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Oscillatory criteria for second-order quasilinear neutral delay difference equations
Abstract
Consider the second-order quasilinear neutral difference equation(*)Δan|Δ(xn+pnxg(n))|α−1Δ(xn+pnxg(n))+f(n,xσ(n))=0.The sufficient conditions are established for oscillation of the solution of this Eq. (*). These generalize and improve some known results about both neutral and delay difference equations.
... Interestingly, one can refer the oscillatory behavior for sublinear neutral delay second and third order diference equations in [3,4]. An investigation of the oscillation of the second-order quasilinear neutral delay difference equations can be seen in [5,6]. Te study of the Oscillation of second order half-linear diference equations has also been given in [7]. ...
... We focus only on the solutions y(ξ) of equation (5), defned for some ξ 0 < ξ 1 satisfying sup |y(ξ)|: ξ > K > 0 for all K > ξ. We assume that a proper solution exists for equation (5). A proper solution y(ξ) of equation (5) is said to be oscillatory if y(ξ)y(ξ + l) < 0 for large ξ and is nonoscillatory otherwise. ...
... A proper solution y(ξ) of equation (5) is said to be oscillatory if y(ξ)y(ξ + l) < 0 for large ξ and is nonoscillatory otherwise. Also, if all the solutions of equation (5) are oscillatory, then the equation itself is oscillatory. ...
In this paper, the authors discuss the oscillatory behaviour of a third order generalized difference equation with multiple neutral terms. We have also applied Riccati transformation and Philo type technique to derive new oscillation criteria for the difference equation in question. Suitable examples are provided to validate our main results.
... The qualitative behavior of solutions of nonlinear neutral type delay difference equations is of particular interest since such equations arise in the problems dealing with vibrating masses attached to an elastic bar and as an Euler equations in some variational problems, see for example [1,2,5,7,8,18] and the references cited therin. ...
... In [6][7][8]14] the authors considered equation (1) with q n ≡ 0 for all n and studied the oscillatory and asymptotic behavior of all solutions of equation (1). In [3] and [4] the authors consider equation (1) with α = 1 and q n ≡ 0 for all n and classified all solutions into the above said four classes and obtain conditions for the existence/ nonexistence of solutions in these classes. ...
In this paper the authors classified all solutions of the fol-lowing equation ∆ an ∆(xn + cnx n−k) α + pnf (x n−l) − qng(xn−m) = 0 into four classes and obtain conditions for the existence/nonexistence of solutions in these classes. Examples are provided to illustrate the results.
... Throughout the paper, we assume that (H 1 ) 0 < α i ≤ 1 for i = 1, 2, ..., m and moreover, each of the α i , γ is a quotient of Furthermore, we assume that (E) is in a canonical form, namely, Neutral type differential/difference equations arise in a number of problems in physics, biology, etc., see [1,16]. Therefore, the qualitative behavior of the solutions of this type of equations has attracted considerable interest and extensive research has been produced on this subject, see for example, [1,2,9,11,12,17,18,20] and the references cited therein. ...
This paper presents new sufficient conditions for the oscillation of second-order difference equations with several sublinear neutral terms. The results obtained here extend and generalize the existing results in the literature. Several examples are provided to illustrate and highlight the importance and novelty of the main results.
... A non trivial solution is said to be oscillatory if it is neither eventually positive nor eventually negative and nonoscillatory otherwise. The oscillatory behavior of nonlinear neutral delay difference equation of second order have been investigated by several authors, see for example [6,9,11,12] and the references quoted therein. Following this trend, in this paper, we obtain some new oscillation criteria for equation (1) which extend some known results. ...
In this paper, we study the oscillatory behavior of solution of second order neutral dierence equation with mixed neutral term of the form(an(zn)) + qnx(n) = 0; n 2 N0; where zn = xn + bnxnô€€€l + cnxn+k and 1Ps=n01as= 1. We obtain some new oscilla-tion criteria for second order neutral dierence equation. Examples are presented to illustrate the main results.
... Recently there has been an increasing interest in the study of the oscillation and non-oscillation of the second order neutral difference equations, see for example [6,7,[10][11][12][13][14] and the references cited there in. In [14], we see that the oscillation criteria for second order non-positive neutral term of the form ...
In this paper, we present some new oscillation criteria for second order neutral type dierence equation of the form (an(zn)) + qnf(xn) = en; n n0 > 0; where zn = xn ô€€€pnxnô€€€l and is ratio of odd positive integers. Examples are provided to illustrate the results.
... Jiang [13], the author discussed the oscillatory behavior of all solutions of the di¤erence equation ...
In this paper, we are concerned with oscillation of a second order nonlinear neutral delay difference equation of the form
Δ(a_{n}Δ(x_{n}-p_{n}x_{τ(n)}))+q_{n}f(x_{σ(n)})=0.
Our results generalize and improve some of the existing results. An example illustrating the results is also given.
... Eq. (1) is said to be oscillatory if all its solutions are oscillatory. Recently, there has been an increasing interest in the study oscillation and asymptotic behavior of solutions of second-order neutral delay difference equations, for example see123456789101112131415161718 and the references therein. To the best of our knowledge, nothing is known regarding the qualitative behavior of solutions of Eq. (1) in the sublinear case. ...
We present new oscillation criteria for the second-order sublinear neutral delay difference equationΔanΔxn+pnxn−τ+qnxn−σγ=0,where 0
... 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 this paper we use the Riccati transformation technique and the Hardy, Littlewood and Pólya [16] and Jianchu [18] ...
In this paper, by using the Riccati transformation technique, chain rule and inequalitywhere A and B are positive constants, we will establish some oscillation criteria for the second-order half-linear dynamic equationon time scales, where γ>1 is an odd positive integer. Our results not only unify the oscillation of half-linear differential and half-linear difference equations but can be applied on different types of time scales and improve some well-known results in the difference equation case. Some examples are considered here to illustrate our main results.
... 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.
The third order non linear difference equations of the form Δ(cnΔ(dnΔxn))+pnΔxn+1+qnf(xn−σ)=0
for n ≥ n0 are considered. Here {cn}, {dn}, {pn} and {qn} are sequences of positive real numbers for n0 ∈ N f is a continuous function such that f(u)u≥K>0
for u ≠ 0. By means of a Riccati transformation technique we obtain some new oscillation criteria. Examples are given to illustrate the importance of the results. Where n0 ∈ N is a fixed integer, Δ denotes to forward difference operator Δxn = xn+1 xn and σ is a nonnegative integer.
Abstract In this paper, the authors obtain some new sufficient conditions for the oscillation of all solutions of the second order neutral difference equation Δ(an(Δzn)β)+qnxn−ℓγ=0,n≥n0, $$\Delta \bigl( a_{n} ( \Delta z_{n} )^{\beta} \bigr) + q_{n}x_{n - \ell}^{\gamma} = 0,\quad n \ge n_{0}, $$ where zn=xn+pnxn−kα $z_{n} = x_{n} + p_{n}x_{n - k}^{\alpha} $. The established results extend, unify and improve some of the results reported in the literature. Examples are provided to illustrate the importance of the main results.
The oscillation of a class of the second order quasilinear neutral variable delay damped functional difference equations was discussed. By using the generalized Riccati transformation combined with the averaging technique and a lot of inequality techniques, some new oscillation criteria for the equations were proposed. The results in this article have extended and improved the results given in other literatures. Further, some examples are given to illustrate the importance of our results.
This paper deals with the oscillation of second order neutral difference equations of the form
The oscillation of all solutions of this equation is established via comparison theorems. Examples are provided to illustrate the main results.
MSC:
39A10.
In this paper, we investigate the oscillatory and non-oscillatory behavior of solutions of second order quasi-linear neutral delay difference equation A a(n) |Delta;(χ(n) +p(n)χ(n -τ)) alpha-1; δ(χ(n)+p(n)χ(n -τ))] + q(n +1)f{χ{n +1-σ)) h{Δ{n + 1)) = 0, where n > no, a > 0, τ > 0 and a > 0 are constants, a(n),q(n),p(n) are positive real sequences and f,h.⋯(R:R).
We obtain sufficient conditions for the oscillation of all solutions to the second-order nonlinear delay difference equation Δ[a n (Δ(x n +p n x g(n) )) α ]+f(n,x σ(n) )=0, where Δx n =x n+1 -x n is the forward difference operator, {p n } is a sequence of nonnegative real numbers and α⩾1 is a quotient of positive odd integers. Our results extend and improve some known results in the literature. An example that dwell up the advantage of our results is included.
The authors examine the oscillatory and non-oscillatory behavior of solutions of a class of second order difference equations of neutral type that includes half-linear equations as a special case. They classify the possible non-oscillatory solutions according to their asymptotic behavior and then prove the existence of solutions of each type using fixed point techniques. Criteria for almost oscillation, i.e., solutions either oscillate or converge to zero, are also proved. The results are illustrated with examples.
By using the Riccati transformation and mathematical analytic methods, some sufficient conditions are obtained for oscillation
of the second-order quasilinear neutral delay difference equations
$
\Delta [r_n |\Delta z_n |^{\alpha - 1} \Delta z_n ] + q_n f(x_{n - \sigma } ) = 0,
$
\Delta [r_n |\Delta z_n |^{\alpha - 1} \Delta z_n ] + q_n f(x_{n - \sigma } ) = 0,
where z
n
= x
n
+ p
n
x
n-τ
$
\sum\limits_{n = 0}^\infty {1/r_n^{\tfrac{1}
{\alpha }} } < \infty
$
\sum\limits_{n = 0}^\infty {1/r_n^{\tfrac{1}
{\alpha }} } < \infty
.
KeywordsOscillation-Riccati transformation-Difference equations
2000 MR Subject Classification34K11-34C10-34K40
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.
In this paper, we consider the oscillatory behavior of the second order neutral delay differential equation
where t ≥ t 0 ,T and σ are positive constants, a,p, q € C(t 0 , ∞), R),f ∊ C[R, R] . Some sufficient conditions are established such that the above equation is oscillatory. The obtained oscillation criteria generalize and improve a number of known results about both neutral and delay differential equations.
In this paper, we obtain some oscillation criteria for the second-order quasi-linear neutral delay difference equation , where α > 0, τ ≥ 0, and σ ≥ 0 are constants, {an}, {pn}, {qn} are nonnegative sequences and f ϵ C(R,R).
Some new oscillation theorems for the difference equations of the form Δ(rnΔun)+qnf(un−k)=0 are established.