Q-difference Systems Research Articles

Existence and Uniqueness of Non-Negative Solution to a Coupled Fractional q-Difference System with Mixed q-Derivative via Mixed Monotone Operator Method

In this paper, we study a nonlinear Riemann-Liouville fractional a q-difference system with multi-strip and multi-point mixed boundary conditions under the Caputo fractional q-derivative, where the nonlinear terms contain two coupled unknown functions and their fractional derivatives. Using the fixed point theorem for mixed monotone operators, we constructe iteration functions for arbitrary initial value and acquire the existence and uniqueness of extremal solutions. Moreover, a related example is given to illustrate our research results.

Solvability Criterion for Fractional q-Integro-Difference System with Riemann-Stieltjes Integrals Conditions

Due to the great application potential of fractional q-difference system in physics, mechanics and aerodynamics, it is very necessary to study fractional q-difference system. The main purpose of this paper is to investigate the solvability of nonlinear fractional q-integro-difference system with the nonlocal boundary conditions involving diverse fractional q-derivatives and Riemann-Stieltjes q-integrals. We acquire the existence results of solutions for the systems by applying Schauder fixed point theorem, Krasnoselskii’s fixed point theorem, Schaefer’s fixed point theorem and nonlinear alternative for single-valued maps, and a uniqueness result is obtained through the Banach contraction mapping principle. Finally, we give some examples to illustrate the main results.

Existence and stability of solutions for coupled fractional delay q-difference systems

Existence and stability of solutions for coupled fractional delay q-difference systems

A q-analogue of the matrix sixth Painlevé system

We derive a q-analogue of the matrix sixth Painlevé system from a connection-preserving deformation of a certain Fuchsian linear q-difference system. In specifying the linear q-difference system, we utilize the correspondence between linear differential systems and linear q-difference systems from the viewpoint of the spectral type. The system of non-linear q-difference equations thus obtained can also be regarded as a non-abelian analogue of Jimbo–Sakai’s q–P VI.

Q-Difference Systems for the Jackson Integral of Symmetric Selberg Type

We provide an explicit expression for the first order q-difference system for the Jackson integral of symmetric Selberg type. The q-difference system gives a generalization of q-analog of contiguous relations for the Gauss hypergeometric function. As a basis of the system we use a set of the symmetric polynomials introduced by Matsuo in his study of the q-KZ equation. Our main result is an explicit expression for the coefficient matrix of the q-difference system in terms of its Gauss matrix decomposition. We introduce a class of symmetric polynomials called interpolation polynomials, which includes Matsuo's polynomials. By repeated use of three-term relations among the interpolation polynomials we compute the coefficient matrix.

Coupled implicit Caputo fractional q-difference systems

This paper deals with some existence, uniqueness, and Ulam stability results for a coupled implicit Caputo fractional q-difference system in Banach and generalized Banach spaces. Some applications are made of some fixed point theorems for the existence and uniqueness of solutions. Next we prove that our problem is generalized Ulam–Hyers–Rassias stable. Some illustrative examples are given in the last section.

Laurent, Rational, and Hypergeometric Solutions of Linear q-Difference Systems of Arbitrary Order with Polynomial Coefficients

Systems of linear q-difference equations with polynomial coefficients are considered. Equations in the system may have arbitrary orders. For such systems, algorithms for searching polynomial, rational, and hypergeometric solutions, as well as solutions in the form of Laurent series, are suggested. Implementations of these algorithms are discussed.

Existence of Solutions for Boundary Value Problems of Ordered Anti-periodic Fractional q-Difference Systems

Existence of Solutions for Boundary Value Problems of Ordered Anti-periodic Fractional q-Difference Systems

On the Riemann–Hilbert problem for difference and q-difference systems

We study an analog of the classical Riemann–Hilbert problem stated for the classes of difference and q-difference systems. We generalize Birkhoff’s existence theorem.

Q-Borel-Laplace summation for q–difference equations with two slopes

In this paper, we consider linear q-difference systems with coefficients that are germs of meromorphic functions, with Newton polygon that has two slopes. Then, we explain how to compute similar meromorphic gauge transformations than those appearing in the work of Bugeaud, using a q-analogue of the Borel–Laplace summation.

Floquet theory based on new periodicity concept for hybrid systems involving q-difference equations

Using the new periodicity concept based on shifts, we construct a unified Floquet theory for homogeneous and nonhomogeneous hybrid periodic systems on domains having continuous, discrete or hybrid structure. New periodicity concept based on shifts enables the construction of Floquet theory on hybrid domains that are not necessarily additive periodic. This makes periodicity and stability analysis of hybrid periodic systems possible on non-additive domains. In particular, this new approach can be useful to know more about Floquet theory for linear q-difference systems defined on qZ¯:={qn:n∈Z}∪{0} where q > 1. By constructing the solution of matrix exponential equation we establish a canonical Floquet decomposition theorem. Determining the relation between Floquet multipliers and Floquet exponents, we give a spectral mapping theorem on closed subsets of reals that are periodic in shifts. Finally, we show how the constructed theory can be utilized for the stability analysis of dynamic systems on periodic time scales in shifts.

Existence of Extremal Solutions for a Nonlinear Fractional q-Difference System

In this paper, we study the boundary value problem of a fractional q-difference system with nonlocal integral boundary conditions involving the fractional q-derivatives of the Riemann–Liouville type. Using the properties of the Green function, and monotone iterative method, the extremal solutions were obtained. Finally, an example is presented to illustrate our main results.

Building Meromorphic Solutions ofq-Difference Equations Using a Borel–Laplace Summation

After introducing q-analogues of the Borel and Laplace transformations, we prove that to every formal power series solution of a linear q-difference equation with rational coefficients, we may apply several q-Borel and Laplace transformations of convenient orders and convenient direction in order to construct a solution of the same equation that is meromorphic on $\mathbb{C}^{*}$. We use this theorem to construct explicitly an invertible matrix solution of a linear q-difference system with rational coefficients, of which entries are meromorphic on $\mathbb{C}^{*}$. Moreover, when the system is put in the Birkhoff-Guenther normal form, we prove that the solutions we compute are exactly the same as the one constructed by Ramis, Sauloy and Zhang.

Positive solutions for nonlinear semipositone fractional q-difference system with coupled integral boundary conditions

This paper investigates the coupled integral boundary value problem for systems of nonlinear fractional q-difference equations. The nonlinear terms are continuous and semipositone. We firstly give the corresponding Green’s function for the boundary value problem and some of its properties. Moreover, by applying the nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorems, we derive an interval of λ such that any λ lying in this interval, the semipositone boundary value problem has one or multiple positive solutions. As applications, some interesting examples are presented to illustrate the main results. Finally, some related boundary value problems are also discussed.

Existence of periodic solutions in shifts [formula omitted] for neutral nonlinear dynamic systems

This paper focuses on the existence of a periodic solution of the delay neutral nonlinear dynamic systemsxΔ(t)=A(t)x(t)+QΔ(t,x(δ-(s,t)))+G(t,x(t),x(δ-(s,t))).In our analysis, we utilize a new periodicity concept in terms of shifts operators, which allows us to extend the concept of periodicity to time scales where the additivity requirement t±T∈T for all t∈T and for a fixed T>0, may not hold. More importantly, the new concept will easily handle time scales that are not periodic in the conventional way such as; qZ‾ and ∪k=1∞3±k,2.3±k∪{0}. Hence, we will develop the tool that enables us to investigate the existence of periodic solutions of q-difference systems. Since we are dealing with systems, in order to convert our equation to an integral systems, we resort to the transition matrix of the homogeneous Floquet system yΔ(t)=A(t)y(t) and then make use of Krasnoselskii’s fixed point theorem to obtain a fixed point.

Positive solutions of nonlinear boundary value problems for delayed fractional q-difference systems

In this paper, we investigate the existence and uniqueness of positive solutions to nonlinear boundary value problems for delayed fractional q-difference systems by applying the properties of the Green function and some well-known fixed-point theorems. As applications, some examples are presented to illustrate the main results. MSC:39A13, 34B18, 34A08.

Three-term relations between interpolation polynomials for a [formula omitted]-type basic hypergeometric series

We present an explicit expression for the q-difference system of rank n+1 satisfied by a BCn-type Jackson integral (q-series) as first order simultaneous q-difference equations with a concrete basis. For this purpose we introduce two types of symmetric Laurent polynomials called the interpolation polynomials. Three-term relations between the interpolation polynomials play an essential role in deriving the q-difference system. As an application, we give another proof for the product formula of the q-integral introduced by Gustafson.

Courbures, groupes de Galois génériques et D-groupoïde de Galois d'un système aux q-différences

Combining the results in Hendriks (1996) [6], Di Vizio (2002) [1] and Di Vizio, Hardouin [2], we prove that the generic, algebraic or differential, Galois group of a q-difference modules over C ( x ) can always be characterized in terms of v-curvatures, in the spirit of the work of Katz (1982) [8]. We use this result to prove that the Malgrange–Granier D-groupoid of a linear q-difference system coincide, in a sense that we specify below, with a sort of Kolchin closure of the dynamics of the linear q-difference system and that the group that fixes a transversal, coincide with the differential generic Galois group.

Un D-groupoïde de Galois local pour les systèmes aux q-différences fuchsiens

We compute the Galois D-groupoid of a constant linear q-difference system and we define, giving its realisations, a local Galois D-groupoid for fuchsian q-difference systems.

A First Order q-Difference System for the BC1-Type Jackson Integral and Its Applications

We present an explicit expression for the q-difference system, which the BC1- type Jackson integral (q-series) satisfies, as first order simultaneous q-difference equations with a concrete basis. As an application, we give a simple proof for the hypergeometric summation formula introduced by Gustafson and the product formula of the q-integral introduced by Nassrallah-Rahman and Gustafson.