Nonoscillation Theorems Research Articles

Partially extended oscillation and nonoscillation theorems for half‐linear Hill‐type differential equations with periodic damping

This study addressed the oscillation problems of half‐linear differential equations with periodic damping. The solution space of any linear equation is homogeneous and additive. Generally, by contrast, the solution space of half‐linear differential equations is homogeneous but not additive. Numerous oscillation and nonoscillation theorems have been devised for half‐linear differential equations featuring periodic functions as coefficients. However, in certain cases, such as applying Mathieu‐type differential equations to control engineering, which is a typical example of the Hill equation, some oscillation theorems cannot be applied. In this study, we established oscillation and nonoscillation theorems for half‐linear Hill‐type differential equations with periodic damping. To prove the results, we used the Riccati technique and the composite function method, which focuses on the composite function of the indefinite integral of the coefficients of the target equation and an appropriate multivalued continuously differentiable function. Furthermore, we discuss the special case of the oscillation constant of a damped half‐linear Mathieu equation.

Wintner-type nonoscillation theorems for conformable linear Sturm-Liouville differential equations

In this study, we addressed the nonoscillation of th Sturm-Liouville differential equation with a differential operator, which corresponds to a proportional-derivative controller. The equation is a conformable linear differential equation. A Wintner-type nonoscillation theorem was established to be applied to such equations. Using this theorem, we provided a sharp nonoscillation condition that guarantees that all nontrivial solutions to Euler-type conformable linear equations do not oscillate. The main nonoscillation theorems can be proven by introducing a Riccati inequality, which corresponds to the conformable linear equation of the Sturm-Liouville type.

Nonoscillation of damped linear differential equations with a proportional derivative controller and its application to Whittaker-Hill-type and Mathieu-type equations

The proportional derivative (PD) controller of a differential operator is commonly referred to as the conformable derivative. In this paper, we derive a nonoscillation theorem for damped linear differential equations with a differential operator using the conformable derivative of control theory. The proof of the nonoscillation theorem utilizes the Riccati inequality corresponding to the equation considered. The provided nonoscillation theorem gives the nonoscillatory condition for a damped Euler-type differential equation with a PD controller. Moreover, the nonoscillation of the equation with a PD controller that can generalize Whittaker-Hill-type equations is also considered in this paper. The Whittaker-Hill-type equation considered in this study also includes the Mathieu-type equation. As a subtopic of this work, we consider the nonoscillation of Mathieu-type equations with a PD controller while making full use of numerical simulations.

Moore-type nonoscillation criteria for half-linear difference equations

This paper deals with the half-liner difference equation $$\begin{aligned} \varDelta (r_n\phi _p(\varDelta x_n))+c_n\phi _p(x_{n+1})=0, \end{aligned}$$ where $$r_n$$ , $$c_n$$ are real-valued sequences, $$r_n>0$$ for $$n \in \mathbb {N} \cup \{0\}$$ , and $$\phi _p(z)=|z|^{p-2}z$$ with $$p>1$$ and $$\mathbb {N}$$ is the set of natural numbers. The purpose of this paper is to use the function transformation and Riccati technology to establish a half-linear difference equations nonoscillation theorem. Our results generalize earlier nonoscillation result of Doslý and Řehak (Comput Math Appl 42:453–464, 2001). Furthermore, in the case of $$p=2$$ , we can present two examples to determine that the solution of the linear difference equation are nonoscillatory even if $$\begin{aligned} \sum ^{n-1}r_{j}^{-1}\sum _{j=n}^{\infty }c_{j} \quad \text {or} \quad \sum _{j=n}^{\infty }r_{j}^{-1}\sum ^{n-1}c_{j} \end{aligned}$$ is less than the lower bound $$-3/4$$ .

Oscillation and nonoscillation theorems for Meissner’s equation

The purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for Meissner’s equation which is the special case of Hill’s equation. Proof is given by means of the Riccati technique. Furthermore, using Liouville transformation and Sturm’s comparison theorem, we show a relation between Meissner equations and Cauchy-Euler equations. Moreover, by numerical computations, we give an approximate value which is the borderline between oscillation and nonoscillation for Meissner equations.

Oscillation and Nonoscillation Theorems for a Class of Fourth Order Quasilinear Generalized α-Difference Equations

Oscillation and Nonoscillation Theorems for a Class of Fourth Order Quasilinear Generalized α-Difference Equations

Oscillation and nonoscillation theorems of neutral dynamic equations on time scales

We present the oscillation criteria for the following neutral dynamic equation on time scales: \t\t\t(y(t)−C(t)y(t−ζ))Δ+P(t)y(t−η)−Q(t)y(t−δ)=0,t∈T,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\bigl(y(t)-C(t)y(t-\\zeta )\\bigr)^{\\Delta }+P(t)y(t-\\eta )-Q(t)y(t-\\delta )=0, \\quad t\\in {\\mathbb{T}}, $$\\end{document} where C, P, Qin C_{mathit{rd}}([t_{0},infty ),{mathbb{R}}^{+}), {mathbb{R}} ^{+}=[0,infty ), gamma , eta , delta in {mathbb{T}} and gamma >0, eta >delta geq 0. New conditions for the existence of nonoscillatory solutions of the given equation are also obtained.

Oscillation problems for Hill's equation with periodic damping

This paper deals with the second-order linear differential equation x″+a(t)x′+b(t)x=0, where a and b are periodic coefficients. The main purpose is to present new criteria which guarantee that all nontrivial solutions are nonoscillatory and that those are oscillatory. Our nonoscillation theorem and oscillation theorem are proved by using the Riccati technique. In our theorem, the composite function of an indefinite integral of b and a suitable multiple-valued continuously differentiable function are focused, and the composite function of them plays an important role. The results obtained here include a result by Kwong and Wong [15] and a result by Sugie and Matsumura [26]. An application to a equation of Whittaker–Hill type is given to show the usefulness of our results. Finally, simulations are also attached to illustrate that our oscillation criterion is sharp.

Sharp oscillation and nonoscillation tests for linear difference equations

We consider difference equations with a single delay term and obtain sufficient conditions for both oscillation and nonoscillation. Moreover, our nonoscillation theorem improves the affirmative answer to Ladas’ corrected conjecture given by Tang and Yu in [Comput. Math. Appl. 38(11–12) (1999), pp. 229–237]. We also present a detailed discussion with examples to emphasize the significance and the sharpness of the new results.

Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation, II

The present paper deals with nonoscillation problem for the second-order linear difference equationcnxn+1+cn−1xn−1=bnxn,n=1,2,…,where {bn} and {cn} are positive sequences. All nontrivial solutions of this equation are nonoscillatory if and only if the Riccati-type difference equationqnzn+1zn−1=1has an eventually positive solution, where qn=cn2/(bnbn+1). Our nonoscillation theorems are proved by using this equivalence relation. In particular, it is focusing on the relation of the triple (q3k−2,q3k−1,q3k) for each k∈N. Our results can also be applied to not only the case that {bn} and {cn} are periodic but also the case that {bn} or {cn} is non-periodic. To compare the obtained results with previous works, we give some concrete examples and those simulations.

Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation

Nonoscillation theorems for second-order linear difference equations via the Riccati-type transformation

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.

Oscillation and Nonoscillation Theorems for a Class of Fourth Order Quasilinear Difference Equations

In this paper, we consider certain quasilinear difference equations$$(A)~~~~~~~~~~~~~~~\Delta^{2}(\mid\Delta^{2}y_{n}\mid^{\alpha-1}\Delta^{2}y_{n})+q_{n}\midy_{\tau(n)}\mid^{\beta-1}y_{\tau(n)}=0$$where \\(a) $\alpha,\beta $ are positive constants; \\(b) $\{q_{n}\}_{n_{0}}^{\infty}$ arepositive real sequences. $n_{0}\in N_{0}=\{1,2,\cdots \}$.Oscillation and nonoscillation theorems of the above equation is obtained.

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

The purpose of this paper is to present a pair of an oscillation theorem and a nonoscillation theorem for the second-order nonlinear difference equation where is continuous on ℝ and satisfies the signum condition if . 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.

Oscillation and nonoscillation of certain second order quasilinear dynamic equations

We establish new oscillation and nonoscillation theorems for the second order quasilinear dynamic equation (r(t)|y^{\Delta}(t)|^{\alpha-1}y^{\Delta}(t))^{\Delta}+f(t,y^{\sigma}(t))=0 on a time scale $\T$. Our results not only extend the results given in J. Wang, On second order quasilinear oscillations, Funkcialaj Ekvacioj, 41 (1998), 25-54, but also unify the oscillation and nonoscillation criteria for second order quasilinear differential equations and difference equations.

Nonoscillation Theorems for a Class of Fourth Order Quasilinear Dynamic Equations on Time Scales

In this paper,some sufficient and necessary conditions for nonoscillation of the fourth order quasilinear dynamic equations on time scales T are established. Our results as special case when T = R and T = N,involve and improve some known results.

On the Spectrum of the Magnetohydrodynamic Mean-Field $\alpha^2$-Dynamo Operator

The existence of magnetohydrodynamic mean-field $\alpha^2$-dynamos with spherically symmetric, isotropic helical turbulence function $\alpha$ is related to a non-self-adjoint spectral problem for a coupled system of two singular second order ordinary differential equations. We establish global estimates for the eigenvalues of this system in terms of the turbulence function $\alpha$ and its derivative $\alpha'$. They allow us to formulate an antidynamo theorem and a nonoscillation theorem. The conditions of these theorems, which again involve $\alpha$ and $\alpha'$, must be violated in order to reach supercritical or oscillatory regimes.

Oscillation and nonoscillation of solutions of second order linear dynamic equations with integrable coefficients on time scales

We obtain Willett–Wong-type oscillation and nonoscillation theorems for second order linear dynamic equations with integrable coefficients on a time scale. The results obtained extend and are motivated by oscillation and nonoscillation results due to Willett [20] and Wong [21] for the second order linear differential equation. As applications of the new results obtained, we give the complete classification of oscillation and nonoscillation for the difference equations Δ 2 x ( n ) + b ( - 1 ) n t c x ( n + 1 ) = 0 and Δ 2 x ( n ) + a t c + 1 + b ( - 1 ) n t c x ( n + 1 ) = 0 for t ∈ N , a , b , c ∈ R . We also improve a nonoscillation result of Mingarelli [17] and extend an oscillation result of Del Medico and Kong [7].

A nonoscillation theorem for half-linear differential equations with delay nonlinear perturbations

This paper deals with the oscillation problem for nonlinear differential equations with delay. A sufficient condition is obtained for the equation to have a nonoscillatory solution. The main result is the best possible in a certain sense. Examples are given to illustrate the main result.

Nonoscillation criteria for second‐order nonlinear differential equations with decaying coefficients

AbstractThis paper is concerned with the problem of deciding conditions on the coefficient q (t) and the nonlinear term g (x) which ensure that all nontrivial solutions of the equation (|x ′|α–1x ′)′ + q (t)g (x) = 0, α > 0, are nonoscillatory. The nonlinear term g (x) is not imposed no assumption except for the continuity and the sign condition xg (x) > 0 if x ≠ 0. In our problem, it is important to examine the relation between the decay of q (t) and the growth of g (x). Our main result extends some nonoscillation theorem for the generalised Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase‐plane analysis. A simple example is includes to show that the monotonicity of g (x) is not essential in our problem. Finally, elliptic equations with the m ‐Laplacian operator are discussed as an application to our results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)