http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/840458We study the existence, multiplicity, and nonexistence of positive solutions for multiparameter semipositone discrete boundary value problems by using nonsmooth critical point theory and subsuper solutions method.Jianshe Yu, Benshi Zhu, and Zhiming Guo© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/867635We first give a theorem on the reducibility of linear system of difference equations of the form x(n+1)=A(n)x(n). Next, by the means of Floquet theory, we obtain some stability results. Moreover, some examples are given to illustrate the importance of the results.Aydin Tiryaki and Adil Misir© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/692713We study the existence of almost periodic solutions for nonlinear discrete Volterra equations with unbounded delay, as a discrete analogue of the results for integro-differential equations by Y. Hamaya (1993).Sung Kyu Choi and Namjip Koo© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/796851This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation.Ravi P. Agarwal, Victoria Otero-Espinar, Kanishka Perera, and Dolores R. Vivero© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/712913We establish WKB estimates for 2×2 linear dynamic systems with a small parameter ε on a time scale unifying continuous and discrete WKB method. We introduce an adiabatic invariant for 2×2 dynamic system on a time scale, which is a generalization of adiabatic invariant of Lorentz's pendulum. As an application we prove that the change of adiabatic invariant is vanishing as ε approaches zero. This result was known before only for a continuous time scale. We show that it is true for the discrete scale only for the appropriate choice of graininess depending on a parameter ε. The proof is based on the truncation of WKB series and WKB estimates.Gro Hovhannisyan© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/586020Consider the multiplicity of solutions to the nonlinear second-order discrete problems with minimum and maximum: Δ2u(k−1)=f(k,u(k),Δu(k)), k∈T, min{u(k):k∈T^}=A, max⁡{u(k):k∈T^}=B, where f:T×ℝ2→ℝ,  a,b∈ℕ are fixed numbers satisfying b≥a+2,  and  A,B∈ℝ are satisfying B>A,   T={a+1,…,b−1},  T^={a,a+1,…,b−1,b}.Ruyun Ma and Chenghua Gao© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/505324The half-linear difference equations with the deviating argument Δ(an|Δxn|αsgn Δxn)+bn|xn+q|αsgn xn+q=0 , q ∈ ℤ are considered. We study the role of the deviating argument q, especially as regards the growth of the nonoscillatory solutions and the oscillation. Moreover, the problem of the existence of the intermediate solutions is completely resolved for the classical half-linear equation (q = 1). Some analogies or discrepancies on the growth of the nonoscillatory solutions for the delayed and advanced equations are presented; and the coexistence with different types of nonoscillatory solutions is studied.M. Cecchi, Z. Došlá, and M. Marini© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/143943We show that every solution of the following system of difference equations xn+1(1)=xn(2)/(xn(2)−1), xn+1(2)=xn(3)/(xn(3)−1),…,xn+1(k)=xn(1)/(xn(1)−1) as well as of the system xn+1(1)=xn(k)/(xn(k)−1), xn+1(2)=xn(1)/(xn(1)−1),…,xn+1(k)=xn(k−1)/(xn(k−1)−1) is periodic with period 2k if k  ≠  0 (mod2), and with period k if k=0 (mod2) where the initial values are nonzero real numbers for x0(1),x0(2),…,x0(k)  ≠  1.İbrahim Yalçinkaya, Cengiz Çinar, and Muhammet Atalay© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/879140This paper is concerned with the existence and nonexistence of positive solutions of the p-Laplacian functional dynamic equation on a time scale, [ϕp(x▵(t))]∇+λa(t)f(x(t),x(u(t)))=0, t∈(0,T), x0(t)=ψ(t), t∈[−τ,0], x(0)−B0(x▵(0))=0, x▵(T)=0. We show that there exists a λ∗>0 such that the above boundary value problem has at least two, one, and no positive solutions for 0<λ<λ∗, λ=λ∗ and λ>λ∗, respectively.Changxiu Song© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/718408It is supposed that the fractional difference equation xn+1=(μ+∑j=0kajxn−j)/(λ+∑j=0kbjxn−j), n=0,1,…, has an equilibrium point x^ and is exposed to additive stochastic perturbations type of σ(xn−x^)ξn+1 that are directly proportional to the deviation of the system state xn from the equilibrium point x^. It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted.Beatrice Paternoster and Leonid Shaikhet© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/816091We investigate the existence of almost-periodic weak solutions of second-order neutral delay-differential equations with piecewise constant argument of the form (x(t)+x(t−1))′′=qx(2[(t+1)/2])+f(t), where [⋅] denotes the greatest integer function, q is a real nonzero constant, and f(t) is almost periodic.Li Wang and Chuanyi Zhang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/317520At the price of sacrificing all suspense, we can already announce that the answer to the question of the title is “no.” It is indeed our belief that one may find counterexamples to all integrability conjectures, unless one constrains the definition of integrability to the point that the integrability criterion becomes tautological. This review is devoted to a critical analysis of the situation.B. Grammaticos, A. Ramani, K. M. Tamizhmani, T. Tamizhmani, and A. S. Carstea© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/868425Nonnegative and compartmental dynamical system models are derived from mass and energy balance considerations that involve dynamic states whose values are nonnegative. These models are widespread in engineering and life sciences, and they typically involve the exchange of nonnegative quantities between subsystems or compartments, wherein each compartment is assumed to be kinetically homogeneous. In this paper, we develop a neuroadaptive control framework for adaptive set-point regulation of discrete-time nonlinear uncertain nonnegative and compartmental systems. The proposed framework is Lyapunov-based and guarantees ultimate boundedness of the error signals corresponding to the physical system states and the neural network weighting gains. In addition, the neuroadaptive controller guarantees that the physical system states remain in the nonnegative orthant of the state space for nonnegative initial conditions.Wassim M. Haddad, VijaySekhar Chellaboina, Qing Hui, and Tomohisa Hayakawa© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/932831This article analyses the asymptotic behaviour of solutions of linear Volterra difference equations. Some sufficient conditions are presented under which the solutions to a general linear equation converge to limits, which are given by a limit formula. This result is then used to obtain the exact asymptotic representation of the solutions of a class of convolution scalar difference equations, which have real characteristic roots. We give examples showing the accuracy of our results.István Győri and László Horváth© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/815750The main purpose of this paper is to study generating functions of the q-Genocchi numbers and polynomials. We prove a new relation for the generalized q-Genocchi numbers, which is related to the q-Genocchi numbers and q-Bernoulli numbers. By applying Mellin transformation and derivative operator to the generating functions, we define q-Genocchi zeta and l-functions, which are interpolated q-Genocchi numbers and polynomials at negative integers. We also give some applications of the generalized q-Genocchi numbers.Yilmaz Simsek, Ismail Naci Cangul, Veli Kurt, and Daeyeoul Kim© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/738603By using p-adic q-integrals on ℤp, we define multiple twisted q-Euler numbers and polynomials. We also find Witt's type formula for multiple twisted q-Euler numbers and discuss some characterizations of multiple twisted q-Euler Zeta functions. In particular, we construct multiple twisted Barnes' type q-Euler polynomials and multiple twisted Barnes' type q-Euler Zeta functions. Finally, we define multiple twisted Dirichlet's type q-Euler numbers and polynomials, and give Witt's type formula for them.Lee-Chae Jang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/670203We examine the various types of stability for the solutions of linear dynamic systems on time scales and give two examples.Sung Kyu Choi, Dong Man Im, and Namjip Koo© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/739602The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases.Elvan Akın-Bohner, Martin Bohner, Smaïl Djebali, and Toufik Moussaoui© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/316207This paper deals with the existence and stability of solutions for semilinear second-order evolution equations on Banach spaces by using recent characterizations of discrete maximal regularity.Claudio Cuevas and Carlos Lizama© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/469815Let T be an integer with T≥3, and let T:={1,…,T}. We study the existence and uniqueness of solutions for the following two-point boundary value problems of second-order difference systems:Δ2u(t−1)+f(t,u(t))=e(t),t∈T, u(0)=u(T+1)=0, where e:T→ℝn  and  f:T×ℝn→ℝn is a potential function satisfying f(t,⋅)∈C1(ℝn) and some nonresonance conditions. The proof of the main result is based upon a mini-max theorem.Ruyun Ma, Hua Luo, and Chenghua Gao© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/496078A new triple fixed-point theorem is applied to investigate the existence of at least triple positive solutions of fourth-order four-point boundary value problems for p-Laplacian dynamic equations on a time scale. The interesting point is that we choose an inversion technique employed by Avery and Peterson in 1998.Mei-Qiang Feng, Xiang-Gui Li, and Wei-Gao Ge© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/143053We investigate the generalized Hyers-Ulam stability of Newton functional equations for logarithmic spirals.Soon-Mo Jung and John Michael Rassias© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/154302The paper presents an existence principle for solving a large class of nonlocal regular discrete boundary value problems with the φ-Laplacian. Applications of the existence principle to singular discrete problems are given.Ravi P. Agarwal, Donal O'Regan, and Svatoslav Stanêk© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/184275We aim to study robust impulsive synchronization problem for uncertain discrete dynamical networks. For the discrete dynamical networks with unknown but bounded network coupling, we will design some robust impulsive controllers which ensure that the state of a discrete dynamical network asymptotically synchronize with an arbitrarily assigned state of an isolate node of the network. Three representative examples are also worked through to illustrate our results.Ming Lei and Bin Liu© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/238068We consider the family of nonlinear difference equations: xn+1=(∑i=13fi(xn,…,xn−k)+f4(xn,…,xn−k)f5(xn,…,xn−k))/(f1(xn,…,xn−k)f2(xn,…,xn−k)+∑i=35fi(xn,…,xn−k)), n=0,1,…, where fi∈C((0,+∞)k+1,(0,+∞)), for i∈{1,2,4,5}, f3∈C([0,+∞)k+1,(0,+∞)), k∈{1,2,…} and the initial values x−k,x−k+1,…,x0∈(0,+∞). We give sufficient conditions under which the unique equilibrium x¯=1 of these equations is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references.Taixiang Sun, Hongjian Xi, and Caihong Han© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/743295We consider Carlitz q-Bernoulli numbers and q-Stirling numbers of the first and the second kinds. From the properties of q-Stirling numbers, we derive many interesting formulas associated with Carlitz q-Bernoulli numbers. Finally, we will prove βn,q=∑m=0n∑k=mn1/(1-q)n+m-k∑d0+⋯+dk=n-kq∑i=0kidis1,q(k,m)(-1)n-m((m+1)/[m+1]q), where βn,q are called Carlitz q-Bernoulli numbers.Taekyun Kim© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/845121This paper investigates the existence of positive solutions for a class of second-order singular m-point Sturm-Liouville-type boundary value problems by using fixed point theorem in cones. The results significantly extend and improve many known results even for nonsingular cases.Xuemei Zhang and Weigao Ge© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/695495We establish a general form of sum-difference inequality in two variables, which includes both two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered by Cheung and Ren (2006). Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.Wu-Sheng Wang© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/143723In this paper, we consider the nonlinear difference equation xn+1=f(xn−l+1,xn−2k+1), n=0,1,…, where k,l∈{1,2,…} with 2k≠l and gcd(2k,l)=1 and the initial values x−α,x−α+1,…,x0∈(0,+∞) with α=max{l−1,2k−1}. We give sufficient conditions under which every positive solution of this equation converges to a ( not necessarily prime ) 2-periodic solution, which extends and includes corresponding results obtained in the recent literature.Taixiang Sun and Hongjian Xi© 2008, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/GetArticle.aspx?doi=10.1155/2008/438130We consider the half-linear second-order difference equation Δ(rkΦ(Δxk))+ckΦ(xk+1)=0, Φ(x):=|x|p−2x, p>1, where r, c are real-valued sequences. We associate with the above-mentioned equation a linear second-order difference equation and we show that oscillatory properties of the above-mentioned one can be investigated using properties of this associated linear equation. The main tool we use is a linearization technique applied to a certain Riccati-type difference equation corresponding to the above-mentioned one.Ondřej Došlý and Simona Fišnarová© 2008, Hindawi Publishing Corporation. All rights reserved.