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Oscillatory and nonoscillatory behavior of second-order neutral delay difference equations II
Abstract
We shall investigate the oscillatory behavior of solutions of second order nonlinear neutral delay difference equations. Several examples which dwell upon the importance of our results are also illustrated.
... In this paper, we are concerned to obtain the conditions for the existence / non-existence solutions of a class M , M , OS, WOS and asymptotic behaviour of solutions of a class of M and M second order nonlinear neutral delay dynamic equations with positive and negative coefficients of the form (r(t)(x(t) p(t)x( (t))) ) q(t)f (x( (t))) h(t)g(x( (t))) 0 T In recent years, there has been much research activity concerning the oscillation, non-oscillation and asymptotic behaviour of solutions of various differential equations, difference equations and dynamic equations. For instance in [7,8,9] etc., authors have been studied by classifying all solutions into four classes such as M , M , OS, WOS and obtained criteria for the existence / nonexistence of solutions. In order to extend and generalize the papers [7,8,9], Rami Reddy et al. [17] were concerned the solutions of existence / nonexistence of a class M , M , OS, WOS of second order nonlinear neutral delay dynamic equation of the form (r(t)(x(t) p(t)x( (t))) ) q(t)f (x( (t))) 0, (1. ...
... For instance in [7,8,9] etc., authors have been studied by classifying all solutions into four classes such as M , M , OS, WOS and obtained criteria for the existence / nonexistence of solutions. In order to extend and generalize the papers [7,8,9], Rami Reddy et al. [17] were concerned the solutions of existence / nonexistence of a class M , M , OS, WOS of second order nonlinear neutral delay dynamic equation of the form (r(t)(x(t) p(t)x( (t))) ) q(t)f (x( (t))) 0, (1. The study of the oscillation and other asymptotic properties of solutions of neutral delay difference / differential / dynamic equations with positive and negative coefficients attracted a good bit of attention in the last several years. ...
... This implies that z(t) as t due to 8 (H ) which is a contradiction. Thus, r(t)z (t) 0 for ...
... Recently, there has been an increasing interest in the study of oscillation for solutions of secondorder difference equations. The papers123456 discuss second-order self-conjugate difference equations , the papers [7,8] discuss second-order neutral difference equations, the paper [9] discusses second-order self-conjugate neutral difference equation. In this paper, we are concerned with the second-order nonlinear delay difference equations with nonlinear neutral term. ...
... Some Riccati type difference inequalities are established for these equations, and using these inequalities, we obtain some oscillation criteria. The results obtained here imply and extend those in [5,9]. ...
... REMARK. Theorems 1-3 all include and extend those in [5,9]. An -1 can obtain their corresponding results. ...
Some Riccati type difference inequalities are given for the second-order nonlinear difference equations with nonlinear neutral term. and using these inequalities, we obtain some oscillation criteria for the above equation.
... Many researches were concerned with the oscillation and nonoscillation of delay and neutral type difference equations of second-order, example [1][2][3][4][5][6][7][8][9][12][13][14][15][16]. The growing interest on delay and neutral type difference equations are due to many applications of these equations in different fields of science, social science and engineering [1,17]. ...
... The proof be found in [3]. ...
In this paper, we discuss the oscillatory behavior of quasilinear second order neutral difference equation Δ(b(ξ)(Δ(u(ξ)+c(ξ)u(ξ−z)))α)+f (ξ)uβ(ξ−σ)=0. By delivering new monotone properties of its nonoscillatory solutions and use them for linearization of studied equations which lead to oscillation criteria. Examples are provided to show the applicability and novelty of the obtained criteria.
... where {α m } m∈N 0 is an arbitrary sequence in [0,1] such that Proof First of all we show (a). Let L ∈ (N, M). ...
... converges to a bounded positive solution x ∈ A(N, M) of Eq. (1.11) and satisfies the error estimate (2.6), where {α m } m∈N 0 is an arbitrary sequence in [0,1] satisfying (2.7); (b) Eq. (1.11) possesses uncountably many bounded positive solutions in A(N, M). ...
This paper deals with the third order nonlinear neutral delay difference equation with a forced term Δ2(a(n)Δ(x(n)+c(n)x(n−τ)))+f(n,x(n−b1(n)),x(n−b2(n)),…,x(n−bk(n)))=d(n),n≥n0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned}& \Delta^{2} \bigl(a(n)\Delta \bigl(x(n)+c(n)x(n-\tau) \bigr) \bigr)+f \bigl(n,x \bigl(n-b_{1}(n) \bigr),x \bigl(n-b_{2}(n) \bigr), \ldots,x \bigl(n-b_{k}(n) \bigr) \bigr) \\& \quad =d(n),\quad n\geq n_{0}. \end{aligned}$$ \end{document} Using the Banach fixed point theorem, we prove the existence of uncountably many bounded positive solutions for the equation, suggest some Mann iterative schemes and obtain the error estimates between these bounded positive solutions and the sequences generated by the iterative schemes. Five nontrivial examples are also included.
... In this paper, we are concerned with the oscillation of the solutions of second Received: September 20, 2014 c 2015 url: www.acadpubl.eu § Correspondence author order neutral advanced equations of the form ∆(r(n)∆[x(n) + p(n)x(τ (n))]) + q(n)x(σ(n)) = 0, n = 0, 1, 2, ..., (1) where ∆ is the forward difference operator given by ∆x(n) = x(n + 1) − x(n), {p(n)} n≥0 , {q(n)} n≥0 and {r(n)} n≥0 assumed to be infinite sequences of real numbers with q(n) > 0, r(n) > 0. Also {τ (n)} n≥0 and {σ(n)} n≥0 are sequences of positive integers. Further the following conditions are assumed for its use in the sequel. ...
... Recently, there has been a lot of interest in studying the oscillatory behavior of difference equations. See, for example123456 10] and references cited there in.They have mainly concerned with the oscillation and nonoscillatory of solutions of (1). The advanced equations have wide use. ...
The aim of this paper is to study the oscillation of the second order neutral advanced difference equations Δ(r(n)Δ[x(n) + p(n)x(τ(n))]) + q(n)x(σ(n)) = 0, n = 0, 1, 2, .... Obtained results are based on the new comparison theorems that enable us to reduce problem of the oscillation of the second order equation the oscillation of the first order equations. Obtained comparison principles essentially simplify the examination of the studied equations.
... Oscillation theory for second order neutral difference equations were discussed by Thandapani etal. [2,3,13], Szafranski and Szmanda [11], Budincevic [5], Grace and Lalli [6], Zafar and Dahiya [15] and Zhon and Zhang [16]. In the delay difference case, that is, equation (E) with p ≡ 0, reference should also be made to Györi and Ladas [8]. ...
... The results in this paper are presented in a form which is essentially new. The results obtained in this paper improves some of the results obtained in [2,3,13,15]. ...
Consider the second order difference equation of the form$\Delta^2(y\n-py_{n-1-k})+q_nf(y_{n-\ell})=0,\quad n=1,2,3,\ldots \hskip 1.9cm\hbox{(E)}$where $ \{q_n\}$ is a nonnegative real sequence, $ f:{\Bbb R}\rightarrow {\Bbb R}$ is continuous such that $ uf(u)>0$ for $ u\not= 0$, $ 0\le p
... The oscillatory behaviour of solutions of difference equations has been the subject of intensive literature during the past few years. For example, we refer the reader to the papers [1]- [3], [5]- [15] and the references cited therein where several particular cases of (1.2) have been discussed. In most cases the function f was assumed as a nondecreasing function (see [1], [6]- [7], [11]- [15].) ...
... For example, we refer the reader to the papers [1]- [3], [5]- [15] and the references cited therein where several particular cases of (1.2) have been discussed. In most cases the function f was assumed as a nondecreasing function (see [1], [6]- [7], [11]- [15].) Therefore our main purpose is to investigate the oscillatory behavior of (1.1) and (1.2) via comparison with certain linear equations particularly when f is not assumed to be a nondecreasing function. ...
In this paper we investigate the oscillatory character of the second order nonlinear difference equations of the forms ∆(cn−1∆(xn−1 + pnxσ n)) + qnf (xτ n) = 0, n = 1, 2, . . . and the corresponding nonhomogeneous equation ∆(cn−1∆(xn−1 + pnxσ n)) + qnf (xτ n) = rn, n = 1, 2, . . . via comparison with certain second order linear difference equations where the function f is not necessarily monotonic. The results of this paper are essentially new and can be extended to more general equa-tions.
... Many authors have investigated the special case of equation (1). For example, Li and Yell [2], Agarwal, Manuel and Thandapani [2,3] have considered the following neutral delay difference equation: A [a,_iA (x,-i + pn--1xn-i-c)] + qnf(%-r) = 0, n = 1,2,3, . . . ...
... Many authors have investigated the special case of equation (1). For example, Li and Yell [2], Agarwal, Manuel and Thandapani [2,3] have considered the following neutral delay difference equation: A [a,_iA (x,-i + pn--1xn-i-c)] + qnf(%-r) = 0, n = 1,2,3, . . . ...
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).
... (H 1 ) α is a ratio of odd positive integers; It is well-known that second order neutral difference equations find applications in so many problems in the field of population dynamics, economics, biology etc. Therefore, there has been much interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of different types of second order difference equations, see for example [1,3,6,7,8,9,10,11,12,13,14,15,16,17,18]. Here, we recall some of the previous works that motivate our study. ...
In this article, some new oscillation criteria are established for the second order neutral difference equation of the form$$\Delta\left(a(n) \Delta(z(n))^\alpha\right)+q(n) x^\alpha(\sigma(n))=0, n \geq n_0$$where $z(n)=x(n)+p(n) x(\tau(n))$. Our results improve and extend some known results in the literature. Some examples are also provided to show the importance of these results.
... The study of oscillation for solutions of second-order neutral difference equations has received much interest. Second-order neutral difference equations are discussed in the articles [14,15], whereas the second-order self-conjugate neutral difference equation is discussed in the work [4]. ...
... Oscillatory and asymptotic behavior of solutions of various types of difference equations are discussed over the past few decades. A large amount of papers and monographs have been devoted to this problem, see for example [1,2,4,6,8,9,18,16,15,10,11,5,7,3] and the references cited therein. There is a significant difference in the structure of nonoscillatory (say positive) solutions between canonical and non-canonical equations. ...
... Oscillatory and asymptotic behavior of solutions of various types of difference equations are discussed over the past few decades. A large amount of papers and monographs have been devoted to this problem, see for example [1,2,4,6,8,9,18,16,15,10,11,5,7,3] and the references cited therein. There is a significant difference in the structure of nonoscillatory (say positive) solutions between canonical and non-canonical equations. ...
Using monotonic properties of nonoscillatory solutions, we obtain new oscillatory criteria for the second-order non-canonical difference equation with retarded argument ∆(µ(ℓ)∆η(ℓ)) + ϕ(ℓ)η(σ(ℓ)) = 0. Our oscillation results improve and extend the earlier ones. Examples illustrating the results are provided.
... There has recently been a surge in interest in deriving sufficient criteria for the oscillatory and nonoscillatory properties of solutions to various classes of difference equations; see, for example, the monographs [2,4,9] and the references cited therein. Many authors were concerned with the oscillatory conditions for first-order and higher-order difference equations; for example, [1,3,12,13,15,18,20]. Many applications of delay and advanced difference equations in various fields of science have attracted attention; one can refer to [7,10,14,17]. ...
We derive new oscillatory conditions for the second-order noncanonical difference equations of the type ∆(r(ν)∆x(ν)) + q(ν)x(ν + σ) = 0, ν ≥ ν0, by creating monotonical properties of nonoscillatory solutions. Our oscillatory outcomes are effectively an extension of the previous ones. We provide several examples to demonstrate the efficacy of the new criteria.
... As is well known that, it is not always easy to solve a functional difference equation and find it's solution in closed form, therefore, qualitative theory of difference equations is developed rapidly, since here we assume that the solutions of the difference equation exist and concentrate to investigate its oscillatory behaviour . Recently, numerous articles on oscillation of solutions of neutral difference equations are published for example (Agarwal et al., 1996;Agarwal and Grace, 1999;Parhi and Tripathy, 2003;Zhou and Huang, 2003;Yildz and Ocalan, 2007;Karpuz et al., 2009a;Karpuz et al., 2009b;Yildiz et al., 2009;Yildiz, 2015) and the references cited therein. Thandapani et al. (1999) found non-oscillation and oscillation criteria for the equation ...
This article, is concerned with finding sufficient conditions for the oscillation and non oscillation of the solutions of a second order neutral difference equation with multiple delays under the forward difference operator, which generalize and extend some existing results.This could be possible by extending an important lemma from the literature.
... A great interest were seen in the recent years for the study of the oscillatory and asymptotic behavior of solutions of second order neutral delay difference equations, see [1][2][3]5,6,9,10,13,[15][16][17][18][19][20][21][22]24] and the references contained therein. In [9,10,19], the authors considered the Eq. ...
In this paper, the authors using summation averaging method and an inequality present some new oscillation criteria for the second order neutral type Emden–Fowler delay difference equation $$\begin{aligned} \varDelta (f_i |\varDelta \chi _i|^{\alpha -1} \varDelta \chi _i) + g_i |\psi _{i-l}|^{\beta -1} \psi _{i-l} = 0, \quad i \ge i_0 >0, \end{aligned}$$
(1)
where \(\chi _i = \psi _i + h_i \psi _{i-k}, \; \alpha > 0\) and \(\beta > 0\). The obtained results improve and extend some known results recorded in the literature. Examples illustrating the significance of our results are provided.
... In [3], Budincevic established that every solution of equation ...
In this paper we establish some sufficient conditions for the oscillations of all solutions of the second order nonlinear neutral advanced functional difference Equation [ ]
... From the review of literature, it is known that there are many results available on the oscillatory behavior of solutions of equation (1.1) when p n ≡ 0 and b n ≤ 0 for n ≥ n 0 , see for example [1,2,3,14,15] and the references cited therein. However few results available on the oscillatory behavior of equation (1.1) when p n ≡ 0 and b n ≥ 0 for n ≥ n 0 , see for example [1,5,6,7,12,15]. ...
This paper deals with oscillatory properties of solutions of a class of second order nonlinear damped neutral difference equation of the form: Δ(an(Δzn)α) + (pn(Δzn)α + qnf(xn-l) = 0, n ≥ n0. where zn = xn - bnxn-k. Some new sufficient conditions are obtained which ensure that all solutions of equation (E) are oscillatory. The results presented in this paper extend and improve some of the related results reported in the literature. Examples are provided to illustrate the importance of the main results.
... Otherwise it is called oscillatory. In recent years, there has been much research activity concerning oscillation of second-order neutral delay difference equations, we refer the reader to the papers [1,2,4,5,7,9,12,16,19,[22][23][24] and the references cited therein. Most of the results in the above references are obtained when p n ≤ 0. Here, we recall some of the results that motivate our study. ...
In this paper, we are concerned with oscillation of a second-order nonlinear neutral delay difference equation of the form ∆ (a n ∆ (x n − p n x n−1) γ) + q n f (x n−τ) = 0. We consider the two cases when γ ≥ 1 and 0 < γ < 1. The main oscillation results will be of Kamenev's type, Philos'type as well as Hille and Nehari types. The results will be established by using the oscillation criteria of the corresponding third order nonlinear difference equation.
... From the review of literature, it is known that there are many results available on the oscillatory and asymptotic behavior of solutions of equation (1) when the neutral term is nonnegative, i.e., p n ≤ 0; see for example [1,2,3,8,14] and the references cited therein. However, there are few results available on the oscillatory behavior of solutions of equation (1) when the neutral term is non-positive; see, for example [5,6,7,10,11,12,13,15,16] and the references cited therein. ...
In this paper, we present some oscillation criteria for second order nonlinear delay difference equation with non-positive neutral term of the form
Δ(an(Δzn)α)+qnf(xn−σ)=0, n≥n0>0,
where zn = xn − pnxn−τ, and α is a ratio of odd positive integers. Examples are provided to illustrate the results. The results obtained in this paper improve and complement to some of the existing results.
... 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. Similar classification have been discussed for the equation (1) when α = 1 and p n ≡ 0 for all n in [13]. ...
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.
... (H 2 ) {p n } is a real sequence with 0 ≤ p n ≤ p < 1 for all n ∈ N(n 0 ); From a review of literature, it is known that there are many results available on the oscillatory and asymptotic behavior of solutions of equation (1.1) when the neutral term is nonnegative, i.e., p n ≤ 0; see for example [1,2,3,9,15] [4,6,7,8,11,12,13,14,16,17] and the references therein. ...
In this paper we obtain some new oscillation criteria for the neutral difference equation (equation presented) where 0 ≥ pn ≥ p < 1, qn > 0 and l and k are positive integers. Examples are presented to illustrate the main results. The results obtained in this paper improve and complement to the existing results.
... A solution of equation (1) is said to be oscillatory if it is neither eventually positive nor eventually negative, and it is nonoscillatory otherwise. From a review of the literature, it is known that there are many results available on the oscillatory and asymptotic behavior of solutions of equation (1) when the neutral term is nonnegative, i.e., p n ≤ 0; see, for example, [1,2,3,4,6,7] and the references cited therein. However, there are very few results available on the oscillatory behavior of solutions of equation (1) when the neutral term is negative; see, for example, [1,2,5,8,9,10,11,12], and the references therein. ...
In this paper the authors obtain some new results on the oscillatory behavior of second order neutral difference equations of the form {equation presented} where 0 ≤ pn ≤ p < 1, qn > 0, and α is a ratio of odd positive integers. Examples are provided to illustrate the main results.
... Recently, there has been an increasing interest in the study of the oscillation of solutions of second-order difference equations such as12345678910. In particular, the oscillation of second-order nonlinear difference equations with a damped term is studied in [7,9,11]. ...
In this paper, we shall establish relations between the oscillation of second-order nonlinear difference equations with damped term and the oscillation of their linear limiting equations.
... In recent years, many authors have shown interest in the oscillation of neutral delay difference equations (NDDEs in short). For recent results and references, see the monograph by Agarwal [4], the papers [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], and the references cited there in. In this paper, neither (H0) nor (H1) is assumed for obtaining positive solution of (1.2). ...
We find necessary conditions for every solution of the neutral delay difference equation Δ(rnΔ(yn − pn yn−m))+ qnG(yn−k) = fn to oscillate or to tend to zero as n →∞ ,w hereΔ is the forward difference operator Δxn = xn+1 − xn ,a ndpn, qn, rn are sequences of real numbers with qn ≥ 0, rn >0. Different ranges of {pn}, including pn =± 1, are considered in this paper. We do not assume that G is Lipschitzian nor nondecreasing with xG(x) >0 for x �= 0. In this way, the results of this paper improve, generalize, and extend recent results. Also, we provide illustrative examples for our results.
... In recent years there has been increasing interest in the study of the qualitative behavior of solutions of equations of the type (1), see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references cited therein. It is important to study second-order nonlinear difference equations in view of their applications as described in [20]. ...
The asymptotic and oscillatory behavior of solutions of some general second-order nonlinear difference equations of the form Δ(anh(yn+1)Δyn)+pnΔyn+qn+1f(yσ(n+1))=0,n∈Z is studied. Oscillation criteria for their solutions, when “pn” is of constant sign, are established. Results are also presented for the damped-forced equation Δ(anh(yn+1)Δyn)+pnΔyn+qn+1f(yσ(n+1))=en,n∈ZExamples are inserted in the text for illustrative purposes.
... Recently, there has been an increasing interest in the study of oscillation of difference equations. Regarding the oscillatory behaviour of solutions, first order difference equations with continuous variable were studied in [1][2][3][4][5] and second order nonlinear difference equations, including neutral and advanced, were investigated in [6][7][8][9][10]. In this paper, we are mainly concerned with the second order nonlinear neutral difference equation ...
In this paper, we are mainly concerned with oscillatory behaviour of solutions for a class of second order nonlinear neutral difference equations with continuous variable. Using an integral transformation, the Riccati transformation and iteration, some oscillation criteria are obtained.
In this paper some new sufficient conditions for the oscillatory behavior of second order nonlinear neutral type difference equation of the form$$\Delta\left(a_n \Delta\left(x_n+p_n x_{n-k}\right)\right)+q_n f\left(x_{\sigma(n+1)}\right)=0$$where $\left\{a_n\right\},\left\{p_n\right\}$ and $\left\{q_n\right\}$ are real sequences, $\{\sigma(n)\}$ is a sequence of integers, $k$ is a positive integer and $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous with $u f(u)>0$ for $u \neq 0$ are established. Examples are provided to illustrate the main results.
The goal of this paper is to look into some criteria for the oscillatory behavior of second order neutral delay difference equations. Starting with a general overview of difference equations, we present some recent findings on the use of neutral delay difference equations based on a comparison principle and summation averaging techniques. The acquired results can be utilised to build and offer theoretical support for the given equations.
In this paper, we want to study the different kinds of solutions of second order nonlinear non-homogeneous neutral delay dynamic equation with mixed type, classifying these solutions into four classes and obtained conditions for the existence/non-existence of solutions in these classes. Also we have mentioned examples to an important results.
In this paper, we want to study the different kinds of solutions of second order nonlinear non-homogeneous neutral delay dynamic equation, classifying these solutions into four classes and obtained conditions for the existence/non-existence of solutions in these classes. Further we have mentioned examples to an important 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.
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.
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.
The existence results of uncountably many bounded positive solutions for a fourth order nonlinear neutral delay difference equation are proved by means of the Krasnoselskii’s fixed point theorem and Schauder’s fixed point theorem. A few examples are included.
This paper deals with the second-order nonlinear neutral delay difference equation
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. Using the Banach fixed point theorem, Mann iterative method with errors, and some new techniques, we prove the existence of uncountably many positive solutions and the convergence of the sequences generated by the Mann iterative method with errors relative to these solutions for the above equation. Six examples are included. Our results extend and improve essentially the known results in this field.
In this paper, we give a classification of non-oscillatory solutions of a second-order neutral delay difference equation of the form Δ2(xn - cnxn-τ) + f(n, xg1(n),..., xgm(n)) = 0 (n ≤ n0 ∈ ℕ). Some existence results for each kind of non-oscillatory solutions are also established.
In this paper, we consider the second-order neutral delay difference equation with positive and negative coefficients Delta (r(n) Delta(x(n) + cs(n-k))) + p(n+1)x(n+1-m) - q(n+1)x(n+1-l) = 0 where c is an element of R, k greater than or equal to 1 and m,l greater than or equal to 0 are integers, {r(n)}(n=n0)(infinity), {p(n)}(n=n0)(infinity) and {qn}(n=n0)(infinity) are sequences of non-negative real numbers. We obtain global results (with respect to c) which are some sufficient conditions for the existences of non-oscillatory solutions.
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 will study the oscillatory properties of the second order half-linear difference equation with distributed deviating arguments. We obtain several new sufficient conditions for the oscillation of all solutions of this equation. Our results improve and extend some known results in the literature. Examples which dwell upon the importance of our results are also included.
The author investigates asymptotic properties of solutions of the second order neutral type difference equation Δ(a n (Δ(x n +c n x n-k )) α )+q n f(x n+1-l )g(Δx n )=0,n∈ℕ(n 0 ), where {a n },{c n },{q n } are real sequences, k and l are non-negative integers, α is a ratio of odd positive integers and f and g are continuous real valued functions. Our results improve and generalize some known results of neutral delay difference equations.
Some new oscillation criteria for the second-order Emden-Fowler type neutral difference equation Δ(a n Δ(x n +p n x n-k ))+q n x σ(n) α =0,n≥n 0 where k is a positive integer, α is a ratio of odd positive integers and σ(n)≤n are established under the condition ∑ n=n 0 ∞ 1 a n <∞. Examples are provided to illustrate the results.
In this paper, the authors establish some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral delay difference equation ∆ 2 (xn − pnx n−k) + qnf (x n−ℓ) = 0, n ≥ n0 where {pn} and {qn} are non-negative sequences with 0 < pn ≤ 1, and k and ℓ are positive integers.
This paper is concerned with solvability of the second-order nonlinear neutral delay difference equation Δ2(xn+anxn-τ)+Δh(n,xh1n,xh2n,…,xhkn)+f(n,xf1n,xf2n,…,xfkn)=bn,∀n≥n0. Utilizing the Banach fixed point theorem and some new techniques, we show the existence of uncountably many unbounded positive solutions for the difference equation, suggest several Mann-type iterative schemes with errors, and discuss the error estimates between the unbounded positive solutions and the sequences generated by the Mann iterative schemes. Four nontrivial examples are given to illustrate the results presented in this paper.
The purpose of this paper is to study solvability of the second-order nonlinear neutral delay difference equation Δ ( a ( n , y a 1 n , … , y a r n ) Δ ( y n + b n y n - τ ) ) + f ( n , y f 1 n , … , y f k n ) = c n , ∀ n ≥ n 0 . By making use of the Rothe fixed point theorem, Leray-Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.
Asymptotic properties of solutions of a difference equation of the form Δmxn = anf (n, xσ(n)) + b n are studied. We present sufficient conditions under which, for any polynomial ℓ(n) of degree at most m- 1 and for any real s ≥ 0, there exists a solution x of the above equation such that xn = ℓ(n) + o(ns). We give also sufficient conditions under which, for given real s ≤ m- 1, all solutions x of the equation satisfy the condition x n = ℓ(n) + o(ns) for some polynomial ℓ(n) of degree at most m- 1.
In this paper, we study the forced oscillation of the higher-order nonlinear difference equation of the form
Δ m [ x ( n ) − p ( n ) x ( n − τ ) ] + q 1 ( n ) Φ α ( n − σ 1 ) + q 2 ( n ) Φ β ( n − σ 2 ) = f ( n ) ,
where m ≥ 1 , τ , σ 1 and σ 2 are integers, 0 < α < 1 < β are constants, Φ ∗ ( u ) = | u | ∗ − 1 u , p ( n ) , q 1 ( n ) , q 2 ( n ) and f ( n ) are real sequences with p ( n ) > 0 . By taking all possible values of τ , σ 1 and σ 2 into consideration, we establish some new oscillation criteria for the above equation in two cases: (i) q 1 = q 1 ( n ) ≤ 0 , q 2 = q 2 ( n ) > 0 ; (ii) q 1 ≥ 0 , q 2 < 0 .
MSC: 39A10.
The authors consider the mth-order neutral difference equation Dm(y(n) + p(n)y(n − k) + q(n)f(y(σ(n))) = e(n), where m ≥ 1, {p(n)}, {q(n)}, {e(n)}, and {a1(n)}, {a2(n)}, …, {am−1(n)} are real sequences, ai(n) > 0 for i = 1,2,…, m−1, am(n) ≡ 1, D0z(n) = y(n)+p(n)y(n − k), Diz(n) = ai(n)ΔDi−1z(n) for i = 1,2, …, m, k is a positive integer, {σ(n)} → ∞ is a sequence of positive integers, and R → R is continuous with uf(u) > 0 for u ≠ 0. In the case where {q(n)} is allowed to oscillate, they obtain sufficient conditions for all bounded nonoscillatory solutions to converge to zero, and if {q(n)} is a nonnegative sequence, they establish sufficient conditions for all nonoscillatory solutions to converge to zero. Examples illustrating the results are included throughout the paper.
Some Riccati type difference inequalities are established for the second-order nonlinear difference equations with negative neutral term
$$\Delta (a(n)\Delta (x(n) - px(n - \tau ))) + f(n,x(\sigma (n))) = 0$$
using these inequalities we obtain some oscillation criteria for the above equation.
This paper is concerned with the oscillation of the bounded solutions of neutral difference equation where Δ is the forward difference operator defined by Δn = n+1 - n.
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.
Asymptotic and oscillatory behavior of solutions of first order nonlinear neutral difference equations are studied. Some illustrative examples are also inserted.
We offer sufficient conditions for the oscillation of all solutions of the perturbed quasilinear differential equation (a(t)|y′|α−1y′)′ + Q(t, y) = P(t, y, y′)as well as for the existence of a positive monotone solution of the damped differential equation (a(t)|y′|α−1y′)′ + b(t)|y′|α−1y′ + H(t, y) = 0, where α > 0. Examples that dwell upon the importance of our results are also included.
Sufficient conditions for the oscillation of all solutions of perturbed difference equations of the form are obtained. Examples are inserted to illustrate the results.
We offer sufficient conditions for the oscillation of all solutions of the perturbed difference equation Δ(an−1(Δyn−1)σ) + F(n, yn) = G(n, yn,Δyn), n ≥ 1, as well as for the existence of a positive monotone solution of the damped difference equation Δ(an(Δyn)σ) + bn(Δyn)σ + H(n, yn, Δyn) = 0, n ≥ 0, where with p, q odd integers, or p even and q odd integers. Examples which dwell upon the importance of our results are also included.
Oscillation and nonoscillation criteria are established for first order linear and nonlinear difference equations with delay. some of the conditions are shown to be sharp. applications are also given to certain nonhomogeneous equations.
We survey almost all the aspects of difference equations which are of current interest. Further, we consistently pose open problems and offer some directions towards possible solutions.
We obtain sufficient conditions for the oscillation of all solutions and existence of positive solutions of the neutral difference equation Δ(xn + cxn − m) + pnxn − k = 0, n = 0, 1, 2, …, where c and pn are real numbers, m and k are integers, and pn, m and k are nonnegative.
A criterion for the existence of a positive solution for the first order difference equation Δ(xn − cxn − m) + pnxn − k = 0, pn ⩾ 0 is established. Results are also obtained for the oscillation and nonoscillation of solutions of a second order difference equation Δ2(xn − cxn − m) = pnxn − k, where the pn are either of constant sign or are oscillatory. Some of the results are sharp.
In this paper some simple inequalities are used to offer sufficient conditions for the oscillation of all solutions of the differential equation[formula]where 0<σ=p/qwithp, qodd integers, orpeven andqodd integers. Several examples which dwell upon the importance of our results are also included.