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Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term
Abstract
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.
... In particular in [6,7,9,11,12], the authors considered equation of the type (1.1) when 0 < α ≤ 1 and either ...
... In all the results the condition p n → 0 as n → ∞ is required to apply the theorems. Further in [11], the authors considered equation of the type (1.1) with α > 1 and studied the oscillatory behavior under the condition that lim n→∞ inf q n > 0. Motivated by this observation, in this paper we examine the other case α ≥ 1 and we do not require that either p n → 0 as n → ∞ or lim n→∞ inf q n > 0. Our method of proof is different from that of in and hence our results are new and complement to that of reported in [6,7,9,11,12]. Examples are presented to illustrate the importance of the main results. ...
... In this paper, we obtain some new oscillation criteria for the equation (1.1) using Riccati type transformation and comparison method which involves α and β. Further the results presented in this paper are new and complement to that of in [6,7,9,11,12]. ...
In this paper, the authors obtain sufficient conditions for the oscillation of all solutions of the equation$$\Delta\left(a_n \Delta\left(x_n+p_n x_{n-k}^\alpha\right)\right)+q_n x_{n+1-l}^\beta=0$$where $\alpha \geq 1$ and $\beta>0$ are ratio of odd positive integers, and $\left\{a_n\right\},\left\{p_n\right\}$ and $\left\{q_n\right\}$ are real positive sequences. Examples are provided to illustrate the importance of the main results.
... In view of the above discussion, in this paper we choose to examine the oscillatory behavior of such type of difference equations because similar properties for difference equations with linear neutral term received great attention of the researchers. The oscillatory behavior of second order difference equations with sub-linear neutral term, that is, x α (nk) with 0 < α < 1, was studied in [2,6,14,17,18], and therefore in this paper we investigate the oscillatory properties of solutions of Eq. (1.1) with super-linear neutral terms, that is, ...
... Motivated by the above observations, in this paper, we have obtained some new sufficient conditions for the oscillation of all solutions of Eq. (1.1) by using a Riccati type transformation, a summation averaging technique, and a comparison method. Thus the results obtained in this paper are new and extend those reported in [5,9,10,[15][16][17][18]. Examples are provided to illustrate the importance of the main results. ...
... We begin with the following lemma. Proof The proof is similar to that of Lemma 1 of [18] and hence the details are omitted. ...
Abstract In this article,we present some new sufficient conditions for the oscillation of all solutions of a second order difference equation with several super-linear neutral terms. The results obtained here extend or complement some of the known results reported in the literature. Examples illustrating the importance of the main results are included.
... A nontrivial solution of equation (1.1) is said to be oscillatory if the terms of the sequence are neither eventually positive nor eventually negative, and nonoscillatory otherwise. In recent years there is a great interest in studying the oscillatory and asymptotic behavior of solutions of various classes of di erence equations, see, for example [1][2][3][4][6][7][8][9][10] and the references cited therein. In particular in [7], the authors considered the equation of the form (1.1) and obtained criteria for the oscillation of equation ( for the case < α ≤ . ...
... To accomplish this is the main purpose of the present paper. Thus the results obtained in this paper are new, and complement to that of in [3,[6][7][8][9][10]. ...
... The results obtained in this paper extend some of the results in [6,7,9,10] for α > . Also the results of [9] cannot be applied to equations (3.1) and (3.2) since ∞ n= an < ∞. ...
This paper deals with the oscillation of solutions of certain class of neutral difference equation ∆(a n ∆(χ n + p n χ αn−k )) + q n χ βn+1−l = 0, where α and β are ratio of odd positive integers. New sufficient conditions are obtained for the oscillation of studied equation and examples illustrating the main results are provided.
... In view of the above observations, the researchers paid attention to the oscillation area for various classes of second-order difference, differential and dynamic equations, see [2,6,7,8,10,12,13,14,15,17,18,21,24,25,26,29,31] and the references cited therein. As far as secondorder difference equations with positive superlinear neutral terms are considered, not many results are known about the oscillation, see [3,4,11,16,19,27,32,33]. A close look at these papers reveals that the neutral coefficient {ρ(ι)} must rectify explicitly or implicitly either ρ(ι) → 0 or ρ(ι) → ∞ as ι → ∞. ...
... Furthermore, the oscillation criteria developed here are novel and add to the findings previously reported in the literature. The neutral coefficient ρ(t) ∈ (0, 1) prevents the results presented in [3,4,11,16,19,27,32,33] from being applicable to our equations (3.1)-(3.3). As a result, our findings constitute a highly valuable addition to the oscillation theory of second-order neutral difference equations with superlinear neutral terms. ...
We obtain oscillation conditions for non-canonical second-order nonlinear delay difference equations with a superlinear neutral term. To cope with non-canonical types of equations, we propose new oscillation criteria for the main equation when the neutral coefficient does not satisfy any of the conditions that call it to either converge to \(0\) or \(\infty\). Our approach differs from others in that we first turn into the non-canonical equation to a canonical form and as a result, we only require one condition to weed out non-oscillatory solutions in order to induce oscillation. The conclusions made here are new and have been condensed significantly from those found in the literature. For the sake of confirmation, we provide examplesthat cannot be included in earlier works.
... Recently, there has been a lot of interest in studying the oscillatory and asymptotic behavior of solutions of various classes of second order nonlinear difference equations, see [1,2,10,12] and the references contained therein. In [4,6,[15][16][17][18], the authors considered equation of the form (1.1) and obtained criteria for the oscillation of (1.1) under the conditions either ...
... The obtained results simplifies the known results in the literature in the sense that we need only one condition instead of two conditions required in [7,9,11] for the oscillation of all solutions of equation (1.1). Further the results presented in this paper extend and generalize the results known [4][5][6][15][16][17][18] for the case 0 < α < 1 to the case α > 1. ...
... The problem of finding sufficient conditions which ensure that all solutions of the neutral type difference equations are oscillatory has been investigated by many authors; see, for example, [1][2][3][4][5][6][7][8][9][10][11][12] and the references cited therein. In all the results the neutral term is linear and few results are available when the neutral term is nonlinear; see [13][14][15][16][17][18][19][20][21]. ...
... Thus the results presented here extend and generalize some of the results in [13,14,16,18,19,21], complement the results in [20], and correct some of the results in [8]. ...
In this paper, some new results are obtained for the even order neutral delay difference equation ΔanΔm-1xn+pnxn-kα+qnxn-lβ=0 , where m≥2 is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results.
... In the last few years there has been a great interest in investigating the oscillatory and asymptotic behavior of neutral type difference equations, see [1,2,4,5,6,7,8,9,10,11,12] and the references cited therein. ...
... In Our technique of proof makes use of some inequalities and Riccati type transformations. The results we obtain here are new and generalize those reported in [4,5,6,11,12]. Examples are provided to illustrate the main results. ...
... In this paper, we are concerned with the oscillatory behavior of second order difference equation with several sub-linear neutral terms of the form The problem of investigating the oscillatory behavior of solutions of second order difference equations received a great interest in the last few decades, see for example [1-3, 11, 15, 16] for recent references. However, there are few results dealing with the oscillation of second order difference equations with a sub-linear neutral term, see [4,8,9,12,13,17,18], even though such equations arise in many applications, see [6]. ...
... 93 In establishing new criteria for the oscillation of solutions of equation (1.1), we reduce the given equation to an equation with linear neutral term by using an inequality. Thus by using an elementary inequality, we obtained some oscillation results, which are new, extend and complement to those established in [1,4,8,13,17,18]. ...
... In [8], the authors discussed the oscillatory behavior of equation (1.1) when ܽ ≡ 1 and lim ୬→ஶ inf ݍ > 0. But in this paper we derived conditions for the oscillation of all solutions of equation (1.1) without such assumptions. Hence our results improve and complement to that of in [7,9,10,8,11]. Examples are included to illustrate the main results. ...
... In this paper we obtained sufficient conditions for the oscillation of all solutions of equation (1.1) using Ricatti type transformation and comparison method. The results presented here improve and complement to that of in [7,9,10,8,11]. ...
... This shows that (19) has an eventually positive solution {z n }. On the other hand, by ...
... In view of (17) and (20), Lemma 1 implies that the inequality (19) has no eventually positive solutions. This contradiction shows that case (i) is impossible. ...
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.
... However, there has been only a few results on the oscillation of secondorder difference equations with sublinear neutral terms [3,5,6,13,14,15,19,22,23]. To the best of our knowledge, explicitly or implicitly, all these results assume that lim n→∞ d i (n) = 0. ...
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.
... There have been numerous studies regarding the second-order neutral functional difference equations, due to the comprehensive use in natural science and theoretical study. Some interesting recent results on the oscillatory and asymptotic behavior of second-order difference equations can be found in [3,[9][10][11][13][14][15][20][21][22][23][24][25][26] and the references cited therein. Recent results concerning the oscillatory and asymptotic behavior of third-order difference equations, one can refer to [4,16,[18][19][20] and the references cited therein. ...
We shall present some new oscillation criteria for third-order nonlinear difference equations with a nonlinear-nonpositive neutral term of the form: Δa(t)Δ2x(t)-p(t)xα(t-k)γ+q(t)xβ(t-m+1)=0,with positive coefficients via comparison with first-order equations whose oscillatory behavior are known, or via comparison with second-order inequalities with solutions having certain properties. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature even for the case of Eq. (1.1) with p (t)=0. Examples are given to illustrate the main results.
... From the review of literature, one can see that many oscillation results are available for the equation when α = 1; see [1, 2, 5, 8-11, 14, 15, 18, 20], and the references cited therein. Also few results available for the oscillation of Eq. (1.1) while β = 1; see [4,12,17,19,21,22]. And as far as the authors knowledge there are no results available in the literature for the oscillatory behavior of Eq. (1.1). ...
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.
... Meanwhile, there also have been numerous research for second order neutral functional difference equations, due to the comprehensive use in natural science and theoretical study. Some interesting recent results on the oscillatory and asymptotic behavior of second order difference equations can be found in [12,13,14,15,16,17,18,19,20,21,22]. However, it seems that there are no known results regarding the oscillation of second order difference equations of type (1). ...
We shall present new oscillation criteria of second order nonlinear difference equations with a non-positive neutral term of the form \(\Delta(a(t)(\Delta(x(t)-p(t)x(t-k)))^{\gamma})+q(t)x^{\beta}(t+1-m)=0,\) with positive coefficients. Examples are given to illustrate the main results.
... 3), we refer to [2]- [12] and the references cited therein. In Section 2, we present some basic lemmas. ...
... ≡ ≡. Then Theorem 2.1 extended and improved Theorem 1 in [19]. By using the inequality in Lemma 2.3, we obtain the following result. ...
... In order to show the application of our results obtained in this paper, let us consider the following second order difference equation with damping term: l Thus Theorem 2.1 asserts that every solution of (3.1) is oscillatory. We should note that the oscillation criteria given in [3] and [12] fail to apply for this difference equation. By Theorem 2.6 every solution of (3.2) is oscillatory. ...
... ≡ ≡. Then Theorem 2.1 extended and improved Theorem 1 in [19]. By using the inequality in Lemma 2.3, we obtain the following result. ...
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.
... In recent years, the oscillation or asymptotic behaviour of second-order difference equations was the subject of invesigation by many authors (see for example [1][2][4][5][6][7][8][9][10][11]). ...
Some oscillation criteria are established by Riccati transformation techniques for the following second-order nonlinear neutral difference equation Δ(p n (Δ(x n +c n x n-τ )) γ )+q n x n-σ β =0,n=0,1,2⋯ which extend and include several oscillation criteria in the paper by B. G. Zhang and S. H. Saker [Indian J. Pure Appl. Math. 34, No. 11, 1571–1584 (2003; Zbl 1061.39009)], and also correct a theorem and its proof in the paper by I. Kubiaczyk et al. [Indian J. Pure Appl. Math. 34, No. 8, 1273–1284 (2003; Zbl 1060.39005)].
... 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.
... Numerous oscillation and nonoscillation criteria have been established for the forms of (1.14) and (1.15); see, for example, [20][21][22][23][24][25][26] and references therein. ...
... where f (n, u) is nondecreasing and continuous with respect to u for fixed n. In papers123, the oscillatory behavior of self-adjoint difference equations has been discussed; the existence of nonoscillatory solutions has been investigated in the papers456; classification and existence of nonoscillatory solutions of neutral difference equations have been studied in [7] and the asymptotic and oscillatory behavior of second order difference equations have been given, see [8]. In this paper, we shall present the asymptotic classification of nonoscillatory solution of second order self-adjoint neutral difference equations. ...
Consider the second order self-adjoint neutral difference equation of form
D(an |D(xn - pn xtn )|a sgnD(xn - pn xtn )) + f(n,xgn ) = 0\Delta (a_n |\Delta (x_n - p_n x_{\tau _n } )|^\alpha sgn\Delta (x_n - p_n x_{\tau _n } )) + f(n,x_{g_n } ) = 0
In this paper, we will give the classification of nonoscillatory solutions of the above equation; and by the fixed point
theorem, we present some existence results for some kinds of nonoscillatory solutions of the equation.
... It is an interesting problem to extend oscillation criteria for second-order nonlinear difference equations of the neutral type to the case of nonlinear two-dimensional difference systems, since such systems include, in particular, secondorder nonlinear, half-linear and quasilinear difference equations as special cases. In this paper, oscillation results obtained by [2][3][4][5][6][7][8][9][10][11] for second-order nonlinear difference equations of the neutral type are extended to nonlinear two-dimensional difference systems. ...
The authors consider the nonlinear difference systems Delta( x(n) - a(n) x(sigma(n)) ) = p(n)g(y(n)) Delta y(n) = delta q(n) f(x(tau (n)) ), n epsilon N (n(0)) where delta = +/- 1, {a(n)}, {p(n)} and {q(n)} are real sequences, {sigma(n)} and {tau(n)} are real non-negative sequences of integers, and f, g : R -> R are continuous with uf(u) > 0 and ug(u) > 0 for u not equal 0. The authors obtain sufficient conditions for all solutions of the system to be oscillatory. Examples to illustrate the results are included. (c) 2007 Elsevier Ltd. All rights reserved.
... 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
... 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.
... Numerous oscillation and nonoscillation criteria have been established for the forms of (1.14) and (1.15); see, for example, [20][21][22][23][24][25][26] and references therein. ...
By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order nonlinear delay dynamic equations (p(t)(xΔ(t))γ)Δ+q(t)f(x(Ä(t)))=0 on a time scale T, here γ≥1 is a quotient of odd positive integers with p and q real-valued positive rd-continuous functions defined on T.
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.
In this paper, the authors establish some new oscillation criteria for the second order nonlinear difference equation with advanced superlinear neutral term. The oscillation results are obtained via only one condition. Further the results improve and simplify some known results reported in the literature. Examples are provided to illustrate the main results.
In this paper, we will study the oscillatory properties of the second order half-linear dynamic equations with distributed deviating arguments on time scales. We obtain several new sufficient conditions for the oscillation of all solutions of this equation. Our results not only unify the oscillation of second order nonlinear differential and difference equations but also can be applied to different types of time scales with sup = ∞ . Our results improve and extend some known results in the literature. Examples which dwell upon the importance of our results are also included.
This paper concerns the oscillatory behavior of the solutions to second-order damped nonlinear difference equations with a superlinear neutral term. We obtain oscillatory criteria by a Riccati type transformation as well as summation averaging conditions. We provide examples, illustrating the results and discuss extensions of this work, for future research.
The theory of difference equations, the methods used, and their wide applications have advanced beyond their adolescent stage to occupy a central position in applicable analysis. In fact, in the last 15 years, the proliferation of the subject has been witnessed by hundreds of research articles, several monographs, many international conferences, and numerous special sessions.
The theory of differential and difference equations forms two extreme representations of real world problems. For example, a simple population model when represented as a differential equation shows the good behavior of solutions whereas the corresponding discrete analogue shows the chaotic behavior. The actual behavior of the population is somewhere in between.
The aim of Advances in Difference Equations is to report mainly the new developments in the field of difference equations, and their applications in all fields. We will also consider research articles emphasizing the qualitative behavior of solutions of ordinary, partial, delay, fractional, abstract, stochastic, fuzzy, and set-valued differential equations.
Advances in Difference Equations will accept high-quality articles containing original research results and survey articles of exceptional merit.
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.
The major components of the time series are the long term trend, the short term trend, cyclic variation and irregular fluctuations. Various attempts have been made to give necessary conditions for processing the specific components. Here we take necessary conditions to predict the asymptotic behavior of the time series using second order difference of the combinations of observations obtained from a general time series. Specific illustrations are given to authenticate our claim.
In this paper we show the existence of bounded nonoscillatory solutions for a second order neutral difference equation and establish convergence of Mann iterative schemes relative to the bounded nonoscillatory solutions.
By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden-Fowler delay dynamic equations (rxδ)δ(t)+p(t)x(τ(t))= 0 on a time scale ; here is a quotient of odd positive integers with r and p as real-valued positive rd-continuous functions defined on T. Our results in this paper not only extend the results given in Agarwal et al. (2005), Akin-Bohner et al. (2007) and Han et al. (2007) but also unify the results about oscillation of the second-order Emden-Fowler delay differential equation and the second-order Emden-Fowler delay difference equation.
The existence of uncountably many positive solutions and convergence of Mann iterative schemes for a fourth order neutral delay difference equation are proved. Seven examples are included.
MSC: 39A10.
The existence, multiplicity, and properties of positive solutions for a third order nonlinear neutral delay difference equation are discussed. Six examples are given to illustrate the results presented in this paper.
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.
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.
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.
By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order Emden–Fowler delay dynamic equationsxΔΔ(t)+p(t)xγ(τ(t))=0 on a time scale T; here γ is a quotient of odd positive integers with p(t) real-valued positive rd-continuous functions defined on T. To the best of our knowledge nothing is known regarding the qualitative behavior of these equations on time scales. Our results in this paper not only extend the results given in [R.P. Agarwal, M. Bohner, S.H. Saker, Oscillation of second-order delay dynamic equations, Can. Appl. Math. Q. 13 (1) (2005) 1–18] but also unify the oscillation of the second-order Emden–Fowler delay differential equation and the second-order Emden–Fowler delay difference equation.
In this paper, the Banach fixed-point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference equationΔa(n)Δx(n)+b(n)x(n-τ)+Δh(n,x(h1(n)),x(h2(n)),…,x(hk(n)))+f(n,x(f1(n)),x(f2(n)),…,x(fk(n)))=c(n),n⩾n0,where τ,k∈N, n0∈N0, a,b,c:Nn0→R with a(n)>0 for n∈Nn0, h,f:Nn0×Rk→R and hl,fl:Nn0→Z withlimn→∞hl(n)=limn→∞fl(n)=+∞,l∈{1,2,…,k}.A few Mann type iterative approximation schemes with errors are suggested, and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed. Seven nontrivial examples are given to illustrate the advantages of our results.
By means of Reccati transformation techniques we present some oscillation criteria for second order-superlinear and sublinear neutral delay 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.
The ridgelet transform was developed over several years to break the limitations of the wavelet transform. In this paper, a novel image denoising algorithm is proposed that incorporates the dual-tree complex wavelets into the ordinary ridgelet transform. The approximate shift invariant property of the dual-tree complex wavelet and the high directional sensitivity of the ridgelet transform make the new method a very good choice for image denoising. We apply the digital complex ridgelet transform to the denoising of some standard images embedded in white noise. A simple hard thresholding of the complex ridgelet coefficients is used. Experimental results show that by using dual-tree complex ridgelets, our algorithms obtain higher peak signal to noise ratio (PSNR) for all the denoised images with different noise levels. The new modified ridgelet denoising algorithm - MRDA is better than Wiener2 and the classical CRD A ridgelet image denoising. Complex ridgelet could be applied to curvelet image denoising as well.
By means of Riccati transformation technique, we establish some new oscillation criteria for the second-order delay dynamic equations. Our results in this paper not only extend the results given in Sahiner [?] but also unify the oscillation of the second order nonlinear delay differential equation and the second order nonlinear delay difference equation.
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.
Sufficient conditions have been obtained for oscillation/nonoscillation of a class of homogeneous nonlinear difference equations. Oscillation of a class of linear nonhomogeneous difference equations has also been considered.
The authors obtain results on the oscillation properties of solutions of second order nonlinear neutral delay difference equations. Some illustrative examples are also included.
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.
We consider the second-order functional difference equation Δ(a(n)Δ(n)) + p(n)x(g(n))=0,where ↶n), p(n) are sequences of integers with a(n) > 0, 0 ≤ g(n) ≤ n, limn→∞g(n) = ∞, p(n) ≥ 0, and p(n) ≢ 0. We obtain some necessary conditions for equation (1) exists nonoscillatory solutions and some sufficient conditions for equation (1) is oscillatory.
Some new oscillation theorems for the difference equations of the form Δ(rnΔun)+qnf(un−k)=0 are established.
Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation
$$\Delta \left( {c_n \Delta \left( {y_n {\text{ + }}p_n y_n - k} \right)} \right) + q_n y_{n + 1 - m}^\beta = 0,n \geqslant n_0 $$
where k, m are positive integers and β is a ratio of odd positive integers are established, under the condition
$${\sum\limits_{n{\text{ = }}n_O }^\infty {1/c_n < } \;\infty }$$
In this paper, we consider the oscillation of the second-order neutral difference equation
D2 ( xn - pxn - t ) + qn f( xn - sn ) = 0\Delta ^2 \left( {x_n - px_{n - \tau } } \right) + q_n f\left( {x_{n - \sigma _n } } \right) = 0
as well as the oscillatory behavior of the corresponding ordinary difference equation
D2 zn + qn f( R( n,l )zn ) = 0\Delta ^2 z_n + q_n f\left( {R\left( {n,\lambda } \right)z_n } \right) = 0
.
We consider the second-order advanced functional difference equation Δ(a(n)Δχ(n)) + p(n)χ(g(n)) = 0,where a(n) > 0, , p(n) ≥ 0, p(n) ≡ 0, g(n) ≥ n + 1, {g(n)} is a monotone increasing integer sequence. We obtain some new oscillation criteria through an appropriate Riccati equation.
We consider the self-adjoint second order scalar difference equation DELTA(r(n)DELTAx(n)) + p(n)x(n+1) = 0, (1) and the matrix system DELTA(R(n)DELTAX(n)) + P(n)X(n+1) = 0, (2) where {r(n)}0infinity, {p(n)}0infinity, ({R(n)}0infinity, {P(n)}0infinity) are sequences of real numbers (d x d Hermitian matrices) with r(n) > 0, (R(n) > 0). We obtain some new oscillation criteria through change of variable techniques combined with consideration of (1) and (2) as perturbations of the cases p(n) = 0 (P(n) = 0).
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.