http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2012, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2011/958602/We consider a rational system of first-order difference equations in the plane with four parameters such that all fractions have a common denominator. We study, for the different values of the parameters, the global and local properties of the system. In particular, we discuss the boundedness and the asymptotic behavior of the solutions, the existence of periodic solutions, and the stability of equilibria.Ignacio Bajo, Daniel Franco, and Juan PeránCopyright © 2011 Ignacio Bajo et al. All rights reserved.http://www.hindawi.com/journals/ade/2011/234215/We investigate the zeros distributions of difference polynomials of meromorphic functions, which can be viewed as the Hayman conjecture as introduced by (Hayman 1967) for difference. And we also study the uniqueness of difference polynomials of meromorphic functions sharing a common value, and obtain uniqueness theorems for difference.Kai Liu, Xinling Liu, and TingBin CaoCopyright © 2011 Kai Liu et al. All rights reserved.http://www.hindawi.com/journals/ade/2011/748608/This paper discusses the disease-free and endemic equilibrium points of a SVEIRS propagation disease model which potentially involves a regular constant vaccination. The positivity of such a model is also discussed as well as the boundedness of the total and partial populations. The model takes also into consideration the natural population growing and the mortality associated to the disease as well as the lost of immunity of newborns. It is assumed that there are two finite delays affecting the susceptible, recovered, exposed, and infected population dynamics. Some extensions are given for the case when impulsive nonconstant vaccination is incorporated at, in general, an aperiodic sequence of time instants. Such an impulsive vaccination consists of a culling or a partial removal action on the susceptible population which is transferred to the vaccinated one. The oscillatory behavior under impulsive vaccination, performed in general, at nonperiodic time intervals, is also discussed.M. De la Sen, Ravi P. Agarwal, A. Ibeas, and S. Alonso-QuesadaCopyright © 2011 M. De la Sen et al. All rights reserved.http://www.hindawi.com/journals/ade/2011/237219/We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation xΔn(t)+f(t,x(t),xΔ(t),…,xΔn-1(t))=0, on an arbitrary time scale T, where the function f is defined on T×Rn. We give sufficient conditions under which every solution x of this equation satisfies one of the following conditions: (1) lim⁡t→∞xΔn-1(t)=0; (2) there exist constants ai    (0≤i≤n-1) with a0≠0, such that lim⁡t→∞x(t)/∑i=0n-1aihn-i-1(t,t0)=1, where hi(t,t0)  (0≤i≤n-1) are as in Main Results.Taixiang Sun, Hongjian Xi, and Xiaofeng PengCopyright © 2011 Taixiang Sun et al. All rights reserved.http://www.hindawi.com/journals/ade/2011/867136/We propose a new discrete version of nonlinear oscillator with damping dynamical system governed by a general maximal monotone operator. We show the weak convergence of solutions and their weighted averages to a zero of a maximal monotone operator A. We also prove some strong convergence theorems with additional assumptions on A. This iterative scheme gives also an extension of the proximal point algorithm for the approximation of a zero of a maximal monotone operator. These results extend previous results by Brézis and Lions (1978), Lions (1978) as well as Djafari Rouhani and H. Khatibzadeh (2008).Hadi KhatibzadehCopyright © 2011 Hadi Khatibzadeh. All rights reserved.http://www.hindawi.com/journals/ade/2011/424809/We first consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to χ. The purpose of this paper is to present a systemic study of some families of higher-order generalized q-Genocchi numbers and polynomials attached to χ by using the generating function of those numbers and polynomials.C. S. Ryoo, T. Kim, J. Choi, and B. LeeCopyright © 2011 C. S. Ryoo et al. All rights reserved.http://www.hindawi.com/journals/ade/2011/283926/We investigate some nonlinear integral inequalities in two independent variables on time scales. Our results unify and extend some integral inequalities and their corresponding discrete analogues which established by Pachpatte. The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales.Wei Nian LiCopyright © 2011 Wei Nian Li. All rights reserved.http://www.hindawi.com/journals/ade/2011/109570/We discuss conditions for the existence of at least one solution of a discontinuous parabolic equation with lower semicontinuous right hand side and a nonlocal initial condition of integral type. Our technique is based on fixed point theorems for multivalued maps.Abdelkader BoucherifCopyright © 2011 Abdelkader Boucherif. All rights reserved.http://www.hindawi.com/journals/ade/2011/513757/This paper is concerned with some oscillation criteria for the second order neutral delay dynamic equations with mixed nonlinearities of the form (r(t)u(t))Δ+q(t)|x(τ(t))|α−1x(τ(t))+∑i=1nqi(t)|x(τi(t))|αi−1x(τi(t))=0, where t∈T and u(t)=|(x(t)+p(t)x(δ(t)))Δ|α-1(x(t)+p(t)x(δ(t)))Δ with α1>α2>⋯>αm>α>αm+1>⋯>αn>0. Further the results obtained here generalize and complement to the results obtained by Han et al. (2010). Examples are provided to illustrate the results.Ethiraju Thandapani, Veeraraghavan Piramanantham, and Sandra PinelasCopyright © 2011 Ethiraju Thandapani et al. All rights reserved.http://www.hindawi.com/journals/ade/2011/806458/By using variational methods and Morse theory, we study the multiplicity of the periodic solutions for a class of difference equations with double resonance at infinity. To the best of our knowledge, investigations on double-resonant difference systems have not been seen in the literature.Xiaosheng Zhang and Duo WangCopyright © 2011 Xiaosheng Zhang and Duo Wang. All rights reserved.http://www.hindawi.com/journals/ade/2011/404917/We study a nonlocal boundary value problem of impulsive fractional differential equations. By means of a fixed point theorem due to O'Regan, we establish sufficient conditions for the existence of at least one solution of the problem. For the illustration of the main result, an example is given.Liu Yang and Haibo ChenCopyright © 2011 Liu Yang and Haibo Chen. All rights reserved.http://www.hindawi.com/journals/ade/2010/281612/This paper discusses a generalized time-varying SEIR propagation disease model subject to delays which potentially involves mixed regular and impulsive vaccination rules. The model takes also into account the natural population growing and the mortality associated to the disease, and the potential presence of disease endemic thresholds for both the infected and infectious population dynamics as well as the lost of immunity of newborns. The presence of outsider infectious is also considered. It is assumed that there is a finite number of time-varying distributed delays in the susceptible-infected coupling dynamics influencing the susceptible and infected differential equations. It is also assumed that there are time-varying point delays for the susceptible-infected coupled dynamics influencing the infected, infectious, and removed-by-immunity differential equations. The proposed regular vaccination control objective is the tracking of a prescribed suited infectious trajectory for a set of given initial conditions. The impulsive vaccination can be used to improve discrepancies between the SEIR model and its suitable reference one.M. De la Sen, Ravi P. Agarwal, A. Ibeas, and S. Alonso-QuesadaCopyright © 2010 M. De la Sen et al. All rights reserved.http://www.hindawi.com/journals/ade/2011/571935/We establish a conjugacy criterion for a 2×2 symplectic difference system by means of the concept of a phase of any basis of this symplectic system. We also describe a construction of a 2×2 symplectic difference system whose recessive solution has the prescribed number of generalized zeros in Z.Ondřej Došlý and Šárka PechancováCopyright © 2011 Ondřej Došlý and Šárka Pechancová. All rights reserved.http://www.hindawi.com/journals/ade/2010/496936/An extension of the uniform invariance principle for ordinary differential equations with finite delay is developed. The uniform invariance principle allows the derivative of the auxiliary scalar function V to be positive in some bounded sets of the state space while the classical invariance principle assumes that V̇≤0. As a consequence, the uniform invariance principle can deal with a larger class of problems. The main difficulty to prove an invariance principle for functional differential equations is the fact that flows are defined on an infinite dimensional space and, in such spaces, bounded solutions may not be precompact. This difficulty is overcome by imposing the vector field taking bounded sets into bounded sets.Marcos Rabelo and L. F. C. AlbertoCopyright © 2010 Marcos Rabelo and L. F. C. Alberto. All rights reserved.http://www.hindawi.com/journals/ade/2010/951764/We present a new generating function related to the q-Bernoulli numbers and q-Bernoulli polynomials. We give a new construction of these numbers and polynomials related to the second-kind Stirling numbers and q-Bernstein polynomials. We also consider the generalized q-Bernoulli polynomials attached to Dirichlet's character χ and have their generating function. We obtain distribution relations for the q-Bernoulli polynomials and have some identities involving q-Bernoulli numbers and polynomials related to the second kind Stirling numbers and q-Bernstein polynomials. Finally, we derive the q-extensions of zeta functions from the Mellin transformation of this generating function which interpolates the q-Bernoulli polynomials at negative integers and is associated with q-Bernstein polynomials.Mehmet Açikgöz, Dilek Erdal, and Serkan AraciCopyright © 2010 Mehmet Açikgöz et al. All rights reserved.http://www.hindawi.com/journals/ade/2010/494607/The mean square BIBO stability is investigated for stochastic control systems with mixed delays and nonlinear perturbations. The system with mixed delays is transformed, then a class of suitable Lyapunov functionals is selected, and some novel delay-dependent BIBO stabilization in mean square criteria for stochastic control systems with mixed delays and nonlinear perturbations are obtained by applying the technique of analyzing controller and the method of existing a positive definite solution to an auxiliary algebraic Riccati matrix equation. A numerical example is given to illustrate the validity of the main results.Xia Zhou and Shouming ZhongCopyright © 2010 Xia Zhou and Shouming Zhong. All rights reserved.http://www.hindawi.com/journals/ade/2010/432796/Let ℝ+ be the set of positive real numbers, B a Banach space, f:ℝ+→B, and ϵ>0, p,q,P,Q∈ℝ with pqPQ≠0. We prove the Hyers-Ulam stability of the Jensen type logarithmic functional inequality ∥f(xpyq)-Pf(x)-Qf(y)∥≤ϵ in restricted domains of the form {(x,y):x>0, y>0, xkys≥d} for fixed k,s∈ℝ with k≠0 or s≠0 and d>0. As consequences of the results we obtain asymptotic behaviors of the inequality as xkys→∞.Jae-Young ChungCopyright © 2010 Jae-Young Chung. All rights reserved.http://www.hindawi.com/journals/ade/2010/862016/We investigate the existence of at least three solutions for a discrete nonlinear Neumann boundary value problem involving the p-Laplacian. Our approach is based on three critical points theorems.Pasquale Candito and Giuseppina D'AguìCopyright © 2010 Pasquale Candito and Giuseppina D'Aguì. All rights reserved.http://www.hindawi.com/journals/ade/2010/324378/We prove the existence of a Gevrey family of invariant curves for analytic reversible mappings under weaker nondegeneracy condition. The index of the Gevrey smoothness of the family could be any number μ>τ+2, where τ>m−1 is the exponent in the small divisors condition and m is the order of degeneracy of the reversible mappings. Moreover, we obtain a Gevrey normal form of the reversible mappings in a neighborhood of the union of the invariant curves.Dongfeng Zhang and Rong ChengCopyright © 2010 Dongfeng Zhang and Rong Cheng. All rights reserved.http://www.hindawi.com/journals/ade/2010/103065/This paper concerns the oscillation of solutions to the second sublinear dynamic equation with damping xΔΔ(t)+q(t)xΔσ(t)+p(t)xα(σ(t))=0, on an isolated time scale T which is unbounded above. In 0<α<1, α is the quotient of odd positive integers. As an application, we get the difference equation Δ2x(n)+n-γΔx(n+1)+[(1/n(ln⁡n)β)+b((-1)n/(ln⁡n)β)]xα(n+1)=0, where γ>0, β>0, and b is any real number, is oscillatory.Quanwen Lin and Baoguo JiaCopyright © 2010 Quanwen Lin and Baoguo Jia. All rights reserved.http://www.hindawi.com/journals/ade/2010/650827/This paper concerns the existence of solutions for two kinds of systems of first-order equations on time scales. Existence results for these problems are obtained with new notions of solution tube adapted to these systems. We consider the general case where the right member of the system is Δ-Carathéodory and, hence, not necessarily continuous.Hugues GilbertCopyright © 2010 Hugues Gilbert. All rights reserved.http://www.hindawi.com/journals/ade/2010/185701/Using the fixed-point theorem, this paper is devoted to study the multiple and single positive solutions of third-order boundary value problems for impulsive differential equations in ordered Banach spaces. The arguments are based on a specially constructed cone. At last, an example is given to illustrate the main results.Jingjing CaiCopyright © 2010 Jingjing Cai. All rights reserved.http://www.hindawi.com/journals/ade/2010/573281/We consider the system in the title where the initial condition (x0,y0)∈R2. We show that the system has exactly two prime period-5 solutions and a unique equilibrium point (0,-1). We also show that every solution of the system is eventually one of the two prime period-5 solutions or else the unique equilibrium point.Wirot Tikjha, Yongwimon Lenbury, and Evelina Giusti LapierreCopyright © 2010 Wirot Tikjha et al. All rights reserved.http://www.hindawi.com/journals/ade/2010/542073/We consider a difference equation involving three parameters and a piecewise constant control function with an additional positive threshold λ. Treating the threshold as a bifurcation parameter that varies between 0 and ∞, we work out a complete asymptotic and bifurcation analysis. Among other things, we show that all solutions either tend to a limit 1-cycle or to a limit 2-cycle and, we find the exact regions of attraction for these cycles depending on the size of the threshold. In particular, we show that when the threshold is either small or large, there is only one corresponding limit 1-cycle which is globally attractive. It is hoped that the results obtained here will be useful in understanding interacting network models involving piecewise constant control functions.Chengmin Hou, Lili Han, and Sui Sun ChengCopyright © 2010 Chengmin Hou et al. All rights reserved.http://www.hindawi.com/journals/ade/2010/101959/The initial-boundary value problem for a class of linear and nonlinear equations in Hilbert space is considers. We prove the existence and uniqueness of solution of this problem. The results of this investigation are applied to solvability of initial-boundary value problems for quasilinear impulsive hyperbolic equations with non-stationary transmission and boundary conditions.Akbar B. Aliev and Ulviya M. MamedovaCopyright © 2010 Akbar B. Aliev and Ulviya M. Mamedova. All rights reserved.http://www.hindawi.com/journals/ade/2010/594783/We discuss in detail the error bounds for asymptotic solutions of second-order linear difference equation y(n+2)+npa(n)y(n+1)+nqb(n)y(n)=0, where p and q are integers, a(n) and b(n) have asymptotic expansions of the form a(n)∼∑s=0∞(as/ns), b(n)∼∑s=0∞(bs/ns), for large values of n, a0≠0, and b0≠0.L. H. Cao and J. M. ZhangCopyright © 2010 L. H. Cao and J. M. Zhang. All rights reserved.http://www.hindawi.com/journals/ade/2010/727486/We present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a class of damped vibration problems on time scale T, uΔ2(t)+w(t)uΔ(σ(t))=∇F،(σ(t),u(σ(t))),  Δ-a.e. t∈[0,T]Tκ, u(0)-u(T)=0,  uΔ(0)-uΔ(T)=0, where uΔ(t) denotes the delta (or Hilger) derivative of u at t, uΔ2(t)=(uΔ)Δ(t), σ is the forward jump operator, T is a positive constant, w∈R+([0,T]T,R), ew(T,0)=1, and F:[0,T]T×RN→R. By establishing a proper variational setting, three existence results are obtained. Finally, three examples are presented to illustrate the feasibility and effectiveness of our results.Yongkun Li and Jianwen ZhouCopyright © 2010 Yongkun Li and Jianwen Zhou. All rights reserved.http://www.hindawi.com/journals/ade/2010/748789/We consider the stability properties for thermoelastic Bresse system which describes the motion of a linear planar shearable thermoelastic beam. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. We use multiplier techniques to prove the exponential stability result when the wave speed of the vertical displacement coincides with the wave speed of the longitudinal or of the shear angle displacement. Moreover, the existence of the global attractor is firstly achieved.Zhiyong MaCopyright © 2010 Zhiyong Ma. All rights reserved.http://www.hindawi.com/journals/ade/2010/798067/We first present several existence results and compactness of solutions set for the following Volterra type integral inclusions of the form: y(t)∈∫0ta(t−s)[Ay(s)+F(s,y(s))]ds,  a.e.  t∈J, where J=[0,b], A is the infinitesimal generator of an integral resolvent family on a separable Banach space E, and F is a set-valued map. Then the Filippov's theorem and a Filippov-Ważewski result are proved.R. P. Agarwal, M. Benchohra, J. J. Nieto, and A. OuahabCopyright © 2010 R. P. Agarwal et al. All rights reserved.http://www.hindawi.com/journals/ade/2010/127093/The authors employs a hybrid fixed point theorem involving the multiplication of two operators for proving an existence result of locally attractive solutions of a nonlinear quadratic Volterra integral equation of fractional (arbitrary) order. Investigations will be carried out in the Banach space of real functions which are defined, continuous, and bounded on the real half axis ℝ+.Mohamed I. AbbasCopyright © 2010 Mohamed I. Abbas. All rights reserved.