Dynamics and Stability of q-Fractional Order Pantograph Equations With Nonlocal Condition

Early works about q -difference calculus or quantum calculus initially read in Jackson (1908; 1910). The difference equations are widely used in mathematical physical problems, sampling theory of signal analysis, dynamical system and quantum models and heat and wave equations. Recently, some researchers have noticed their attention to discrimination research of the fractional q-difference calculus, we refer readers to (Agarwal et al., 2014; Liang and Zhang, 2012; Al-Yami, 2016; Stankovic et al., 2009). For a long time, in many areas the fractional differential equations are very popular. For example, engineers and scientists have developed new methods that include fractional equations; we refer to (Hilfer, 1999; Podlubny, 1999). Since the beginning of the last decade, the fractional q-differential equations have become an important mathematical tool. The pantograph equations have been widely studied (Balachandran et al., 2013; Liu and Li, 2004) and references therein)since they can be utilized to depict many phenomena that arise in electro dynamics, probability, quantum mechanics, dynamical systems and number theory. Recently, fractional pantograph differential equations have been considered by many researchers. One of the motivating topics in this area is the research of the existence of solutions by fixed point theorems, we refer to (Balachandran et al., 2013). The Ulam stability of functional equation, which was Ulam founded for a speech to a conference at the University of Wisconsin in 1940, is one of the important subjects in the mathematical analysis area. The finding of Ulam stability plays a vital role in regard to this subject. For detailed study on the progress of Ulam type (U-H) stability, readers refer to (Andras and Kolumban, 2013; Jung, 2004; Muniyappan and Rajan, 2015) and the references therein. The credit of solving this problem partially goes to Hyers. To study U-H stability of fractional differential equations, different researchers studied their works with different methods, see (Ibrahim, 2012; Wang et al., 2011; Wang and Zhou, 2012; Wang and Zhang, 2014). Koca (2015) proved local asymptotics stability of q-fractional nonlinear dynamics systems. Inspired by the above discussion, we initiate the existence and U-H-q-Mittag-Leffler stability for qfractional pantograph equations. Consider the following system represented by the qfractional order pantograph equation with nonlocal condition of the form:


Introduction
Early works about q -difference calculus or quantum calculus initially read in Jackson (1908;1910). The difference equations are widely used in mathematical physical problems, sampling theory of signal analysis, dynamical system and quantum models and heat and wave equations. Recently, some researchers have noticed their attention to discrimination research of the fractional q-difference calculus, we refer readers to (Agarwal et al., 2014;Liang and Zhang, 2012;Al-Yami, 2016;Stankovic et al., 2009). For a long time, in many areas the fractional differential equations are very popular. For example, engineers and scientists have developed new methods that include fractional equations; we refer to (Hilfer, 1999;Podlubny, 1999). Since the beginning of the last decade, the fractional q-differential equations have become an important mathematical tool.
The pantograph equations have been widely studied (Balachandran et al., 2013;Liu and Li, 2004) and references therein)since they can be utilized to depict many phenomena that arise in electro dynamics, probability, quantum mechanics, dynamical systems and number theory. Recently, fractional pantograph differential equations have been considered by many researchers. One of the motivating topics in this area is the research of the existence of solutions by fixed point theorems, we refer to (Balachandran et al., 2013).
The Ulam stability of functional equation, which was Ulam founded for a speech to a conference at the University of Wisconsin in 1940, is one of the important subjects in the mathematical analysis area. The finding of Ulam stability plays a vital role in regard to this subject. For detailed study on the progress of Ulam type (U-H) stability, readers refer to (Andras and Kolumban, 2013;Jung, 2004;Muniyappan and Rajan, 2015) and the references therein. The credit of solving this problem partially goes to Hyers. To study U-H stability of fractional differential equations, different researchers studied their works with different methods, see (Ibrahim, 2012;Wang et al., 2011;Wang and Zhou, 2012;Wang and Zhang, 2014). Koca (2015) proved local asymptotics stability of q-fractional nonlinear dynamics systems. Inspired by the above discussion, we initiate the existence and U-H-q-Mittag-Leffler stability for qfractional pantograph equations. Consider the following system represented by the qfractional order pantograph equation with nonlocal condition of the form: where, c q D α is the Caputo fractional q-derivative, q∈ (0,1). Let 0 < α < 1, 0 < λ < 1 and Let C (J; X) be the Banach space of continuous function x(t) with x(t) ∈ X for t ∈ J and ||x|| C (J; X) = max t ∈ J ||x(t)||.
In passing, we note that the application of nonlinear condition x(0) + g(x) = x 0 in physical problems yeilds better effect than the initial condition x(0) = x 0 (Bashir and Sivasundaram, 2008).
The outline of the paper is as follows. In section 2, we give some basic definitions and results con-cerning the fractional q-calculus. In section 3, we present our main results by fixed point theorems. Stability analysis is discussed in section 4.
Let q ∈ (0, 1) and define: The q-analogue of the Pochhammer symbol was presented as follow: The q-gamma function is defined by: The q-derivative of a function F(x) is here defined by: The q-integral of a function F defined in the interval [0, T] is provided by: now, it can be defined an operator n q I , as follows: We can point the basic formula which will be used at a later time: where, s D q denotes the q-derivative with respect to variable s.
The fractional q-derivative of the Caputo type of order α > 0 is defined by: is the smallest integer greater than or equal to α.

Remark 2.6
The q-Mittag-Leffler function will tend to the classical one when q → 1.
Theorem 2.7. (Darbo's Fixed Point Theorem (Lakshmikantham, 1994),p.no.21) Let K be a bounded, closed convex set of a Banach space X. Suppose that T and S are two mappings from K to X satisfying: • Tx + Sy ∈ K for any x, y ∈ K • T is a contraction mapping • S is a completely continuous on K Then T + S has atleast a fixxed point on K.

Main Results
Let us list some hypotheses to prove our existence results: exists a constant b > 0, such that: There exists a function µ ∈ L 1 (J) such that: We are now ready to present our results. The existence results are based on Darbo's fixed point theorem.

Proof
Let P and Q the two operators defined on B r by: Indeed it is easy to check the inequality: By (A3), it is also clear that Q is a contraction mapping. Produced from continuity of x, the operator (Px)(t) is continuous in accordance with (A1). Also we observe that: Then P is uniformly bounded on B r . Now let's prove that (Px)(t) is equicontinuous. Let t 1 , t 2 ∈ J, t 2 ≤ t 1 and x ∈ B r . Using the fact F is bounded on the compact set J × B r (thus max (t,x,y)∈J × Br |F(t, x, y)|:= C 0 < ∞).
We will get: , , which is autonomous of x and head for zero as t 1 − t 2 → 0 consequently P is equicontinuous. Thus, P is relatively compact on B r . By the Arzela-Ascoli theorem, P is compact. We now conclude the result of the theorem based on the Darbo's fixed point. Thus, the problem (1) has at least one fixed point on J.

Stability Analysis
In this section, we define some basic concepts of U-H-q-Mittag-Leffler stability. We adopt some ideas in (Otrocol and Ilea, 2013).

Definition 4.1
The Equation (1) is U-H-q-Mittag-Leffler stable with respect to e α (t α ;q) if there exists e C α such that for each ϵ > 0 and for each solution z ∈ C (J; X) of the inequality: There exists a solution x ∈ C (J, X) of Equation (1) with: where, e α (t α ;q) is the q-Mittag-Leffler function.

Remark 4.2
A function z ∈ C (J, X) is a solution of the inequality: if and only if there exists a function h ∈ C (J, X) such that: If a function z ∈ C (J, X ) is a solution of the inequality:
We are now in a position to state and prove our stability results for problem (1). The arguments are based on the Banach contraction principle with respect to Chebyshev norm.

Proof
The operator P: C (J, X) → C (J, X): Then we can show that PB r ⊂ B r . So let x ∈ B r and set G = max x∈ C (J, X) |g(x)|. Then we get: By the choice of L and r. Now take x, y ∈ C (J, X). Then we get: depends only on the parameters of the problem and since Ω b,L,T,α,q < 1, the result follows in view of the contraction mapping principle due to Chebyshev norm.

Proof
Let ϵ > 0 and let z ∈ C (J,X) be a function which satisfies the inequality: and let x ∈ C (J, X) be the unique solution of the following q-fractional order pantograph equation: x t x g x t qs F s x s x s d s , , t q q z t x g z e t q t qs F s z s z s d s For each t ∈ J, we have: For u ∈ C (J, X) we consider the operator A: C (J, X) → C (J, X) defined by: Au t e t q bu t t qs u s d s L t t qs u s d s Next, we verify that A is a Picard operator. For all t ∈ J, it follows (A2): Thus, A is a contraction via the Chebyshev norm ||⋅|| on C (J, X) due to (3).
Applying the Banach contraction principle to A, we derive that A is a Picard operator and F A = {u * }. Then for t ∈ J: u t e t q bu t t qs u s d s L t qs u s d s It remains to verify that the solution u * is increasing. Indeed, for 0 ≤ t 1 < t 2 ≤ b and denote  ; In particular, if u = |z − x|, from (6), u ≤ Au and applying the Lemma 4.5 we obtain u ≤ u * , where A is a picard and increasing operator. As a result, we know: Thus, the Equation (1) is U-H-q-Mittag-Leffler stable.

Remark 4.10
Theorem 3.1 and 4.9 can easily be extended to the generalized q-fractional multi-pantograph of the form: where, c q D α is the Caputo q-fractional derivative, α ∈ (0, 1). Now we give an example to illustrate our results.