Browsing by Subject "oscillation"

Now showing 1 - 20 of 22

Access status: Open Access ,
General solutions of second-order linear difference equations of Euler type
(2017) Hongyo, Akane; Yamaoka, Naoto
The purpose of this paper is to give general solutions of linear difference equations which are related to the Euler-Cauchy differential equation $y^{\prime\prime}+(\lambda/t^2)y=0$ or more general linear differential equations. We also show that the asymptotic behavior of solutions of the linear difference equations are similar to solutions of the linear differential equations.
Access status: Open Access ,
Kneser-type oscillation criteria for second-order half-linear advanced difference equations
(Wydawnictwa AGH, 2022) Indrajith, N.; Graef, John R.; Thandapani, Ethiraju
The authors present Kneser-type oscillation criteria for a class of advanced type second-order difference equations. The results obtained are new and they improve and complement known results in the literature. Two examples are provided to illustrate the importance of the main results.
Access status: Open Access ,
Monotonic properties of Kneser solutions of second order linear differential equations with delayed argument
(Wydawnictwa AGH, 2025) Baculíková, Blanka
In this paper new monotonic properties of nonoscillatory solutions for second order linear functional differential equations with delayed argument $\textcolor{white}\$y{''}(t)=p(t)y(\tau(t))\textcolor{white}\$$ have been established. New properties are used to introduce criteria for elimination of bounded nonoscillatory solutions for studied equations.
Access status: Open Access ,
New aspects for the oscillation of first-order difference equations with deviating arguments
(Wydawnictwa AGH, 2022) Attia, Emad R.; El-Matary, Bassant M.
We study the oscillation of first-order linear difference equations with non-monotone deviating arguments. Iterative oscillation criteria are obtained which essentially improve, extend, and simplify some known conditions. These results will be applied to some numerical examples.
Access status: Open Access ,
New oscillation conditions for first-order linear retarded difference equations with non-monotone arguments
(Wydawnictwa AGH, 2022) Attia, Emad R.; El-Matary, Bassant M.; Chatzarakis, George E.
In this paper, we study the oscillatory behavior of the solutions of a first-order difference equation with non-monotone retarded argument and nonnegative coefficients, based on an iterative procedure. We establish some oscillation criteria, involving $\lim \sup$, which achieve a marked improvement on several known conditions in the literature. Two examples, numerically solved in MAPLE software, are also given to illustrate the applicability and strength of the obtained conditions.
Access status: Open Access ,
New oscillation constraints for even-order delay differential equations
(Wydawnictwa AGH, 2023) Moaaz, Osama; Anis, Mona; El-Deeb, Ahmed A.; Elshenhab, Ahmed M.
The purpose of this paper is to study the oscillatory properties of solutions to a class of delay differential equations of even order. We focus on criteria that exclude decreasing positive solutions. As in this paper, this type of solution emerges when considering the noncanonical case of even equations. By finding a better estimate of the ratio between the Kneser solution with and without delay, we obtain new constraints that ensure that all solutions to the considered equation oscillate. The new findings improve some previous findings in the literature.
Access status: Open Access ,
Novel oscillation criteria for third-order semi-canonical differential equations with an advanced neutral term
(Wydawnictwa AGH, 2025) Vidhyaa, Kumar S.; Thandapani, Ethiraju; Tunç, Ercan
The main purpose of this paper is to present new oscillation results for nonlinear semi-canonical third-order differential equations with an advanced neutral term. The main idea is first by reducing the studied semi-canonical equation into standard canonical type equation without assuming any extra conditions. Then, by using the comparison method and integral averaging technique, sufficient conditions are established to ensure the oscillation of the reduced canonical equation, which in turn leads to the oscillation of the original equation. Therefore, the technique used here is very useful since the results already known for the canonical equations can be applied to obtain the oscillation of the semi-canonical equations. Two examples are provided to illustrate the importance of the main results
Access status: Open Access ,
On nonoscillatory solutions of two dimensional nonlinear delay dynamical systems
(2016) Öztürk, Özkan; Akın, Elvan
We study the classification schemes for nonoscillatory solutions of a class of nonlinear two dimensional systems of first order delay dynamic equations on time scales. Necessary and sufficient conditions are also given in order to show the existence and nonexistence of such solutions and some of our results are new for the discrete case. Examples will be given to illustrate some of our results.
Access status: Open Access ,
On oscillatory behaviour of third-order half-linear dynamic equations on time scales
(Wydawnictwa AGH, 2022) Grace, Said R.; Chhatria, Gokula Nanda
In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case $\mathbb{T}=\mathbb{R}$ or $\mathbb{T}=\mathbb{Z}$.
Access status: Open Access ,
On oscillatory solutions of certain difference equations
(2006) Grzegorczyk, Grzegorz; Werbowski, Jarosław
Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions foroscillation of solutions of the equations discussed.
Access status: Open Access ,
Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments
(2012) Elabbasy, Elmetwally M.; Hassan Abdelmonem, Taher Saleh; Moaaz, O.
Oscillation criteria are established for second order nonlinear neutral differential equations with deviating arguments of the form $r(t)\psi(x(t))|z'(t)|^{\alpha -1} z'(t)+ \int_a^b q(t,\xi)f(x(g(t,\phi)))d\sigma (\xi) =0,\quad t\gt t_0,$ where $\alpha \gt 0$ and $z(t)= x(t)+p(t)x(t-\tau)$. Our results improve and extend some known results in the literature. Some illustrating examples are also provided to show the importance of our results
Access status: Open Access ,
Oscillation conditions for difference equations with several variable delays
(Wydawnictwa AGH, 2023) El-Matary, Bassant M.; El-Morshedy, Hassan A.; Benekas, Vasileios; Stavroulakis, Ioannis P.
A technique is developed to establish a new oscillation criterion for a first-order linear difference equation with several delays and non-negative coefficients. Our result improves recent oscillation criteria and covers the cases of monotone and non-monotone delays. Moreover, the paper is concluded with an illustrative example to show the applicability and strength of our result.
Access status: Open Access ,
Oscillation criteria for even order neutral difference equations
(Wydawnictwa AGH, 2019) Selvarangam, Srinivasan; Rupadevi, S. A.; Thandapani, Ethiraju; Pinelas, Sandra
In this paper, we present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form $\Delta^m(x_n+ax_{n-\tau_1}+bx_{n+\tau_2})+p_nx_{n-\sigma_1}^{\alpha}+q_nx_{n+\sigma_2}^{\beta}=0,\quad n\geq n_0\gt0,$ where $m\geq 2$ is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results.
Access status: Open Access ,
Oscillation criteria for linear difference equations with several variable delays
(Wydawnictwa AGH, 2021) Benekas, Vasileios; Garab, Ábel; Kashkynbayev, Ardak; Stavroulakis, Ioannis P.
We obtain new sufficient criteria for the oscillation of all solutions of linear delay difference equations with several (variable) finite delays. Our results relax numerous well-known limes inferior-type oscillation criteria from the literature by letting the limes inferior be replaced by the limes superior under some additional assumptions related to slow variation. On the other hand, our findings generalize an oscillation criterion recently given for the case of a constant, single delay.
Access status: Open Access ,
Oscillation criteria for third order nonlinear delay differential equations with damping
(2015) Grace, Said R.
This note is concerned with the oscillation of third order nonlinear delay differential equations of the form $\left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{$\ast$}$ $(*)$ In the papers [A.Tiryaki, M.F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M.F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear-functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation $(*)$ oscillates or converges to zero, provided that the second order equation $\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{$\ast\ast$}$ $(**)$ is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation $(*)$ oscillates if equation $(**)$ is nonoscillatory. We also establish results for the oscillation of equation $(*)$ when equation $(**)$ is oscillatory.
Access status: Open Access ,
Oscillation of even order linear functional differential equations with mixed deviating arguments
(Wydawnictwa AGH, 2022) Baculíková, Blanka
In the paper, we study oscillation and asymptotic properties for even order linear functional differential equations $y^{(n)}(t)=p(t)y(\tau(t))$ with mixed deviating arguments, i.e. when both delayed and advanced parts of $\tau(t)$ are significant. The presented results essentially improve existing ones.
Access status: Open Access ,
Oscillation theorems concerning non-linear differential equations of the second order
(2011) Elabbasy, Elmetwally M.; Elzeiny, Sh. R.
This paper concerns the oscillation of solutions of the differential eq. $\left[ r\left( t\right) \psi \left(x\left( t\right) \right) f\text{ }( \overset{\cdot }{x}(t))\right]^{\cdot }+q\left( t\right) \varphi (g\left( x\left( t\right) \right), r\left( t\right) \psi \left( x\left( t\right) \right) f(\overset{\cdot }{x}(t)))=0,$ where $u\varphi \left( u,v\right) \gt 0$ for all $u\neq 0$, $xg\left( x\right) \gt 0$, $xf\left( x\right)\gt 0$ for all $x\neq 0$, $\psi \left( x\right) \gt 0$ for all $x\in \mathbb{R}$, $r\left( t\right) \gt 0$ for $t\geq t_{0}\gt 0$ and $q$ is of arbitrary sign. Our results complement the results in [A.G. Kartsatos, On oscillation of nonlinear equations of second order, J. Math. Anal. Appl. 24 (1968), 665–668], and improve a number of existing oscillation criteria. Our main results are illustrated with examples.
Access status: Open Access ,
Oscillatory behavior of second-order damped differential equations with a superlinear neutral term
(Wydawnictwa AGH, 2020) Tunç, Ercan; Özdemir, Osman
This article concerns the oscillatory behavior of solutions to second-order damped nonlinear differential equations with a superlinear neutral term. The results are obtained by a Riccati type transformation as well as by an integral criterion. Examples illustrating the results are provided and some suggestions for further research are indicated.
Access status: Open Access ,
Oscillatory criteria for second order differential equations with several sublinear neutral terms
(Wydawnictwa AGH, 2019) Baculíková, Blanka
In this paper, sufficient conditions for oscillation of the second order differential equations with several sublinear neutral terms are established. The results obtained generalize and extend those reported in the literature. Several examples are included to illustrate the importance and novelty of the presented results.
Access status: Open Access ,
Oscillatory criteria via linearization of half-linear second order delay differential equations
(Wydawnictwa AGH, 2020) Baculíková, Blanka; Džurina, Jozef
In the paper, we study oscillation of the half-linear second order delay differential equations of the form $\left(r(t)(y'(t))^{\alpha}\right)'+p(t)y^{\alpha}(\tau(t))=0.$ We introduce new monotonic properties of its nonoscillatory solutions and use them for linearization of considered equation which leads to new oscillatory criteria. The presented results essentially improve existing ones.