Complex Boundary Value Problems of Nonlinear Differential Equations 2014

Xinguang Zhang, Yong Hong Wu, Dragoș-Pãtru Covei, Xinan Hao
2014 Abstract and Applied Analysis  
Complex boundary value problems of nonlinear differential equations have merged as an interesting and fascinating branch of applied mathematics and pure mathematics with a wide range of applications in industry, economics, biology, physics, chemistry, social, and pure and applied sciences. The aim of this special issue is to present new approaches and theories for solving complex boundary value problems of nonlinear differential equations arising in the relative field. This special issue
more » ... s 33 high quality peer-reviewed papers which deal with different aspects of nonlinear differential equations. These papers contain some new, novel, and innovative techniques and ideas. We hope that all the papers published in this special issue can motivate and foster further scientific works and development of the research in the area of theory and applications of nonlinear differential equations. Here, we are very grateful to all the authors and reviewers of the papers for their excellent contributions. In the following we summarize briefly the content of the special issue. In the paper titled "A new proof of central limit theorem for i.i.d. random variables, " the authors give a new proof of central limit theorem for independent identically distributed random variables by using the viscosity solution theory of partial differential equation. In the paper titled "Exit problems for jump processes having double-sided jumps with rational Laplace transforms, " the authors consider the two-sided first exit problem for a jump process having jumps with rational Laplace transform and derive the joint distribution of the first passage time to two-sided barriers and the value of process at the first passage time. As applications, the explicit expressions of the dividend formulae for barrier strategy and threshold strategy are presented. In the paper titled "Dynamic analysis and chaos of the 4D fractional-order power system, " the authors study the dynamic analysis of a fractional-order power system with parameter Q1 and firstly report about bifurcation analysis of the fractional order power system. In this work, the authors also discuss the dynamic analysis with different fractional order and different parameters and establish its numerical simulations which are provided to demonstrate the feasibility and efficacy of the analysis. In the paper titled "Pricing of American put option under a jump diffusion process with stochastic volatility in an incomplete market, " the authors study the pricing of American options in an incomplete market in which the dynamics of the underlying risky asset are driven by a jump diffusion process with stochastic volatility. By employing a risk-minimization criterion, the authors obtain the Radon-Nikodym derivative for the minimal martingale measure and consequently a linear complementarity problem for American option price. An iterative method is then established to solve the LCP Abstract and Applied Analysis problem for American put option price. The numerical results show that the model and numerical scheme are robust in capturing the feature of incomplete finance market, particularly the influence of market volatility on the price of American options. In the paper titled "A two-grid finite element method for a second-order nonlinear hyperbolic equation, " based on two conforming piecewise linear finite element spaces on one coarse grid with grid size H and one fine grid with grid size h, respectively, the authors consider the two-grid finite element discretization techniques for the second-order nonlinear hyperbolic problems. With the proposed techniques, solving the nonlinear problems on the fine-grid space is reduced to solving a linear system on the fine-grid space and a nonlinear system on a much smaller space. In the paper titled "Positive solutions for systems of nonlinear higher order differential equations with integral boundary conditions, " by constructing some general type conditions and using fixed point theorem of cone, this paper investigates the existence of at least one and at least two positive solutions for systems of nonlinear higher order differential equations with integral boundary conditions. As application, some examples are given. In the paper titled "The existence of solutions for four-point coupled boundary value problems of fractional differential equations at resonance," a four-point coupled boundary value problem of fractional differential equations is studied. Based on Mawhin's coincidence degree theory, some existence theorems are obtained in the case of resonance. In the paper titled "Boundary stabilization of a semilinear wave equation with variable coefficients under the timevarying and nonlinear feedback," the authors investigate the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term are obtained. In the paper titled "Existence and uniqueness of positive solutions for a fractional switched system, " the authors discuss the existence and uniqueness of positive solutions for a class of fractional switched system; the main results are based on a fixed point theorem of a sum operator and contraction mapping principle. Furthermore, two examples are also given to illustrate the results. In the paper titled "On the study of global solutions for a nonlinear equation, " the well-posedness of global strong solutions for a nonlinear partial differential equation including the Novikov equation is established provided that its initial value satisfies a sign condition and further conditions. If the mean function satisfies the sign condition, it is proved that there exists at least one global weak solution in the sense of distribution. In the paper titled "Positive solutions for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions, " the author's objective is to study the existence of positive solution for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions by using known Krasnosel'skii's fixed point theorem. Some sufficient conditions that guarantee the existence of at least one positive solution for eigenvalues sufficiently small and sufficiently large are established. In the paper titled "Multiple results to some biharmonic problems, " the authors study a nonlinear elliptic problem defined in a bounded domain involving biharmonic operator together with an asymptotically linear term. The results of at least three nontrivial solutions are established by using the topological 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 damping terms by using multiplier method and the estimation techniques of the energy and shows 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. In the paper titled "The local stability of solutions for a nonlinear equation, " 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 by assuming that the initial value only lies in a suitable space. In the paper titled "The uniqueness of solution for a class of fractional order nonlinear systems with p-Laplacian operator, " the authors are concerned with the uniqueness of solutions for a class of p-Laplacian fractional order nonlinear systems with nonlocal boundary conditions. Based on some properties of the p-Laplacian operator, the criterion of uniqueness for solutions is established. In the paper titled "Exponential stabilization of impulsive switched systems with time delays using guaranteed cost control, " the authors investigate the stabilization problem for impulsive switched systems with time delays. First, exponential stability criteriaof the delayed impulsive switched systems are established by use of the Lyapunov-Krasovskii functional method. Based on these results, sufficient conditions for the existence of a guaranteed cost control are also given. Subject to these sufficient conditions, the closed-loop impulsive switched system under the guaranteed cost control law will be exponentially stable with a guaranteed cost value. In the paper titled "Delta-nabla type maximum principles for second-order dynamic equations on time scales and applications, " some delta-nabla type maximum principles for second-order dynamic equations on time scales are proved. By using these maximum principles, the uniqueness theorems of the solutions, the approximation theorems of the solutions, the existence theorem, and construction techniques of the lower and upper solutions for second-order linear and nonlinear initial value problems and boundary value problems on time scales are proved, the oscillation of second-order mixed delta-nabla differential equations is discussed, and some maximum principles for second-order mixed forward and backward difference dynamic system are proved. In the paper titled "Some existence results of positive solution to second-order boundary value problems, " the authors study the existence of positive and monotone solution to Abstract and Applied Analysis 3
doi:10.1155/2014/496350 fatcat:vohr34wmsndohhpsun74csnarm