonlinear systems , 1976 .

[9]  J. Wong.  A nonoscillation theorem for Emden-Fowler equations , 2002 .

[10]  J. Sugie,et al.  Growth conditions for oscillation of nonlinear differential equations with p-Laplacian , 2005 .

[11]  C. Yeh,et al.  Sturmian comparison theorem for half-linear second-order differential equations , 1995, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[12]  Á. Elbert,et al.  Oscillation and non-oscillation theorems for a class of second order quasilinear differential equations , 1990 .

[13]  Á. Elbert,et al.  Perturbations of the Half-Linear Euler Differential Equation , 2000 .

[14]  J. Wong.  Nonoscillation theorems for second order nonlinear differential equations , 1999 .

[15]  J. Sugie,et al.  Oscillation constant of second-order non-linear self-adjoint differential equations , 2002 .