Naoto Yamaoka

A comparison theorem and oscillation criteria for second-order nonlinear differential equations

Abstract This paper presents a new comparison theorem for the oscillation of solutions of second-order nonlinear differential equations of the form ( ϕ p ( x ′ ) ) ′ + 1 t p f ( x ) = 0 , where p > 1 and ϕ p ( x ) = | x | p − 2 x and f ( x ) satisfies the signum condition x f ( x ) > 0 if x ≠ 0 , but is not assumed to be monotone. Combining the comparison theorem and known oscillation criteria, we can also derive new oscillation criteria for the equations. Proof is established by means of phase plane analysis of systems.

Oscillation criteria for second-order nonlinear difference equations of Euler type

AbstractThe purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for the second-order nonlinear difference equation Δ2x(n)+1n(n+1)f(x(n))=0, where f(x) is continuous on ℝ and satisfies the signum condition xf(x)>0 if x≠0. The obtained results are best possible in a certain sense. Proof is given by means of the Riccati technique and phase plane analysis of a system. A discrete version of the Riemann-Weber generalization of Euler-Cauchy differential equation plays an important role in proving our results.MSC:39A12, 39A21.

Applications of phase plane analysis of a Liénard system to positive solutions of Schrödinger equations

This paper deals with semilinear elliptic equations in an exterior domain of R N with N > 3. Sufficient conditions are obtained for the equation to have a positive solution which decays at infinity. The main result is proved by means of a supersolution-subsolution method presented by Noussair and Swanson. By using phase plane analysis of a system of Lienard type, a suitable positive supersolution is found out. Asymptotic decay estimation on a solution of the Lienard system gains a positive subsolution. Examples are given to illustrate the main result.

Oscillation constants for second-order nonlinear dynamic equations of Euler type on time scales

Abstract We are concerned with the oscillation problem for second-order nonlinear dynamic equations on time scales of the form , where f(x) satisfies if . By means of Riccati technique and phase plane analysis of a system, (non)oscillation criteria are established. A necessary and sufficient condition for all nontrivial solutions of the Euler–Cauchy dynamic equation to be oscillatory plays a crucial role in proving our results.

Influence of nonlinear perturbed terms on the oscillation of elliptic equations

Our concern is to solve the nonlinear perturbation problem for the semilinear elliptic equation Δu + p(x)u + Φ(x, u) = 0 in an exterior domain of R N with N > 3. The lower limit of the nonlinear perturbed term Φ(x, u) is given for all nontrivial solutions to be oscillatory. The tools for obtaining our theorems are the so-called supersolution-subsolution method and some results concerning the oscillation and nonoscillation of solutions of the ordinary differential equation associated with the elliptic equation. A simple example is given to illustrate the main results.

Decaying positive solutions of quasilinear elliptic equations in exterior domains in R2

Abstract The existence of decaying, positive solutions of quasilinear elliptic equations of the form Δu+φ(x,u)+x·∇u/|x|2=0 is established in an exterior domain Ω⊂ R 2 , under suitable smoothness and growth conditions. The main result is proved by means of a supersolution–subsolution method given by Noussair and Swanson. By using phase plane analysis of a system of Lienard type, a supersolution and a subsolution of the above equation are found out. An extension of the main result to more general case is also attempted. Finally, some examples are attached.

Comparison Theorems for Second-Order Damped Nonlinear Differential Equations

In this paper, we present comparison theorems for the oscillation of solutions of second-order damped nonlinear differential equations with p-Laplacian. Proof is given by means of phase plane analysis of systems. Moreover, combining the comparison theorem and (non)oscillation criteria for the generalized Euler differential equation, we give new (non)oscillation criteria for the damped equations.

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

This paper deals with an equivalent system to the nonlinear differential equation of Lienard type (φ(ẋ))˙+ f(x)φ(ẋ) + g(x) = 0, where the range of the function φ 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 Poincare–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.