Browsing by Author "Grace, Said R."

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Access status: Open Access ,
Asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations
(2014) Bohner, Martin; Grace, Said R.; Sultana, Nasrin
In this paper, we establish some new criteria on the asymptotic behavior of nonoscillatory solutions of higher-order integro-dynamic equations on time scales.
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 the asymptotic behavior of nonoscillatory solutions of certain fractional differential equations with positive and negative terms
(Wydawnictwa AGH, 2020) Graef, John R.; Grace, Said R.; Tunç, Ercan
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.
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 ,
Oscillatory and asymptotic behavior of a third-order nonlinear neutral differential equation
(Wydawnictwa AGH, 2017) Graef, John R.; Tunҫ, Ercan; Grace, Said R.
This paper discusses oscillatory and asymptotic properties of solutions of a class of third-order nonlinear neutral differential equations. Some new sufficient conditions for a solution of the equation to be either oscillatory or to converges to zero are presented. The results obtained can easily be extended to more general neutral differential equations as well as to neutral dynamic equations on time scales. Two examples are provided to illustrate the results.
Access status: Open Access ,
Oscillatory behavior of even-order nonlinear differential equations with a sublinear neutral term
(Wydawnictwa AGH, 2019) Graef, John R.; Grace, Said R.; Tunç, Ercan
The authors present a new technique for the linearization of even-order nonlinear differential equations with a sublinear neutral term. They establish some new oscillation criteria via comparison with higher-order linear delay differential inequalities as well as with first-order linear delay differential equations whose oscillatory characters are known. Examples are provided to illustrate the theorems.