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Abstract
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.
... The problem of oscillation and nonoscillation of all solutions of the neutral difference equation with constant delays ∆(a n + ca n−m ) + p n a n−k = 0, n = 0, 1, 2, ... has been investigated in many papers (see [2], [3], [4], [8] and references therein), where ∆ denotes the forward difference operator: ∆a n = a n+1 − a n . ...
... If the values a n = φ n , for n = n −1 , n −1 + 1, ..., n 0 , φ n ∈ R . (4) are given, then equation (1) has a unique solution satisfying the initial conditions (4). ...
... If the values a n = φ n , for n = n −1 , n −1 + 1, ..., n 0 , φ n ∈ R . (4) are given, then equation (1) has a unique solution satisfying the initial conditions (4). ...
... It is well-known that difference equation ∆(x n + δx n−τ ) + α(n)x n−σ = 0, (1) where n ∈ N, the operator ∆ is defined as ∆x n = x n+1 − x n , the function α(n) is defined on N, δ is a constant, τ is a positive integer and σ is a nonnegative integer, was first considered by Brayton and Willoughby from the numerical point of view (see [1]). In recent years, the asymptotic behavior of solutions of this equation has been studied extensively (see [2][3][4][5][6][7]). In [4,6,7], the oscillation of solutions of the difference equation (1) was discussed. ...
... which contradicts condition (4). Hence, (5) has no eventually positive solution. ...
... which contradicts condition (8). Hence, (5) has no eventually positive solution. ...
... In most of the papers [1,2,6,7,10,11], the authors established conditions for the oscillation and nonoscillation of solutions of equation of type (1) with α = 1 and treating the deviations are constant. In [1,3,5,8,9,12], the authors consider the particular cases of equation (1) in the form ∆(a n (∆y n ) α ) − g n f (y α(n) )) = 0 (2) or ∆(a n ∆y n ) − g n f (y n+1 ) = 0 ...
... From Theorem 3.1 of [6] it is clear that it is not possible to find criteria for all the solutions of equation (1) to be oscillatory when {σ(n)} is increasing with σ(n) ≤ n. Proof. ...
... Remark 1. If α = 1, a n ≡ 1 and σ(n) = n − l, then Theorem 1 reduces to Theorem 4.1 of Lalli and Zhang [6]. ...
This paper deals with the oscillatory behavior of all bounded/ unbounded solutions of second order neutral type difference equation of the form Δ(an(Δcyn + Pyn-k)) α) - gnf(yσ(n))) = 0, where p is real, α is a ratio of odd positive integers, k is a positive integer and {σ(n)} is a sequence of integers. Examples are provided to illustrate the results.
... In a number of recent papers [2][3][4][5][6][7][8][9][10], the oscillation and nonoscillation of solutions of delay difference equations are being extensively investigated. In paticular the oscillation of solutions of the neutral difference equation ...
... (1.1) has been investigated in [3,6,8,9,10], where Pn > 0, c E (0, 1), 6. denotes the forward difference operator 6.· yn = Yn+I -Yn· Equation (1.1) was considered by Brayron and Willoughby [1] from the numerical analysis point of view. ...
In this paper we study qualitative properties of solutions of the neutral difference equation $$ \Delta(y_n-py_{n-k})+\sum_{i=1}^m q_n^i y_{n-k_i} =0 $$ $$ y_n=A_n \quad \text{ for } n=-M, \cdots, -1, 0$$ where $p \ge 1$, $M =\max\{k, k_1, \cdots, k_m\}$, and $k$, $k_i$, $i =1, \cdots, m$, are nonnegative integers. Riccati techniques are used.
... 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 ...
... (7) [or (8)] and for which u(n) = x(n) − p(n)x(σ(n)) is of degree ℓ. One can observe that if (7) holds that the condition (4) is fulfilled with δ = 1. Since m = 3, so (−1) m+ℓ−1 δ = 1 if ℓ is even. ...
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.
... Since vi -k n, Xn1 _k <max{xn : n 1 < n -k} <max{x n : n 1 5 ...
... Theorem 2, it follows that all bounded solutions of equation(5)are either oscillatory or tend to zero as n -* oo. In particular, {x,,} = {} is a solution of equation(5) tending to zero. n(2n + 1) -(n + 1)(2n + 3) ...
Some new oscillation results for certain class of forced nonlinear difference equations of the form Δ(an Δ(xn + pnxn-k)) + qn f(xn+1-1) = en (n ∈ ℕ0) are established. Examples which dwell upon the importance of the results are also given.
... Recently, there has been a great deal of work on the oscillation of solutions of difference equations (see, for example,123456789101112131415 and the references cited therein). Ladas, Philos and Sficas [6], Philos [10,12], and Philos and Purnaras [13,14] investigated the oscillation problem for linear difference equations with variable coefficients, which are assumed to be nonnegative, and obtained necessary conditions and also sufficient conditions for the oscillation of all solutions. ...
... Ladas, Philos and Sficas [6], Philos [10,12], and Philos and Purnaras [13,14] investigated the oscillation problem for linear difference equations with variable coefficients, which are assumed to be nonnegative, and obtained necessary conditions and also sufficient conditions for the oscillation of all solutions. Some oscillation results for neutral difference equations have given by Chen, Lalli and Yu [1], Georgiou, Grove and Ladas [2], and Lalli and Zhang [7,8]. In the special case of (homogeneous) linear difference equations with constant coefficients, it is known that all solutions are oscillatory if and only if the associated characteristic equation has no positive roots. ...
Two classes of neutral linear difference equations with periodic coefficients are considered, and necessary conditions and also sufficient conditions for the oscillation of all solutions are established. These conditions are referred to the roots of the associated characteristic equations.
... is a sequence of positive real numbers; Recently, there has been much interest in studying the oscillatory and asymptotic behavior of second order functional difference equations; see for example [3,4,6,8,9,[12][13][14][15][16][17][18][19][20][21][22][23][24]. For the general theory of difference equations, one can refer to [1,2,7]. ...
In this paper, we discuss a class of second order neutral delay difference equation of the form Where We determine sufficient conditions under which every solution of (*) is either oscillatory or tends to zero. Our results improve a number of related results reported in the literature.
... However, the impulsive difference equations of neutral type especially (E 1 )/(E h ) is not well studied. In this direction, we refer the reader to some works [3], [4], [7]- [13] and the references cited there in. ...
In this work, we have established sufficient conditions for oscillation
and nonoscillation of a class of forced first order neutral impulsive difference
equations with deviating arguments and fixed moments of impulsive effect.
... Recently, there has been much interest in studying the oscillatory and asymptotic behavior of second order functional difference equations; see for example [3,4,6,8,9,[12][13][14][15][16][17][18][19][20][21][22][23][24]. For the general theory of difference equations, one can refer to [1,2,7]. ...
... By a solution of (E) we mean a sequence x : N → R satisfying (E), for all large n. Convergent solutions to difference equations were investigated by many authors, see, for example, [1]- [6] or [12]- [14]. The aim of this paper is to investigate, for a given b ∈ R and by the assumption of absolute convergence of the series ∑ ∞ n=1 u n , the properties of a function ϕ sufficient for the existence of a solution of (E) convergent to b. ...
In this paper the difference equation
Δxn=unφ(xσ (n))
is considered. For a given b ∈ ℝ, and assuming the absolute convergence of the series ∑ un, we investigate the properties of a function φ sufficient for the existence of a solution convergent to b. It is known that the continuity of φ on some neighborhood of b is sufficient for the existence of such solution. The main result of this paper is the example in which we show that the continuity at the point b is not sufficient.
... Difference equations are used in mathematical models in diverse areas such as economy, biology, computer science, see, for example [1], [7]. In the past thirty years, oscillation, nonoscillation, the asymptotic behaviour and existence of bounded solutions to many types second-order difference equation have been widely examined, see for example [2], [4], [6], [9], [10], [11], [13], [14], [17], [18], [19], [20], [27], [28], [29], [30], [31], [32], and references therein. ...
This work is devoted to the study of the nonlinear second-order neutral difference equations with quasi-differences of the form $$ \Delta \left( r_{n} \Delta \left( x_{n}+q_{n}x_{n-\tau}\right)\right)= a_{n}f(x_{n-\sigma})+b_n%, \ n\geq n_0 $$ with respect to $(q_n)$. For $q_n\to1$, $q_n\in(0,1)$ the standard fixed point approach is not sufficed to get the existence of the bounded solution, so we combine this method with an approximation technique to achieve our goal. Moreover, for $p\ge 1$ and $\sup|q_n|<2^{1-p}$ using Krasnoselskii's fixed point theorem we obtain sufficient conditions of the existence of the solution which belongs to $l^p$ space.
... In recent years, there has been an increasing interest in studying the oscillatory and nonoscillatory behavior of various classes of differential, difference, and dynamic equations; see, for instance, the monographs [1, 2], papers [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13], and the references cited therein. Candan [3] investigated a higher-order nonlinear neutral differential equation ...
We study a class of second-order neutral delay difference equations with positive and negative coeffcients (formula presented) Some suffcient conditions for the existence of a nonoscillatory solution of the studied equation expressed in terms of (formula presented). © 2015 International Scientific Research Publications. All rights reserved.
... 6 denotes the forward difference operator 6.xn = Xn+1 -Xn. However, the results for the existence of positive solutions of Eq.(1) are relatively scarce in the literature, we refer to [10,11], see also Cyori and Lada.s's book [4]. ...
Consider the neutral difference equation \[\Delta(x_n- cx_{n-m})+p_nx_{n-k}=0, n\ge N\qquad (*) \] where $c$ and $p_n$ are real numbers, $k$ and $N$ are nonnegative integers, and $m$ is positive integer. We show that if \[\sum_{n=N}^\infty |p_n|<\infty \qquad (**) \] then Eq.(*) has a positive solution when $c \neq 1$. However, an interesting example is also given which shows that (**) does not imply that (*) has a positive solution when $c =1$.
... Difference equations of neutral type have been studied by a number of authors in recent years, for example, see [2][3][4][5][6][7][8][9][10][11]13] authors have obtained results guaranteeing the oscillation of equation (1), and we cite the papers [2,[6][7][8]. In this paper we are interested in obtaining necessary and sufficient conditions for the existence of nonoscillatory solutions of (1). ...
Necessary and sufficient conditions for the existence of nonoscillatory solutions of the forced nonlinear difference equation Δ(x n -p n x τ(n) )+f(n,x σ(n) )=q n are obtained. Examples are included to illustrate the results.
... Since (zn) is a nonincreasing solution of (1), hence (13) has no eventually negative solution, then the equation (1) is oscillatory. ...
We obtain sufficient conditions for the oscillation of all solutions of some linear difference equations with variable coefficients.
... Some special cases of the equation (1.1) have been studied by several authors [1]- [4]. In particular, Chen and Zhang [2] and Lalli and Zhang [3] discussed the oscillation, the bounded oscillation and the existence of positive solutions of the first order neutral delay difference equation: ...
In this paper, we study the first order nonlinear neutral difference equation: ∆ x(n) + px(n − τ) + f n, x(n − c), x(n − d) = r(n), n ≥ n 0 . Using the Banach fixed point theorem, we prove the existence of bounded positive solutions of the equation, suggest Mann iterative schemes of bounded positive solutions, and discuss the error esti-mates between bounded positive solutions and sequences generated by Mann iterative schemes.
... [2] [3] [58] [10] [12] [15] [16] [18] [19] [22] [2431] ...
In this paper, we study the asymptotic behavior of the solutions of a neutral type difference equation of the form Δ[x(n) + ∑j=1wcjx(τj(n))] + p(n)x(σ(n)) = 0, n ≥ 0 where τj(n), j = 1, . . . , w, are general retarded arguments, σ(n) is a general deviated argument (retarded or advanced), cj ∈ ℝ, j = 1, . . . , w, (p(n))n≥0 is a sequence of positive real numbers such that p(n) ≥ p, p ∈ ℝR+, and Δ denotes the forward difference operator Δx(n) = x(n + 1) - x(n). We also examine the convergence of the solutions when these are continuous and differentiable with respect to cj, j = 1, . . . , w.
... In the last few decades, the asymptotic and oscillatory behavior of neutral difference equations has been extensively studied. See, for example [2][3][4][5][6][7][8][9][10][11][12][14][15][16][17][19][20][21][22][23][24][25][26][27] and the references cited therein. Most of these papers concern the special case where the delay n À sðnÞ ð Þ nP0 is constant, while a small number of these papers are dealing with the general case of Eq. ...
In this paper, we present several sufficient conditions for the existence of nonoscillatory solutions to the following third order neutral type difference equation
\Delta^3(x_n+a_n x_{n-l} +b_n x_{n+m})+p_n x_{n-k} - q_n x_{n+r}=0,\quad n\geq n_0
via Banach contraction principle. Examples are provided to illustrate the main results. The results obtained in this paper extend and complement some of the existing results.
A second-order nonlinear neutral difference equation with a quasi-difference is studied. Sufficient conditions are established under which for every real constant there exists a solution of the considered equation convergent to this constant.
In this paper, we derive several comparison and oscillation criteria for solutions of a class of nonlinear difference equations with a nonlinear neutral term. Existence criteria are also derived for the eventually positive solutions of this equation.
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.
By means of Riccati technique, we establish some new oscillation criteria for difference equations of neutral type in terms of the coefficients. The results are illustrated by examples.
Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form
\[\Delta \left( r_{n}\left( \Delta \left( x_{n}+p_{n}x_{n-k}\right) \right)
^{\gamma }\right) +q_{n}x_{n}^{\alpha }+a_{n}f(x_{n+1})=0,\]
where $x\colon{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$, $a,p,q\colon {\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $r\colon {\mathbb{N}}_{0}\rightarrow {\mathbb{R}} \setminus \{0\}$, $f\colon {\mathbb{R}}\rightarrow {\mathbb{R}}$ is continuous and $k$ is a given positive integer, $\alpha \geq 1$ is a ratio of positive integers with odd denominator, and $\gamma \leq 1$ is ratio of odd positive integers; ${\mathbb{N}}_{k}:=\left\{ k,k+1,\dots \right\}$. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability is studied.
Autism spectrum disorders (ASD) are a group of poorly understood behavioural disorders, which have increased in prevalence in the past two decades. Animal models offer the opportunity to understand the biological basis of these disorders. Studies comparing different mouse strains have identified the inbred BTBR T + tf/J (BTBR) strain as a mouse model of ASD based on its anti-social and repetitive behaviours. Adult BTBR mice have complete agenesis of the corpus callosum, reduced cortical thickness and changes in early neurogenesis. However, little is known about the development or ultimate organisation of cortical areas devoted to specific sensory and motor functions in these mice that may also contribute to their behavioural phenotype.
In this study, we performed diffusion tensor imaging and tractography, together with histological analyses to investigate the emergence of functional areas in the cerebral cortex and their connections in BTBR mice and age-matched C57Bl/6 control mice. We found evidence that neither the anterior commissure nor the hippocampal commissure compensate for the loss of callosal connections, indicating that no interhemispheric neocortical connectivity is present in BTBR mice. We also found that both the primary visual and somatosensory cortical areas are shifted medially in BTBR mice compared to controls and that cortical thickness is differentially altered in BTBR mice between cortical areas and throughout development.
We demonstrate that interhemispheric connectivity and cortical area formation are altered in an age- and region-specific manner in BTBR mice, which may contribute to the behavioural deficits previously observed in this strain. Some of these developmental patterns of change are also present in human ASD patients, and elucidating the aetiology driving cortical changes in BTBR mice may therefore help to increase our understanding of this disorder.
There is converging preclinical and clinical evidence to suggest that the extracellular signal-regulated kinase (ERK) signaling pathway may be dysregulated in autism spectrum disorders.
We evaluated Mapk/Erk1/2, cellular proliferation and apoptosis in BTBR mice, as a preclinical model of Autism. We had previously generated 410 F2 mice from the cross of BTBR with B6. At that time, six different social behaviors in all F2 mice were evaluated and scored. In this study, eight mice at each extreme of the social behavioral spectrum were selected and the expression and activity levels of Mapk/Erk in the prefrontal cortex and cerebellum of these mice were compared. Finally, we compared the Mapk/Erk signaling pathway in brain and lymphocytes of the same mice, testing for correlation in the degree of kinase activation across these separate tissues.
Levels of phosphorylated Erk (p-Erk) were significantly increased in the brains of BTBR versus control mice. We also observed a significant association between juvenile social behavior and phosphorylated mitogen-activated protein kinase kinase (p-Mek) and p-Erk levels in the prefrontal cortex but not in the cerebellum. In contrast, we did not find a significant association between social behavior and total protein levels of either Mek or Erk. We also tested whether steady-state levels of Erk activation in the cerebral cortex in individual animals correlated with levels of Erk activation in lymphocytes, finding a significant relationship for this signaling pathway.
These observations suggest that dysregulation of the ERK signaling pathway may be an important mediator of social behavior, and that measuring activation of this pathway in peripheral lymphocytes may serve as a surrogate marker for central nervous system (CNS) ERK activity, and possibly autistic behavior.
An important diagnostic criterion for social communication deficits in autism spectrum disorders (ASD) are difficulties in adjusting behavior to suit different social contexts. While the BTBR T+tf/J (BTBR) inbred strain of mice is one of the most commonly used mouse models for ASD, little is known about whether BTBR mice display deficits in detecting changes in social context and their ability to adjust to them. Here, it was tested therefore whether the emission of isolation-induced ultrasonic vocalizations (USV) in BTBR mouse pups is affected by the social odor context, in comparison to the standard control strain with high sociability, C57BL/6J (B6). It is known that the presence of odors from mothers and littermates leads to a calming of the isolated mouse pup, and hence to a reduction in isolation-induced USV emission. In accordance with their behavioral phenotypes with relevance to all diagnostic core symptoms of ASD, it was predicted that BTBR mouse pups would not display a calming response when tested under soiled bedding conditions with home cage bedding material containing maternal odors, and that similar isolation-induced USV emission rates would be seen in BTBR mice tested under clean and soiled bedding conditions. Unexpectedly, however, the present findings show that BTBR mouse pups display such a calming response and emit fewer isolation-induced USV when tested under soiled as compared to clean bedding conditions, similar to B6 mouse pups. Yet, in contrast to B6 mouse pups, which emitted isolation-induced USV with shorter call durations and lower levels of frequency modulation under soiled bedding conditions, social odor context had no effect on acoustic call features in BTBR mouse pups. This indicates that the BTBR mouse model for ASD does not display deficits in detecting changes in social context, but has a limited ability and/or reduced motivation to adjust to them.
Autism spectrum disorders are characterized by impaired social and communicative skills and repetitive behaviors. Emerging evidence supported the hypothesis that these neurodevelopmental disorders may result from a combination of genetic susceptibility and exposure to environmental toxins in early developmental phases. This study assessed the effects of prenatal exposure to chlorpyrifos (CPF), a widely diffused organophosphate insecticide endowed with developmental neurotoxicity at sub-toxic doses, in the BTBR T+tf/J mouse strain, a validated model of idiopathic autism that displays several behavioral traits relevant to the autism spectrum. To this aim, pregnant BTBR mice were administered from gestational day 14 to 17 with either vehicle or CPF at a dose of 6 mg/kg/bw by oral gavages. Offspring of both sexes underwent assessment of early developmental milestones, including somatic growth, motor behavior and ultrasound vocalization. To evaluate the potential long-term effects of CPF, two different social behavior patterns typically altered in the BTBR strain (free social interaction with a same-sex companion in females, or interaction with a sexually receptive female in males) were also examined in the two sexes at adulthood. Our findings indicate significant effects of CPF on somatic growth and neonatal motor patterns. CPF treated pups showed reduced weight gain, delayed motor maturation (i.e., persistency of immature patterns such as pivoting at the expenses of coordinated locomotion) and a trend to enhanced ultrasound vocalization. At adulthood, CPF associated alterations were found in males only: the altered pattern of investigation of a sexual partner, previously described in BTBR mice, was enhanced in CPF males, and associated to increased ultrasonic vocalization rate. These findings strengthen the need of future studies to evaluate the role of environmental chemicals in the etiology of neurodevelopment disorders.
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.
We study the neutral difference equation of the form Δ m(xn - pxn-τ)=qnx n-σ. Existence of a positive unbounded solution is established. Sufficient conditions for all bounded solutions to be oscillatory are obtained.
Asymptotic and oscillatory behavior of solutions of a second order nonlinear neutral difference equation of the formΔ(a n(Δ(x n+p nx n-k))α)-q nf(x n+1-1)=0 is studied. Examples are included to illustrate the results.
The authors investigate asymptotic properties of solutions of the perturbed second order nonlinear difference equation Δa n-1 ψ (y n-1 ) Δ y n-1 +Q(n,y σ(n) )=P(n,y σ(n) ,Δy n )· Examples illustrating the necessity of some of the hypotheses are included.
Some new oscillation criteria for parabolic neutral delay difference equations corresponding to two sets of boundary conditions are obtained. Our results improve the well known results in the literature.
In this paper, we derive several nonexistence criteria for eventually positive solutions of a class of nonlinear recurrence relation. These criteria are then used to obtain oscillation criteria for an associated nonlinear neutral difference equation with delay.
In this paper,we will establish some sufficient conditions for the existence of nonoscillatory solutions of the first order delay dynamic equation (x(t) + cx(t τ))Δ +p(t)x(t - γ) = 0 on a time scale T, where c ∈ ℝ, τ, γ ≥ 0, σ = max{τ, γ}, p(t) is a rd-continuous function defined on T. Our results as a special case when T = ℝ and T = ℕ, involve and improve some nonoscillation results.
Autism spectrum disorder (ASD) may arise from increased ratio of excitatory to inhibitory neurotransmission in the brain. Many pharmacological treatments have been tested in ASD, but only limited success has been achieved. Here we report that BTBR T(+)Itpr3(tf)/J (BTBR) mice, a model of idiopathic autism, have reduced spontaneous GABAergic neurotransmission. Treatment with low nonsedating/nonanxiolytic doses of benzodiazepines, which increase inhibitory neurotransmission through positive allosteric modulation of postsynaptic GABAA receptors, improved deficits in social interaction, repetitive behavior, and spatial learning. Moreover, negative allosteric modulation of GABAA receptors impaired social behavior in C57BL/6J and 129SvJ wild-type mice, suggesting that reduced inhibitory neurotransmission may contribute to social and cognitive deficits. The dramatic behavioral improvement after low-dose benzodiazepine treatment was subunit specific-the α2,3-subunit-selective positive allosteric modulator L-838,417 was effective, but the α1-subunit-selective drug zolpidem exacerbated social deficits. Impaired GABAergic neurotransmission may contribute to ASD, and α2,3-subunit-selective positive GABAA receptor modulation may be an effective treatment.
In this paper, we study the asymptotic behavior of the solutions of a neutral difference equation of the form Delta[x(n) + cx(tau(n))] - p(n)x(sigma(n)) = 0, where tau(n) is a general retarded argument, sigma(n) is a general deviated argument, c a a"e, (-p(n)) (na parts per thousand yen0) is a sequence of negative real numbers such that p(n) a parts per thousand yen p, p a a"e(+), and Delta denotes the forward difference operator Delta x(n) = x(n+1)-x(n).
In this paper, we investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form
where τj (n), j = 1, . . . , w are general retarded arguments, σ(n) is a general deviated argument, [###] is a sequence of positive real numbers such that p(n) ≥ p, p ∈ R+, and Δ denotes the forward difference operator Δx(n) = x(n + 1) − x(n).
The asymptotic behavior of nonoscillatory solutions to the first-order neutral difference equation of the form Δ[x(n) + cx(σ(n))] + p(n)x(∈(n)) = 0, where τ(n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ R, (p(n))n≥0 is a sequence of real numbers, and Δ denotes the forward difference operator Δx(n) = x(n+1)-x(n), is studied. Examples are provided to illustrate the results.
We study the asymptotic behavior of solutions of the higher-order neutral difference equation (Formula presented.) where τ (n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ ℝ, (p(n))n ≥ 0 is a sequence of real numbers, {increment} denotes the forward difference operator {increment}x(n) = x(n+1) - x(n); and {increment}j denotes the jth forward difference operator {increment}j (x(n) = {increment} ({increment}j-1(x(n))) for j = 2, 3,...,m. Examples illustrating the results are also given.
Suppose that { p n } is a nonnegative sequence of real numbers and let k be a positive integer. We prove that lim n → ∞ inf [ 1 k ∑ i = n − k n − 1 p i ] > k k ( k + 1 ) k + 1 is a sufficient condition for the oscillation of all solutions of the delay difference equation A n + 1 − A n + p n A n − k = 0 , n = 0 , 1 , 2 , … . This result is sharp in that the lower bound k k / ( k + 1 ) k + 1 in the condition cannot be improved. Some results on difference inequalities and the existence of positive solutions are also presented.
We present discrete analogues of Taylor’s formula, 1’Hospital’s rule, Kneser’s theorem etc., and use these to study qualitative properties of solutions of higher order difference equations. The proofs are based on some simple inequalities.
Let [·] denote the greatest-integer function and consider the nonlinear equation with piecewise constant arguments (1)y ˙(t)+∑ i=1 m P i (t)f i (y[t-k i ]))=0,t≥0· We obtain sufficient and also necessary and sufficient conditions for the oscillation of all solutions of (1) in terms of the oscillation of all solutions of an associated linear equation with piecewise constant arguments.
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 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.
This course text fills a gap for first-year graduate-level students reading applied functional analysis or advanced engineering analysis and modern control theory. Containing 100 problem-exercises, answers, and tutorial hints, the first edition is often cited as a standard reference. Making a unique contribution to numerical analysis for operator equations, it introduces interval analysis into the mainstream of computational functional analysis, and discusses the elegant techniques for reproducing Kernel Hilbert spaces. There is discussion of a successful ''hybrid'' method for difficult real-life problems, with a balance between coverage of linear and non-linear operator equations. The authors successful teaching philosophy: ''We learn by doing'' is reflected throughout the book.
We obtain sufficient conditions for the oscillatons of all solutions of the neutral difference equation where p and q are real numbers and k and l are integers.
We obtain sufficient conditions for the oscillation of all solutions of the difference equation , where the pi's are real numbers and the ki's are integers. The conditions are given explicitly in terms of the pi's and the ki's.
We establish a necessary and sufficient condition for the existence of positive solutions and obtain Sturm-type comparison theorem for the difference equation . (∗) We also obtain the following comparison result: Equation (∗) oscillates if and only if . (∗∗) oscillates provided .