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Oscillation of second-order perturbed nonlinear difference equations
Abstract
, 144 (2-3) (2003), 305-324.
... In fact, in the last few years several monographs and hundreds of research papers have been written, see for example the monographs [1, 2] and the papers345 and the references therein. Recently, Saker [17] considered the second-order perturbed nonlinear difference equation ...
... Using the Riccati transformation technique, the author presented some new oscillation criteria for Eq.(1.1) under the condition (1.2) or (1.3) which improved many known criteria discussed in [1, 3, 5, 8, 14, 15, 26, 27, 28, 30, 31]. Like [17], throughout this paper we will assume that there exist two real sequences {q n } ∞ n=1 and {p n } ∞ n=1 such that q n − p n ≥ 0, and ...
... If every solution of Eq.(1.1) is oscillatory, we say Eq.(1.1) is oscillatory. In the recent paper of Saker [17], we note that the main results, Theorem 2.1 and Theorem 2.5, hold only for the case when γ ≥ β and for the case when γ > 1, respectively. It is natural to ask whether Eq.(1.1) is oscillatory for the cases when γ < β and 0 < γ < 1. ...
Using the Riccati transformation techniques we will establish some new oscillation criteria for the second-order perturbed nonlinear difference equationwhere γ>0 is a quotient of odd positive integers. Some comparison between our theorems and those previously known results are indicated. Examples are interested in the text to illustrate the relevance of our results.
... By a solution of equation (1) we mean a sequence x : N k → R which satisfies (1) for every n ∈ N k . There has been an interest of many authors to study properties of solutions of the second-order neutral difference equations attract attention; see the papers [6], [8]- [10], [14]- [17], [20]- [22], [25]- [26] and the references therein. The interesting oscillatory results for first order and even order neutral difference equations can be found in [13], [18] and [19]. ...
... which is obtained from (22). We find that ...
\noindent We consider the neutral difference equation of the following form
\begin{equation*} \Delta \left(r_{n}\left(\Delta
\left(x_{n}+p_{n}x_{n-k}\right) \right) ^{\gamma}\right)
+q_{n}x_{n}^{\alpha}+a_{n}f(x_{n})=0. \end{equation*}% where
$x:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$,
$a,p,q:{\mathbb{N}}%_{0}\rightarrow {\mathbb{R}}$,
$r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}% \setminus \{0\}$, $f\colon
{\mathbb{R}}\rightarrow {\mathbb{R}}$ is a continuous function, and $k$ is a
given positive integer, $\gamma \leq 1$ is ratio of odd positive integers,
$\alpha $ is a nonnegative constant. %$\sum a_{n}\left(t\right)$ converges
uniformly on ${\mathbb{R}}$. %Here $\bN_0\colon =\left\{0,1,2, \dots \right\}$
and $\bN_k \colon = \left\{k, k+1, -k+2, \dots \right\}$ where $k$ is a given
positive integer. Sufficient conditions for the existence of a bounded solution
are obtained. Also, stability and asymptotic stability are studied. Some
earlier results are generalized. {\small \textbf{Keywords} Difference equation,
measures of noncompactness, Darbo's fixed point theorem, boundedness,
stability} {\small \textbf{AMS Subject classification} 39A10, 39A22, 39A30}
... Some important studies of them are as follows: stability, controllability, solutions of dynamic equation on time scale (DETS) of several orders, oscillation, dynamic inequality and qualitative analysis of dynamic equations on time scales, etc. We can follow all those contributions of the researchers by their papers that basically a time scale is an arbitrary non-empty closed subset of the real numbers i.e., R, which is denoted by T and the inherits the standard topology of R. It may be in any of these forms: [1,2,3,4,5,6,7,11,12,13,15,16,17,19,21,23,26,27,28,29,30,31,33,35,36,37,38,41,42,43,45,46,48,49,50,51,52,53,54,55,57,61,62,63,64,65,67,68,71,72,74,75,76,82,84,89,90,93,94,95,96,97,98,99,104,105,106,109,111,113,115,116,117,119,121,122,124,126,130,132,133,134,135,136,138,141,142,145,150,152,154,155,156,157,158,159,162,163] and references given therein. ...
By the dynamic equations on time scale, we mean an equation that keeps the equations of continuous, discrete and quantum calculus within themselves in the same equation. Here, the time scale is a non-empty closed subset of real numbers. Moreover, the significance of this equation is very much evident in those circumstances where we need to deal with differential and difference equations together. By keeping the applications under observation, a lot of studies of several orders of this equation have been done by many authors. Through this thesis, we present several oscillatory results for the first and second-order dynamic equations on a time scale. Our studies are more general to the studies given in the literature. Furthermore, we establish the results by using less restrictive conditions as compared to the existing conditions in the literature. These conditions are easy to verify and implement. As an application, by means of coincidence degree theory, we establish the existence of a positive periodic solution of the general N-prey and M-predator model on the time scale. On the other hand, we also establish a few sufficient conditions for oscillation of the p-Laplacian dynamic equation on the time scale for which we use a relaxed technique that compliments the existing techniques to prove oscillatory results. Further, we also use the Riccati transformation technique to transform second-order dynamic equations into the first-order dynamic equation. Furthermore, we derive some important inequalities and directly utilize the use of a well-known Young’s inequality for some of the oscillatory results. In addition, we make the conditions of our findings such that we can easily demonstrate the well- known Kamenev and Philos-type oscillation criteria for our dynamic equations on the time scale. Besides, our contribution is not limited to this, we provide a new trend of finding a derivative of continuous, discrete, and quantum calculus. This derivative is denoted by the name “black-delta” (symbol N) -derivative on time scale. Through this derivative, we put together the usual and discrete derivatives at the same time. Further, it has several applications in various fields, for instance, engineering, population dynamics, biology, economics, social sciences, and quantum physics, etc. Some fundamental results, associated with this derivative, are presented. From the point of application purposes, a necessary and sufficient condition for this derivative is also provided. Moreover, a drastic connection between the well-known Hilger derivative and new derivative on the time scale is demonstrated which makes our outcome better in comparison to Hilger-derivative on some time scale. Furthermore, some crucial examples are presented so that we could shed light on the practicability and effectiveness of our results.
... and studied the oscillation by using Riccati techneque and under some assumptions on forcing term H such that |H ∆ (t, u, v)| ≥ p(t)|u| and uH ∆ (t, u, v) < for u ≠ . Saker [14] has established some new oscillation criteria for the second-order perturbed nonlinear di erence equations having form ...
By using generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of the second-order dynamic equation on time scales. Some oscillatory results
of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.
... Difference equations are used in mathematical models in diverse areas such as economy, biology, computer science, see, for example [1], [7]. In the past thirty years, oscillation, nonoscillation, the asymptotic behaviour and existence of bounded solutions to many types second-order difference equation have been widely examined, see for example [2], [4], [6], [9], [10], [11], [13], [14], [17], [18], [19], [20], [27], [28], [29], [30], [31], [32], and references therein. ...
This work is devoted to the study of the nonlinear second-order neutral difference equations with quasi-differences of the form $$ \Delta \left( r_{n} \Delta \left( x_{n}+q_{n}x_{n-\tau}\right)\right)= a_{n}f(x_{n-\sigma})+b_n%, \ n\geq n_0 $$ with respect to $(q_n)$. For $q_n\to1$, $q_n\in(0,1)$ the standard fixed point approach is not sufficed to get the existence of the bounded solution, so we combine this method with an approximation technique to achieve our goal. Moreover, for $p\ge 1$ and $\sup|q_n|<2^{1-p}$ using Krasnoselskii's fixed point theorem we obtain sufficient conditions of the existence of the solution which belongs to $l^p$ space.
... In [3,5,6,7,8,9,10,11], the authors considered the equation of the type (1) and established sufficient conditions for the oscillation of all solutions of equation (1) (1) using Philos type oscillation criteria. ...
Some oscillation criteria for the second-order nonlinear neutral delay difference equation Δ 2 (y n +p n y n -k)+q n f(y n -ℓ)=0,n=0,1,2,⋯ are established.
... A solution x n of Eq. (1) is said to be oscillatory, if for every integer N > 0, there exists n P N such that x n x n+1 6 0; otherwise, it is called nonoscillatory. In the literature, there are many papers dealing with the oscillation of solutions of Eq. (1) with e n 0, fox example, see123467891011121314151617 and references therein. However, when e n 6 0, there are few papers devoted to the oscillatory behavior of Eq. (1) except [8,9]. ...
This paper considers the oscillation problem for forced nonlinear difference equations of the formΔmxn+qnf(xn-τ)=en.We study three cases: qn ⩾ 0, qn < 0 and qn is oscillatory. No restriction assumed in known literatures is imposed on the forcing term en.
... In recent years, there has been an increasing interest in studying the oscillation and nonoscillation of solutions of the second-order neutral delay difference equations. For example, see the monographs [1,2] and the papers [3,4,6789101112131415161718 and the references therein. Speaking of oscillation theory of secondorder neutral delay difference equations, most of the previous studies have been restricted to the linear and nonlinear cases in which c ¼ 1 and f ðn; uÞ ¼ q n f ðuÞ, where f ðuÞ is a continuous function in R. Recently, Jiang [7] and Saker [10] studied the oscillatory behavior of solutions of Eq. (1), respectively. ...
In this paper we will establish some new oscillation criteria for the second-order nonlinear neutral delay difference equation where γ>0 is a quotient of odd positive integers.
By means of Riccati technique, we establish some new oscillation criteria for difference equations of neutral type in terms of the coefficients. The results are illustrated by examples.
Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form
\[\Delta \left( r_{n}\left( \Delta \left( x_{n}+p_{n}x_{n-k}\right) \right)
^{\gamma }\right) +q_{n}x_{n}^{\alpha }+a_{n}f(x_{n+1})=0,\]
where $x\colon{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$, $a,p,q\colon {\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $r\colon {\mathbb{N}}_{0}\rightarrow {\mathbb{R}} \setminus \{0\}$, $f\colon {\mathbb{R}}\rightarrow {\mathbb{R}}$ is continuous and $k$ is a given positive integer, $\alpha \geq 1$ is a ratio of positive integers with odd denominator, and $\gamma \leq 1$ is ratio of odd positive integers; ${\mathbb{N}}_{k}:=\left\{ k,k+1,\dots \right\}$. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability is studied.
Using the Riccati transformation techniques, we will extend some almost
oscillation criteria for the second-order nonlinear neutral difference equation
with quasidifferences $$\Delta\left(r_n\left(\Delta \left(x_n+c
x_{n-k}\right)\right)^{\gamma}\right)+q_nx_{n+1}^{\alpha}=e_n.$$
Some new oscillation theorems for even-order difference equations of the form
$$\Delta(|\Delta^{k-1}x_{n}|^{\alpha-1}\Delta ^{k-1}x_{n})+f(n,x_{\sigma(n)})=0$$
are obtained. Our results generalize some well-known results for second order difference equations.
Using the Riccati transformation techniques, we will extend some oscillation criteria of [Appl. Math. Comput. 146 (2003) 791] and [Appl. Math. Comput. 142 (2003) 99] to the second-order nonlinear neutral delay difference equationin the case when 0<γ<1, which answers a question posed by Saker [Appl. Math. Comput. 142 (2003) 99]. Two examples are considered to illustrate our main results.
Some new oscillation criteria for nonlinear delay difference equation with damping
(∗)$$
\begin{array}{*{20}c}
{{\Delta ^{2} x_{n} + p_{n} \Delta x_{n} + F{\left( {n,x_{{n - r}} ,\Delta x_{{n - \sigma }} } \right)} = 0,}} & {{n = 0,1,2,...,}} \\
\end{array}
$$
are given. Our results partially solve the open problem posed in [Math. Bohemica, 125 (2000), 421– 430]. Also, we will establish some new oscillation criteria for special cases of (∗), which improve some of the well–known results in the literature.
In this paper we will establish Kamenev-type criteria for oscillation of the second order delay dynamic equation (r(t)x (t)) + p(t)x( (t)) = 0, t 2 T, where T is a time scale. Our results are not only new for dierential and dierence equations, but are also new for the generalized dierence and q-dierence equations and many other dynamic equations on time scales. Our results are new for delay equations and extend some recent results of Medico and Kong. An example is given to illustrate the main results.
The half-linear difference equations with the deviating argument Δ(an|Δxn|αsgn Δxn)+bn|xn+q|αsgn xn+q=0 , q ∈ ℤ are considered. We study the role of the deviating argument q, especially as regards the growth of the nonoscillatory solutions and the oscillation. Moreover, the problem of the existence of the intermediate solutions is completely resolved for the classical half-linear equation (q = 1). Some analogies or discrepancies on the growth of the nonoscillatory solutions for the delayed and advanced equations are presented; and the coexistence with different types of nonoscillatory solutions is studied.
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