A NONOSCILLATION THEOREM FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DECAYING COEFFICIENTS

The purpose of this paper is to give sufficient conditions for all nontrivial solutions of the nonlinear differential equation x″ +a(t)g(x) = 0 to be nonoscillatory. Here, g(x) satisfies the sign condition xg(x) > 0 if x ≠ 0, but is not assumed to be monotone increasing. This differential equation includes the generalized Emden–Fowler equation as a special case. Our main result extends some nonoscillation theorems for the generalized Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase-plane analysis.

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500005096 ◽

2006 ◽

Vol 136 (3) ◽

pp. 633-647 ◽

Cited By ~ 17

Author(s):

Jitsuro Sugie ◽

Masakazu Onitsuka

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Asymptotic Behaviour ◽

Linear Differential Equation ◽

Phase Plane ◽

Nonlinear Differential Equations ◽

Phase Plane Analysis ◽

Oscillation Theorem ◽

Plane Analysis ◽

Linear Differential

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.

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Existence and nonexistence of limit cycles for Liénard-type equations with bounded nonlinearities and φ-Laplacian

Communications in Contemporary Mathematics ◽

10.1142/s0219199716500577 ◽

2017 ◽

Vol 19 (06) ◽

pp. 1650057 ◽

Cited By ~ 3

Author(s):

Kōdai Fujimoto ◽

Naoto Yamaoka

Keyword(s):

Differential Equation ◽

Limit Cycle ◽

Nonlinear Differential Equation ◽

Limit Cycles ◽

Phase Plane ◽

Sufficient Conditions ◽

Phase Plane Analysis ◽

Equivalent System ◽

Plane Analysis ◽

Existence And Nonexistence

This paper deals with an equivalent system to the nonlinear differential equation of Liénard type [Formula: see text], where the range of the function [Formula: see text] is bounded. Sufficient conditions are obtained for the system to have at least one limit cycle. The proofs of our results are based on phase plane analysis of the system with the Poincaré–Bendixon theorem. Moreover, to show that these sufficient conditions are suitable in some sense, we also establish the results that the system has no limit cycles. Finally, some examples are given to illustrate our results.

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Phase Plane Analysis of a Host a nd Commensal Ecological Model – A Special Case Study

International Journal of Advanced Science and Technology ◽

10.14257/ijast.2015.78.05 ◽

2015 ◽

Vol 78 ◽

pp. 59-66

Author(s):

Dr. N. Seshagiri Rao ◽

Dr. K. V. L. N. Acharyulu ◽

K. Kalyani

Keyword(s):

Phase Plane ◽

Ecological Model ◽

Phase Plane Analysis ◽

Plane Analysis ◽

Special Case

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Infinitely Many Rotationally Symmetric Solutions to a Class of Semilinear Laplace–Beltrami Equations on Spheres

Canadian Mathematical Bulletin ◽

10.4153/cmb-2015-056-7 ◽

2015 ◽

Vol 58 (4) ◽

pp. 723-729 ◽

Cited By ~ 2

Author(s):

Alfonso Castro ◽

Emily M. Fischer

Keyword(s):

Differential Equation ◽

Phase Plane ◽

Phase Plane Analysis ◽

Point Boundary ◽

Initial Value ◽

Beltrami Equations ◽

Plane Analysis ◽

Singular Ordinary Differential Equation ◽

Rotationally Symmetric ◽

Pohozaev Type Identity

AbstractWe show that a class of semilinear Laplace–Beltrami equations on the unit sphere in ℝn has inûnitely many rotationally symmetric solutions. The solutions to these equations are the solutions to a two point boundary value problem for a singular ordinary differential equation. We prove the existence of such solutions using energy and phase plane analysis. We derive a Pohozaev-type identity in order to prove that the energy to an associated initial value problem tends to infinity as the energy at the singularity tends to infinity. The nonlinearity is allowed to grow as fast as |s|p-1s for |s| large with 1 < p < (n + 5)/(n − 3).

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