http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2013 , Hindawi Publishing Corporation . All rights reserved. Mon, 20 May 2013 15:13:34 +0000 http://www.hindawi.com/journals/ddns/2013/104173/ In fractional calculus, there are two approaches to obtain fractional derivatives. The first approach is by iterating the integral and then defining a fractional order by using Cauchy formula to obtain Riemann fractional integrals and derivatives. The second approach is by iterating the derivative and then defining a fractional order by making use of the binomial theorem to obtain Grünwald-Letnikov fractional derivatives. In this paper we formulate the delta and nabla discrete versions for left and right fractional integrals and derivatives representing the second approach. Then, we use the discrete version of the Q-operator and some discrete fractional dual identities to prove that the presented fractional differences and sums coincide with the discrete Riemann ones describing the first approach. Thabet Abdeljawad, Dumitru Baleanu, Fahd Jarad, and Ravi P. Agarwal Copyright © 2013 Thabet Abdeljawad et al. All rights reserved. Mon, 20 May 2013 11:56:27 +0000 http://www.hindawi.com/journals/ddns/2013/179401/ We focus on a system of a rational -order difference equation , , , where . We investigate the dynamical behavior of positive solution for the system. Banyat Sroysang Copyright © 2013 Banyat Sroysang. All rights reserved. Sun, 19 May 2013 13:56:41 +0000 http://www.hindawi.com/journals/ddns/2013/681689/ Social distancing, vaccination, and medical treatments have been extensively studied and widely used to control the spread of infectious diseases. However, it is still a difficult task for health administrators to determine the optimal combination of these strategies when confronting disease outbreaks with limited resources, especially in the case of interconnected populations, where the flow of individuals is usually restricted with the hope of avoiding further contamination. We consider two coupled populations and examine them independently under two variants of well-known discrete time disease models. In both examples we compute approximations for the control levels necessary to minimize costs and quickly contain outbreaks. The main technique used is simulated annealing, a stochastic search optimization tool that, in contrast with traditional analytical methods, allows easy implementation to any number of patches with different kinds of couplings and internal dynamics. Geisel Alpízar and Luis F. Gordillo Copyright © 2013 Geisel Alpízar and Luis F. Gordillo. All rights reserved. Sun, 19 May 2013 11:28:12 +0000 http://www.hindawi.com/journals/ddns/2013/816382/ Xiang Li, Chen Zhou, and Xiaochen Sun Copyright © 2013 Xiang Li et al. All rights reserved. Thu, 16 May 2013 18:03:43 +0000 http://www.hindawi.com/journals/ddns/2013/474852/ We present a new evolutionary algorithm to solve the 0-1 multidimensional knapsack problem. We tackle the problem using duality concept, differently from traditional approaches. Our method is based on Lagrangian relaxation. Lagrange multipliers transform the problem, keeping the optimality as well as decreasing the complexity. However, it is not easy to find Lagrange multipliers nearest to the capacity constraints of the problem. Through empirical investigation of Lagrangian space, we can see the potentiality of using a memetic algorithm. So we use a memetic algorithm to find the optimal Lagrange multipliers. We show the efficiency of the proposed method by the experiments on well-known benchmark data. Yourim Yoon and Yong-Hyuk Kim Copyright © 2013 Yourim Yoon and Yong-Hyuk Kim. All rights reserved. Wed, 15 May 2013 16:40:49 +0000 http://www.hindawi.com/journals/ddns/2013/183420/ Combining the exponential dichotomy of evolution family, composition theorems for almost automorphic functions with Banach fixed point theorem, we establish new existence and uniqueness theorems for almost automorphic mild solutions to neutral parabolic nonautonomous evolution equations with nondense domain. A unified framework is set up to investigate the existence and uniqueness of almost automorphic mild solutions to some classes of parabolic partial differential equations and neutral functional differential equations. Zhanrong Hu and Zhen Jin Copyright © 2013 Zhanrong Hu and Zhen Jin. All rights reserved. Wed, 15 May 2013 12:03:52 +0000 http://www.hindawi.com/journals/ddns/2013/943961/ This paper is devoted to introduce a new approach to investigate the existence of solutions for a three-point boundary value problem of fractional difference equations as fllows: , , and = We present an existence result at resonance case. The proof relies on coincidence degree theory. Zhongmin Huang and Chengmin Hou Copyright © 2013 Zhongmin Huang and Chengmin Hou. All rights reserved. Wed, 15 May 2013 10:23:01 +0000 http://www.hindawi.com/journals/ddns/2013/864707/ The investigations are made on the exponential synchronization of the stochastic Lur'e system with nonlinear coupling and impulsive disturbance. The impulsive effects in complex networks could play a positive or a negative role for synchronization. For the sake of simplification and efficiency, a single impulsive controller is designed to realize the synchronization of the impulsive dynamical network with respect to stabilizing and destabilizing impulsive effects. Sufficient conditions are derived to guarantee the realization of the exponential synchronization for all initial values by means of the Lyapunov stability theorem, the comparison principle, and the linear matrix inequalities (LMIs). Numerical simulations are given to support the validity of the analytical results. Yi Zhao, Ze Tang, and Jianwen Feng Copyright © 2013 Yi Zhao et al. All rights reserved. Tue, 14 May 2013 19:13:52 +0000 http://www.hindawi.com/journals/ddns/2013/841764/ This study is based on an early brain tumor growth that is modeled as a hybrid system such as (A): , where the parameters , and denote positive numbers, and are negative numbers and is the integer part of . Equation (A) explains a brain tumor growth, where is embedded to show the drug effect on the tumor and is a rate that causes a negative effect by the immune system on the tumor population. Using (A), we have constructed two models of a tumor growth: one is (A) and the other one is a population model at low density by incorporating an Allee function to (A) at time . To consider the global behavior of (A), we investigate the discrete solutions of (A). Examination of the characterization of the stability shows that increase of the population growth rate decreases the local stability of the positive equilibrium point of (A). The simulations give a detailed description of the behavior of solutions of (A) with and without Allee effect. F. Bozkurt Copyright © 2013 F. Bozkurt. All rights reserved. Tue, 14 May 2013 17:34:33 +0000 http://www.hindawi.com/journals/ddns/2013/903593/ We deal with the form of the solutions for the following systems of rational difference equations , , with nonzero real numbers initial conditions. Also we investigate some properties of the obtained solutions and present some numerical examples. H. El-Metwally Copyright © 2013 H. El-Metwally. All rights reserved. Mon, 13 May 2013 12:08:34 +0000 http://www.hindawi.com/journals/ddns/2013/495838/ We investigate the global convergence, boundedness, and periodicity of solutions of the recursive sequence , where the parameters and are positive real numbers, and the initial conditions and are positive real numbers where . M. M. El-Dessoky Copyright © 2013 M. M. El-Dessoky. All rights reserved. Sun, 12 May 2013 14:45:54 +0000 http://www.hindawi.com/journals/ddns/2013/604105/ By introducing new definitions of convex and concave quasioperator and quasilower and quasiupper, by means of the monotone iterative techniques without any compactness conditions, we obtain the iterative unique solution of nonlinear mixed monotone Fredholm-type integral equations in Banach spaces. Our results are even new to convex and concave quasi operator, and then we apply these results to the two-point boundary value problem of second-order nonlinear ordinary differential equations in the ordered Banach spaces. Hua Su Copyright © 2013 Hua Su. All rights reserved. Sun, 12 May 2013 13:18:05 +0000 http://www.hindawi.com/journals/ddns/2013/408904/ We propose a price game model of irrigation water in a coastal irrigation district. Then, we discuss the stability and codimension-two period-doubling (flip) bifurcation. Then, the MATLAB package Cl_MatContM is employed to illustrate its numerical bifurcations-based continuation methods. Lastly, the 0-1 test algorithm is used to compute the median value of correlation coefficient which can indicate whether the underlying dynamics is regular or chaotic. Baogui Xin and Yuting Li Copyright © 2013 Baogui Xin and Yuting Li. All rights reserved. Sun, 12 May 2013 07:58:11 +0000 http://www.hindawi.com/journals/ddns/2013/385419/ We study the bifurcation of limit cycles from periodic orbits of a four-dimensional system when the perturbation is piecewise linear with two switching boundaries. Our main result shows that when the parameter is sufficiently small at most, six limit cycles can bifurcate from periodic orbits in a class of asymmetric piecewise linear perturbed systems, and, at most, three limit cycles can bifurcate from periodic orbits in another class of asymmetric piecewise linear perturbed systems. Moreover, there are perturbed systems having six limit cycles. The main technique is the averaging method. Yanyan Cheng Copyright © 2013 Yanyan Cheng. All rights reserved. Thu, 09 May 2013 15:20:50 +0000 http://www.hindawi.com/journals/ddns/2013/756251/ We consider a single-machine scheduling problem with upper bounded actual processing time and upper bounded maintenance time under deteriorating effect. The actual processing time of a job is a position-dependent power function. If the actual processing time of a job exceeds the upper bound, tardiness penalty of the job should be paid. And if the maintenance time exceeds the corresponding upper bound, tardiness penalty of the maintenance should also be paid. The maintenance duration studied in the paper is a position-dependent exponential function. The objective is to find jointly the optimal maintenance frequency and the optimal job sequence to minimize the total cost, which is a linear function of the makespan and the total tardiness. We show that the studied scheduling problem can be transformed as a classic assignment problem to solve. There is also shown that a special case of the scheduling problem can be optimally solved by a lower order algorithm. Pengfei Xue and Yulin Zhang Copyright © 2013 Pengfei Xue and Yulin Zhang. All rights reserved. Thu, 09 May 2013 15:10:21 +0000 http://www.hindawi.com/journals/ddns/2013/480560/ Being inspired by the Hopfield neural networks (Hopfield (1982) and Hopfield and Tank (1985)) and the nonlinear sigmoid power control algorithm for cellular radio systems in Uykan and Koivo (2004), in this paper, we present a novel discrete recurrent nonlinear system and extend the results in Uykan (2009), which are for autonomous linear systems, to nonlinear case. The proposed system can be viewed as a discrete-time realization of a recently proposed continuous-time network in Uykan (2013). In this paper, we focus on discrete-time analysis and provide various novel key results concerning the discrete-time dynamics of the proposed system, some of which are as follows: (i) the proposed system is shown to be stable in synchronous and asynchronous work mode in discrete time; (ii) a novel concept called Pseudo-SINR (pseudo-signal-to-interference-noise ratio) is introduced for discrete-time nonlinear systems; (iii) it is shown that when the system states approach an equilibrium point, the instantaneous Pseudo-SINRs are balanced; that is, they are equal to a target value. The simulation results confirm the novel results presented and show the effectiveness of the proposed discrete-time network as applied to various associative memory systems and clustering problems. Zekeriya Uykan Copyright © 2013 Zekeriya Uykan. All rights reserved. Wed, 08 May 2013 15:17:25 +0000 http://www.hindawi.com/journals/ddns/2013/705601/ The discrete-time epidemic model is investigated, which is obtained using the Euler method. It is verified that there exist some dynamical behaviors in this model, such as transcritical bifurcation, flip bifurcation, Hopf bifurcation, and chaos. The numerical simulations, including bifurcation diagrams and computation of Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors. Junhong Li and Ning Cui Copyright © 2013 Junhong Li and Ning Cui. All rights reserved. Tue, 07 May 2013 18:42:22 +0000 http://www.hindawi.com/journals/ddns/2013/704962/ Iterating an orientation-preserving piecewise isometry of -dimensional Euclidean space, the phase space can be partitioned with full measure into the union of the rational set consisting of periodically coded points, and the complement of the rational set is usually called the exceptional set. The tangencies between the periodic cells have been studied in some previous papers, and the results showed that almost all disk packings for certain families of planar piecewise isometries have no tangencies. In this paper, the authors further investigate the structure of any periodic cells for a general piecewise isometry of even dimensional Euclidean space and the tangencies between the periodic cells. First, we show that each periodic cell is a symmetrical body to a center if the piecewise isometry is irrational; this result is a generalization of the results in some previously published papers. Second, we show that the periodic cell packing induced by an invertible irrational planar piecewise rotation, such as the Sigma-Delta map and the overflow map, has no tangencies. And furthermore, we generalize the result to general even dimensional Euclidean spaces. Our results generalize and strengthen former research results on this topic. Rongzhong Yu and Xinchu Fu Copyright © 2013 Rongzhong Yu and Xinchu Fu. All rights reserved. Tue, 07 May 2013 10:45:56 +0000 http://www.hindawi.com/journals/ddns/2013/798961/ A Nicholson's blowflies model with feedback control and time delay is studied. By applying the comparison theorem of the differential equation and fluctuation lemma and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the permanence, extinction, and existence of a unique globally attractive positive almost periodic solution of the system are obtained. It is proved that the feedback control variable and time delay have no influence on the permanence and extinction of the system. Haihui Wu and Shengbin Yu Copyright © 2013 Haihui Wu and Shengbin Yu. All rights reserved. Thu, 02 May 2013 09:57:10 +0000 http://www.hindawi.com/journals/ddns/2013/630545/ A criterion of stabilization for the switched large-scale discrete-time system is deduced by employing state-driven switching method and the Lyapunov stability theorem. This particular method especially can be applied to cases when all individual subsystems are unstable. Chi-Jo Wang and Juing-Shian Chiou Copyright © 2013 Chi-Jo Wang and Juing-Shian Chiou. All rights reserved. Sun, 28 Apr 2013 13:25:28 +0000 http://www.hindawi.com/journals/ddns/2013/430963/ This paper studies a small neural network with three neurons. First, the activation function takes the sign function. Although the network is a simple hybrid system with all subsystems being exponentially stable, we find that it can exhibit very complex dynamics such as limit cycles and chaos. Since the sign function is a limit case of sigmoidal functions, we find that chaos robustly exists with some different activation functions, which implies that such chaos in this network is more related to its weight matrix than the type of activation functions. For chaos, we present a rigorous computer-assisted study by virtue of topological horseshoe theory. Fangyan Yang, Song Tang, and Guilan Xu Copyright © 2013 Fangyan Yang et al. All rights reserved. Wed, 24 Apr 2013 15:03:13 +0000 http://www.hindawi.com/journals/ddns/2013/808262/ Some results are obtained on finite unions of -spaces. It is proved that if a space is the union of finitely many locally compact -subspaces, then it is a -space. It follows that a space is a -space if it is the union of finitely many locally compact submetacompact subspaces. And a space is a -space if it is the union of a -subspace with a locally compact -subspace. This partially answers one problem raised by Arhangel’skii. At last, some examples are given to exhibit the applications of nearly good relation to discover -classes. Xin Zhang, Hongfeng Guo, and Yuming Xu Copyright © 2013 Xin Zhang et al. All rights reserved. Tue, 23 Apr 2013 14:25:42 +0000 http://www.hindawi.com/journals/ddns/2013/418123/ A function is said to be a Fibonacci function if for all . In 2012, some properties on the Fibonacci functions were presented. In this paper, for any positive integer , a function is said to be a Fibonacci function with period if for all ; we present some properties on the Fibonacci functions with period . Banyat Sroysang Copyright © 2013 Banyat Sroysang. All rights reserved. Tue, 23 Apr 2013 13:44:39 +0000 http://www.hindawi.com/journals/ddns/2013/298032/ We introduce some local properties by soft semi-open sets. For example, soft semi-neighborhoods of the soft point, soft semi-first-countable spaces and soft semi-pu-continuous at the soft point are given. Furthermore, we define soft semi-connectedness and prove that a soft topological space is soft semiconnected if and only if both soft semi-open and soft semi-closed sets are only and . Bin Chen Copyright © 2013 Bin Chen. All rights reserved. Tue, 23 Apr 2013 09:43:57 +0000 http://www.hindawi.com/journals/ddns/2013/741901/ The limiting behavior of the maximum partial sums is investigated, and some new results are obtained, where is an array of rowwise AANA random variables and is a sequence of positive real numbers. As an application, the Chung-type strong law of large numbers for arrays of rowwise AANA random variables is obtained. The results extend and improve the corresponding ones of Hu and Taylor (1997) for arrays of rowwise independent random variables. Aiting Shen, Ranchao Wu, Yan Chen, and Yu Zhou Copyright © 2013 Aiting Shen et al. All rights reserved. Mon, 22 Apr 2013 09:51:14 +0000 http://www.hindawi.com/journals/ddns/2013/185612/ We study the effect of the parking on heterogeneous commuters' travel choice in a competitive transportation system which consists of a subway and a parallel road with a bottleneck of limited service capacity. Every morning, commuters either use their private cars only or drive their cars to the bottleneck, park there, and then take the subway to the destination. Considering the effects caused by body congestion in carriage and the parking fees, we developed a bottleneck model to describe the commuters' travel choice. There exist several types of equilibrium that corresponds to user equilibrium. We investigated the influence of the capacity of the bottleneck and the total travel demand on the travel behaviors and on the total social cost. It is shown that there exists a scheme with suitable subway fare and parking fees to implement the minimum total social cost. Chengjuan Zhu, Bin Jia, Linghui Han, and Ziyou Gao Copyright © 2013 Chengjuan Zhu et al. All rights reserved. Mon, 22 Apr 2013 09:43:21 +0000 http://www.hindawi.com/journals/ddns/2013/932058/ This paper investigates the mean-square exponential synchronization issues of delayed stochastic complex dynamical networks with switching topology and impulsive control. By using the Lyapunov functional method, impulsive control theory, and linear matrix inequality (LMI) approaches, some sufficient conditions are derived to guarantee the mean-square exponential synchronization of delay complex dynamical network with switch topology, which are independent of the network size and switch topology. Numerical simulations are given to illustrate the effectiveness of the obtained results in the end. Xuefei Wu and Chen Xu Copyright © 2013 Xuefei Wu and Chen Xu. All rights reserved. Sun, 21 Apr 2013 13:33:30 +0000 http://www.hindawi.com/journals/ddns/2013/301718/ First, we make the Jain's derivative-free method optimal and subsequently increase its efficiency index from 1.442 to 1.587. Then, a novel three-step computational family of iterative schemes for solving single variable nonlinear equations is given. The schemes are free from derivative calculation per full iteration. The optimal family is constructed by applying the weight function approach alongside an approximation for the first derivative of the function in the last step in which the first two steps are the optimized derivative-free form of Jain's method. The convergence rate of the proposed optimal method and the optimal family is studied. The efficiency index for each method of the family is 1.682. The superiority of the proposed contributions is illustrated by solving numerical examples and comparing them with some of the existing methods in the literature. In the end, we provide the basins of attraction for some methods to observe the beauty of iterative nonlinear solvers in providing fractals and also choose the best method in case of larger attraction basins. Farahnaz Soleimani, Fazlollah Soleymani, and Stanford Shateyi Copyright © 2013 Farahnaz Soleimani et al. All rights reserved. Sun, 21 Apr 2013 11:38:49 +0000 http://www.hindawi.com/journals/ddns/2013/134296/ We study the stabilization problem of discrete-time planar switched linear systems with impulse. When all subsystems are controllable, based on an explicit estimation on the state transition matrix, we establish a sufficient condition such that the switched impulsive system is stabilizable under arbitrary switching signal with given switching frequency. When there exists at least one uncontrollable subsystem, a sufficient condition is also given to guarantee the stabilization of the switched impulsive system under appropriate switching signal. Yanli Zhu and Yuangong Sun Copyright © 2013 Yanli Zhu and Yuangong Sun. All rights reserved. Sun, 21 Apr 2013 08:45:04 +0000 http://www.hindawi.com/journals/ddns/2013/492014/ The study of recurrent neural networks with piecewise constant transition or control functions has attracted much attention recently because they can be used to simulate many physical phenomena. A recurrent and discontinuous two-state dynamical system involving a nonnegative bifurcation parameter is studied. By elementary but novel arguments, we are able to give a complete analysis on its asymptotic behavior when the parameter varies from 0 to . It is hoped that our analysis will provide motivation for further results on large-scale recurrent McCulloch-Pitts-type neural networks and piecewise continuous discrete-time dynamical systems. Chenmin Hou and Sui Sun Cheng Copyright © 2013 Chenmin Hou and Sui Sun Cheng. All rights reserved.