Browsing by Subject "second order"

Now showing 1 - 6 of 6

Access status: Open Access ,
About sign-constancy of Green's functions for impulsive second order delay equations
(2014) Domoshnitsky, Alexander; Landsman, Guy; Yanetz, Shlomo
We consider the following second order differential equation with delay $\begin{cases} (Lx)(t)\equiv{x''(t)+\sum_{j=1}^p {b_{j}(t)x(t-\theta_{j}(t))}}=f(t), \quad t\in[0,\omega],\\ x(t_j)=\gamma_{j}x(t_j-0), x'(t_j)=\delta_{j}x'(t_j-0), \quad j=1,2,\ldots,r. \end{cases}$ In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality $\sum_{i=1}^p{b_i(t)\left(\frac{1}{4}+r\right)}\lt \frac{2}{\omega^2}$ is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case $0\lt \gamma_i\leq{1}$, $0\lt \delta_i\leq{1}$ for $i=1,\ldots ,p$.
Access status: Open Access ,
An abstract nonlocal second order evolution problem
(2012) Byszewski, Ludwik; Winiarska, Teresa
The aim of the paper is to prove theorems on the existence and uniqueness of mild and classical solutions of a semilinear evolution second order equation together with nonlocal conditions. The theory of strongly continuous cosine families of linear operators in a Banach space is applied.
Access status: Open Access ,
Forced oscillation and asymptotic behavior of solutions of linear differential equations of second order
(Wydawnictwa AGH, 2022) Shoukaku, Yutaka
The paper deals with the second order nonhomogeneous linear differential equation $(p(t) y'(t))' + q(t) y(t) = f(t),$ which is oscillatory under the assumption that $p(t)$ and $q(t)$ are positive, continuously differentiable and monotone functions on $[0,\infty)$. Throughout this paper we shall use pairs of quadratic forms, which obtained by different methods than Kusano and Yoshida. This form will lead to a property of qualitative behavior, including amplitudes and slopes, of oscillatory solutions of the above equation. In addition, we will discuss the existence of three types (moderately bounded, small, large) of oscillatory solutions, which are based on results due to Kusano and Yoshida.
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 ,
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 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.