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Oscillatory and asymptotic behavior of second-order neutral difference equations with maxima
Abstract
In this paper, the asymptotic behavior of the non-oscillatory solutions of neutral difference equations with maxima is considered. Sufficient conditions for oscillation of all solutions are also obtained.
... under the condition ̸ = − 1. Luo and Bainov [13] and M. Migda and J. Migda [16] considered the asymptotic behaviors of nonoscillatory solutions for the second-order neutral difference equation with maxima: ...
... Set ∈ ( , ). It follows from (13), (14), and (15) that there exist ∈ (0, 1) and ...
... which together with Lemma 2 yield that (13) and (14) hold. It follows from Theorem 3 that (52) possesses uncountably many unbounded positive solutions in ( , ). ...
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.
... In recent years there has been increasing interest in the study of the qualitative theory of neutral difference equations. For example, the first and second order difference equations of neutral type have been investigated in [5], [6], [8], [9], [12], [14]. For higher order difference equations we refer to [4], [10], [11], [13], [15]. ...
... For higher order difference equations we refer to [4], [10], [11], [13], [15]. In most of the papers [5], [6], [7], [8], [10], [11] it is assumed that the coefficient q satisfies the divergent condition of the series ...
We consider the difference equation of neutral type Δ 3 [x(n)-p(n)x(σ(n))]+q(n)f(x(τ(n)))=0,n∈ℕ(n 0 ), where p,qℕ(n 0 )→ℝ + ; σ,τℕ→ℤ, σ is strictly increasing and lim n→∞ σ(n)=∞; τ is nondecreasing and lim n→∞ τ(n)=∞, fℝ→ℝ, xf(x)>0. We examine the following two cases: 0<p(n)≤λ * <1,σ(n)=n-k,τ(n)=n-l, and 1<λ * ≤p(n),σ(n)=n+k,τ(n)=n+l, where k,l are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as n→∞ with a weaker assumption on q than the usual assumption ∑ i=n 0 ∞ q(i)=∞ that is used in the literature.
... Equation (1.1) is a quantum version of ∆ 2 x n + p n x n−k + q n max {n−,··· ,} x s = 0, (1.2) studied by Luo and Bainov [5]; there the usual forward difference operator ∆y n := y n+1 − y n was used. For more results on differential and difference equations related to (1.1) and (1.2), please see the work by Bainov, Petrov, and Proytcheva [1,2,3], Luo and Bainov [5], Luo and Petrov [6], and Petrov [7]. ...
... Equation (1.1) is a quantum version of ∆ 2 x n + p n x n−k + q n max {n−,··· ,} x s = 0, (1.2) studied by Luo and Bainov [5]; there the usual forward difference operator ∆y n := y n+1 − y n was used. For more results on differential and difference equations related to (1.1) and (1.2), please see the work by Bainov, Petrov, and Proytcheva [1,2,3], Luo and Bainov [5], Luo and Petrov [6], and Petrov [7]. The particular appeal of (1.1) is that it is still a discrete problem, but with non-constant step size between domain points. ...
In this study, the behavior of solutions to certain second order quantum (q-difference) equations with maxima are considered. In particular, the asymptotic behavior of non-oscillatory solutions is described, and sufficient conditions for os- cillation of all solutions are obtained.
... The existence of solutions, asymptotic behavior, oscillation and non-oscillation for second-order difference equations were extensively analyzed in many research papers over the last three decades; see, for example, [4,7,8,9,10,11,17,18,20] and the cited references. Many areas of applied mathematics use neutral difference and differential equations, such as bifurcation analysis [5], stability theory [22,23], the dynamical behavior of delayed network systems [24], circuit theory [6], population dynamics [14] and so on. ...
... From the review of literature it is well known that there is a lot of results available on the oscillatory and asymptotic behavior of solutions of neutral difference equations without maxima, see [1,2,9,10], and the references cited therein. But very few results available in the literature dealing with the oscillatory and asymptotic behavior of solutions of neutral difference equations with "maxima", see [3,4,7,8] and the references cited therein. Therefore, in this paper, we investigate the oscillatory and asymptotic behavior of all solutions of equation (1.1). ...
... 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.
... Neutral difference equations were studied in many other papers by Grace and Lalli [11] and [14], Lalli and Zhang [15], Migda and Migda [20], Luo and Bainov [16], and Luo and Yu [17]. ...
The purpose of this paper is to investigate a nonlinear second-order neutral difference equation of the form Δ(r n Δ(x n + p n x n−k)) + a n f (x n) = 0, where x : N 0 → R, a : N 0 → R, p, r : N 0 → R \ {0}, f : R → R is a continuous function, and k is a given positive integer. Sufficient conditions for the existence of a bounded solution of this equation are obtained. Also, stability and asymptotic stability of this equation are studied. Additionally, the Emden-Fowler difference equation is considered as a special case of the above equation. The obtained results are illustrated by examples.
... Recently there has been an increasing interest in the study of the qualitative behavior of solutions of neutral difference equations (see the monographs [1]- [3], [6]). Particularly, the oscillation and nonoscillation of solutions of the second-order neutral difference equations attract attention; see the papers [4], [5], [7]- [9], [11], [13]- [16] and the references therein. The interesting oscillatory results for first order and even order neutral difference equations can be found in [10] and [12]. ...
Using the Riccati transformation techniques, we will extend some almost
oscillation criteria for the second-order nonlinear neutral difference equation
with quasidifferences $$\Delta\left(r_n\left(\Delta \left(x_n+c
x_{n-k}\right)\right)^{\gamma}\right)+q_nx_{n+1}^{\alpha}=e_n.$$
... Let (y (p) ) be a sequence in B such that y (p) − x → 0 as p → ∞. Since B is closed, x ∈ B. By (16) and (3), we get ...
\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}
... 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 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.
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.
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.
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.
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.
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 this paper, we study the asymptotic behavior of nonoscillatory solutions of neutral difference equations with "maxima" of the form Examples are given to illustrate the main result.
For the second order functional difference equation, Δ 2x(t-1) = f(t, x(t), x(w(t))), t E {1,⋯,T+ 1), subject to the initial condition x(j) = v(j),j E {-r,⋯,-1}, and the boundary conditions, x(0) = c1 and x(T + 2) = C2, where w : {1,⋯,T + 1} -v {-r,⋯,-1}, v {-r,. . . , -1) - ℝ, and c1, c 2 E R, we consider differences of solutions with respect to the functional w, as well as the smooth dependence of solutions with respect to the initial value v and with respect to the boundary values c, and c2.
In this paper we consider the neutral delay difference equation Δm (xn + pnxn-k) + f(n,xn-l) = 0, where f (n, u) is nondecreasing in u, and k and l are positive integers. We establish some sufficient conditions for the oscillations of all solutions of this equation, and obtain the result for monotone solutions.
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.
The purpose of this paper is to investigate a nonlinear second-order neutral difference equation of the form
where
,
,
,
is a continuous function, and k is a given positive integer. Sufficient conditions for the existence of a bounded solution of this equation are obtained. Also, stability and asymptotic stability of this equation are studied. Additionally, the Emden-Fowler difference equation is considered as a special case of the above equation. The obtained results are illustrated by examples.
MSC:
39A10, 39A22, 39A30.
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.
In this paper, we study asymptotic behavior of solutions of second-order neutral difference equationΔ2(xn+pxn-k)+f(n,xn)=0.We present conditions under which all nonoscillatory solutions of the above equation have the property xn=cn+o(n) for some c∈R as well as sufficient conditions under which all nonoscillatory solutions are asymptotically linear.
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.
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 asymptotic behaviour of nonoscillatory solutions of neutral equations with “maxima” is considered. Examples are given showing the difference between the equations with “maxima” and the corresponding neutral equations without “maxima”.
First-order neutral differential equations with “maxima” are considered. Sufficient conditions for oscillation of all solutions are obtained. The asymptotic behavior of the nonoscillatory solutions is investigated.
The author establishes some new criteria for the asymptotic behavior of nonoscillatory solutions of neutral equations with “maxima” of the form ((a(t)x(t)+p(t)x(t-τ)) ' ) ' +q(t)max [t-σ,t] x α (s)=0,t≥t 0 ≥0, where a(t)>0, q(t)>0, p(t)≥0, τ≥0, σ≥0, α is the ratio of odd positive integers and ∫ 0 ∞ 1 a(t)dt=∞. Sufficient conditions for oscillation of all solutions are obtained. Examples are provided to illustrate the main results.
Sufficient conditions for oscillation of all solutions of second order neutral differential equations with maxima are obtained. The main result is new even for neutral equations without maxima.
The asymptotic behavior of the nonoscillatory solutions of neutral equations with “maxima” is considered. Sufficient conditions for oscillation of all solutions are obtained.
The authors discuss the oscillation of the second-order nonlinear neutral differential equation [a(t)(x(t)+p(t)x(t-τ)) ' ] ' +q(t)f(x(t-σ))=0·(E) It is sufficient to discuss the oscillation of the second-order linear ordinary differential equation x '' (t)+b(t)x(t)=0, where b(t) is related to the coefficients of (E). Some new and explicit oscillation criteria are presented.
In the paper two conjectures of Ladas and Sficas on the asymptotic behaviour of the nonoscillating solutions of neutral differential equations are confirmed. A conjecture of Grammatikopoulos, Grove, and Ladas of 1986 is also proved.
Verifiable sufficient conditions are obtained respectively for the oscillation and nonoscillation of the neutral differential equation x(t) − cx(t − τ) + px(t − σ) = 0.
Bainov, D.D., V. Petrov and V. Proytcheva, Existence and asymptotic behaviour of nonoscillatory solutions of second order neutral differential equations with “maxima”, Journal of Computational and Applied Mathematics.In the paper, sufficient conditions for existence of positive and negative solutions of neutral equations with “maxima” are obtained and their asymptotic behaviour is investigated.