三阶柯西差分方程在几类群上的解

在几类群上讨论了三阶柯西差分方程解的存在性问题,将二阶柯西差分方程的已有结论进一步推广到三阶的情况,并给出在不同群上的一般解.     

广义Cauchy中值定理及其逆定理

将C auchy中值定理的条件进行适当减弱,得到了广义C auchy中值定理,从而推广了C auchy中值定理,并在凸函数的条件下,证明了其逆定理亦成立.     

The Cauchy problem for semilinear hyperbolic equations with cross discontinuous data

This paper is devoted to study Cauchy problems of multidimensional semilinear strictly hyperbolic equations of second order with strongly singular initial data, where the derivatives of the initial data have discontinuity on two smooth curves transversally intersecting each other. The existence of the solution is proved, meanwhile, it is precisely discribed the flowery structure of the singularity of the solution.     

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

We study the Cauchy problem in the layer Π _T =ℝ ⁿ ×[0,T] for the equationu _t =cGΔu _t +Δϕ(u), wherec is a positive constant and the functionϕ(p) belongs toC ¹(ℝ₊) and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functionsu(x,t) ∈C _x,t ^2,1 (Π _T ) with the properties: , . Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 356–362, September, 1996. This research was partially supported by the International Science Foundation under grant No. MX6000.     

Generalized implicit Euler method for hyperbolic functional differential equations

Nonlinear hyperbolic functional differential equations with initial boundary conditions are considered. Theorems on the convergence of difference schemes and error estimates of approximate solutions are presented. The proof of the stability of the difference functional problem is based on a comparison technique. Nonlinear estimates of the Perron type with respect to the functional variable for given functions are used. Numerical examples are given (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the stability of the orthogonal Pexiderized Cauchy equation

We investigate the stability of Pexiderized mappings in Banach modules over a unital Banach algebra. As a consequence, we establish the Hyers-Ulam stability of the orthogonal Cauchy functional equation of Pexider type f₁(x+y)=f₂(x)+f₃(y), x⊥y in which ⊥ is the orthogonality in the sense of Rätz.     

Existence of periodic and subharmonic solutions for second order functional difference equations

In this paper, by using the critical point theory, the existence of periodic and subharmonic solutions to a class of second order functional difference equations is obtained. The main approach used in our paper is variational technique and the Saddle Point Theorem. The problem is to solve the existence of periodic and subharmonic solutions of second order forward and backward difference equations.     

Generalized stability of the Cauchy and the Jensen functional equations on spheres

We study a generalized stability problem for Cauchy and Jensen functional equations satisfied for all pairs of vectors x,y from a linear space such that γ(x)=γ(y) or γ(x+y)=γ(x−y) with a given function γ.     

Topological simplification of nonautonomous difference equations

In this paper, we continue the study of geometric properties of nonautonomous difference equations in arbitrary Banach spaces which was begun in [2 Aulbach, B. 1998. The fundamental existence theorem on invariant fiber bundles. Journal of Difference Equations and Applications, 3(5–6): 501–537.  [Google Scholar],3 Aulbach, B. and Wanner, T. 2003. Invariant foliations and decoupling of non-autonomous difference equations. Journal of Difference Equations and Applications, 9(5): 459–472. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. Building on previous results on invariant fiber bundles and foliations, this paper addresses the problem of topological simplifications via continuous conjugacies and semiconjugacies. In particular, we establish a reduction principle for not necessarily invertible difference equations, as well as a generalized Hartman–Grobman theorem for systems with not necessarily invertible linear part.     

Convergent solutions of linear functional difference equations in phase space

By using weighted summable dichotomies and Schauder's fixed point theorem, we prove the existence of convergent solutions of linear functional difference equations. We apply our result to Volterra difference equations with infinite delay.     

Lebesgue regularity for differential difference equations with fractional damping

We provide necessary and sufficient conditions for the existence and uniqueness of solutions belonging to the vector‐valued space of sequences for equations that can be modeled in the form where X is a Banach space, A is a closed linear operator with domain D(A) defined on X, and G is a nonlinear function. The operator Δ^γ denotes the fractional difference operator of order γ>0 in the sense of Grünwald‐Letnikov. Our class of models includes the discrete time Klein‐Gordon, telegraph, and Basset equations, among other differential difference equations of interest. We prove a simple criterion that shows the existence of solutions assuming that f is small and that G is a nonlinear term.     

Periodic solutions of certain hybrid delay-differential equations and their corresponding difference equations

In this paper we use a method due to Carvalho (A method to investigate bifurcation of periodic solution in retarded differential equations, J. Differ. Equ. Appl. 4 (1998), pp. 17–27) to obtain conditions for the existence of nonconstant periodic solutions of certain systems of hybrid delay-differential equations. We first deal with a scalar equation of Lotka–Valterra type; then a system of two equations in two unknowns that could model the interactions of two identical neurons. It will be seen that such solutions are determined by solutions of corresponding difference equations. Another paper in which this method is used is by Cooke and Ladeira (Applying Carvalho's method to find periodic solutions of difference equations, J. Differ. Equ. Appl. 2 (1996), pp. 105–115).

We first state Carvalho's result.     

Positive periodic solutions for nonlinear repulsive singular difference equations

The existence and multiplicity of positive solutions are established to the periodic boundary value problems for repulsive singular nonlinear difference equations. The proof relies on a nonlinear alternative of Leray–Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.     

On the asymptotics of some systems of difference equations

We prove an inclusion theorem regarding a system of difference equations and apply it in getting the asymptotics of some solutions of some concrete difference equations.     

The Cauchy problem for the system of thermoelasticity equations in space

We consider the problem of analytic continuation of the solution of the system of thermoelasticity equations in a bounded three-dimensional domain on the basis of known values of the solution and the corresponding stress on a part of the boundary, i.e., the Cauchy problem. We construct an approximate solution of the problem based on the method of Carleman's function.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 212–217, August, 1998.In conclusion, the authors wish to thank Professor M. M. Lavrent'ev and Professor Sh. Ya. Yarmukhamedov for setting the problem and for discussions in the course of the solution.     

On a symmetric bilinear system of difference equations

We show in an elegant way how the main result in the recent paper Matsunaga and Suzuki (2018) follows from a known result, and discuss the system appearing therein.