Editorial Complex Boundary Value Problems of …downloads.hindawi.com/journals/aaa/2014/496350.pdf · Editorial Complex Boundary Value Problems of Nonlinear Differential Equations

EditorialComplex Boundary Value Problems of NonlinearDifferential Equations 2014

Xinguang Zhang,1 Yong Hong Wu,2 Dragos,-Pãtru Covei,3 and Xinan Hao4

1 School of Mathematical and Informational Sciences, Yantai University, Yantai, Shandong 264005, China2Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia3 Department of Applied Mathematics, The Bucharest University of Economic Studies, Piata Romana, 1st District, 22,010374 Bucharest, Romania

4 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Correspondence should be addressed to Xinguang Zhang; [email protected]

Received 21 July 2014; Accepted 21 July 2014; Published 14 August 2014

Copyright © 2014 Xinguang Zhang et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Complex boundary value problems of nonlinear differentialequations have merged as an interesting and fascinatingbranch of applied mathematics and pure mathematics witha wide range of applications in industry, economics, biology,physics, chemistry, social, and pure and applied sciences.Theaim of this special issue is to present new approaches andtheories for solving complex boundary value problems ofnonlinear differential equations arising in the relative field.This special issue includes 33 high quality peer-reviewedpapers which deal with different aspects of nonlinear differ-ential equations. These papers contain some new, novel, andinnovative techniques and ideas. We hope that all the paperspublished in this special issue canmotivate and foster furtherscientific works and development of the research in the areaof theory and applications of nonlinear differential equations.

Here, we are very grateful to all the authors and reviewersof the papers for their excellent contributions.

In the following we summarize briefly the content of thespecial issue.

In the paper titled “A new proof of central limit theoremfor i.i.d. random variables,” the authors give a new proof ofcentral limit theorem for independent identically distributedrandom variables by using the viscosity solution theory ofpartial differential equation.

In the paper titled “Exit problems for jump processeshaving double-sided jumps with rational Laplace transforms,”

the authors consider the two-sided first exit problem for ajump process having jumps with rational Laplace transformand derive the joint distribution of the first passage timeto two-sided barriers and the value of process at the firstpassage time. As applications, the explicit expressions of thedividend formulae for barrier strategy and threshold strategyare presented.

In the paper titled “Dynamic analysis and chaos of the4D fractional-order power system,” the authors study thedynamic analysis of a fractional-order power system withparameter Q1 and firstly report about bifurcation analysis ofthe fractional order power system. In this work, the authorsalso discuss the dynamic analysis with different fractionalorder and different parameters and establish its numericalsimulations which are provided to demonstrate the feasibilityand efficacy of the analysis.

In the paper titled “Pricing of American put optionunder a jump diffusion process with stochastic volatility in anincompletemarket,” the authors study the pricing of Americanoptions in an incompletemarket inwhich the dynamics of theunderlying risky asset are driven by a jump diffusion processwith stochastic volatility. By employing a risk-minimizationcriterion, the authors obtain the Radon-Nikodym derivativefor the minimal martingale measure and consequently alinear complementarity problem for American option price.An iterative method is then established to solve the LCP

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 496350, 4 pageshttp://dx.doi.org/10.1155/2014/496350

2 Abstract and Applied Analysis

problem forAmerican put option price.Thenumerical resultsshow that themodel and numerical scheme are robust in cap-turing the feature of incomplete finance market, particularlythe influence of market volatility on the price of Americanoptions.

In the paper titled “A two-grid finite element method fora second-order nonlinear hyperbolic equation,” based on twoconforming piecewise linear finite element spaces on onecoarse grid with grid sizeH and one fine grid with grid size h,respectively, the authors consider the two-grid finite elementdiscretization techniques for the second-order nonlinearhyperbolic problems. With the proposed techniques, solvingthe nonlinear problems on the fine-grid space is reduced tosolving a linear system on the fine-grid space and a nonlinearsystem on a much smaller space.

In the paper titled “Positive solutions for systems of nonlin-ear higher order differential equations with integral boundaryconditions,” by constructing some general type conditions andusing fixed point theorem of cone, this paper investigates theexistence of at least one and at least two positive solutions forsystems of nonlinear higher order differential equations withintegral boundary conditions. As application, some examplesare given.

In the paper titled “The existence of solutions for four-pointcoupled boundary value problems of fractional differentialequations at resonance,” a four-point coupled boundary valueproblem of fractional differential equations is studied. Basedon Mawhin’s coincidence degree theory, some existencetheorems are obtained in the case of resonance.

In the paper titled “Boundary stabilization of a semilinearwave equation with variable coefficients under the time-varying and nonlinear feedback,” the authors investigate theboundary stabilization of a semilinear wave equation withvariable coefficients under the time-varying and nonlinearfeedback. By the Riemannian geometrymethods, the stabilityresults of the system under suitable assumptions of thebound of the time-varying term and the nonlinearity of thenonlinear term are obtained.

In the paper titled “Existence and uniqueness of positivesolutions for a fractional switched system,” the authors discussthe existence and uniqueness of positive solutions for a classof fractional switched system; the main results are based ona fixed point theorem of a sum operator and contractionmapping principle. Furthermore, two examples are also givento illustrate the results.

In the paper titled “On the study of global solutions for anonlinear equation,” the well-posedness of global strong solu-tions for a nonlinear partial differential equation includingthe Novikov equation is established provided that its initialvalue satisfies a sign condition and further conditions. If themean function satisfies the sign condition, it is proved thatthere exists at least one global weak solution in the sense ofdistribution.

In the paper titled “Positive solutions for the eigenvalueproblem of semipositone fractional order differential equationwith multipoint boundary conditions,” the author’s objectiveis to study the existence of positive solution for the eigen-value problem of semipositone fractional order differentialequation with multipoint boundary conditions by using

known Krasnosel’skii’s fixed point theorem. Some sufficientconditions that guarantee the existence of at least one positivesolution for eigenvalues sufficiently small and sufficientlylarge are established.

In the paper titled “Multiple results to some biharmonicproblems,” the authors study a nonlinear elliptic problemdefined in a bounded domain involving biharmonic operatortogether with an asymptotically linear term. The results of atleast three nontrivial solutions are established by using thetopological degree theory and the critical groups.

In the paper titled “Stability of a class of coupled systems,”the author deals with a class of coupled systems with damp-ing terms by using multiplier method and the estimationtechniques of the energy and shows that even if the kernelfunction is nonincreasing and integrable without additionalconditions, the energy of the system decays also to zero in agood rate.

In the paper titled “The local stability of solutions fora nonlinear equation,” the approach of Kruzkov’s deviceof doubling the variables is applied to establish the localstability of strong solutions for a nonlinear partial differentialequation by assuming that the initial value only lies in asuitable space.

In the paper titled “The uniqueness of solution for a class offractional order nonlinear systems with p-Laplacian operator,”the authors are concerned with the uniqueness of solutionsfor a class of p-Laplacian fractional order nonlinear systemswith nonlocal boundary conditions. Based on some proper-ties of the p-Laplacian operator, the criterion of uniquenessfor solutions is established.

In the paper titled “Exponential stabilization of impulsiveswitched systems with time delays using guaranteed costcontrol,” the authors investigate the stabilization problem forimpulsive switched systems with time delays. First, exponen-tial stability criteriaof the delayed impulsive switched systemsare established by use of the Lyapunov-Krasovskii functionalmethod. Based on these results, sufficient conditions for theexistence of a guaranteed cost control are also given. Subjectto these sufficient conditions, the closed-loop impulsiveswitched system under the guaranteed cost control law willbe exponentially stable with a guaranteed cost value.

In the paper titled “Delta-nabla type maximum princi-ples for second-order dynamic equations on time scales andapplications,” some delta-nabla type maximum principlesfor second-order dynamic equations on time scales areproved. By using these maximum principles, the uniquenesstheorems of the solutions, the approximation theorems ofthe solutions, the existence theorem, and construction tech-niques of the lower and upper solutions for second-orderlinear and nonlinear initial value problems and boundaryvalue problems on time scales are proved, the oscillationof second-order mixed delta-nabla differential equations isdiscussed, and some maximum principles for second-ordermixed forward and backward difference dynamic system areproved.

In the paper titled “Some existence results of positive solu-tion to second-order boundary value problems,” the authorsstudy the existence of positive and monotone solution to

Abstract and Applied Analysis 3

a class of boundary value problems by using the fixed pointtheorem of cone expansion and compression of functionaltype by Avery, Henderson, and O’Regan. Finally, four exam-ples are provided to demonstrate the availability of the mainresults.

In the paper titled “Crank-Nicolson fully discrete H1-Galerkin mixed finite element approximation of one nonlin-ear integrodifferential model,” the author considers a fullydiscrete H1-Galerkin mixed finite element approximation ofone nonlinear integrodifferential model whichoften arisesin mathematical modeling of the process of a magneticfield penetrating into a substance. By adopting the Crank-Nicolson discretization for time derivative, optimal ordera priori error estimates for the unknown function and itsgradient function are presented. A numerical example isgiven to verify the theoretical results.

In the paper titled “Positive solutions of a singular nonlocalfractional order differential system via Schauder’s fixed pointtheorem,” the authors establish the existence of positivesolutions to a class of singular nonlocal fractional orderdifferential system depending on two parameters. The mainmethods are based on Schauder’s fixed point theorem.

In the paper titled “Dynamics of a predator-prey sys-tem with Beddington-DeAngelis functional response anddelays,” the authors consider a predator-prey system withBeddington-DeAngelis functional response and delays, inwhich not only the stage structure on prey but also thedelay due to digestion is considered. First, a sufficient andnecessary condition for the existence of a unique positiveequilibrium by analyzing the corresponding locations of ahyperbolic curve and a line is given. Then, by constructingan appropriate Lyapunov function, the authors prove that thepositive equilibrium is locally asymptotically stable under asufficient condition. Finally, by using comparison theoremand the w-limit set theory, the global asymptotic stabilityof the boundary equilibrium and the positive equilibrium, asufficient condition to assure the global asymptotic stability,respectively, are studied.

In the paper titled “Existence of positive solutions ofLotka-Volterra competition model with nonlinear boundaryconditions,” by using upper and lower solutions method fornonlinear boundary problems, a Lotka-Volterra competitionmodel with nonlinear boundary conditions is considered.First, the authors investigate the existence of positive solu-tions inweak competition case and then prove that the systemhas no positive solutionwhen one of the diffusion coefficientsis sufficiently large.

In the paper titled “Characteristics weak Galerkin finiteelement methods for convection-dominated diffusion prob-lems,” the weak Galerkin finite element method is combinedwith the method of characteristics to treat the convection-diffusion problems on the triangularmesh.The optimal ordererror estimates are derived for the corresponding characteris-tics weak Galerkin finite element procedure. Numerical testsare performed and reported.

In the paper titled “Positive solutions of a nonlinearparabolic partial differential equation,” the authors deal withthe existence and uniqueness of positive solutions to a class

of nonlinear parabolic partial differential equations, by usingsome fixed point theorems for mixed monotone operatorswith perturbation.

In the paper titled “Optimal tracking for a divergent-typeparabolic PDE system in current profile control,” the authorsconsider the optimal tracking problem for a divergent-typeparabolic PDE system, which can be used to model thespatial-temporal evolution of the magnetic diffusion processin a tokamak plasma. With the feed-forward trajectoriesgenerated by numerical optimization, a linear system todescribe the error evolution is derived. The weak variationapproach is then extended to cylindrical coordinates toobtain a Riccati-type PDE for feedback kernel synthesis.Then, a finite difference method is used to give numericalsolutions for the Riccati-type PDE. The obtained feedbackkernel is finally used to simulate the closed-loop system. Thesimulation results demonstrate that the proposed trackingmethod can attenuate the tracking error effectively.

In the paper titled “Solution of time periodic electroosmosisflow with slip boundary,” an exact solution is derived forthe time periodic electroosmotic flow in two-dimensionalstraight channels under slip boundary conditions.

In the paper titled “Solving a class of singularly perturbedpartial differential equation by using the perturbation methodand reproducing kernel method,” the authors give the analyt-ical solution and the series expansion solution of a class ofsingularly perturbed partial differential equationby combin-ing traditional perturbation method and reproducing kernelmethod. The numerical example is studied to demonstratethe accuracy of the present method. Results obtained by themethod indicate the method is simple and effective.

In the paper titled “Transient flows of Newtonian fluidthrough a rectangular microchannel with slip boundary,” theauthors study the transient flow of a Newtonian fluid inrectangular microchannels taking into account boundaryslip. An exact solution is derived by using the separation ofvariables in space and Fourier series expansion in time.

In the paper titled “Hybrid intelligence model based onimage features for the prediction of flotation concentrate grade,”a hybrid intelligent model combining the two kinds ofmodels is proposed in this paper. A slide window scheme isemployed to update the hybrid model in order to improveits adaptability. The industrial practical data testing resultsshow that the performance of the hybrid model is better thaneither of the two single models and it satisfies the accuracyand stability requirements in industrial applications.

In the paper titled “Different approximations to the solu-tion of upper-convected Maxwell fluid over a porous stretch-ing plate,” the authors consider an incompressible magne-tohydrodynamic flow of two-dimensional upper-convectedMaxwell fluid over a porous stretching plate with suctionand injection.The nonlinear partial differential equations arereduced to an ordinary differential equation by the similar-ity transformations and taking into account the boundarylayer approximations. This equation is solved approximatelyby means of the optimal homotopy asymptotic method(OHAM).

In the paper titled “Positive solutions for impulsive differ-ential equationswithmixedmonotonicity and optimal control,”

4 Abstract and Applied Analysis

the authors consider positive solutions and optimal controlproblem for a second-order impulsive differential equationwith mixed monotone terms. Some results on the existenceof an optimal control and its stability are established. Inaddition, related examples are given for illustrations.

In the paper titled “Existence of positive solutions to non-linear fractional boundary value problem with changing signnonlinearity and advanced arguments,” the authors discussthe existence of positive solutions to a class of fractionalboundary value problem with changing sign nonlinearityand advanced arguments; the analysis relies on a nonlinearalternative of Leray-Schauder type. An example is given toillustrate their results.

In the paper titled “A comparative study of marketingchannel multiagent Stackelberg model based on perfect ratio-nality and fairness preference,” the authors study channelconsisting of a manufacturer and two retailers. As a basisfor comparison, the first, multiagent Stackelberg modelhas been structured based on perfect rationality. Further,fairness preference theory will be embedded in marketingchannel multiagent Stackelberg model, and the results showthat, if the retailers have a jealous fairness preference, themanufacturer will reduce the wholesale price, retailers willincrease the effort level, product sales will be increased, andthe total channel utility and manufacturers’ utility will bePareto improvement, but the Pareto improvement of retailers’utility is associated with the interval of jealousy fairnesspreference coefficient. If the retailers have a sympatheticfairness preference, the manufacturer increases wholesaleprice, retailers reduce the effort level, and the total channelutility, manufacturer’s utility, and retailers’ utility are less thanthat of the no fairness preference utility.

Xinguang ZhangYong Hong Wu

Dragos,-Patru CoveiXinan Hao

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Editorial Complex Boundary Value Problems of …downloads.hindawi.com/journals/aaa/2014/496350.pdf · Editorial Complex Boundary Value Problems of Nonlinear Differential Equations