© Hindawi Publishing Corp. C m SOLUTIONS OF SYSTEMS OF FINITE DIFFERENCE EQUATIONS

Let ℝ be the real number axis. Suppose that G , H are C m maps from ℝ 2 n + 3 to ℝ . In this note, we discuss the system of finite difference equations G ( x , f ( x ) , f ( x + 1 ) , … , f ( x + n ) , g ( x ) , g ( x + 1 ) , … , g ( x + n ) ) + 0 and H ( x , g ( x ) , g ( x + 1 ) , … , g ( x + n ) , f ( x ) , f ( x + 1 ) , … , f ( x + n ) ) = 0 for all x ∈ ℝ , and give some relatively weak conditions for the above system of equations to have unique C m solutions ( m ≥ 0 ) .


Introduction.
In [4,5,6], the iterative functional equations f 2 (z)(= f (f (z))) = az 2 + bz + c and n k=0 c k f k = 0 were considered, respectively.Zhang [7,8] showed the existence and uniqueness of C 0 , C 1 solutions of the equation F(x) − n k=1 λ k f k (x) = 0.In [3], the authors studied more general iterative functional equation G(x,f (x),...,f n (x)) = 0 and showed the existence, uniqueness, and stability of C m solutions (m ≥ 0) of the equation.The C m solutions (m ≥ 0) of the equation n i=0 c i f (x +i) = F(x) were discussed in [2].In this note, we discuss the following system of finite difference equations: G x, f (x), f (x + 1),... ,f (x + n), g(x), g(x + 1),... ,g(x + n) = 0, H x, g(x), g(x + 1),...,g(x + n), f (x), f (x + 1),... ,f (x + n) = 0, ( for all x ∈ R, where G, H ∈ C m (R 2n+3 , R) are given functions and f ,g ∈ C m (R, R) are unknown functions to be solved.Using the method of approximating fixed points by a small shift of maps, we give some relatively weak conditions for the above system of equations to have unique C m solutions for any integer m ≥ 0. Denote by Z + the set of all nonnegative integers.For m ∈ Z + and k ∈ N, write Z m = {0, 1,...,m} and N k = {1,...,k}.For f ,g ∈ C 0 (R, R) and r ,s ∈ R, define the map r f + sg : R → R by (r f + sg)(x) = r f (x) + sg(x) (for any x ∈ R).Then, under this operation, C 0 (R, R) is a linear space.
Let m ≥ k > 0. For g ∈ C m (R, R), denote by g (k) the kth derivative of g.Then g (k) ∈ C m−k (R, R).Usually, g (1) and g (2) are written as g and g .In addition, for any g ∈ C 0 (R, R), we put g (0) = g and call g (0) the 0th derivative of g.Now we introduce some symbols which are defined as in [3].For any two points x = y in R, (g(x) − g(y))/(x − y) is called a difference quotient of g.Let The set Λ g is called the set of difference quotients of g.
it is easy to verify that where Write that is, Λ g is bounded, then g is said to be Lipschitz continuous and λ g is called the (smallest) Lipschitz constant of g.Let m ≥ j ≥ 0 be integers and let r ≥ 0 be a real number.Suppose that K, K 0 , K 1 ,...,K j are all connected closed subsets of R. Write for any (y 0 ,y 1 ,...,y 2n+2 ) ∈ R 2n+3 .Obviously, In addition, we also write (1.10) Let K 0 ,K 1 ,...,K 2n+2 be all connected closed subsets of R and m ≥ 0. Write (1.12) 3), we have For convenience, we write for all f ,g ∈ C 0 (R, R) and all x ∈ R.
Let m ≥ 0 and G, H ∈ C m (R 2n+3 , R).For real number δ = 0, define where for all x ∈ R. It is easy to see that (f , g) is a fixed point of the map Ψ δGH if and only if (f , g) is a C m solution of (1.1).Thus, the problem of solutions of (1.1) can be translated into that of fixed points of Ψ δGH .In order to decide the existence of the fixed points of Ψ δGH , we need the following theorem which can be found in [1, page 74].
Theorem 1.1 (Schauder and Tychonoff).Let X be a compact convex set in a locally convex linear topological space.Then each continuous map Ψ : X → X has a fixed point.

Define a metric ρ
(1.17) Denote by 0 * the function on R which is identical to 0. For Analogous to the proof of [3, Proposition 3.2], we can obtain the following lemma.
In general, for k = 2,...,m, it is easy to see that where , and g (p) (x + q − 1) (where p ∈ N k−1 , q ∈ Z n+1 ) whose coefficients are all positive integers.The functional relation ξ k itself is related only to the rules of partial derivatives of general functions of several variables and the rules of derivatives of compositions and products of functions, but not related to specific G, f , or g.Therefore, ξ k is still well defined for k > m.If G ∈ C m+1 (R 2n+3 , R) and f ,g ∈ C m+1 (R, R), then (3.4) also holds for k = m + 1.
For k = 0, 1,...,m, let b k = max λ f (k) ,λ g (k) , (1.8), (1.9), (1.10), and (1.13), we have where whose coefficients are all positive integers.The functional relation η k itself is determined by ξ k and is independent of specific G, H, f , and g.Therefore, η k is still well defined for k > m.
For k = 2,...,m, noting that λ(f ) and (3.9) are also true for k = m + 1. Adopting the method that is used in the proof of Theorem 2.1 to show that (2.7) still holds when 2) and (3.11), we get (3.12) Similarly, we can obtain that Therefore, it follows from (3.12) and (3.13) that Noting that C m (R, R) ⊂ C 0 (R, R), we have the following proposition.(1.1) has at most a solution in F, then (1.1) By Proposition 3.2, after obtaining Theorem 2.2, we need not discuss the uniqueness of C m solutions of (1.1), for m ≥ 1, in detail unless we can give some conditions weaker than those in Theorem 2.2 (or, at least, they do not imply each other) or we can discuss the uniqueness of solutions in a subspace of C 0 (R, R)×C 0 (R, R) larger than the subspace of BC 0 (R, R; f m )×BC 0 (R, R; g m ).

Example.
Let a ≥ 10 be a real number.Suppose that the system of equations is for all x ∈ R. Then the representatives of the corresponding G, H : R 5 → R are for all (x 0 ,x 1 ,x 2 ,x 3 ,x 4 ) ∈ R 5 .We can easily calculate the derivatives of G and H, and then obtain the supremums and infimums of Λ iG and Λ iH (i = 0, 1, 2, 3, 4); hence we can take Consider the following system of equations: ), and it is the unique solution of (4.3)

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: by the upper bounds B k−1 and b p−1 of their absolute values, from (3.4) we get k= 2,...,m + 1. (3.10) Then a k−1 only depends on G and H. Since η k (B k−1 ,b 0 ,b 1 ,...,b k−2 ) is a monotone increasing function of b 0 ,b 1 ,...,b k−2 , and 1−δµ > δµ > 0, from (3.9) and (3.10) it follows that if b