http://www.hindawi.comThe latest articles from Hindawi Publishing Corporation© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/728484.htmlThis paper is devoted to derive some sufficient conditions for the existence and uniqueness of positive solutions to a singular second-order dynamic equation with Dirichlet boundary conditions.Ana Gómez González and Victoria Otero-Espinar© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/623932.htmlLet T be a time scale such that 0,T∈T. By using a monotone iterative method, we present some existence criteria for positive solution of a multiple point general Dirichlet-Robin BVP on time scales with the singular sign-changing nonlinearity. These results are even new for the corresponding differential (T=ℝ) and difference equation (T=ℤ) as well as in general time scales setting. As an application, an example is given to illustrate the results. The interesting point here is that the sign-changing nonlinear term is involved with the first-order derivative explicitly, and the singularity may occur at u=0, t=0, and t=T.You-Hui Su, Subei Li, and Can-Yun Huang© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/143175.htmlThe spectrum containment of almost periodic solution of second-order neutral delay differential equations with piecewise constant of argument (EPCA, for short) of the form (x(t)+px(t-1))′′=qx(2[(t+1)/2])+f(t) is considered. The main result obtained in this paper is different from that given by some authors for ordinary differential equations (ODE, for short) and clearly shows the differences between ODE and EPCA. Moreover, it is also different from that given for equation (x(t)+px(t-1))′′=qx([t])+f(t) because of the difference between [t] and 2[(t+1)/2].Li Wang and Chuanyi Zhang© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/798685.htmlThis paper considers a new class of additive neural networks where the neuron activations are modelled by discontinuous functions with nonlinear growth. By Leray-Schauder alternative theorem in differential inclusion theory, matrix theory, and generalized Lyapunov approach, a general result is derived which ensures the existence and global asymptotical stability of a unique periodic solution for such neural networks. The obtained results can be applied to neural networks with a broad range of activation functions assuming neither boundedness nor monotonicity, and also show that Forti's conjecture for discontinuous neural networks with nonlinear growth activations is true.Yingwei Li and Huaiqin Wu© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/859832.htmlThe impulsive control method is developed to stabilize a class of neural networks with both time-varying and distributed delays. Some exponential stability criteria are obtained by using Lyapunov functionals, stability theory, and control by impulses. A numerical example is also provided to show the effectiveness and feasibility of the impulsive control method.Lizi Yin and Xiaodi Li© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/394635.htmlThis paper studies the global asymptotic stability for positive solutions to the higher order rational difference equation xn=(∏j=1m(xn−kj+1)+∏j=1m(xn−kj−1))/(∏j=1m(xn−kj+1)−∏j=1m(xn−kj−1)), n=0,1,2,…, where m is odd and x−km,x−km+1,…,x−1∈(0,∞). Our main result generalizes several others in the recent literature and confirms a conjecture by Berenhaut et al., 2007.Maoxin Liao, Xianhua Tang, and Changjin Xu© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/784935.htmlPlanar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turns (after several steps) into a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced.J. Diblík, D. Ya. Khusainov, and Z. Šmarda© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/104310.htmlWe obtain new explicit exponential stability results for difference equations with several variable delays and variable coefficients. Several known results, such as Clark's asymptotic stability criterion, are generalized and extended to a new class of equations.Leonid Berezansky and Elena Braverman© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/981728.htmlWe establish sufficient conditions for the existence of mild solutions for some densely defined semilinear functional differential equations and inclusions involving the Riemann-Liouville fractional derivative. Our approach is based on the 𝒞0-semigroups theory combined with some suitable fixed point theorems.Ravi P. Agarwal, Mohammed Belmekki, and Mouffak Benchohra© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/829735.htmlCriteria are established for existence of least one or three positive solutions for boundary value problems of second-order functional dynamic equations on time scales. By this purpose, we use a fixed-point index theorem in cones and Leggett-Williams fixed-point theorem.Ilkay Yaslan Karaca© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/290625.htmlWe give a representation of the solution for the best approximation of the harmonic oscillator equation formulated in a general Banach space setting, and a characterization of lp-maximal regularity—or well posedness—solely in terms of R-boundedness properties of the resolvent operator involved in the equation.Airton Castro, Claudio Cuevas, and Carlos Lizama© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/896218.htmlSwitching controller design for a class of Markovian jump nonlinear systems with unmodeled dynamics is considered in this paper. Based on the differential equation and infinitesimal generator of jump systems, the concept of Jump Input-to-State practical Stability (JISpS) in probability and stochastic Lyapunov stability criterion are put forward. By using backsetpping technology and stochastic small-gain theorem, a switching controller is proposed which ensures JISpS in probability for the jump nonlinear system. A simulation example illustrates the validity of this design.Jin Zhu, Junhong Park, Kwan-Soo Lee, and Maksym Spiryagin© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/195049.htmlWe investigate the properties of the Genocchi functions and the Genocchi polynomials. We obtain the Fourier transform on the Genocchi function. We have the generating function of (h,q)-Genocchi polynomials. We define the Cangul-Ozden-Simsek's type twisted (h,q)-Genocchi polynomials and numbers. We also have the generalized twisted (h,q)-Genocchi numbers attached to the Dirichlet's character χ. Finally, we define zeta functions related to (h,q)-Genocchi polynomials and have the generating function of the generalized (h,q)-Genocchi numbers attached to χ.Kyoung Ho Park and Young-Hee Kim© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/141589.htmlThis paper investigates the existence of periodic solutions of a ratio-dependent predator-prey diffusion system with Michaelis-Menten functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain suffcient criteria for the existence of periodic solutions for the system. Moreover, when the time scale 𝕋 is chosen as ℝ or ℤ, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.Zhenjie Liu© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/708587.htmlWe investigate certain sum-difference inequalities in two variables which provide explicit bounds on unknown functions. Our result enables us to solve those discrete inequalities considered by Sheng and Li (2008). Furthermore, we apply our result to a boundary value problem of a partial difference equation for estimation.Wu-Sheng Wang© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/147979.htmlWe consider an n-dimensional version of the functional equations of Aczél and Chung in the spaces of generalized functions such as the Schwartz distributions and Gelfand generalized functions. As a result, we prove that the solutions of the distributional version of the equation coincide with those of classical functional equation.Jae-Young Chung© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/654267.htmlWe establish Pontryagin Maximum Principles in the strong form for infinite horizon optimal control problems for bounded processes, for systems governed by difference equations. Results due to Ioffe and Tihomirov are among the tools used to prove our theorems. We write necessary conditions with weakened hypotheses of concavity and without invertibility, and we provide new results on the adjoint variable. We show links between bounded problems and nonbounded ones. We also give sufficient conditions of optimality.Joël Blot and Naïla Hayek© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/838347.htmlWe consider the general solution of quartic functional equations and prove the Hyers-Ulam-Rassias stability. Moreover, using the pullbacks and the heat kernels we reformulate and prove the stability results of quartic functional equations in the spaces of tempered distributions and Fourier hyperfunctions.Young-Su Lee and Soon-Yeong Chung© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2009/781579.htmlThis paper deals with singularly perturbed initial value problem for linear first-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first-order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem.Fevzi Erdogan© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/458687.htmlWe obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations. Examples, some of which coincide with well-known results on particular time scales, are provided to illustrate the applicability of our results.M. Bohner, B. Karpuz, and Ö. Öcalan© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/598964.htmlFor positive linear Volterra difference equations in Banach lattices, the uniform asymptotic stability of the zero solution is studied in connection with the summability of the fundamental solution and the invertibility of the characteristic operator associated with the equations. Moreover, the robust stability is discussed and some stability radii are given explicitly.Satoru Murakami and Yutaka Nagabuchi© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/840458.htmlWe 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© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/867635.htmlWe 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© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/692713.htmlWe 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© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/796851.htmlThis 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© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/712913.htmlWe 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© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/586020.htmlConsider 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© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/505324.htmlThe 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© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/143943.htmlWe 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© 2009, Hindawi Publishing Corporation. All rights reserved.http://www.hindawi.com/journals/ade/2008/879140.htmlThis 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© 2009, Hindawi Publishing Corporation. All rights reserved.