Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives

1 Introduction and preliminaries

In recent years, the fractional differential equations have been applied in various areas of engineering, mathematics, physics, and other applied sciences because of their ability to describe memory effects (see, e.g., [1–3]).

The discrete versions of continuous-type problems in science can be made from the point of view of the so-called q -calculus. The Caputo q -fractional derivative has been introduced on the base of the fractional q -integral and fractional q -derivative, always with the lower limit of integration equal to 0. However, in some considerations, such as solving q -differential equation of fractional order with initial values at a nonzero point, it is of interest to allow that the lower limit of integration is variable. Recently, many researchers got much interested in looking at fractional q -differential equations as new model equations for many physical problems. For example, some researchers obtained q -analogue of the integral and differential fractional operators properties, such as the q -Laplace transform, q -Taylor’s formula, and q -Mittage-Leffler function. For some fundamental results in the theory of fractional calculus and fractional differential equations, see, e.g., [4–9]. In addition, many scholars have paid much attention to the fractional quantum calculus, which has lots of applications in different areas of mathematics, such as combinatorics, number theory, basic hypergeometric functions, and other sciences. Recently, considerable attention has been given to the existence and stability of solutions for differential equations and systems with fractional quantum calculus. For more details, we refer the reader to the monographs [10–12] and the references cited therein. Hybrid differential equations have been the object of many researchers, see [13–15]. The hybrid differential equation of first order

was studied in [16] under the hypotheses that the functions ϕ ∈ C ( [ 0 , T ] × R , R \ { 0 } ) and ψ ∈ C ( [ 0 , T ] × R , R ) . The hybrid differential equations with fractional derivative have been addressed extensively by several researchers, for which the reader can consult [17–20] and the references cited therein. On the other hand, several papers including the hybrid differential equations with fractional q-derivative have been raised, for example [21–23] and the references therein. In [21], Ahmad and Ntouyas studied the existence of solutions for the fractional hybrid q-differential equation as follows:

where D q δ C is the Caputo fractional q-derivative of order δ and the functions ϕ ∈ C ( [ 0 , 1 ] × R , R \ { 0 } ) , and ψ ∈ C ( [ 0 , 1 ] × R , R ) . In [22], the author studied the existence, uniqueness, and Ulam-Hyers-Rassias stability for a class of hybrid Caputo fractional q-differential pantograph equations described as follows:

where 0 < q , δ < 1 , 0 < λ , μ < 1 , D q δ C is the Caputo fractional q-derivative, ς ∈ [ 0 , T ] , ϕ i ∈ C ( J × R 2 , R − { 0 } ) , and φ i ∈ C ( J × R 3 , R ) , i = 1 , … , k , k ∈ N ∗ . Samei and Ranjbar [23] discussed the existence and uniqueness of solutions for the fractional hybrid q-differential inclusions with the boundary conditions of the form

where 1 < δ ≤ 2 , q ∈ ( 0 , 1 ) , I q β denotes the Riemann-Liouville-type q-integral of order β > 0 , β ∈ { α i , γ j } , i = 1 , … , n , j = 1 , … , m , n , m ∈ N , D q δ C denotes Caputo-type q-derivative of order δ , ϕ : [ 0 , 1 ] × R n → ( 0 , ∞ ) is continuous, and φ : [ 0 , 1 ] × R m → P ( R ) is a multifunction. Recently, sequential hybrid fractional differential equations have also been studied by several scholars, see, e.g., [24–27] and the references cited therein. In [24], the authors considered the fractional sequential type of the hybrid differential equation

where 0 < δ , γ ≤ 1 , 1 < δ + γ ≤ 2 , I β is the Riemann-Liouville fractional integral of order β > 0 , β ∈ { α , γ i } , ϕ i ∈ C ( [ 0 , T ] × R , R ) , i = 1 , 2 , … , k , ψ ∈ C ( [ 0 , T ] × R , R \ { 0 } ) , and φ ∈ C ( [ 0 , T ] × R 2 , R ) and a , b , c are real constants with a + b ≠ 0 . The existence and uniqueness results were obtained by applying a generalization of Krasnoselskii’s fixed point theorem. In [25], the authors studied the existence, uniqueness, and stability analysis for a class of sequential hybrid fractional differential equations described as follows:

where 0 < δ ≤ 1 , 1 < γ ≤ 2 , 0 < ε < 1 , I β is the Riemann-Liouville fractional integral of order β > 0 , β ∈ { α , λ i } , the functions ψ : [ 0 , 1 ] × R 2 → R \ { 0 } , ϕ i : [ 0 , 1 ] × R → R , i = 1 , 2 , … , k , and φ ∈ C ( [ 0 , 1 ] × R 2 , R ) satisfy the Carathéodory conditions, the boundary functions g 1 , g 2 : R → R are non linear, and R represents the set of real numbers. To the best of our knowledge, there is no article discussing the coupled system of fractional hybrid q-differential equations in the literature. The objective of this article is to study the sequential coupled system of fractional hybrid q-differential equations of the following form:

where D q α C is the Caputo fractional q-derivative of order α , where α ∈ { δ i , θ i } , ϕ i : [ 0 , 1 ] × R 2 → R − { 0 } and ψ i : [ 0 , 1 ] × R 2 → R , i = 1 , 2 are continuous functions. The operator D q α C is the fractional q-derivative in the sense of the Caputo, which is define as follows:

where n is the smallest integer greater than or equal to α . The fractional q-integral of the Riemann-Liouville-type [28–30] is given as follows:

where the q-gamma function is defined by Γ q ( α ) = ( 1 − q ) ( α − 1 ) ( 1 − q ) α − 1 , α ∈ R \ { 0 , − 1 , − 2 , … } and satisfies

We need the following essential lemmas.

The rest of the article is organized as follows. In Section 2, we establish sufficient conditions for the existence and uniqueness of solutions for the main system. The stability of solutions is discussed in Section 4. We present numerical examples to illustrate and validate the effectiveness of the main results in Section 5.

2 Existence and uniqueness results

In view of Lemma 4, we can define operator T : W × Z → W × Z as follows:

where

and

We impose the following hypotheses:

1. ψ i : [ 0 , 1 ] × R 2 → R are continuous functions and there exist constants η i > 0 such that for all ς ∈ J and κ i , ν i ∈ R , i = 1 , 2 ,

   ∣ ψ 1 ( ς , κ 1 , ν 1 ) − ψ 1 ( ς , κ 2 , ν 2 ) ∣ ≤ η 1 ( ∣ κ 1 − κ 2 ∣ + ∣ ν 1 − ν 2 ∣ ) , ∣ ψ 2 ( ς , κ 1 , ν 1 ) − ψ 2 ( ς , κ 2 , ν 2 ) ∣ ≤ η 2 ( ∣ κ 1 − κ 2 ∣ + ∣ ν 1 − ν 2 ∣ ) .

2. ϕ i : [ 0 , 1 ] × R 2 → R − { 0 } are continuous functions and there exist positive constants Λ i , i = 1 , 2 , such that for all ς ∈ J and κ , ν ∈ R ,

   ∣ ϕ 1 ( ς , κ , ν ) ∣ ≤ Λ 1 , ∣ ϕ 2 ( ς , κ , ν ) ∣ ≤ Λ 2 .

In the following, we present the existence and uniqueness of solutions of problem (1) using Banach’s fixed point theorem.