On a class of third-order nonlinear difference equations

This paper studies the boundedness character of the positive solutions of the difference equation

     

On the solutions of a class of difference equations

In this note we investigate the solutions of a class of difference equations and prove that Conjectures 4.8.2, 4.8.3, 5.4.6 and 6.10.3 proposed by M. Kulenovic and G. Ladas in [M. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, Chapman & Hall/CRC Press, 2002] are true.     

On a class of equations of monge-ampere type

We prove that the Dirichlet problem is solvable in a generalized sense for a class of nonlinear elliptic equations and related equations of Monge-Ampère type. Bibliography: 14 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 89–106.     

On a class of impulsive functional-differential equations with nonatomic difference operator

We establish conditions for the existence and uniqueness of the solutions of nonlinear functional-differential equations with impulsive action in a Banach space. The equation under consideration is not solved for the derivative. It is assumed that the characteristic operator pencil corresponding to the linear part of the equation satisfies a constraint of parabolic type in the right half-plane. Applications to partial functional-differential equations not of Kovalevskaya type are considered.     

On difference of two monotone operators

In this paper, we introduce and consider the problem of finding zeroes of difference of two monotone operators in a Hilbert space. Using the resolvent operator technique, we show that this problem is equivalent to the fixed point problem. This equivalence is used to suggest and analyze an iterative method for finding a zero of difference of two monotone operators. We also discuss the convergence of the iterative method under suitable conditions. Our method of proof is very simple as compared with other techniques.     

On the eigenvalue problem of a class of linear partial difference equations

In this paper, the eigenvalue problem of a class of linear partial difference equations is studied. The results concern the existence of eigenvalues, their character (real, positive), as well as the behavior of its eigenfunctions (positivity, oscillation). Moreover a theorem is given concerning the existence of a unique solution of an associated non-homogeneous partial difference equation. The results generalize previously known results for ordinary linear difference equations. The method used is a functional-analytic one, which transforms the eigenvalue problem for the difference equation into the equivalent problem of the eigenvalues of an operator defined on an abstract separable Hilbert space.     

On a class of Volterra nonlinear equations of parabolic type

In this paper, we show the backward uniqueness in time of solution for a class of Volterra nonlinear equations of parabolic type. We also give reasonable physical interpretation for our conclusion.     

Accelerated monotone iterative methods for finite difference equations of reaction-diffusion

This paper is concerned with numerical methods for a finite difference system of reaction-diffusion-convection equation under nonlinear boundary condition. Various monotone iterative methods are presented, and each of these methods leads to an existence-comparison theorem as well as a computational algorithm for numerical solutions. The monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration, and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function. Application is given to a model problem from chemical engineering, and some numerical results, including a test problem with known analytical solution, are presented to illustrate the various rates of convergence of the iterations. Received November 2, 1995 / Revised version received February 10, 1997     

On a class of two-dimensional adjoint integral equations of Volterra type

In a rectangular domain, we consider a two-dimensional integral equation of Volterra type with fixed singular kernels. For various values of the parameters occurring in this equation, we prove its unique solvability and establish the asymptotic behavior of the obtained solution.     

On preimages of a class of generalized monotone operators

In this paper we first provide a geometric interpretation of the Minty-Browder monotonicity which allows us to extend this concept to the so called h-monotonicity, still formulated in an analytic way. A topological concept of monotonicity is also known in the literature: it requires the connectedness of all preimages of the operator involved. This fact is important since combined with the local injectivity, it ensures global injectivity. When a linear structure is present on the source space, one can ask for the preimages to even be convex. In an earlier paper, the authors have shown that Minty-Browder monotone operators defined on convex open sets do have convex preimages, obtaining as a by-product global injectivity theorems. In this paper we study the preimages of h-monotone operators, by showing that they are not divisible by closed connected hypersurfaces, and investigate them from the dimensional point of view. As a consequence we deduce that h-monotone local homeomorphisms are actually global homeomorphisms, as the proved properties of their preimages combined with local injectivity still produce global injectivity.     

一类周期差分微分方程的解

运用一些分析技巧 ,讨论纯量周期差分微分方程x(t) =a(t) x(t) +b(t) x(t- kω)的解 ,其中 a(t) ,b(t)是以ω为周期的连续周期函数 ,k为某自然数 .得到了该方程具有趋于零快于任何指数但最终不恒等于零的解的充分必要条件 .同时给出解的具体的表达式