Tomoyuki Tanigawa

Oscillation and nonoscillation theorems for a class of even-order quasilinear functional differential equations

We are concerned with the oscillatory and nonoscillatory behavior of solutions of even-order quasilinear functional differential equations of the type, where and are positive constants, and are positive continuous functions on, and is a continuously differentiable function such that,. We first give criteria for the existence of nonoscillatory solutions with specific asymptotic behavior, and then derive conditions (sufficient as well as necessary and sufficient) for all solutions to be oscillatory by comparing the above equation with the related differential equation without deviating argument.

Existence and precise asymptotics of positive solutions for a class of nonlinear differential equations of the third order

Abstract. Our purpose here is to show that an application of the theory of regular variation (in the sense of Karamata) gives the possibility of determining the existence and precise asymptotic behavior of positive solutions of the third-order nonlinear differential equation , where and are positive constants satisfying , is a continuous regularly varying function and , .

Nonoscillatory solutions of planar half-linear differential systems: a Riccati equation approach

In this paper an attempt is made to depict a clear picture of the overall structure of nonoscillatory solutions of the first order half-linear differential system x′ − p(t)φ1/α(y) = 0, y′ + q(t)φα(x) = 0, (A) where α > 0 is a constant, p(t) and q(t) are positive continuous functions on [0, ∞), and φγ(u) = |u|γsgn u, u ∈ R, γ > 0. A systematic analysis of the existence and asymptotic behavior of solutions of (A) is proposed for this purpose. A special mention should be made of the fact that all possible types of nonoscillatory solutions of (A) can be constructed by solving the Riccati type differential equations associated with (A). Worthy of attention is that all the results for (A) can be applied to the second order half-linear differential equation (p(t)φα(x)) + q(t)φα(x) = 0, (E) to build automatically a nonoscillation theory for (E).

Regularly Varying Solutions of Half-Linear Diffferential Equations with Retarded and Advanced Arguments

Abstract Sharp conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the half-linear functional differential equation with both retarded and advanced arguments of the form (|x′(t)|α sgn x′(t))′ ± p(t)|x(g(t))|α sgn x(g(t)) ± q(t)|x(h(t))|α sgn x(h(t)) = 0, where α > 0 is a constant, p, q: [a,∞) → (0,∞), a ≧ 0 are continuous functions, g, h are continuous and increasing with g(t) < t, h(t) > t for t ≥ a and .

Effect of nonlinear perturbations on second order linear nonoscillatory differential equations

Abstract. The aim of this paper is to show that any second order nonoscillatory linear differential equation can be converted into an oscillating system by applying a “sufficiently large”nonlinear perturbation. This can be achieved through a detailed analysis of possible nonoscillatory solutions of the perturbed differential equation which may exist when the perturbation is “sufficiently small”. As a consequence the class of oscillation-generating perturbations is determined precisely with respect to the original nonoscillatory linear equation.