We obtain conditions for the existence of continuous and N -periodic solutions, where N is a positive integer number, for systems of linear difference equations with continuous argument and investigate the structure of the set of these solutions.Consider the system of linear difference equationswhere t ∈ R + = [0, ∞), A i (t), i = 0, 1, . . . , k, are certain real n × n matrices, and F (t) : R + → R n , x(t) is an unknown vector function of dimension n. Under various assumptions for the matrices A i (t), i = 0, 1, . . . , k, and the vector F (t), these systems of equations were studied by many mathematicians, and some important problems in their theory are well studied (see [1][2][3][4][5] and references therein). Among them, there are problems of the existence of continuous and periodic solutions, which have been extensively studied in recent years. In particular, in [4, 5], a general continuous solution of the system of equations (1) was constructed for the case where k = 0 and F (t) = 0. and its structure was investigated. It is quite natural to investigate problems of the existence of continuous and N -periodic solutions (N is a positive integer number) of the system of equations (1) in the general case. This and the investigation of the structure of a general continuous solution of system (1) are the main objectives of the present paper. A solution of the system of equations (1) is understood as a vector function x(t) uniquely defined for t ≥ −k that transforms this system into the identity. Continuous SolutionsFirst, we investigate the problem of the existence of solutions of the system of equations (1) that are continuous for t ≥ −k. Assume that the following condition is satisfied:(i) all elements of the matrices A i (t), i = 0, 1, . . . , k, and the vector F (t) are continuous functions for t ≥ 0.For arbitrary real t ≥ 0, one has t − [t] = τ ∈ [0, 1), where [t] is the integer part of t. Setting

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We establish conditions for the existence of continuous bounded solutions of systems of nonlinear functional difference equations and study their properties.Functional difference equations of the formwhere A is a certain real n n matrix, q D const; and f W R R n R n ! R n ; were studied by many mathematicians, and, at present, some types of them are fairly well investigated. In particular, in [1][2][3][4][5], the foundations of the theory of systems of equations of the formwere developed in the case where the elements of the matrix A.t/ are holomorphic functions in a neighborhood of the point t D 1: In [6, 7], the structure of the set of continuous solutions of system (1) was studied in the case where the vector function f .t; x; y/ is linear with respect to x and y: In [8-10], sufficient conditions for the existence of continuous periodic solutions of systems of the form (1) were established and properties of these solutions were investigated. In the present paper, we establish conditions for the existence of solutions of nonlinear systems of the form (1) continuous for t 2 R C .t 2 R / and study the structure of the set of these solutions.1. First, we consider the system of equationswhere f W R R n ! R n and q is a certain real constant, under the assumption that the following conditions are satisfied:(i) det A ¤ 0; a D jAj < 1; q > 0I(ii) the vector function f .t; x/ is continuous and bounded for all t 2 R and x 2 R n and jf .t; x/ f .t; y/j Ä ljx yj;where l is a certain positive constant such that jAj C l < 1I(iii) sup t jf .t; 0/j D M < C1:

We establish conditions for the existence of continuous solutions of systems of linear functional difference equations with linearly transformed argument and develop a method for the construction of these solutions. This paper is devoted to the investigation of systems of linear equations of the formwhere t ∈ R, A(t) and B(t) are real n × n matrices, F (t) is a real vector of dimension n, and q is a certain real constant. Under various assumptions on the matrices A(t) and B(t) and the vector F (t), special cases of these systems of equations have been extensively investigated by many mathematicians (see [1-10] and the bibliography therein), and, at present, a number of problems of the theory of these equations are studied in detail. This is especially the case for the problem of the existence of various (analytic, continuous, etc.) solutions and the investigation of their properties. In the present paper, we also consider the problem of the existence of solutions (mainly continuous and bounded) and investigate the structure of the set of these solutions. The main aim of the present paper is to establish conditions for the existence of continuous solutions and to develop a method for their construction. First, we consider a system of equations of the formwhere A and B are real constant n × n matrices and q is a real constant, and show that, under certain conditions, this system has continuous solutions that can be constructed. Assume that the eigenvalues λ i , i = 1, . . . , n, of the matrix A satisfy the conditions λ i = λ j , |λ i | = 0, 1, i, j = 1, . . . , n.As is known, there exists a change of variableswhere C is a certain constant nonsingular n × n matrix, that reduces the system of equations (2) to the form y(t + 1) = Λy(t) + By(qt),