Oscillatory Behavior Of Solutions Research Articles

ON OSCILLATIONS OF PERTURBED SECOND ORDER DIFFERENTIAL EQUATION

A perturbed second order linear differential equation is considered. A criteria concerning the oscillatory behaviour of solutions of the perturbed equation based on the perturbing function and the solutions of the associated unperturbed equation is given.

Asymptotic behavior of oscillatory solutions of a differential equation with deviating arguments

Asymptotic behavior of oscillatory solutions of a differential equation with deviating arguments

Oscillation criteria for a class of nonlinear partial differential equations

There is much current interest in studying the oscillatory behavior of solutions of hyperbolic equations. The problem of establishing oscillation criteria for characteristic initial value problems for hyperbolic equations has been investigated by several authors. The reader is referred to Kreith [3] and Pagan [8,9] for linear hyperbolic equations and to Travis and Yoshida [lo] and Yoshida [ 1 l] for nonlinear hyperbolic equations. Our objective here is to provide oscillation theorems for the characteristic initial value problem for the differential operator L defined by

Oscillation criteria for a class of nonlinear vector delay-differential equations

Oscillation criteria are obtained for a class of vector delay differential equation of the form $$u''\left( t \right) + F_1 \left( {t,u\left( t \right),u\left( {g\left( t \right)} \right)} \right)u\left( t \right) + F_2 \left( {t,u\left( t \right),u\left( {g\left( t \right)} \right)} \right)u\left( {g\left( t \right)} \right) = 0$$ t≥0, u e C2 ([0, ∞),Rn), where F1 and F2 are continuous nonnegative valued functionals in [0, ∞)×Rn×Rn, and g is a continuous real valued function with g(t)≤t, g(t)→∞ as t→ ∞. A solution u: [0, ∞)→Rn is called h-oscillatory in [0, ∞) whenever the scalar product [u(t), h] (│h│=1) has zeros in [a, ∞) with a arbitrary large. The method involves oscillatory behaviour of solutions of a nonlinear scalar delay differential inequality satisfied by h-nonoscillatory solutions of the above equation.

Oscillation theorems for second-order functional differential equations

This paper presents some comparison theorems on the oscillatory behavior of solutions of second-order functional differential equations. Here we state one of the main results in a simplified form: Let q, τ 1, τ 2 be nonnegative continuous functions on (0, ∞) such that τ 1 − τ 2 is a bounded function on [1, ∞) and t − τ 1( t) → ∞ if t → ∞. Then y ̈ (t) + q(t) y(t − τ 1(t)) = 0 is oscillatory if and only if y ̈ (t) + q(t) y(t − τ 2(t)) = 0 is oscillatory.

Oscillation, nonoscillation, and growth of solutions of nonlinear functional differential equations of arbitrary order

This paper discusses the growth and oscillatory behavior of solutions of n-th-order nonlinear functional differential equations. Sufficient conditions for all solutions or all bounded solutions to be nonoscillatory are given. An extensive bibliography is also included.

On the oscillation of a class of fourth order differential equations

1. Introduction. This paper is concerned with fourth order dif­ferential equations of the form (L) (p(x)y) - q(x)y - r(x)y = 0, where p, q and r are assumed to be continuous, real-valued functions on the interval [a, °° ). In addition, it will be assumed throughout that p > 0, q ^ 0 and r^Oon [a, o° ), with r not identically zero on any subinterval. If q is a (non-negative) constant, then (L) is self-adjoint; otherwise (L) is non-self-adjoint. The objective of the paper is to study the oscillatory behavior of the solutions of (L). A non-trivial y is oscillatory if the set of zeros of y is not bounded above. If the set of zeros of y is bounded above, which implies y has only finitely many zeros, then y is non-oscillatory. Hereafter, the term solution will be interpreted to mean non-trivial solution. Various special cases of (L) have been studied in detail. In par­ticular, we refer to the fundamental work of W. Leigh ton and Z. Nehari [5, Part I] on the self-adjoint equation (1) (P(x)y) - r(x)y = 0. M. Keener [3, Part I] continued the investigation of (1), concentrat­ing on the oscillatory behavior of solutions. S. Hastings and A. Lazer [2] considered the self-adjoint equation (2) y(4) - r(x)y = 0, showing that (2) has a linearly independent pair of bounded oscillatory solutions when it is assumed that r E C ' [a, «> ), with r > 0 and r ' ^ 0 on [a, oo ). S. Ahmad [1] has also studied (2), giving necessary and sufficient conditions for the existence of a linearly independent pair of oscillatory solutions. Finally, we refer to the work of V. Pudei [6], [7] in which the equation (3) yW - q(x)y - r(x)y = 0 is considered.

Remarks on the oscillatory behavior of solutions of functional differential equations with deviating argument

Remarks on the oscillatory behavior of solutions of functional differential equations with deviating argument

6.—Oscillation Theory for Semilinear Schrödinger Equations and Inequalities

SynopsisSufficient conditions are derived for every solution of a nonlinear Schrödinger equation (or inequality) to be oscillatory in an exterior domain ofEn. Such results apply in particular to then-dimensional Emden-Fowler equation. The method involves oscillatory behaviour of solutions of a nonlinear ordinary differential inequality satisfied by the spherical mean of a positive solution of the Schrödinger equation.

Oscillatory behavior of solutions of y″ + p( x) y = f( x)

Oscillatory behavior of solutions of y″ + p( x) y = f( x)

On Oscillatory Solutions of Certain Fourth Order Linear Differential Equations

We consider the fourth order linear homogeneous differential equation $y^{(4)} + p_3 (x)y''' + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = 0$ with continuous coefficients. Our studies center on the oscillatory behavior of solutions of the above equation under the separate disconjugacy conditions $r_{22} = r_{31} = \infty $ and $r_{22} = r_{13} = \infty $.

The Behavior of Oscillatory Solutions of $x''(t) + p(t)g(x(t)) = 0$

Various quantitative properties of oscillatory solutions of the scalar second order nonlinear differential equation $x'' + p(t)g(x) = 0$ are obtained under appropriate hypotheses on p and g. In particular, letting $\{ {t_i } \}_{i = 1}^\infty $, $0 < t_i < t_{i + 1} $, $t_i \to \infty $, as $i \to \infty $ be the zeros of any solution $x(t)$, we obtain inequalities on $J_i^{{\operatorname{def}}} \equiv \int _{t_i }^{t_{i + 1} } g(x(t))dt$ which yield asymptotic behavior on $x(t)$. For example, it is shown that $\lim_{t \to \infty } \int _0^t g(x(s))ds$ exists and is finite; moreover, assuming an added growth condition on ${{g(x)} / x}$, we have then that $\lim _{t \to \infty } \int _0^t x(s)ds$ exists and is finite.

On solutions of certain self-adjoint differential equations of fourth order

In this paper the author considers the differential equation [ r( x) y″]″ − p( x) y = 0, where r( x) and p( x) are continuous, r( x) > 0, and p( x) ≠ 0 on an interval [ϵ, ∞), ϵ > 0. The cases for which p( x) is positive and p( x) is negative on [ϵ, ∞) are treated separately. In the first part of the paper it is assumed that p( x) is positive on [ϵ, ∞). A study is made of the general properties of solutions of the equation, particularly, oscillatory solutions. Special emphasis is given the case when all oscillatory solutions are bounded. A rather simple representation for all bounded oscillatory solutions is given. The function p( x) is assumed to be negative in the second part of the paper. Some of Marko Svec's results on the behavior of oscillatory solutions are extended to this more general equation. Also two theorems analogus to the Bôcher-Osgood theorem for second-order equations are given. Finally, the author gives some necessary conditions that certain types of nonoscillatory solutions exist.

Oscillation theorems for linear second order ordinary differential equations

Abstract : The report describes the oscillatory behavior of solutions of the equation (1) x double prime + a(t)x=0, where a(t) is locally integrable on the interval from zero to, but not including, infinity. The main result is an extension of a nonoscillation theorem due to Hartman, the contrapositive of which is a useful criterion for equation (1) to be oscillatory. (Author)