一类多方渗流方程正解的存在性和爆破性

该文研究了一类具有非局部Neumann边界条件和非线性吸收项的多方渗流方程解的全局存在性和爆破情况.首先针对所研究方程定义了其上下解,并建立和证明了比较原理;然后通过构造函数以及利用微分不等式、特征值特征函数、常微分方程的解和椭圆第二边值的解等方法对方程进行了研究,得到了对于不同取值范围的参数、权函数和初始值时,方程非负解的全局存在性和在有限时间内爆破的充分条件.     

Nonlocal Problems for Differential Inclusions in Hilbert Spaces

An existence theorem for differential inclusions in Hilbert spaces with nonlocal conditions is proved. Periodic, anti-periodic, mean value and multipoint conditions are included in this study. The investigation is based on a combination of the approximation solvability method with Hartman-type inequalities. A feedback control problem associated to a first order partial differential equation completes this discussion.     

半线性方程ROBIN问题的角层解(英文)

本文讨论了一类半线性方程Robin边值问题.利用微分不等式理论,研究了边值问题角层解的存在性和渐近性态.     

Differential Games of Mixed Type with Control and Stopping Times

We study a zero sum differential game of mixed type where each player uses both control and stopping times. Under certain conditions we show that the value function for this problem exists and is the unique viscosity solution of the corresponding variational inequalities. We also show the existence of saddle point equilibrium for a special case of differential game.      

THE CORNER LAYER SOLUTION TO ROBIN PROBLEM FOR REACTION DIFFUSION EQUATION

A class of Robin boundary value problem for reaction diffusion equation is considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of the corner layer solution to the initial boundary value problem are studied.     

一类具有摄动边界的弱非线性反应扩散方程

研究了具有摄动边界的非线性反应扩散方程的问题.在适当的条件下,利用微分不等式理论,讨论了问题的渐近解.     

On the stability and convergence of nonlocal difference schemes

We study nonlocal boundary value problems of the first and second kind for the heat equation with variable coefficients in the differential and finite-difference settings. By using the method of energy inequalities, we obtain a priori estimates for the corresponding differential and finite-difference problems.     

Singularly perturbed solution for third order nonlinear equations with two parameters

A class of singularly perturbed boundary value problems for nonlinear equation of the third order with two parameters is considered. Under suitable conditions, using the theory of differential inequalities the existence and asymptotic behavior of the solution for boundary value problem are studied.     

具有奇性方程的非线性奇摄动问题的正解(英文)

本文讨论了一类具有奇性方程的奇摄动初值问题.在适当条件下,利用微分不等式理论,研究了初值问题解的存在性及其渐近性态,并且得到了具有初始层的一致有效解的渐近展开式.     

一类具有边界摄动的非线性泛函椭圆型方程奇摄动问题

研究了具有边界摄动的非线性泛函椭圆型方程奇摄动边值问题．在适当的条件下．利用伸长变量、微分不等式理论，讨论了问题解的渐近性态和原问题解的存在唯一性。     

Nonlocal boundary value problems in differential and difference settings

We study nonlocal boundary value problems of the first and second kind for the heat equation with variable coefficients in the differential and difference settings. By the method of energy inequalities, we find a priori estimates for the differential and difference problems.     

Generalized Lyapunov inequalities involving critical Sobolev exponents

We consider Lyapunov-type inequalities generalizing the famous inequality that gives a necessary condition for the existence of solutions to a boundary value problem for a second order ordinary differential equation. For certain critical cases, when the inequalities are strict, we study the asymptotic behavior of minimizing sequences.     

A Nonlinear Singularly Perturbed Problem for Reaction Diffusion Equations with Boundary Perturbation

A nonlinear singularly perturbed problems for reaction diffusion equation with boundary perturbation is considered. Under suitable conditions, the asymptotic behavior of solution for the initial boundary value problems of reaction diffusion equations is studied using the theory of differential inequalities.     

New optimality conditions for average-payoff continuous-time Markov games in Polish spaces

This paper concerns two-person zero-sum games for a class of average-payoff continuous-time Markov processes in Polish spaces.The underlying processes are determined by transition rates that are allowed to be unbounded,and the payoff function may have neither upper nor lower bounds.We use two optimality inequalities to replace the so-called optimality equation in the previous literature.Under more general conditions,these optimality inequalities yield the existence of the value of the game and of a pair of ...     

Blow-up solutions of second-order differential inequalities with a nonlinear memory term

We consider the Cauchy problem for second-order nonlinear ordinary differential inequalities with a nonlinear memory term. We obtain blow-up results under some conditions on the initial data. We also give an application to a semilinear hyperbolic equation in a bounded domain.     

The singular perturbation of boundary value problem for third-order nonlinear vector integro-differential equation and its application

In this paper, the singular perturbation of boundary value problem to a class of third-order nonlinear vector integro-differential equation is studied. Using the method of differential inequalities, under certain conditions, the existence of perturbed solution is proved, the uniformly valid asymptotic expansion for arbitrary order and the estimation of remainder term are given. Finally, the results are applied to study singularly perturbed boundary value problem to a nonlinear vector fourth-order differential equation. The existence of solution and its asymptotic estimation can be obtained conveniently.     

Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks

We present a method to solve boundary value problems using artificial neural networks (ANN). A trial solution of the differential equation is written as a feed-forward neural network containing adjustable parameters (the weights and biases). From the differential equation and its boundary conditions we prepare the energy function which is used in the back-propagation method with momentum term to update the network parameters. We improved energy function of ANN which is derived from Schrodinger equation and the boundary conditions. With this improvement of energy function we can use unsupervised training method in the ANN for solving the equation. Unsupervised training aims to minimize a non-negative energy function. We used the ANN method to solve Schrodinger equation for few quantum systems. Eigenfunctions and energy eigenvalues are calculated. Our numerical results are in agreement with their corresponding analytical solution and show the efficiency of ANN method for solving eigenvalue problems.     

Exact solution to the periodic boundary value problem for a first-order linear fuzzy differential equation with impulses

Using the expression of the exact solution to a periodic boundary value problem for an impulsive first-order linear differential equation, we consider an extension to the fuzzy case and prove the existence and uniqueness of solution for a first-order linear fuzzy differential equation with impulses subject to boundary value conditions. We obtain the explicit solution by calculating the solutions on each level set and justify that the parametric functions obtained define a proper fuzzy function. Our results prove that the solution of the fuzzy differential equation of interest is determined, under the appropriate conditions, by the same Green’s function obtained for the real case. Thus, the results proved extend some theorems given for ordinary differential equations.     

Existence of Value and Saddle Point in Infinite-Dimensional Differential Games

We study two-player zero-sum differential games of finite duration in a Hilbert space. Following the Berkovitz notion of strategies, we prove the existence of value and saddle-point equilibrium. We characterize the value as the unique viscosity solution of the associated Hamilton–Jacobi–Isaacs equation using dynamic programming inequalities.     

A CLASS OF NONLINEAR NONLOCAL SINGULARLY PERTURBED PROBLEMS FOR ELLIPTIC EQUATION WITH BOUNDARY PERTURBATION

A class of nonlinear nonlocal singularly perturbed boundary value problems for elliptic equation with boundary perturbation is considered. Under suitable conditions, firstly, the outer solution of the original problem is obtained, secondly, using the stretched variable, the composing expansion method and the expanding theory of power series, the boundary layer is constructed, finally, using the theory of differential inequalities the asymptotic behavior of solution for the boundary value problems is studied and educing some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation is discussed.