Asymptotic Behavior Of Nonoscillatory Solutions Research Articles

A note on the asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

A note on the asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

We consider the half-linear differential equation \[(|x'|^{\alpha}\mathrm{sgn}\,x')' + q(t)|x|^{\alpha}\mathrm{sgn}\,x = 0, \quad t \geq t_{0},\] under the condition \[\lim_{t\to\infty}t^{\alpha}\int_{t}^{\infty}q(s)ds = \frac{\alpha^{\alpha}}{(\alpha+1)^{\alpha+1}}.\] It is shown that if certain additional conditions are satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as \(t\to\infty\).

New results for the non-oscillatory asymptotic behavior of high order differential equations of Poincaré type

<abstract><p>This paper discusses the study of asymptotic behavior of non-oscillatory solutions for high order differential equations of Poincaré type. We present two new and weaker hypotheses on the coefficients, which implies a well posedness result and a characterization of asymptotic behavior for the solution of the Poincaré equation. In our discussion, we use the scalar method: we define a change of variable to reduce the order of the Poincaré equation and thus demonstrate that a new variable can satisfies a nonlinear differential equation; we apply the method of variation of parameters and the Banach fixed-point theorem to obtain the well posedness and asymptotic behavior of the non-linear equation; and we establish the existence of a fundamental system of solutions and formulas for the asymptotic behavior of the Poincaré type equation by rewriting the results in terms of the original variable. Moreover we present an example to show that the results introduced in this paper can be used in class of functions where classical theorems fail to be applied.</p></abstract>

Asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations

This paper deals with the asymptotic behavior of solutions of the half-linear differential equation $$\begin{aligned} (p(t)|x'|^{\alpha }\mathrm {sgn}\,x')' + q(t)|x|^{\alpha }\mathrm {sgn}\,x = 0, \quad t \ge t_{0}. \end{aligned}$$ It will be shown that, for any solution x(t) of this equation, there are only two types of asymptotic behavior as $$t\rightarrow \infty $$ .

On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms

This paper is concerned with the asymptotic behavior of the nonoscillatory solutions of the forced fractional differential equation with positive and negative terms of the form \[^{C}D_{c}^{\alpha}y(t)+f(t,x(t))=e(t)+k(t)x^{\eta}(t)+h(t,x(t)),\] where \(t\geq c \geq 1\), \(\alpha \in (0,1)\), \(\eta \geq 1\) is the ratio of positive odd integers, and \(^{C}D_{c}^{\alpha}y\) denotes the Caputo fractional derivative of \(y\) of order \(\alpha\). The cases \[y(t)=(a(t)(x^{\prime}(t))^{\eta})^{\prime} \quad \text{and} \quad y(t)=a(t)(x^{\prime}(t))^{\eta}\] are considered. The approach taken here can be applied to other related fractional differential equations. Examples are provided to illustrate the relevance of the results obtained.

Asymptotic behavior of solutions of forced third-order dynamic equations

AbstractThis paper deals with asymptotic behavior of nonoscillatory solutions of certain third-order forced dynamic equations on time scales. The main goal is to investigate when all solutions behave at infinity like certain nontrivial nonlinear functions.

Nonoscillatory Solutions to Second-Order Neutral Difference Equations

We study asymptotic behavior of nonoscillatory solutions to second-order neutral difference equation of the form: Δ ( r n Δ ( x n + p n x n − τ ) ) = a n f ( n , x n ) + b n . The obtained results are based on the discrete Bihari type lemma and a Stolz type lemma.

On the Asymptotic Behavior of Nonoscillatory Solutions of Certain Fractional Differential Equations

We present the conditions under which every nonoscillator solution x(t) of the forced fractional differential equation $$\begin{aligned} ^{\mathrm{C}}D_{\mathrm{c}}^{\alpha } y ( t ) = e ( t ) +f ( {t, x ( t )} ), c > 1,\alpha \in ( {0,1} ), \quad \mathrm{{and}} \,\, \delta \ge 1, \end{aligned}$$ where $$y(t)= ( {a(t) ( {{x}'(t)} )^{\delta }})^{\prime },c_0 =\frac{y(c)}{\Gamma (1)}= y(c)$$ , is a real constant which satisfies $$\begin{aligned} |x(t)|=O\left( {t^{1/\delta }e^{t}\int _{\mathrm{c}}^{t} {a^{-1/\delta }} (s)\mathrm{d}s} \right) , \quad t \rightarrow \infty \end{aligned}$$ It is shown that the technique can be applied to some related fractional differential equations. Examples are inserted to illustrate the relevance of the obtained results.

On the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations

This paper deals with the asymptotic behavior of nonoscillatory solutions of fractional differential equations of the form CDaαy = e(t) + f(t,x), t ≥ a where 0 < α < 1, a ≥ 0, CDaαy denotes the Caputo fractional derivative of order α of y. The following particular cases are considered: y = (r(t)|x′|δ-1x′)′, (δ ≥ 1), y = x′, y = x. We offer a method that can be applied to investigate more general class of fractional differential equations as well.

On the existence of nonoscillatory solutions of three-dimensional time scale systems

We consider a three-dimensional nonlinear system of first order dynamic equations on time scales and show the existence and asymptotic behavior of nonoscillatory solutions by using the most well-known fixed point theorems. Examples are also provided, which validates our theoretical claims.

Properties of Kneser solutions for third-order differential equations

We study asymptotic behavior of nonoscillatory solutions for the third-order differential equations of the form[r(t)(y′(t))γ]′′+p(t)y(t)=0.For property A of studied equations we extend the information about the asymptotic properties of positive decreasing solutions.

On nonoscillatory solutions of Emden-Fowler dynamic systems on time scales

We study the existence and asymptotic behavior of nonoscillatory solutions of Emden -Fowler dynamic sytems on time scales. In order to show the existence, we use Schauder , Knaster and Tychonoff Fixed Point Theorems. Some examples are illustrated as well.

COMPORTAMIENTO ASINTÓTICO DE SOLUCIONES NO OSCILATORIAS DE CUARTO ORDEN DE ECUACIONES DIFERENCIALES LINEALES

This article deals with the asymptotic behavior of nonoscillatory solutions of fourth order linear differential equation where the coefficients are perturbations of linear constant coefficient equation. We define a change of variable and deduce that the new variable satisfies a third order nonlinear differential equation. We assume three hypotheses. The first hypothesis is related to the constant coefficients and set up that the characteristic polynomial associated with the fourth order linear equation has simple and real roots. The other two hypotheses are related to the behavior of the perturbation functions and establish asymptotic integral smallness conditions of the perturbations. Under these general hypotheses, we obtain four main results. The first two results are related to the application of a fixed point argument to prove that the nonlinear third order equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the nonlinear third order equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise the formulas for the asymptotic behavior of the linear fourth order differential equation. In addition, we present an example to show that the results introduced in this paper can be applied in situations where the assumptions of some classical theorems are not satisfied.

Asymptotic Behavior of Certain Integrodifferential Equations

This paper deals with asymptotic behavior of nonoscillatory solutions of certain forced integrodifferential equations of the form:atx′t′=e(t)+∫ct‍(t-s)α-1k(t,s)f(s,x(s))ds, c&gt;1, 0&lt;α&lt;1.From the obtained results, we derive a technique which can be applied to some related integrodifferential as well as integral equations.

Asymptotic integration of a linear fourth order differential equation of Poincaré type

This article deals with the asymptotic behavior of nonoscillatory solutions of fourth order linear differential equation where the coefficients are perturbations of con- stants. We define a change of variable and deduce that the new variable satisfies a third order nonlinear differential equation. We assume three hypotheses. The first hypoth- esis is related to the constant coefficients and set up that the characteristic polynomial associated with the fourth order linear equation has simple and real roots. The other two hypotheses are related to the behavior of the perturbation functions and establish asymptotic integral smallness conditions of the perturbations. Under these general hy- potheses, we obtain four main results. The first two results are related to the application of a fixed point argument to prove that the nonlinear third order equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the nonlinear third order equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise the formulas for the asymptotic behavior of the linear fourth order differential equation. In addition, we present an example to show that the results introduced in this paper can be applied in situations where the assumptions of some classical theorems are not satisfied.

Asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations

In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales.

Dynamics of Third-Order Nonlinear Neutral Equations

The aim of this paper is to study oscillatory and asymptotic properties of the third-order nonlinear neutral equation with continuously distributed delays of the form(r(t)([xt+∫abpt,μxτt,μdμ]′′)α)′+∫cdqt,ξfαxgt,ξdξ=0. Applying suitable generalized Riccati transformation and integral averaging technique, we present new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of this equation. Obtained results essentially improve and complement earlier ones.

Asymptotic Behavior of Non-oscillatory Solutions of Second order Integro-Dynamic Equations on Time Scales

In this paper, we investigate some new criteria on the asymptotic behavior of non-oscillatory solutions of second order integro-dynamic equations on time-scales. We also provide some numerical examples to illustrate the relevance of the obtained results.

On the Existence and the Asymptotic Behavior of Nonoscillatory Solutions of Second Order Quasilinear Difference Equations

On the Existence and the Asymptotic Behavior of Nonoscillatory Solutions of Second Order Quasilinear Difference Equations

Nonoscillatory Solutions of Second‐Order Superlinear Dynamic Equations with Integrable Coefficients

The asymptotic behavior of nonoscillatory solutions of the superlinear dynamic equation on time scales , γ &gt; 1, is discussed under the condition that exists and P(t) ≥ 0 for large t.