http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/289340.htmlSome new oscillation criteria for the second-order neutral delay differential equation (r(t)z′(t))′+q(t)x(σ(t))=0, t≥t0 are established, where ∫t0∞(1/r(t))dt=∞, z(t)=x(t)+p(t)x(τ(t)), 0≤p(t)≤p0<∞, q(t)>0. These oscillation criteria extend and improve some known results. An example is considered to illustrate the main results.Zhenlai Han, Tongxing Li, Shurong Sun, and Weisong Chen© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/347670.htmlWe provide a complete diagram of the relation between the admissibility of pairs of Banach function spaces and the exponential dichotomy of evolution families on the real line. We prove that if W∈ℋ(ℝ) and V∈𝒯(ℝ) are two Banach function spaces with the property that either W∈𝒲(ℝ) or V∈𝒱(ℝ), then the admissibility of the pair (W(ℝ,X),V(ℝ,X)) implies the existence of the exponential dichotomy. We study when the converse implication holds and show that the hypotheses on the underlying function spaces cannot be dropped and that the obtained results are the most general in this topic. Finally, our results are applied to the study of exponential dichotomy of C0-semigroups.Adina Luminiţa Sasu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/795145.htmlDelay discrete inequalities with more than one nonlinear term are discussed, which generalize some known results and can be used in the analysis of various problems in the theory of certain classes of discrete equations. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given.Yu Wu, Xiaopei Li, and Shengfu Deng© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/287861.htmlThis paper is concerned with the existence and uniqueness of mild solution of the fractional differential equations with nonlocal conditions dqx(t)/dtq=−Ax(t)+f(t,x(t),Gx(t)),  t∈[0,T],   and  x(0)+g(x)=x0, in a Banach space X, where 0<q<1. General existence and uniqueness theorem, which extends many previous results, are given.Fang Li© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/674630.htmlWe deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: Dαx(t)+Ax(t)=f(t,xt), t∈[0,T], x(t)=ϕ(t), t∈]−∞,0], with T>0 and 0<α<1. We prove the existence (and uniqueness) of solutions, assuming that −A is a linear closed operator which generates an analytic semigroup (T(t))t≥0 on a Banach space 𝕏 by means of the Banach's fixed point theorem. This generalizes some recent results.Gisle M. Mophou and Gaston M. N'Guérékata© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/938492.abs.htmlConsidering the linear delay difference system x(n+1)=ax(n)+Bx(n-k), where a∈(0,1), B is a p×p real matrix, and k is a positive integer, the stability domain of the null solution is completely characterized in terms of the eigenvalues of the matrix B. It is also shown that the stability domain becomes smaller as the delay increases. These results may be successfully applied in the stability analysis of a large class of nonlinear difference systems, including discrete-time Hopfield neural networks.Eva Kaslik© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/375486.abs.htmlThe permanence and the existence of periodic solution for a discrete nonautonomous competitive system with multidelays are considered. Also the stability of the periodic solution is discussed. Numerical examples are given to confirm the theoretical results.Chunqing Wu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/586312.htmlWe will establish some oscillation criteria for the third-order Emden-Fowler neutral delay dynamic equations (r(t)(x(t)−a(t)x(τ(t)))ΔΔ)Δ+p(t)xγ(δ(t))=0 on a time scale 𝕋, where γ>0 is a quotient of odd positive integers with r, a, and p 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, so this paper initiates the study. Some examples are considered to illustrate the main results.Zhenlai Han, Tongxing Li, Shurong Sun, and Chenghui Zhang© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/395693.htmlIn this paper, we consider the additive-cubic-quartic functional equation 11[f(x+2y)+f(x−2y)]=44[f(x+y)+f(x−y)]+12f(3y)−48f(2y)+60f(y)−66f(x) and prove the generalized Hyers-Ulam stability of the additive-cubic-quartic functional equation in Banach spaces.M. Eshaghi-Gordji, S. Kaboli-Gharetapeh, Choonkil Park, and Somayyeh Zolfaghari© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/262461.htmlBy using Mawhin's continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of at least 2n positive periodic solutions for n-species nonautonomous Lotka-Volterra type food chains with harvesting terms. An example is given to illustrate the effectiveness of our results.Yongkun Li and Kaihong Zhao© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/360871.htmlLet T∈ℕ be an integer with T>2, and let 𝕋:={1,…,T}. We study the existence of solutions of nonlinear discrete problems Δ2u(t−1)+λka(t)u(t)+g(t,u(t))=h(t),  t∈𝕋,  u(0)=u(T),  u(1)=u(T+1), where a,h:𝕋→ℝ with a>0,  λk is the kth eigenvalue of the corresponding linear eigenvalue problem.Ruyun Ma and Huili Ma© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/132802.htmlWe investigate global dynamics of the following systems of difference equations xn+1=(α1+β1xn)/yn, yn+1=(α2+γ2yn)/(A2+xn), n=0,1,2,…, where the parameters α1, β1, α2, γ2, and A2 are positive numbers and initial conditions x0 and y0 are arbitrary nonnegative numbers such that y0>0. We show that this system has rich dynamics which depend on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points.S. Kalabušić, M. R. S. Kulenović, and E. Pilav© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/243245.htmlSzekeley observed that the dynamic pattern of the locomotion of salamanders can be explained by periodic vector sequences generated by logical neural networks. Such sequences can mathematically be described by “doubly periodic traveling waves” and therefore it is of interest to propose dynamic models that may produce such waves. One such dynamic network model is built here based on reaction-diffusion principles and a complete discussion is given for the existence of doubly periodic waves as outputs. Since there are 2 parameters in our model and 4 a priori unknown parameters involved in our search of solutions, our results are nontrivial. The reaction term in our model is a linear function and hence our results can also be interpreted as existence criteria for solutions of a nontrivial linear problem depending on 6 parameters.Jian Jhong Lin and Sui Sun Cheng© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/362627.htmlWe consider eigenvalue problems for self-adjoint Sturm-Liouville difference equations of any even order. It is well known that such problems with Dirichlet boundary conditions can be transformed into an algebraic eigenvalue problem for a banded, real-symmetric matrix, and vice versa. In this article it is shown that such a transform exists for general separated, self-adjoint boundary conditions also. But the main result is an explicit procedure (algorithm) for the numerical computation of this banded, real-symmetric matrix. This construction can be used for numerical purposes, since in the recent paper by Kratz and Tentler (2008) there is given a stable and superfast algorithm to compute the eigenvalues of banded, real-symmetric matrices. Hence, the Sturm-Liouville problems considered here may now be treated by this algorithm.Werner Kratz© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/840386.htmlThis paper deals with variational optimal-control problems on time scales in the presence of delay in the state variables. The problem is considered on a time scale unifying the discrete, the continuous, and the quantum cases. Two examples in the discrete and quantum cases are analyzed to illustrate our results.Thabet Abdeljawad (Maraaba), Fahd Jarad, and Dumitru Baleanu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/262719.htmlAlberto Cabada and Victoria Otero-Espinar© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/569803.htmlWe consider special basic difference equations which are related to discretizations of Schrödinger equations on time scales with special symmetry properties, namely, so-called basic discrete grids. These grids are of an adaptive grid type. Solving the boundary value problem of suitable Schrödinger equations on these grids leads to completely new and unexpected analytic properties of the underlying function spaces. Some of them are presented in this work.Andreas Ruffing, Maria Meiler, and Andrea Bruder© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/698463.htmlBy using a multiple fixed point theorem (Avery-Peterson fixed point theorem) for cones, some criteria are established for the existence of three positive periodic solutions for a class of higher-dimensional functional differential equations with impulses on time scales of the following form: xΔ(t)=A(t)x(t)+f(t,xt), t≠tj, t∈𝕋, x(tj+)=x(tj−)+Ij(x(tj)), where A(t)=(aij(t))n×n is a nonsingular matrix with continuous real-valued functions as its elements. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.Yongkun Li and Meng Hu© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/584375.htmlWe systematically explore the periodicity of Liénard type p-Laplacian equations on time scales. Sufficient criteria are established for the existence of periodic solutions for such equations, which generalize many known results for differential equations when the time scale is chosen as the set of the real numbers. The main method is based on the Mawhin's continuation theorem.Fengjuan Cao, Zhenlai Han, and Shurong Sun© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2010/143298.htmlThe existence and uniqueness of solutions and a representation of solution formulas are studied for the following initial value problem: x˙(t)+∫t0tK(t,s)x(h(s))ds=f(t),  t≥t0,  x∈ℝn, x(t)=φ(t), t<t0. Such problems are obtained by transforming second-order delay differential equations x¨(t)+a(t)x˙(g(t))+b(t)x(h(t))=0 to first-order differential equations.Leonid Berezansky, Josef Diblík, and Zdenĕk Šmarda© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/214309.htmlWe study the asymptotic behavior of the solutions for the following nonlinear difference equation xn+1=∑i=1sAkixn−ki/(B0+∑j=1tBljxn−lj),n=0,1,…, where the initial conditions x−r,x−r+1,…,x1,x0 are arbitrary nonnegative real numbers, k1,…,ks,l1,…,lt are nonnegative integers, r=max{k1,…,ks,l1,…,lt}, and Ak1,…,Aks,B0,Bl1,…,Blt are positive constants. Moreover, some numerical simulations to the equation are given to illustrate our results.Chang-you Wang, Fei Gong, Shu Wang, Lin-rui Li, and Qi-hong Shi© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/503724.htmlWe will investigate the superstability of the (hyperbolic) trigonometric functional equation from the following functional equations: f(x+y)±g(x−y)=λf(x)g(y), f(x+y)±g(x−y)=λg(x)f(y), f(x+y)±g(x−y)=λf(x)f(y), f(x+y)±g(x−y)=λg(x)g(y), which can be considered the mixed functional equations of the sine function and cosine function, of the hyperbolic sine function and hyperbolic cosine function, and of the exponential functions, respectively.Gwang Hui Kim© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/521058.htmlWe show that the recessive solution of the second-order half-linear difference equation Δ(rkΦ(Δxk))+ckΦ(xk+1)=0,   Φ(x):=|x|p−2x,  p>1, where r,c are real-valued sequences, is closely related to the divergence of the infinite series ∑∞(rkxkxk+1|Δxk|p−2)−1.Ondřej Došlý and Simona Fišnarová© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/262857.htmlThis paper is concerned with the existence of multiple positive solutions for the third-order p-Laplacian dynamic equation (ϕp(uΔ∇(t)))∇+a(t)f(t,u(t),uΔ(t))=0,t∈[0,T]𝕋 with the multipoint boundary conditions uΔ(0)=uΔ∇(0)=0,u(T)+B0(∑i=1m−2biuΔ(ξi))=0, where ϕp(u)=|u|p−2u with p>1. Using the fixed point theorem due to Avery and Peterson, we establish the existence criteria of at least three positive solutions to the problem. As an application, an example is given to illustrate the result. The interesting points are that not only do we consider third-order p-Laplacian dynamic equation but also the nonlinear term f is involved with the first-order delta derivative of the unknown function.Li-Hua Bian, Xi-Ping He, and Hong-Rui Sun© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/137084.htmlBy using variational methods directly, we establish the existence of periodic solutions for a class of nonautonomous differential delay equations which are superlinear both at zero and at infinity.Rong Cheng© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/730484.htmlThe variational method and critical point theory are employed to investigate the existence of solutions for 2nth-order difference equation Δn(pk−nΔnyk−n)+(−1)n+1f(k,yk)=0 for k∈[1,N] with boundary value condition y1−n=y2−n=⋯=y0=0,  yN+1=⋯=yN+n=0 by constructing a functional, which transforms the existence of solutions of the boundary value problem (BVP) to the existence of critical points for the functional. Some criteria for the existence of at least one solution and two solutions are established which is the generalization for BVP of the even-order difference equations.Qingrong Zou and Peixuan Weng© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/389624.htmlBy using the critical point theory, we establish various sets of sufficient conditions on the nonexistence and existence of solutions for the boundary value problems of a class of fourth-order difference equations.Shenghuai Huang and Zhan Zhou© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/737461.htmlBy constructing available upper and lower solutions and combining the Schauder's fixed point theorem with maximum principle, this paper establishes sufficient and necessary conditions to guarantee the existence of Cld[0,1]𝕋 as well as CldΔ[0,1]𝕋 positive solutions for a class of singular boundary value problems on time scales. The results significantly extend and improve many known results for both the continuous case and more general time scales. We illustrate our results by one example.Meiqiang Feng, Xuemei Zhang, Xianggui Li, and Weigao Ge© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/235691.htmlWe mainly study the global behavior of the nonlinear difference equation in the title, that is, the global asymptotical stability of zero equilibrium, the existence of unbounded solutions, the existence of period two solutions, the existence of oscillatory solutions, the existence, and asymptotic behavior of non-oscillatory solutions of the equation. Our results extend and generalize the known ones.Dongmei Chen, Xianyi Li, and Yanqin Wang© 2010, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/410823.htmlStochastic effects on convergence dynamics of reaction-diffusion Cohen-Grossberg neural networks (CGNNs) with delays are studied. By utilizing Poincaré inequality, constructing suitable Lyapunov functionals, and employing the method of stochastic analysis and nonnegative semimartingale convergence theorem, some sufficient conditions ensuring almost sure exponential stability and mean square exponential stability are derived. Diffusion term has played an important role in the sufficient conditions, which is a preeminent feature that distinguishes the present research from the previous. Two numerical examples and comparison are given to illustrate our results.Jie Pan and Shouming Zhong© 2010, Hindawi Publishing Corporation. All rights reserved.