Abstract and Applied Analysis http://www.hindawi.com The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Attractor for a Reaction-Diffusion System Modeling Cancer Network Thu, 17 Apr 2014 16:55:01 +0000 http://www.hindawi.com/journals/aaa/2014/420386/ A reaction-diffusion cancer network regulated by microRNA is considered in this paper. We study the asymptotic behavior of solution and show the existence of global uniformly bounded solution to the system in a bounded domain . Some estimates and asymptotic compactness of the solutions are proved. As a result, we establish the existence of the global attractor in and prove that the solution converges to stable steady states. These results can help to understand the dynamical character of cancer network and propose a new insight to study the mechanism of cancer. In the end, the numerical simulation shows that the analytical results agree with numerical simulation. Xueyong Chen, Jianwei Shen, and Hongxian Zhou Copyright © 2014 Xueyong Chen et al. All rights reserved. Lipschitz Spaces and Fractional Integral Operators Associated with Nonhomogeneous Metric Measure Spaces Thu, 17 Apr 2014 16:45:55 +0000 http://www.hindawi.com/journals/aaa/2014/174010/ The fractional operator on nonhomogeneous metric measure spaces is introduced, which is a bounded operator from into the space . Moreover, the Lipschitz spaces on nonhomogeneous metric measure spaces are also introduced, which contain the classical Lipschitz spaces. The authors establish some equivalent characterizations for the Lipschitz spaces, and some results of the boundedness of fractional operator in Lipschitz spaces are also presented. Jiang Zhou and Dinghuai Wang Copyright © 2014 Jiang Zhou and Dinghuai Wang. All rights reserved. Stability Analysis of a Multigroup SEIR Epidemic Model with General Latency Distributions Thu, 17 Apr 2014 14:02:37 +0000 http://www.hindawi.com/journals/aaa/2014/740256/ The global stability of a multigroup SEIR epidemic model with general latency distribution and general incidence rate is investigated. Under the given assumptions, the basic reproduction number is defined and proved as the role of a threshold; that is, the disease-free equilibrium is globally asymptotically stable if , while an endemic equilibrium exists uniquely and is globally asymptotically stable if . For the proofs, we apply the classical method of Lyapunov functionals and a recently developed graph-theoretic approach. Nan Wang, Jingmei Pang, and Jinliang Wang Copyright © 2014 Nan Wang et al. All rights reserved. An Approach for Solving Discrete Game Problems with Total Constraints on Controls Thu, 17 Apr 2014 13:43:22 +0000 http://www.hindawi.com/journals/aaa/2014/674651/ We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Controls of the players satisfy total constraints. Terminal set is a subset of and it is assumed to have nonempty interior. Game is said to be completed if at some step . To construct the control of the pursuer, at each step , we use positions of the players from step 1 to step and the value of the control parameter of the evader at the step . We give sufficient conditions of completion of pursuit and construct the control for the pursuer in explicit form. This control forces the evader to expend some amount of his resources on a period consisting of finite steps. As a result, after several such periods the evader exhausted his energy and then pursuit will be completed. Asqar Raxmonov and Gafurjan I. Ibragimov Copyright © 2014 Asqar Raxmonov and Gafurjan I. Ibragimov. All rights reserved. A Note on Some Numerical Approaches to Solve a Neuron Networks Model Thu, 17 Apr 2014 13:31:54 +0000 http://www.hindawi.com/journals/aaa/2014/863842/ Space time integration plays an important role in analyzing scientific and engineering models. In this paper, we consider an integrodifferential equation that comes from modeling neuron networks. Here, we investigate various schemes for time discretization of a theta-neuron model. We use collocation and midpoint quadrature formula for space integration and then apply various time integration schemes to get a full discrete system. We present some computational results to demonstrate the schemes. Samir Kumar Bhowmik, Feras M. Al Faqih, and Md. Nazmul Islam Copyright © 2014 Samir Kumar Bhowmik et al. All rights reserved. A Korovkin Type Approximation Theorem and Its Applications Thu, 17 Apr 2014 13:11:52 +0000 http://www.hindawi.com/journals/aaa/2014/859696/ We present a Korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic functions via A-statistical approximation, for the rate of the third order Ditzian-Totik modulus of smoothness. Finally, we obtain an interleave between Riesz's representation theory and Lebesgue-Stieltjes integral-i, for Riesz's functional supremum formula via statistical limit. Malik Saad Al-Muhja Copyright © 2014 Malik Saad Al-Muhja. All rights reserved. A New Super Extension of Dirac Hierarchy Thu, 17 Apr 2014 13:07:38 +0000 http://www.hindawi.com/journals/aaa/2014/472101/ We derive a new super extension of the Dirac hierarchy associated with a matrix super spectral problem with the help of the zero-curvature equation. Super trace identity is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable hierarchy. Jiao Zhang, Fucai You, and Yan Zhao Copyright © 2014 Jiao Zhang et al. All rights reserved. Combinatorial Properties and Characterization of Glued Semigroups Thu, 17 Apr 2014 10:19:15 +0000 http://www.hindawi.com/journals/aaa/2014/436417/ This work focuses on the combinatorial properties of glued semigroups and provides its combinatorial characterization. Some classical results for affine glued semigroups are generalized and some methods to obtain glued semigroups are developed. J. I. García-García, M. A. Moreno-Frías, and A. Vigneron-Tenorio Copyright © 2014 J. I. García-García et al. All rights reserved. Weighted Differentiation Composition Operator from Logarithmic Bloch Spaces to Zygmund-Type Spaces Thu, 17 Apr 2014 10:08:08 +0000 http://www.hindawi.com/journals/aaa/2014/832713/ Let denote the space of all holomorphic functions on the unit disk of , and let  n be a positive integer, a holomorphic self-map of , and a weight. In this paper, we investigate the boundedness and compactness of a weighted differentiation composition operator , from the logarithmic Bloch spaces to the Zygmund-type spaces. Huiying Qu, Yongmin Liu, and Shulei Cheng Copyright © 2014 Huiying Qu et al. All rights reserved. The Regularity of Functions on Dual Split Quaternions in Clifford Analysis Thu, 17 Apr 2014 08:46:59 +0000 http://www.hindawi.com/journals/aaa/2014/369430/ This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function on that has a dual split Cauchy-Riemann system in dual split quaternions. Ji Eun Kim and Kwang Ho Shon Copyright © 2014 Ji Eun Kim and Kwang Ho Shon. All rights reserved. Spatiotemporal Patterns in a Ratio-Dependent Food Chain Model with Reaction-Diffusion Thu, 17 Apr 2014 08:00:41 +0000 http://www.hindawi.com/journals/aaa/2014/130851/ Predator-prey models describe biological phenomena of pursuit-evasion interaction. And this interaction exists widely in the world for the necessary energy supplement of species. In this paper, we have investigated a ratio-dependent spatially extended food chain model. Based on the bifurcation analysis (Hopf and Turing), we give the spatial pattern formation via numerical simulation, that is, the evolution process of the system near the coexistence equilibrium point , and find that the model dynamics exhibits complex pattern replication. For fixed parameters, on increasing the control parameter , the sequence “holes holes-stripe mixtures stripes spots-stripe mixtures spots” pattern is observed. And in the case of pure Hopf instability, the model exhibits chaotic wave pattern replication. Furthermore, we consider the pattern formation in the case of which the top predator is extinct, that is, the evolution process of the system near the equilibrium point , and find that the model dynamics exhibits stripes-spots pattern replication. Our results show that reaction-diffusion model is an appropriate tool for investigating fundamental mechanism of complex spatiotemporal dynamics. It will be useful for studying the dynamic complexity of ecosystems. Lei Zhang Copyright © 2014 Lei Zhang. All rights reserved. Existence and Uniqueness of Almost Periodic Solutions for Neural Networks with Neutral Delays Thu, 17 Apr 2014 06:08:55 +0000 http://www.hindawi.com/journals/aaa/2014/642685/ A class of neural networks system with neutral delays is investigated. The existence and uniqueness of almost periodic solution for the system are obtained by using fixed point theorem; we extend some results in the references. Min Xu, Zengji Du, and Kaige Zhuang Copyright © 2014 Min Xu et al. All rights reserved. Stability to a Kind of Functional Differential Equations of Second Order with Multiple Delays by Fixed Points Thu, 17 Apr 2014 00:00:00 +0000 http://www.hindawi.com/journals/aaa/2014/413037/ We discuss the stability of solutions to a kind of scalar Liénard type equations with multiple variable delays by means of the fixed point technique under an exponentially weighted metric. By this work, we improve some related results from one delay to multiple variable delays. Cemil Tunç and Emel Biçer Copyright © 2014 Cemil Tunç and Emel Biçer. All rights reserved. Enhanced Multistage Homotopy Perturbation Method: Approximate Solutions of Nonlinear Dynamic Systems Thu, 17 Apr 2014 00:00:00 +0000 http://www.hindawi.com/journals/aaa/2014/486509/ We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameter h by following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves. Daniel Olvera and Alex Elías-Zúñiga Copyright © 2014 Daniel Olvera and Alex Elías-Zúñiga. All rights reserved. The Local Stability of Solutions for a Nonlinear Equation Thu, 17 Apr 2014 00:00:00 +0000 http://www.hindawi.com/journals/aaa/2014/781813/ The approach of Kruzkov’s device of doubling the variables is applied to establish the local stability of strong solutions for a nonlinear partial differential equation in the space by assuming that the initial value only lies in the space . Haibo Yan and Ls Yong Copyright © 2014 Haibo Yan and Ls Yong. All rights reserved. LP Well-Posedness for Bilevel Vector Equilibrium and Optimization Problems with Equilibrium Constraints Thu, 17 Apr 2014 00:00:00 +0000 http://www.hindawi.com/journals/aaa/2014/792984/ The purpose of this paper is introduce several types of Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Base on criterion and characterizations for these types of Levitin-Polyak well-posedness we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescus measures of noncompactness of approximate solution sets under suitable conditions, and we prove the Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Obtain a gap function for bilevel vector equilibrium problems with equilibrium constraints using the nonlinear scalarization function and consider relations between these types of LP well-posedness for bilevel vector optimization problems with equilibrium constraints and these types of Levitin-Polyak well-posedness for bilevel vector equilibrium problems with equilibrium constraints under suitable conditions; we prove the Levitin-Polyak well-posedness for bilevel equilibrium and optimization problems with equilibrium constraints. Phan Quoc Khanh, Somyot Plubtieng, and Kamonrat Sombut Copyright © 2014 Phan Quoc Khanh et al. All rights reserved. Bifurcations of Tumor-Immune Competition Systems with Delay Wed, 16 Apr 2014 17:26:31 +0000 http://www.hindawi.com/journals/aaa/2014/723159/ A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results. Ping Bi and Heying Xiao Copyright © 2014 Ping Bi and Heying Xiao. All rights reserved. Representation of the Solutions of Linear Discrete Systems with Constant Coefficients and Two Delays Wed, 16 Apr 2014 14:28:41 +0000 http://www.hindawi.com/journals/aaa/2014/320476/ The purpose of this paper is to develop a method for the construction of solutions to initial problems of linear discrete systems with constant coefficients and with two delays where , are fixed, , , are constant matrices, is a given vector, and is an unknown vector. Solutions are expressed with the aid of a special function called the discrete matrix delayed exponential for two delays. Such approach results in a possibility to express an initial Cauchy problem in a closed form. Examples are shown illustrating the results obtained. Josef Diblík and Blanka Morávková Copyright © 2014 Josef Diblík and Blanka Morávková. All rights reserved. On Hölder and Minkowski Type Inequalities Wed, 16 Apr 2014 14:25:07 +0000 http://www.hindawi.com/journals/aaa/2014/915635/ We obtain inequalities of Hölder and Minkowski type with weights generalizing both the case of weights with alternating signs and the classical case of nonnegative weights. Petr Chunaev, Ljiljanka Kvesić, and Josip Pečarić Copyright © 2014 Petr Chunaev et al. All rights reserved. Modelling the Influence of Awareness Programs by Media on the Drinking Dynamics Wed, 16 Apr 2014 14:14:25 +0000 http://www.hindawi.com/journals/aaa/2014/938080/ We develop a nonlinear mathematical model with the effect of awareness programs on the binge drinking. Due to the fact that awareness programs are capable of inducing behavioral changes in nondrinkers, we introduce a separate class by avoiding contacts with the heavy drinkers. Furthermore we assume that cumulative density of awareness programs increases at a rate proportional to the number of heavy drinkers. We establish some sufficient conditions for the stability of the alcohol free and the alcohol present equilibria and give some numerical simulations to explain our main result. Our results show that awareness programs is an effective measure in reducing alcohol problems. Hai-Feng Huo and Qian Wang Copyright © 2014 Hai-Feng Huo and Qian Wang. All rights reserved. Focusing Modeling of OPFC Linear Array Transducer by Using Distributed Point Source Method Wed, 16 Apr 2014 14:14:00 +0000 http://www.hindawi.com/journals/aaa/2014/840748/ The improvement of ultrasonic phased array detection technology is a major concern of engineering community. Orthotropic piezoelectric fiber composite (OPFC) can be constructed to multielement linear array which may be applied conveniently to actuators and sensors. The phased array transducers can generate special directional strong actuator power and high sensitivity for its orthotropic performance. Focusing beam of the linear phased array transducer is obtained simply only by adjusting a parabolic time delay. In this work, the distributed point source method (DPSM) is used to model the ultrasonic field. DPSM is a newly developed mesh-free numerical technique that has been developed for solving a variety of engineering problems. This work gives the basic theory of this method and solves the problems from the application of new OPFC phased array transducer. Compared with traditional transducer, the interaction effect of two OPFC linear phased array transducers is also modeled in the same medium, which shows that the pressure beam produced by the new transducer is narrower or more collimated than that produced by the conventional transducer at different angles. DPSM can be used to analyze and optimally design the OPFC linear phased array transducer. Ziping Wang and Ying Luo Copyright © 2014 Ziping Wang and Ying Luo. All rights reserved. Stability of a Class of Coupled Systems Wed, 16 Apr 2014 14:13:35 +0000 http://www.hindawi.com/journals/aaa/2014/835765/ We consider a class of coupled systems with damping terms. By using multiplier method and the estimation techniques of the energy, we show that even if the kernel function is nonincreasing and integrable without additional conditions, the energy of the system decays also to zero in a good rate. Kun-Peng Jin Copyright © 2014 Kun-Peng Jin. All rights reserved. On the Analyticity for the Generalized Quadratic Derivative Complex Ginzburg-Landau Equation Wed, 16 Apr 2014 14:11:09 +0000 http://www.hindawi.com/journals/aaa/2014/607028/ We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation in any spatial dimension with rough initial data. For , we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces . For , we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in . The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup to overcome the derivative in the nonlinear term. Chunyan Huang Copyright © 2014 Chunyan Huang. All rights reserved. New Rational Homoclinic and Rogue Waves for Davey-Stewartson Equation Wed, 16 Apr 2014 14:09:56 +0000 http://www.hindawi.com/journals/aaa/2014/572863/ A new method, homoclinic breather limit method (HBLM), for seeking rogue wave solution of nonlinear evolution equation is proposed. A new family of homoclinic breather wave solution, and rational homoclinic solution (homoclinic rogue wave) for DSI and DSII equations are obtained using the extended homoclinic test method and homoclinic breather limit method (HBLM), respectively. Moreover, rogue wave solution is exhibited as period of periodic wave in homoclinic breather wave approaches to infinite. This result shows that rogue wave can be generated by extreme behavior of homoclinic breather wave for higher dimensional nonlinear wave fields. Changfu Liu, Chuanjian Wang, Zhengde Dai, and Jun Liu Copyright © 2014 Changfu Liu et al. All rights reserved. Nontrivial Solutions for Asymmetric Kirchhoff Type Problems Wed, 16 Apr 2014 14:05:18 +0000 http://www.hindawi.com/journals/aaa/2014/163645/ We consider a class of particular Kirchhoff type problems with a right-hand side nonlinearity which exhibits an asymmetric growth at and in . Namely, it is 4-linear at and 4-superlinear at . However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semiaxis. Some existence results for nontrivial solution are established by combining Mountain Pass Theorem and a variant version of Mountain Pass Theorem with Moser-Trudinger inequality. Ruichang Pei and Jihui Zhang Copyright © 2014 Ruichang Pei and Jihui Zhang. All rights reserved. A New System of Multivalued Mixed Variational Inequality Problem Wed, 16 Apr 2014 13:45:15 +0000 http://www.hindawi.com/journals/aaa/2014/982606/ We consider a new system of multivalued mixed variational inequality problem, which includes some known systems of variational inequalities as special cases. Under suitable conditions, the existence of solutions for the system of multivalued mixed variational inequality problem and the convergence of iterative sequences generated by the generalized -projection algorithm are proved. A perturbational algorithm for solving a special case of multivalued mixed variational inequality problem is formally constructed. The results concerned with the existence of solutions and the convergence of iterative sequences generated by the perturbational algorithm are also given. Some known results are improved and generalized. Xi Li and Xue-song Li Copyright © 2014 Xi Li and Xue-song Li. All rights reserved. Stability Analysis of a Population Model with Maturation Delay and Ricker Birth Function Wed, 16 Apr 2014 10:59:08 +0000 http://www.hindawi.com/journals/aaa/2014/136707/ A single species population model is investigated, where the discrete maturation delay and the Ricker birth function are incorporated. The threshold determining the global stability of the trivial equilibrium and the existence of the positive equilibrium is obtained. The necessary and sufficient conditions ensuring the local asymptotical stability of the positive equilibrium are given by applying the Pontryagin's method. The effect of all the parameter values on the local stability of the positive equilibrium is analyzed. The obtained results show the existence of stability switch and provide a method of computing maturation times at which the stability switch occurs. Numerical simulations illustrate that chaos may occur for the model, and the associated parameter bifurcation diagrams are given for certain values of the parameters. Chongwu Zheng, Fengqin Zhang, and Jianquan Li Copyright © 2014 Chongwu Zheng et al. All rights reserved. Infinitely Many Periodic Solutions of Duffing Equations with Singularities via Time Map Wed, 16 Apr 2014 09:33:46 +0000 http://www.hindawi.com/journals/aaa/2014/398512/ We study the periodic solutions of Duffing equations with singularities . By using Poincaré-Birkhoff twist theorem, we prove that the given equation possesses infinitely many positive periodic solutions provided that satisfies the singular condition and the time map related to autonomous system tends to zero. Tiantian Ma and Zaihong Wang Copyright © 2014 Tiantian Ma and Zaihong Wang. All rights reserved. Delay-Dependent Robust Filtering for a Class of Fuzzy Stochastic Systems Wed, 16 Apr 2014 09:09:51 +0000 http://www.hindawi.com/journals/aaa/2014/673956/ This paper is concerned with the filtering problem for a kind of Takagi-Sugeno (T-S) fuzzy stochastic system with time-varying delay and parameter uncertainties. Parameter uncertainties in the system are assumed to satisfy global Lipschitz conditions. And the attention of this paper is focused on the stochastically mean-square stability of the filtering error system, and the performance level of the output error with the disturbance input. The method designed for the delay-dependent filter is developed based on linear matrix inequalities. Finally, the effectiveness of the proposed method is substantiated with an illustrative example. Ze Li and Xinhao Yang Copyright © 2014 Ze Li and Xinhao Yang. All rights reserved. The Stability of Solutions for a Fractional Predator-Prey System Wed, 16 Apr 2014 08:45:58 +0000 http://www.hindawi.com/journals/aaa/2014/124145/ We study a class of fractional predator-prey systems with Holling II functional response. A unique positive solution of this system is obtained. In order to prove the asymptotical stability of positive equilibrium for this system, we study the Lyapunov stability theory of a fractional system. Yingjia Guo Copyright © 2014 Yingjia Guo. All rights reserved.