Uniqueness and Ulam–Hyers–Rassias stability results for sequential fractional pantograph q -differential equations

We study sequential fractional pantograph q -diﬀerential equations. We establish the uniqueness of solutions via Banach’s contraction mapping principle. Further, we deﬁne and study the Ulam–Hyers stability and Ulam–Hyers–Rassias stability of solutions. We also discuss an illustrative example.


Introduction
Differential equations involving q-difference calculus have become a strong tool in modeling many problems in engineering, physics, and mathematics [1-3]. Differential equations with fractional q-difference calculus have been studied by different researchers [4][5][6][7][8]. Many interesting topics concerning fractional q-differential equations (FqDEs) are devoted to the existence and stability of the solutions. In recent years, several scholars have studied the existence, uniqueness, and different types of Ulam stability (US) of solutions of FqDEs; see, for example, [9][10][11][12]. Recently, sequential fractional differential equations has been studied by many scholars [13][14][15].
The outline of the paper is the following. In Sect. 2, we discuss the main definitions and lemmas by providing a necessary background of q-calculus, including the q-derivative and q-integral. In Sect. 3, we investigate the uniqueness for the FPqDE (1). In Sect. 5, we present an example to apply our outcomes.

Let us now define the space
equipped with the norm It is clear that (W, w W ) is a Banach space.

Uniqueness results
We prove the following auxiliary lemma, which is pivotal to define the solution for Problem (1).
where r > 0, 1 < ν ≤ 2, 0 < σ ≤ 1 and η ∈ , is given by Proof We have Now we write the linear sequential FDE (6) as By taking the fractional q-integral of order σ for (7) we get where a 0 and b 0 are arbitrary constants. By the boundary condition w(0) = 0 we conclude that b 0 = 0. Using the boundary condition λ 1 w(T)λ 2 w(η) = , we obtain that Substituting the values of a 0 and b 0 into (8), we obtain solution (5). This completes the proof.
In view of Lemma 3.1, we can define the operator: G : W → W by For convenience, we denote Our first result is based on Banach's fixed point theorem.
Proof Let us fix = sup s∈J ϕ(s, 0, 0, 0), choose Using this estimate, we get We also have Thus we obtain From the definition of · W we have which implies that GB ⊂ B . For w,ẃ ∈ B and for all s ∈ , we have Using (C1), we get We also have By (C1) we can write Consequently, we obtain By (11) we see that G is a contractive operator. Consequently, by the Banach fixed point theorem, G has a fixed point, which is a solution of problem (1). This completes the proof.
Proof Letẃ ∈ W be a solution of inequality (12). Let us denote by w ∈ W the unique solution of the problem According to Lemma 3.1, we have where ψ w (s) = ϕ * w (s) for s ∈ . By integration of (12) we obtain Then, for any s ∈J, we havé By (C1) and (15) we can write This implies that from which it follows that Thus problem (1) is UHS.

So by Lemma 3.1 we have
Then we get From (C1) and (16) we can write
Hence problem (1) is stable in the UHR sense.