Journal of Applied Mathematics The latest articles from Hindawi Publishing Corporation © 2014 , Hindawi Publishing Corporation . All rights reserved. Fast Stiffness Matrix Calculation for Nonlinear Finite Element Method Thu, 28 Aug 2014 11:36:36 +0000 We propose a fast stiffness matrix calculation technique for nonlinear finite element method (FEM). Nonlinear stiffness matrices are constructed using Green-Lagrange strains, which are derived from infinitesimal strains by adding the nonlinear terms discarded from small deformations. We implemented a linear and a nonlinear finite element method with the same material properties to examine the differences between them. We verified our nonlinear formulation with different applications and achieved considerable speedups in solving the system of equations using our nonlinear FEM compared to a state-of-the-art nonlinear FEM. Emir Gülümser, Uğur Güdükbay, and Sinan Filiz Copyright © 2014 Emir Gülümser et al. All rights reserved. Application of a New Membership Function in Nonlinear Fuzzy PID Controllers with Variable Gains Thu, 28 Aug 2014 11:36:01 +0000 This paper proposes a nonlinear fuzzy PID control algorithm, whose membership function (MF) is adjustable, is universal, and has a wide adjustable range. Appling this function to fuzzy control theory will increase system’s tunability. The continuity of this function is proved. This method was employed in the simulation and HIL experiments. Effectiveness and feasibility of this function are demonstrated in the results. Xuda Zhang, Hong Bao, Jingli Du, and Congsi Wang Copyright © 2014 Xuda Zhang et al. All rights reserved. Time-Dependent Global Attractor for a Class of Nonclassical Parabolic Equations Thu, 28 Aug 2014 11:32:32 +0000 Based on the recent theory of time-dependent global attractors in the works of Conti et al. (2013) and di Plinio et al. (2011), we prove the existence of time-dependent global attractors as well as the regularity of the time-dependent global attractor for a class of nonclassical parabolic equations. Fang-hong Zhang Copyright © 2014 Fang-hong Zhang. All rights reserved. A Hierarchy of Discrete Integrable Coupling System with Self-Consistent Sources Wed, 27 Aug 2014 11:54:33 +0000 Integrable coupling system of a lattice soliton equation hierarchy is deduced. The Hamiltonian structure of the integrable coupling is constructed by using the discrete quadratic-form identity. The Liouville integrability of the integrable coupling is demonstrated. Finally, the discrete integrable coupling system with self-consistent sources is deduced. Yuqing Li, Huanhe Dong, and Baoshu Yin Copyright © 2014 Yuqing Li et al. All rights reserved. A Poisson Cluster Stochastic Rainfall Generator That Accounts for the Interannual Variability of Rainfall Statistics: Validation at Various Geographic Locations across the United States Wed, 27 Aug 2014 08:50:03 +0000 A novel approach for a Poisson cluster stochastic rainfall generator was validated in its ability to reproduce important rainfall and watershed response characteristics at 104 locations in the United States. The suggested novel approach, The Hybrid Model (THM), as compared to the traditional Poisson cluster rainfall modeling approaches, has an additional capability to account for the interannual variability of rainfall statistics. THM and a traditional approach of Poisson cluster rainfall model (modified Bartlett-Lewis rectangular pulse model) were compared in their ability to reproduce the characteristics of extreme rainfall and watershed response variables such as runoff and peak flow. The results of the comparison indicate that THM generally outperforms the traditional approach in reproducing the distributions of peak rainfall, peak flow, and runoff volume. In addition, THM significantly outperformed the traditional approach in reproducing extreme rainfall by 2.3% to 66% and extreme flow values by 32% to 71%. Dongkyun Kim, Jongho Kim, and Yong-Sik Cho Copyright © 2014 Dongkyun Kim et al. All rights reserved. Hybrid Iterations for the Fixed Point Problem and Variational Inequalities Wed, 27 Aug 2014 07:51:59 +0000 A hybrid iterative algorithm with Meir-Keeler contraction is presented for solving the fixed point problem of the pseudocontractive mappings and the variational inequalities. Strong convergence analysis is given as . Li-Jun Zhu, Shin Min Kang, Zhangsong Yao, and Young Chel Kwun Copyright © 2014 Li-Jun Zhu et al. All rights reserved. A Multistep Framework for Vision Based Vehicle Detection Wed, 27 Aug 2014 07:29:44 +0000 Vision based vehicle detection is a critical technology that plays an important role in not only vehicle active safety but also road video surveillance application. In this work, a multistep framework for vision based vehicle detection is proposed. In the first step, for vehicle candidate generation, a novel geometrical and coarse depth information based method is proposed. In the second step, for candidate verification, a deep architecture of deep belief network (DBN) for vehicle classification is trained. In the last step, a temporal analysis method based on the complexity and spatial information is used to further reduce miss and false detection. Experiments demonstrate that this framework is with high true positive (TP) rate as well as low false positive (FP) rate. On road experimental results demonstrate that the algorithm performs better than state-of-the-art vehicle detection algorithm in testing data sets. Hai Wang and Yingfeng Cai Copyright © 2014 Hai Wang and Yingfeng Cai. All rights reserved. Real-Time Control Strategy of Elman Neural Network for the Parallel Hybrid Electric Vehicle Tue, 26 Aug 2014 13:00:05 +0000 Through researching the instantaneous control strategy and Elman neural network, the paper established equivalent fuel consumption functions under the charging and discharging conditions of power batteries, deduced the optimal control objective function of instantaneous equivalent consumption, established the instantaneous optimal control model, and designs the Elman neural network controller. Based on the ADVISOR 2002 platform, the instantaneous optimal control strategy and the Elman neural network control strategy were simulated on a parallel HEV. The simulation results were analyzed in the end. The contribution of the paper is that the trained Elman neural network control strategy can reduce the simulation time by 96% and improve the real-time performance of energy control, which also ensures the good performance of power and fuel economy. Ruijun Liu, Dapai Shi, and Chao Ma Copyright © 2014 Ruijun Liu et al. All rights reserved. Continuous Nondestructive Monitoring Method Using the Reconstructed Three-Dimensional Conductivity Images via GREIT for Tissue Engineering Tue, 26 Aug 2014 12:41:48 +0000 A continuous Nondestructive monitoring method is required to apply proper feedback controls during tissue regeneration. Conductivity is one of valuable information to assess the physiological function and structural formation of regenerated tissues or cultured cells. However, conductivity imaging methods suffered from inherited ill-posed characteristics in image reconstruction, unknown boundary geometry, uncertainty in electrode position, and systematic artifacts. In order to overcome the limitation of microscopic electrical impedance tomography (micro-EIT), we applied a 3D-specific container with a fixed boundary geometry and electrode configuration to maximize the performance of Graz consensus reconstruction algorithm for EIT (GREIT). The separation of driving and sensing electrodes allows us to simplify the hardware complexity and obtain higher measurement accuracy from a large number of small sensing electrodes. We investigated the applicability of the GREIT to 3D micro-EIT images via numerical simulations and large-scale phantom experiments. We could reconstruct multiple objects regardless of the location. The resolution was 5 mm3 with 30 dB SNR and the position error was less than 2.54 mm. This shows that the new micro-EIT system integrated with GREIT is robust with the intended resolution. With further refinement and scaling down to a microscale container, it may be a continuous nondestructive monitoring tool for tissue engineering applications. Sujin Ahn, Hun Wi, Tong In Oh, Alistair Lee McEwan, Sung Chan Jun, and Eung Je Woo Copyright © 2014 Sujin Ahn et al. All rights reserved. The Interval-Valued Triangular Fuzzy Soft Set and Its Method of Dynamic Decision Making Tue, 26 Aug 2014 12:26:40 +0000 A concept of interval-valued triangular fuzzy soft set is presented, and some operations of “AND,” “OR,” intersection, union and complement, and so forth are defined. Then some relative properties are discussed and several conclusions are drawn. A dynamic decision making model is built based on the definition of interval-valued triangular fuzzy soft set, in which period weight is determined by the exponential decay method. The arithmetic weighted average operator of interval-valued triangular fuzzy soft set is given by the aggregating thought, thereby aggregating interval-valued triangular fuzzy soft sets of different time-series into a collective interval-valued triangular fuzzy soft set. The formulas of selection and decision values of different objects are given; therefore the optimal decision making is achieved according to the decision values. Finally, the steps of this method are concluded, and one example is given to explain the application of the method. Xiaoguo Chen, Hong Du, and Yue Yang Copyright © 2014 Xiaoguo Chen et al. All rights reserved. SCondi: A Smart Context Distribution Framework Based on a Messaging Service for the Internet of Things Tue, 26 Aug 2014 11:12:00 +0000 When developing IoT (Internet of Things) applications, context distribution is a key feature to support effective delivery of related contextual data obtained from things to all interested entities. With the advent of the IoT era, multiple billion devices can generate huge amounts of data that might be used in IoT applications. In this paper, we present a context distribution framework named SCondi utilizing the messaging service which supports MQTT—an OASIS standard IoT messaging protocol. SCondi provides the notion of context channel as a core feature to support efficient and reliable mechanism for distributing huge context information in the IoT environment. The context channel provides a pluggable filter mechanism that supports effective extraction, tailoring, authentication, and security of information. Jongmoon Park and Myung-Joon Lee Copyright © 2014 Jongmoon Park and Myung-Joon Lee. All rights reserved. A Nonlinear Weakly Singular Retarded Henry-Gronwall Type Integral Inequality and Its Application Tue, 26 Aug 2014 10:39:02 +0000 We establish a class of new nonlinear retarded weakly singular integral inequality. Under several practical assumptions, the inequality is solved by adopting novel analysis techniques, and explicit bounds for the unknown functions are given clearly. An application of our result to the fractional differential equations with delay is shown at the end of the paper. Yuanhua Lin, Shanhe Wu, and Wu-Sheng Wang Copyright © 2014 Yuanhua Lin et al. All rights reserved. A New Version of the Accelerated Overrelaxation Iterative Method Tue, 26 Aug 2014 10:12:23 +0000 Hadjidimos (1978) proposed a classical accelerated overrelaxation (AOR) iterative method to solve the system of linear equations, and discussed its convergence under the conditions that the coefficient matrices are irreducible diagonal dominant, L-matrices, and consistently orders matrices. In this paper, a new version of the AOR method is presented. Some convergence results are derived when the coefficient matrices are irreducible diagonal dominant, H-matrices, symmetric positive definite matrices, and L-matrices. A relational graph for the new AOR method and the original AOR method is presented. Finally, a numerical example is presented to illustrate the efficiency of the proposed method. Shi-Liang Wu and Yu-Jun Liu Copyright © 2014 Shi-Liang Wu and Yu-Jun Liu. All rights reserved. Existence of Periodic Fixed Point Theorems in the Setting of Generalized Quasi-Metric Spaces Tue, 26 Aug 2014 08:43:37 +0000 We introduce the notions of -weaker Meir-Keeler contractive mappings and -stronger Meir-Keeler contractive mappings. We discuss the existence of periodic points in the setting of generalized quasi-metric spaces. Our results improve, extend, and generalize several results in the literature. Chi-Ming Chen, Erdal Karapınar, and Vladimir Rakočević Copyright © 2014 Chi-Ming Chen et al. All rights reserved. Seismic Waveform Inversion Using the Finite-Difference Contrast Source Inversion Method Tue, 26 Aug 2014 06:54:41 +0000 This paper extends the finite-difference contrast source inversion method to reconstruct the mass density for two-dimensional elastic wave inversion in the framework of the full-waveform inversion. The contrast source inversion method is a nonlinear iterative method that alternatively reconstructs contrast sources and contrast function. One of the most outstanding advantages of this inversion method is the highly computational efficiency, since it does not need to simulate a full forward problem for each inversion iteration. Another attractive feature of the inversion method is that it is of strong capability in dealing with nonlinear inverse problems in an inhomogeneous background medium, because a finite-difference operator is used to represent the differential operator governing the two-dimensional elastic wave propagation. Additionally, the techniques of a multiplicative regularization and a sequential multifrequency inversion are employed to enhance the quality of reconstructions for this inversion method. Numerical reconstruction results show that the inversion method has an excellent performance for reconstructing the objects embedded inside a homogeneous or an inhomogeneous background medium. Bo Han, Qinglong He, Yong Chen, and Yixin Dou Copyright © 2014 Bo Han et al. All rights reserved. Modeling of Water Quality, Quantity, and Sustainability Tue, 26 Aug 2014 05:54:23 +0000 Yongping Li, Guohe Huang, Yuefei Huang, and Xiaosheng Qin Copyright © 2014 Yongping Li et al. All rights reserved. A Postverification Method for Solving Forced Duffing Oscillator Problems without Prescribed Periods Tue, 26 Aug 2014 00:00:00 +0000 This paper proposes a postverification method (PVM) for solving forced Duffing oscillator problems without prescribed periods. Comprising a postverification procedure and small random perturbation, the proposed PVM improves the sensitivity of the convergence of Newton’s iteration. Numerical simulations revealed that the PVM is more accurate and robust than Kubíček’s approach. We applied the PVM to previous research on bifurcation problems. Hong-Yen Lin, Chien-Chang Yen, Kuo-Ching Jen, and Kang C. Jea Copyright © 2014 Hong-Yen Lin et al. All rights reserved. Dynamical Behavior for a Food-Chain Model with Impulsive Harvest and Digest Delay Mon, 25 Aug 2014 09:17:43 +0000 We investigate the dynamics of a food-chain system with digest delay and periodic harvesting for the prey. By using the comparison theorem, small amplitude skills in the impulsive differential equation, and a special qualitative analysis method in the delay differential equation, we prove that there exists a predators-eradication periodic solution which is globally attractive and show that the pest population can be controlled under the economic threshold level and the system can be uniformly permanent when the harvest period is long enough or the harvesting rate is not too large. Furthermore, we perform a series of numerical simulations to display the effects of the digest delay and periodic harvesting on the dynamic behavior of the food-chain system. Baodan Tian, Shouming Zhong, Ning Chen, and Yanhong Qiu Copyright © 2014 Baodan Tian et al. All rights reserved. Symmetry Reductions of (2 + 1)-Dimensional CDGKS Equation and Its Reduced Lax Pairs Mon, 25 Aug 2014 08:32:43 +0000 With the aid of symbolic computation, we obtain the symmetry transformations of the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation by Lou’s direct method which is based on Lax pairs. Moreover, we use the classical Lie group method to seek the symmetry groups of both the CDGKS equation and its Lax pair and then reduce them by the obtained symmetries. In particular, we consider the reductions of the Lax pair completely. As a result, three reduced (1 + 1)-dimensional equations with their new Lax pairs are presented and some group-invariant solutions of the equation are given. Na Lv, Xuegang Yuan, and Jinzhi Wang Copyright © 2014 Na Lv et al. All rights reserved. Double Girder Bridge Crane with Double Cycling: Scheduling Strategy and Performance Evaluation Mon, 25 Aug 2014 00:00:00 +0000 This paper introduces a novel quay crane design, double girder bridge crane (DGBC). DGBC is capable of handling containers of two adjacent bays simultaneously, avoiding crane collisions, saving travelling and reposition cost, and eventually improving terminal efficiency. This problem is formulated as a resource-constrained project scheduling with objective to minimize the maximum completion time. A two-stage heuristic algorithm is proposed in which an operating sequences on each bay is obtained by double cycling, and the integrated timetable for both bays is constructed by solving resource conflicts using the proposed minimum cost strategy. We examine effectiveness and performance of applying DGBC with double cycling. A case study is presented to illustrate how DGBC works with the two-stage method. Three extreme cases with respective conflict types are investigated to develop the performance bounds of DGBC with double cycling. The results show that DGBC can significantly improve terminal productivity, and outperforms single girder crane in both makespan and the lift operation percentage. The highest DGBC efficiency does not require maximum double cycles in two bay schedules; rather the integrated timetable for two bays is the main contribution to the DGBC performance as it yields better cooperation between two spreaders and the driver. Dandan Wang, Anne Goodchild, Xiaoping Li, and Zun Wang Copyright © 2014 Dandan Wang et al. All rights reserved. Trapezoidal Intuitionistic Fuzzy Multiattribute Decision Making Method Based on Cumulative Prospect Theory and Dempster-Shafer Theory Sun, 24 Aug 2014 11:09:28 +0000 With respect to decision making problems under uncertainty, a trapezoidal intuitionistic fuzzy multiattribute decision making method based on cumulative prospect theory and Dempster-Shafer theory is developed. The proposed method reflects behavioral characteristics of decision makers, information fuzziness under uncertainty, and uncertain attribute weight information. Firstly, distance measurement and comparison rule of trapezoidal intuitionistic fuzzy numbers are used to derive value function under trapezoidal intuitionistic fuzzy environment. Secondly, the value function and decision weight function are used to calculate prospect values of attributes for each alternative. Then considering uncertain attribute weight information, Dempster-Shafer theory is used to aggregate prospect values for each alternative, and overall prospect values are obtained and thus the alternatives are sorted consequently. Finally, an illustrative example shows the feasibility of the proposed method. Xihua Li, Fuqiang Wang, and Xiaohong Chen Copyright © 2014 Xihua Li et al. All rights reserved. The Burgers Equation for a New Continuum Model with Consideration of Driver’s Forecast Effect Sun, 24 Aug 2014 09:43:22 +0000 A new continuum model with consideration of driver’s forecast effect is obtained to study the density wave problem and the stop-and-go phenomena. The stability condition of the new model is derived by using linear analysis. The triangular shock wave, one type of density wave, which is determined by Burgers equation in the stable region, is discussed in great detail with reductive perturbation method. The local cluster appears when we perform the numerical simulations for the new model. It also proves that the driver’s forecast effect has the positive effect of reducing the local cluster. Lei Yu and Bingchang Zhou Copyright © 2014 Lei Yu and Bingchang Zhou. All rights reserved. Bifurcation of Travelling Wave Solutions of the Generalized Zakharov Equation Sun, 24 Aug 2014 08:48:58 +0000 By using bifurcation theory of planar ordinary differential equations all different bounded travelling wave solutions of the generalized Zakharov equation are classified in to different parametric regions. In each of these parametric regions the exact explicit parametric representation of all solitary, kink (antikink), and periodic wave solutions as well as their numerical simulation and their corresponding phase portraits are obtained. Masoud Mosaddeghi Copyright © 2014 Masoud Mosaddeghi. All rights reserved. Nonlinear Torsional Vibration Dynamics Behaviors of Rolling Mill’s Multi-DOF Main Drive System under Parametric Excitation Sun, 24 Aug 2014 07:39:27 +0000 Considering the influence caused by joint angle, nonlinear damping, and nonlinear rigidity, the nonlinear torsional vibration dynamical modeling of the multi-DOF rolling mill’s main drive system is established. To analyze the coupled equations by analytic method, the equations are decoupled by transforming them into principal coordinates. The amplitude-frequency characteristic equations are obtained by multiscale method. Furthermore, numerical example based on the 1780 rolling mill’s main drive system of some Steel Co. is given to illustrate the effects of the resonance on the response of the system. The relationship between amplitude and frequency varies according to the parameters changes of nonlinear stiffness, nonlinear friction damping, torque disturbance, and joint angle. During the rolling process, the limited joint angles range is obtained and the variation rules of the joint angle caused by the nonlinear damping, nonlinear stiffness, and the disturbance torque are gained. The results present that the rolling mill can work more stably with the joint angle at a range from 2° to 8° by controlling the value of parameters. The research results provide a theoretical basis and reference for analyzing torsional vibration of rolling mill’s transmission system caused by joint angle. Dongying Han, Peiming Shi, and Kewei Xia Copyright © 2014 Dongying Han et al. All rights reserved. Weighted Cache Location Problem with Identical Servers Sun, 24 Aug 2014 07:34:15 +0000 This paper extends the well-known -CLP with one server to -CLP with identical servers, denoted by -CLP. We propose the closest server orienting protocol (CSOP), under which every client connects to the closest server to itself via a shortest route on given network. We abbreviate -CLP under CSOP to -CSOP CLP and investigate that -CSOP CLP on a general network is equivalent to that on a forest and further to multiple CLPs on trees. The case of is the focus of this paper. We first devise an improved -time parallel exact algorithm for -CLP on a tree and then present a parallel exact algorithm with at most time in the worst case for -CSOP CLP on a general network. Furthermore, we extend the idea of parallel algorithm to the cases of to obtain a worst-case -time exact algorithm. At the end of the paper, we first give an example to illustrate our algorithms and then make a series of numerical experiments to compare the running times of our algorithms. Hongfa Wang and Wei Ding Copyright © 2014 Hongfa Wang and Wei Ding. All rights reserved. A Prediction System Using a P2P Overlay Network for a Bus Arrival System Sun, 24 Aug 2014 07:21:32 +0000 Along with the evolution of times and the surge of metropolitan populations, government agencies often promote the construction of public transport. Unlike rail transportation or rapid transit systems, it is often difficult to estimate the vehicle arrival times at each station in a bus transportation system due to metropolitan transportation congestion. Traffic status is often monitored using wireless sensor networks (WSNs). However, WSNs are always separated from one another spatially. Recent studies have considered the connection of multiple sensor networks. This study considers a combination view of peer-to-peer (P2P) overlay networks and WSN architecture to predict bus arrival times. Each bus station, which is also a P2P overlay peer, is connected in a P2P overlay network. A sensor installed in each bus can receive data via peers to obtain the moving speed of a bus. Then, each peer can exchange its data to predict bus arrival times at bus stations. This method can considerably increase the accuracy with which bus arrival times can be predicted and can provide traffic status with high precision. Furthermore, these data can also be used to plan new bus routes according to the information gathered. Ssu-Hsuan Lu and Yu-Wei Chan Copyright © 2014 Ssu-Hsuan Lu and Yu-Wei Chan. All rights reserved. Joint Pricing and Purchasing Decisions for the Dual-Channel Newsvendor Model with Partial Information Sun, 24 Aug 2014 06:34:14 +0000 We investigate a joint pricing and purchasing problem for the dual-channel newsvendor model with the assumption that only the mean and variance of the demand are known. The newsvendor in our model simultaneously distributes a single product through traditional retail and Internet. A robust optimization approach that maximizes the worst-case profit is adapted under the aforementioned conditions to model demand uncertainty and linear clearing functions that characterize the relationship between demand and prices. We obtain a close-form expression for the robust optimal policy. Illustrative simulations and numerical experiments show the effects of several parameters on the optimal policy and on newsvendor performance. Finally, we determine that the gap between newsvendor performance under demand certainty and uncertainty is minimal, which shows that the robust approach can significantly improve performance. Jixiang Zhou, Yong Wang, and Xiaoming Yan Copyright © 2014 Jixiang Zhou et al. All rights reserved. The Coupled Nonlinear Schrödinger Equations Describing Power and Phase for Modeling Phase-Sensitive Parametric Amplification in Silicon Waveguides Sun, 24 Aug 2014 05:44:31 +0000 The coupled nonlinear Schrödinger (NLS) equations describing power and phase of the optical waves are used to model phase-sensitive (PS) parametric amplification in a width-modulated silicon-on-insulator (SOI) channel waveguide. Through solving the coupled NLS equations by the split-step Fourier and Runge-Kutta integration methods, the numerical results show that the coupled NLS equations can perfectly describe and character the PS amplification process in silicon waveguides. Xuefeng Li, Zhaolu Wang, and Hongjun Liu Copyright © 2014 Xuefeng Li et al. All rights reserved. Three-Dimensional Flow and Heat Transfer Past a Permeable Exponentially Stretching/Shrinking Sheet in a Nanofluid Sun, 24 Aug 2014 00:00:00 +0000 The three-dimensional flow and heat transfer of a nanofluid over a stretching/shrinking sheet is investigated. Numerical results are obtained using bvp4c in MATLAB. The results show nonunique solutions for the shrinking case. The effects of the stretching/shrinking parameter, suction parameter, Brownian motion parameter, thermophoresis parameter, and Lewis number on the local skin friction coefficient and the local Nusselt number are studied. Suction increases the solution domain. Furthermore, as the sheet is shrunk in the -direction, suction increases the skin friction coefficient in the same direction while decreasing the skin friction coefficient in the -direction. The local Nusselt number is consistently lower for higher values of thermophoresis parameter and Lewis number. On the other hand, the local Nusselt number increases as the Brownian motion parameter increases. Syahira Mansur, Anuar Ishak, and Ioan Pop Copyright © 2014 Syahira Mansur et al. All rights reserved. Positive Solutions for a Nonlinear Higher Order Differential System with Coupled Integral Boundary Conditions Sun, 24 Aug 2014 00:00:00 +0000 We investigate the existence of positive solutions for a nonlinear higher order differential system, where the differential system is coupled not only in the differential system but also through the boundary conditions. By constructing a special cone and using the fixed point theorem of cone expansion and compression of norm type, the existence of single and multiple positive solutions is established. As an application, we give some examples to demonstrate our results. Yaohong Li and Haiyan Zhang Copyright © 2014 Yaohong Li and Haiyan Zhang. All rights reserved.