On asymptotic equivalence of linear and quasilinear difference equations

The asymptotic equivalence of systems of difference equations of linear and quasilinear type is investigated. The first result on the asymptotic equivalence of linear systems is a discrete analog of an improved version of the Levinson's well-known theorem on asymptotic equivalence of linear differential equations, while the second one providing conditions for asymptotic equivalence of linear and quasilinear systems is related to that of Yakubovich in differential equations case.     

On asymptotic equivalence of perturbed linear systems of differential and difference equations

Standard results on asymptotic integration of systems of linear differential equations give sufficient conditions which imply that a system is strongly asymptotically equivalent to its principal diagonal part. These involve certain dichotomy conditions on the diagonal part as well as growth conditions on the off-diagonal perturbation terms. Here, we study perturbations with a triangularly-induced structure and see that growth conditions can be substantially weakened. In addition, we give results for not necessarily triangular perturbations which in some sense “interpolate” between the classical theorems of Levinson and Hartman-Wintner. Some analogous results for systems of linear difference equations are also given.     

Smooth equivalence of differential equations and linear automorphisms

For systems of ordinary differential equations admitting linear automorphisms, we consider the problems of smooth equivalence and linearization preserving these automorphisms. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 567–578, October, 1999.     

On the equivalence of linear complementarity problems

We show that the Extended Linear Complementarity Problem (ELCP) can be recast as a standard Linear Complementarity Problem (LCP) provided that the surplus variables or the feasible set of the ELCP are bounded. Since many extensions of the LCP are special cases of the ELCP, this implies that these extensions can be rewritten as an LCP as well. Our equivalence proof is constructive and leads to three possible numerical solution methods for a given ELCP: regular ELCP algorithms, mixed integer linear programming algorithms, and regular LCP algorithms.     

On the solvability of certain spatially nonlocal boundary-value problems for linear hyperbolic equations of second order

The present paper studies the solvability of spatially nonlocal boundary-value problems with Samarskii boundary condition with variable coefficients for linear hyperbolic equations of second order in the one-dimensional case. In the case of boundary conditions with constant coefficients for the equation $$u_{tt} - u_{xx} + c(x)u = f(x,t),$$ similar problems were studied earlier by other authors; a significant aspect of their papers was the use of the Fourier method, which dictated a special form of the equation as well as the constancy of the coefficients of the boundary conditions. The method used here does not involve such constraints and allows us to study more general problems.     

On a class of nonlocal parabolic equations

We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painlevé singular nonfixed points theorem is proved. In this case, there is an alternative—either a solution exists for all t ≥ 0 or it goes to infinity in a finite time t = T (blowup mode). Sufficient conditions for the existence of a blowup mode are given.     

On matrix equivalence and matrix equations

Roth's theorem on the solvability of matrix equations of the form AX?YB=C is proved for matrices with coefficients from a division ring or a ring which is module-finite over its center.     

Conditions for the well-posedness of nonlocal problems for second-order linear differential equations

For second-order linear differential equations, we obtain sharp sufficient conditions for the well-posedness of nonlocal problems with functional and multipoint boundary conditions.     

On the equivalence of estimations under a general linear model and its transformed models

Some necessary and sufficient conditions are given for two equalities of ordinary least-squares estimators and best linear unbiased estimators of an estimable vector of parametric functions under a general linear model and its transformed linear model to hold     

A nonlocal problem for singular linear systems of ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite half-interval. This system is supplemented by the boundedness condition for solutions and a nonlocal linear condition specified by the Stieltjes integral. A method for approximating the resulting problem by a problem posed on a finite interval is proposed, and the properties of the latter are investigated. A numerically stable method for solving this problem is examined. This method uses an auxiliary boundary value problem with separated boundary conditions.     

On meaningful transformation equations

Aequationes mathematicae - The meaningfulness condition, applied to scientific or geometric laws, requires that the mathematical form of an equation does not change when we change the units of its...     

On nonlocal solutions of semilinear equations of the Sobolev type

We prove the existence and uniqueness of a classical solution of the Cauchy problem and the Showalter problem for some semilinear equations of the Sobolev type on a given interval. The abstract results are used to study initial-boundary value problems for some systems of equations unsolvable for the time derivative.     

Discontinuous reaction-diffusion equations under discontinuous and nonlocal flux conditions

In this paper, we consider a quasilinear parabolic equation with discontinuous source term in a bounded cylindrical domain under nonlocal and discontinuous flux conditions. Our main goal is to prove the existence of extremal solutions within a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, an abstract fixed-point result in partially ordered sets, compact embeddings, comparison, and truncation techniques.     

On some nonlocal evolution equations in Banach spaces

This note is concerned with the initial value problem for the abstract nonlocal equation where A is a maximal monotone operator from a reflexive Banach space E to its dual E^*, while B is a nonlocal maximal monotone operator from . Under proper boundedness and coercivity assumptions on the operators, a solution is achieved by means of a discretization argument. Uniqueness and continuous dependence are also discussed and we prove some estimates for the discretization error. Finally, we deal with the approximation of linear Volterra integrodifferential operators.     

On a nonlocal problem for nonlinear pseudoparabolic equations

For a nonlinear pseudoparabolic equation with one space dimension we consider its initial boundary value problem on an interval. The boundary condition on the left end is of Dirichlet type, the right end condition is replaced by a nonlocal one. Because it is given by an integral, the function involved could exhibit singularities, which distinguishes this nonlocal condition from its Dirichlet counterpart. Based on an elliptic estimate and an iteration method we established the well-posedness of solutions in a weighted Sobolev space.