On The Solutions of Some Difference Equations Systems and Analytical Properties

Absract. In this study, we investigated the global asymptotic behaviors of their solutions by taking the second-order difference equation system. According to the given conditions, we obtained some asymptotic results for the positive balance of the system. We have also worked on q-fast changing functions. Such functions form the class of q-Caramate functions. We have applied q-Caramate functions to linear q-difference equations and We have also learned about the asymptotic behavior of solutions. In addition, we have studied the problem of initial and boundary value for the q-difference equation


Introduction
Difference equations (ordinary differential and q-differential) since ancient times are equations of interest to mathematicians and physicists.These equations solve the differential equations by shaping them into mathematical models of many other practical phenomena.Difference equations can be easily algorithmized and solved on a computer they are very suitable for.
Studies on q-difference equations appeared already at the beginning of the last century in intensive works especially by Adams [1], Atkinson [2], Bairamov [3,4], Jackson [5], Elaydi [6], Guseinov [7,8], Carmichael, Trjitzinsky and others.Unfortunately, from the thirties up to the beginning of the eighties, only nonsignificant interest in the area was observed.Since years eighties, an intensive and somewhat surprising interest in the subject reappeared in many areas of mathematics and applications including mainly new difference calculus and orthogonal polynomials, q-combinatorics, q-arithmetics, q-integrable systems and variational q calculus.

Preliminaries
We have investigated the global asymptotic behavior of the solutions of the system of the difference equation (1) , where N ≥ 2 is a fixed integer and q> 1 is a fixed real number. (2) find the solution and real parametres, complex, equation ( 2) is the real number of the limit condition , meets the following conditions.
If a solution to the problem (1), ( 3) , (2) boundary conditions taking into account, they are obtained, (5) This system, together with the initial conditions (3), , , we can write.Thus, the initial boundary value problem ( 1) -( 3), the initial value problem (8), ( 9) the problem becomes equivalent [9].Let (10) be a non-trivial solution of the equation, where α is the complex constant,  .Equation ( 10), (8) as well, they are obtained Let us record (11) the equation (13) it is equivalent to the boundary value problem.If the solution of the problem ( 12), (13) is , the vector provides the equation (11) .

Asymptotics of regular solutions of q-difference equations
For 1> q > 0, we define the q-derivate of a real valued function u as (14) The higher order qderivatives are given by (15) we will describe asymptotics of solutions of q-difference equations.A q-difference equation for a sequence (y(1), y(2), y(3), . ..) of smooth functions of q has the form: (16 where are smooth functions and .In the usual analytic theory of q-difference equations, q is a complex number inside or outside the unit disk.With this in mind, ǫ-difference equations are obtained from q-difference equations by the substitution where ǫ is a small nonnegative real number, that plays the role of Planck's constant.[10].The characteristic polynomial of the q-difference equation ( 16) is We will say that equation ( 16) is regular if for all v ∈ S 1 , where is the discriminant of ), which is a polynomial in the coefficients of ).Let 1 denote the roots of the characteristic polynomial, which we call the eigenvalues of eq. ( 16).It turns out that eq.( 16) is regular iff the eigenvalues 1 . . ., never collide and never vanish, for every v ∈ S 1 .Moreover, it follows by the implicit function theorem that the roots are smooth functions of v ∈ S 1 .Let S 1 = denote a partition of S 1 into a finite union of closed arcs (with nonoverlapping interiors), such that the magnitude of the eigenvalues does not change in each arc.In other words, for each , there is a permutation σ p of the set {1, .., d} such that The following definition introduces a locally fundamental set of solutions of q-difference equations.
Fix a partition of I as above.A set {ψ 1 , . .., ψ d } is a locally fundamental set of solutions of eq.( 16) iff for every solution ψ for every p ∈ P and for every there exist smooth functions c p m such that for all (k, q) : q k ∈ I p .

Theorem 1.([11]
) Assume that eq.( 16) is regular.Then, there exists a locally fundamental set of solutions {ψ 1 , . .., ψ d } such that • For every and such that q k ∈ I p we have ψ m ( k, ) = exp( Φ m ( , )). • For some smooth functions Φ m with uniform (with respect to ) asymptotic expansion We find this last one by continuous limit (or "confluence") of coefficients.when q → 1, with the constraints We'll see further in which precise direction one can confuse the functional equation ( 1) it self to a differential equation with .
it admits for singularities 0, 1 and ∞ only and that these singularities are regular.The Frobenius Method provides fundamental systems of solutions in 0 and in ∞: The analytical extension of a fundamental system solutions in the neighborhood of a singularity still provides a fundamental system solutions and therefore expresses itself from another fundamental system by an array of constants.
To determine the behavior of the solutions we give the following lemmas.Lemma 2. For define the operator by Then and equation ( 6) can be written as .
Proof.From the q-derived formula q
(17) where φ m,s ∈ C ∞ (I) for all s • and leading term φ m,0 (x)= (e 2πit ))dt where we have chosen a branch for the logarithm of λ m .Definition 1. ([12]) We say that a formal series is a solution to (8) iff ∞ (18) It is easy to see that if is a formal solution to (8), then the leading term φ 0 satisfies the equation where λ(x) is an eigenvalue of (8).The hypergeometrical basic series: (19) is a convergent series, and it is the solution of the q-difference equation rational linear order two, ITM Web of Conferences 22, 01050 (2018) https://doi.org/10.1051/itmconf/20182201050CMES-2018 with .It's a q-analogue (and a deformation) of Euler Gauss's classical hypergeometric symmetry, =

Lemma 1 .
Define the function f : (0, ∞) → R by Then is strictly increasing for and strictly decreasing for .Proof.The statement follows from the equality f ′ (t) = -).