Communications on Applied Nonlinear Analysis 

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The Cauchy Problems for Fractional Q-Difference Equations with Integral 

Conditions in Banach Spaces 

Faouzi Hireche 

Abdelhamid Ibn Badis University, Mostaganem 27000, Algeria. 

faouzi.hireche.ma@gmail.com;  faouzi.hireche@univ-mosta.dz 

 

Article History: 

Received: 19-08-2025 

Revised: 24-09-2025 

Accepted: 15-10-2025 

Abstract: In  In this paper, we investigate the existence of solutions to the fractional q-

difference equation of the type 𝐷𝑐 𝑞
𝛼𝑢(𝑡) = 𝐴𝑢(𝑡) + 𝑓(𝑡, 𝑢(𝑡)), 𝑡 ∈ 𝐽 ≔ [0,1]  with  

𝑢(0) = 𝑎 ∫ 𝑢(𝑠)𝑑𝑞𝑠 + 𝑏,
1

0
  where 0 < 𝛼 < 1, 𝑓 ∈ 𝐶(𝐽 × 𝑋, 𝑋), and 𝐴(𝑡) is a bounded 

linear operator on a Banach space 𝑋 The operator 𝐷𝑐 𝑞
𝛼𝑢(𝑡) denotes the Caputo 

fractional q-derivative of order 𝛼. Our existence result are obtained by using the 

Banach fixed point theorem  and the Schaefer fixed point theorem. 

Keywords: Fixed point, Existence, Caputo fractional q-derivative, Fractional q-

difference equations. 

1. Introduction 

Over the past years, the theory of fractional calculus has received increasing attention from 

researchers and has become one of the most active areas of research due to its significant 

importance and wide applications on many subjects. The importance of this theory lies in its 

ability to contribute to mathematical modeling in various fields such as technical sciences, physics, 

engineering, biophysics and biomathematics. For more details, see [9–11, 13, 16]. 

At the beginning of the twentieth century, Jackson was the first to develop quantum calculus, also 

known as q-difference calculus, by introducing the concept of the q- integral along with several 

other fundamental notions in this theory. For further details on this topic, see references [8, 12]. 

In the late 1960s, a new branch known as fractional q-difference calculus emerged as a 

generalization of the q-difference calculus. This development is attributed to Al-Salam [6] and 

Agarwal [2]. This branch has received considerable attention in the academic community due to 

its wide range of applications in modeling mathematical phenomena across various scientific 

fields. 

Recently, Several researchers have studied the fractional q-difference equations in- volving the 

Caputo fractional q-derivative by using all kinds of fixed point theorems and obtained many 

interesting results , for example, by Abbas et al [1], Ahmad and al. [5]. 

In [3], N. Allouch et al studied the existence of solutions to the following fractional 

q-difference equations with nonlinear integral conditions: 

𝐷𝑞
𝛼𝑐 𝑢(𝑡) = 𝑓(𝑡, 𝑢(𝑡)),      𝑡 ∈ 𝐼 = [0, 𝑇],     1 < 𝛼 ≤ 2, 

𝑢(0) − 𝑢′(0) = ∫ 𝑔(𝑠, 𝑢(𝑠))𝑑𝑠,
𝑇

0

 

mailto:faouzi.hireche.ma@gmail.com


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𝑢(𝑇) + 𝑢′(𝑇) = ∫ ℎ(𝑠, 𝑢(𝑠))𝑑𝑠,
𝑇

0

 

where 𝑇 > 0, 𝑞 ∈]0,1[, 𝐷𝑐 𝑞
𝛼  denotes the Caputo fractional q-difference derivative of order 

1 < 𝛼 ≤ 2, and  𝑓, 𝑔, ℎ ∈ 𝐶(𝐼 × 𝑋,𝑋). 

In [4], N. Allouch et al applied some standard fixed point theorems and investigated the 

existence of solutions of fractional q-difference equations of the type: 

𝐷𝑞
𝛼𝑐 𝑢(𝑡) = 𝑓(𝑡, 𝑢(𝑡)),      𝑡 ∈ 𝐼 = [0, 𝑇],     1 < 𝛼 ≤ 1, 

𝑎𝑢(0) + 𝑏𝑢(𝑇) = 𝑐, 

where 𝑇 > 0, 𝑞 ∈]0,1[, 𝐷𝑐 𝑞
𝛼  denotes the Caputo fractional q-difference derivative of order 

𝛼, 𝑓 ∈ 𝐶(𝐼 × 𝑋, 𝑋) an d  𝑎 + 𝑏 ≠ 0. 

In this paper, we establish the existence of solutions to the fractional q-difference equations 

of the type: 

𝐷𝑞
𝛼𝑐 𝑢(𝑡) = 𝐴(𝑡) + 𝑓(𝑡, 𝑢(𝑡)),      𝑡 ∈ 𝐽 ≔ [0,1],                                             (1.1) 

𝑢(0) = 𝑎 ∫ 𝑢(𝑠)𝑑𝑞𝑠 + 𝑏,
1

0
                                                              (1.2)  

where 0 < 𝛼 < 1, 𝑓 ∈ 𝐶(𝐽 × 𝑋, 𝑋), and 𝐴(𝑡) is a bounded linear operator on a Banach space 𝑋. 

The operator 𝐷𝑐 𝑞
𝛼 denotes the Caputo fractional q-difference derivative of order 𝛼. The existence 

result is based on the fixed point theorem and the Schaefer fixed point theorem. 

The paper is structured as follows. In Section 2, we present the notations and definitions required 

for the study, and we review essential preliminaries from fractional q-calculus. Section 3 contains 

the principal results: the first derived from the Banach fixed point theorem, and the second from 

Schaefer’s fixed point theorem. Section 4 is devoted to an illustrative example highlighting the 

applicability of these results. 

2.   Preliminaries 

This section is concerned with presenting basic definitions together with auxiliary results required 

in the later parts of this paper. 

Assume that 𝑋 is a Banach space. Define 𝐽: = [0,1] and let 𝐶(𝐽, 𝑋) denote the Banach space 

of continuous functions 𝑢 from 𝐽 into 𝑋 with the norm 

‖𝑢‖∞ = 𝑠𝑢𝑝𝑡∈𝐽|𝑢(𝑡)|. 

Now, we introduce the essential definitions and relevant properties of the fractional 

q-calculus. For more details, see [8, 12]. 

We assume that 𝑞 ∈]0,1[. For every 𝑎 ∈ ℝ, we define 

[𝑎]𝑞 =
1 − 𝑞𝑎

1 − 𝑞
. 



Communications on Applied Nonlinear Analysis 

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Let 𝑎, 𝑏 ∈ ℝ. The q-analogue of (𝑎 − 𝑏)(𝑛) is given by: 

(𝑎 − 𝑏)(𝑛) = {

1                            𝑖𝑓 𝑛 = 0

∏(𝑎 − 𝑏𝑞𝑖
𝑛−1

𝑖=0

)       𝑖𝑓 𝑛 ∈ ℕ∗
 

For 𝛽 ∈ ℝ, we have 

(𝑎 − 𝑏)(𝛽) = 𝑎𝛽∏(
𝑎 − 𝑏𝑞𝑖

𝑎 − 𝑏𝑞𝑖+𝛽
) ,    𝑎, 𝑏 ∈ ℝ.

∞

𝑖=0

 

Note that, if 𝑏 = 0, then 𝑎(𝛽) = 𝑎𝛽 . 

Definition 1 [12] The q-gamma function is defined as follows: 

Γ𝑞(𝛽) =
(1 − 𝑞)(𝛽−1)

(1 − 𝑞)𝛽−1
,   𝛽 > 0. 

Observe that the q-gamma function verifies Γ𝑞(𝛽 + 1) = [𝛽]𝑞Γ𝑞(𝛽). 

Definition 2 [12] Let 𝑓: 𝐽 → ℝ. The q-derivative of order 𝑛 ∈ ℕ is given by: 

(𝐷𝑞
0𝑓)(𝑡) = 𝑓(𝑡),       (𝐷𝑞

1𝑓)(𝑡) =
𝑓(𝑡) − 𝑓(𝑞𝑡)

(1 − 𝑞)𝑡
, 

𝑎𝑛𝑑 

(𝐷𝑞
𝑛𝑓)(𝑡) = (𝐷𝑞

1𝐷𝑞
𝑛−1𝑓)(𝑡),   𝑛 ∈ ℕ∗. 

Definition 3 [12] Let 𝐽𝑡 = {𝑡𝑞
𝑛:  𝑛 ∈ ℕ}⋃{0}. The q-integral of a function 𝑓: 𝐽𝑡 → ℝ is 

defined by: 

(𝐼𝑞𝑓)(𝑡) = ∫ 𝑓(𝑠)𝑑𝑞𝑠 = ∑𝑡(1 − 𝑞)𝑞𝑛𝑓(𝑡𝑞𝑛),

∝

𝑛=0

1

0

 

under the assumption that the series converges. 

Note that (𝐷𝑞𝐼𝑞𝑓)(𝑡) = 𝑓(𝑡), Furthermore, if 𝑓 is continuous at 0, then  

(𝐼𝑞𝐷𝑞𝑓)(𝑡) = 𝑓(𝑡) − 𝑓(0). 

Definition 4 [2] Let 𝑓: 𝐽 → ℝ. the Riemann-Liouville fractional q-integral of order 𝛼 ≥
0 is defined as: 

(𝐼𝑞
𝛼𝑓)(𝑡) = {

𝑓(𝑡)                                            𝑖𝑓 𝛼 = 0,

∫
(𝑡 − 𝑞𝑠)(𝛼−1)

Γ𝑞(𝛼)
𝑓(𝑠)𝑑𝑞𝑠

𝑡

0

       𝑖𝑓 𝛼 > 0.
 

Observe that when 𝛼 = 1, we have (𝐼𝑞
1𝑓)(𝑡) = (𝐼𝑞𝑓)(𝑡). 



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Lemma 5 [15] For all 𝛼 ≥ 0 and 𝛽 ∈] − 1,+∞[, we have 

(𝐼𝑞
𝛼(𝑡 − 𝑎)(𝛽))(𝑡) =

Γ𝑞(𝛽 + 1)

Γ𝑞(𝛼 + 𝛽 + 1)
(𝑡 − 𝑎)(𝛼+𝛽),      0 < 𝛼 < 𝑡 < 1. 

In particular, 

(𝐼𝑞
𝛼1)(𝑡) =

1

Γ𝑞(𝛼 + 1)
𝑡(𝛼). 

Definition 6 [14] The Riemann-Liouville fractional q-derivative of order 𝛼 ≥ 0 for a 

function 𝑓: 𝐽 → ℝ is defined as follows: 

(𝐷𝑞
0𝑓)(𝑡) = 𝑓(𝑡)      and        (𝐷𝑞

𝛼𝑓)(𝑡) = (𝐷𝑞
[𝛼]𝐼𝑞

[𝛼]−𝛼𝑓) (𝑡),   𝑡 ∈ 𝐽, 

where [𝛼] is the integer part of 𝛼. 

Definition 7 [14]  Consider a function  𝑓: 𝐽 → ℝ and let 𝛼 ≥ 0. The Caputo fractional q-

derivative of order 𝛼 is defined as follows: 

(𝐷𝑞
0𝑓)(𝑡) = 𝑓(𝑡)     and       ( 𝐷𝑞

𝛼𝑐 𝑓)(𝑡) = (𝐼𝑞
[𝛼]−𝛼

𝐷𝑞
[𝛼]
 𝑓) (𝑡),   𝑡 ∈ 𝐽, 

where [𝛼] is the integer part of 𝛼. 

Lemma 8 [14] Suppose that 𝛼,𝛽 ≥ 0, and let 𝑓: 𝐽 → ℝ be a given function. Then, the 

following identities hold: 

(i) (𝐼𝑞
𝛼𝐼𝑞
𝛽
𝑓)(𝑡) = (𝐼𝑞

𝛼+𝛽
𝑓)(𝑡), 

(ii) (𝐷𝑞
𝛼𝐼𝑞
𝛽
𝑓)(𝑡) = 𝑓(𝑡). 

Lemma 9 [14] Assume 𝛼 ≥ 0, and let 𝑓 be a function defined on the interval 𝐽. The 

following identity holds 

(𝐼𝑞
𝛼 𝐷𝑞

𝛼𝑐 𝑓)(𝑡) = 𝑓(𝑡) − ∑
𝑡𝑘

Γ𝑞(𝑘 + 1)
(𝐷𝑞

𝛼𝑓)(0).

[𝛼]−1

𝑘=0

 

If 𝛼 ∈]0,1[, we have 

(𝐼𝑞
𝛼 𝐷𝑞

𝛼𝑐 𝑓)(𝑡) = 𝑓(𝑡) − 𝑓(0). 

Theorem 10 (Banach contraction principle) [7] Suppose that 𝐶 is a non-empty closed 

subset of a Banach space 𝑋. If 𝐻: 𝐶 → 𝐶 is a contraction, then 𝐻 admits a unique fixed 

point in 𝐶. 

Theorem 11 (Schaefer) [17] Let 𝑋 be a Banach space and let 𝐻:𝑋 → 𝑋 be a completely 

continuous operator. Assume that the set 

ℰ ≔ {𝑢 ∈ 𝑋 |𝑢 = 𝜆𝐻(𝑢),   𝜆 ∈]0,1[} 

is bounded. Then 𝐻 admits a fixed point in 𝑋. 



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3.   Existence 

In this section, we investigate the existence of solutions for the fractional qqq-difference problem 

given by (1.1)-(1.2). 

Definition 12 A function 𝑢 ∈ 𝐶([0,1], 𝑋) is called a solution of the fractional q-difference 

problem (1.1)-(1.2) if 𝑢 satisfies the equation 𝐷𝑐 𝑞
𝛼𝑢(𝑡) = 𝐴𝑢(𝑡) + 𝑓(𝑡, 𝑢(𝑡)) on 𝐽 =

[0,1], and the condition 𝑢(0) = 𝑎 ∫ 𝑢(𝑠)𝑑𝑞𝑠 + 𝑏.
1

0
 

To establish the existence of solutions for the fractional problem (1.1)-(1.2), we  require 

the following lemma: 

Lemma 13 Let 𝛼 ∈]0,1[ and let ℎ: [0,1] × 𝑋 → 𝑋 be continuous function. The solution of 

the fractional q-difference problem 

𝐷𝑞
𝛼𝑐 𝑢(𝑡) = 𝐴(𝑡) + 𝑓(𝑡, 𝑢(𝑡)),      𝑡 ∈ 𝐽 = [0,1], 

𝑢(0) = 𝑎 ∫ 𝑢(𝑠)𝑑𝑞𝑠 + 𝑏,
1

0
     

is given by        

𝑢(𝑡) = 𝑎∫ 𝑢(𝑠)𝑑𝑞𝑠 + 𝑏 +
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

𝑓(𝑠, 𝑢(𝑠))𝑑𝑞𝑠
1

0

+
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠.
𝑡

0

 

The first result is obtained by applying the Banach fixed point theorem. 

Theorem 14 Suppose that: 

(𝐻1)  There exist 𝑘 > 0 such that 

∀𝑡 ∈ 𝐽,   ∀𝑢, 𝑣 ∈ 𝑋,   |𝑓(𝑡, 𝑢) − 𝑓(𝑡, 𝑣)| ≤ 𝑘|𝑢 − 𝑣|. 

If  

                                                      |𝛼| +
𝑘 + ‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
< 1.                                                      (3.1) 

Then, the fractional q-difference problem (1.1)-(1.2) admits a unique solution on [0,1]. 

Proof 15 The fractional q-difference problem (1.1)-(1.2) can be reformulated in terms of a fixed 

point problem. For this purpose, we define the operator 

𝐹: 𝐶([0,1], 𝑋) → 𝐶([0,1], 𝑋), 

where 



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          𝐹(𝑢(𝑡)) ≔ 𝑎∫ 𝑢(𝑠)𝑑𝑞𝑠 + 𝑏 +
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

𝑓(𝑠, 𝑢(𝑠))𝑑𝑞𝑠
1

0

                         (3.2)  

+
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠.
𝑡

0

                                                             

It is evident that the fixed points of the operator 𝐹 correspond to solutions of the fractional q-

difference problem (1.1)-(1.2). We shall use the Banach contraction mapping principle to 

demonstrate that 𝐹 defined by (3.2) has a fixed point. We shall demonstrate that 𝐹 is a 

contraction. 

Let 𝑢, 𝑣 ∈ 𝐶([0,1], 𝑋) and 𝑡 ∈ [0,1], we obtain  

      |𝐹(𝑢)(𝑡) − 𝐹(𝑣)(𝑡)|                                                                                                                          

≤ |𝑎|∫ |𝑢(𝑠) − 𝑣(𝑠)|𝑑𝑞𝑠 +
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

|𝑓(𝑠, 𝑢(𝑠)) − 𝑓(𝑠, 𝑣(𝑠))|𝑑𝑞𝑠
1

0

+
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)‖𝐴‖ℒ(𝑋)|𝑢(𝑠)
𝑡

0

− 𝑣(𝑠)|𝑑𝑞𝑠                                                                                                                             

≤ |𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠) − 𝑣(𝑠)| +
𝑘

Γ𝑞(𝛼)
𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠) − 𝑣(𝑠)|∫ (𝑡 − 𝑞𝑠)

(𝛼−1)𝑑𝑞𝑠
𝑡

0

 

+  
‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼)
𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠) − 𝑣(𝑠)|∫ (𝑡 − 𝑞𝑠)

(𝛼−1)𝑑𝑞𝑠
𝑡

0

                                                      

≤ |𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠) − 𝑣(𝑠)| +
𝑘

Γ𝑞(𝛼 + 1)
𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠) − 𝑣(𝑠)|                                    

+  
‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
𝑠𝑢𝑝𝑠𝜖[0,1]|𝑢(𝑠) − 𝑣(𝑠)|                                                                                        

= (|𝑎| +
𝑘 + ‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
)𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠) − 𝑣(𝑠)|. 

Hence, 

‖𝐹(𝑢) − 𝐹(𝑣)‖∞ ≤ [|𝑎| +
𝑘 + ‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
] ‖𝑢 − 𝑣‖∞. 

 

According to (3.2), the operator 𝐹 satisfies the contraction condition. Hence, by the Banach fixed 

point theorem, 𝐹  admits a  fixed point, representing the solution of the fractional q-difference 

problem (1.1)-(1.2). 

Theorem 16 Suppose that: 

(𝐻2) The function 𝑓: [0,1] × 𝑋 → 𝑋 is continuous. 

(𝐻3) ∃𝑀 > 0, ∀𝑡 ∈ 𝐽, ∀𝑢 ∈ 𝑋,   |𝑓(𝑡, 𝑢)| ≤ 𝑀. 

Hence, the fractional q-difference problem (1.1)-(1.2) admits at least one solution on [0,1]. 



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Proof 17  We apply Schaefer’s fixed point theorem to verify that the operator 𝐹, defined in (3.2), 

admits a fixed point. The proof is organized in four steps. 

Step 1: 𝐹 is continuous operator. 

Consider a sequence {𝑢𝑛} such that 𝑢𝑛 → 𝑢 in 𝐶([0,1], 𝑋). Then for all 𝑡 ∈ [0,1]: 

|𝐹(𝑢𝑛)(𝑡) − 𝐹(𝑢)(𝑡)|

≤ |𝑎|∫ |𝑢𝑛(𝑠) − 𝑢(𝑠)|𝑑𝑞𝑠
1

0

+
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)|𝑓(𝑠, 𝑢𝑛(𝑠)) − 𝑓(𝑠, 𝑢(𝑠))|𝑑𝑞𝑠
1

0

+
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)‖𝐴‖ℒ(𝑋)|𝑢𝑛(𝑠) − 𝑢(𝑠)|𝑑𝑞𝑠
𝑡

0

≤ |𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠) − 𝑢(𝑠)|

+
1

Γ𝑞(𝛼)
𝑠𝑢𝑝𝑠∈[0,1]|𝑓(𝑠, 𝑢𝑛(𝑠)) − 𝑓(𝑠, 𝑢(𝑠))|∫ (𝑡 − 𝑞𝑠)

(𝛼−1)
𝑡

0

𝑑𝑞𝑠

+
‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼)
𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠) − 𝑢(𝑠)|∫ (𝑡 − 𝑞𝑠)

(𝛼−1)𝑑𝑞𝑠
𝑡

0

≤ |𝑎|𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠) − 𝑢(𝑠)| +
1

Γ𝑞(𝛼 + 1)
𝑠𝑢𝑝𝑠∈[0,1]|𝑓(𝑠, 𝑢𝑛(𝑠)) − 𝑓(𝑠, 𝑢(𝑠))|

+
‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠) − 𝑢(𝑠)|

= (|𝑎| +
‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
) 𝑠𝑢𝑝𝑠∈[0,1]|𝑢𝑛(𝑠) − 𝑢(𝑠)|

+
1

Γ𝑞(𝛼 + 1)
𝑠𝑢𝑝𝑠∈[0.1]|𝑓(𝑠, 𝑢𝑛(𝑠)) − 𝑓(𝑠, 𝑢(𝑠))|. 

Since the function 𝑓 is a continuous, we have 

‖𝐹(𝑢𝑛) − 𝐹(𝑢)‖∞

≤ (|𝑎| +
‖𝐴‖𝓛(𝑋)

Γ𝑞(𝛼 + 1)
) ‖𝑢𝑛(∙) − 𝑢(∙)‖∞ +

1

Γ𝑞(𝛼 + 1)
‖𝑓(∙, 𝑢𝑛(∙)) − 𝑓(∙, 𝑢(∙))‖∞

𝑛→+∞
→    0. 

Step 2: 𝐹 maps bounded sets into bounded sets in 𝐶([0,1], 𝑋). 

Indeed, it is enough to prove that for every 𝜂∗ > 0, there exists a constant ℓ > 0 such that for all   

𝑢 ∈ 𝐵𝜂∗ = {𝑢 ∈ 𝐶([0,1], ℝ):   ‖𝑢‖∞ ≤ 𝜂
∗}, we have ‖𝐹(𝑢)‖∞ ≤  ℓ. 

By (𝐻3) we have for every 𝑡 ∈ [0,1]: 



Communications on Applied Nonlinear Analysis 

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|𝐹(𝑢)(𝑡)| ≤ |𝑎| ∫ |𝑢(𝑠)|𝑑𝑞𝑠
1

0

+ |𝑏| +
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

|𝑓(𝑠, 𝑢(𝑠))|𝑑𝑞𝑠

+
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

‖𝐴‖ℒ(𝑋)|𝑢(𝑠)|𝑑𝑞𝑠

≤ |𝑎|𝜂∗ + |𝑏| +
𝑀

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)𝑑𝑞𝑠
𝑡

0

+
‖𝐴‖ℒ(𝑋)𝜂

∗

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

𝑑𝑞𝑠

≤ |𝑎|𝜂∗ + |𝑏| +
𝑀

Γ𝑞(𝛼 + 1)
+
‖𝐴‖ℒ(𝑋)𝜂

∗

Γ𝑞(𝛼 + 1)
. 

Hence 

‖𝐹(𝑢)‖∞ ≤ |𝑎|𝜂
∗ + |𝑏| +

𝑀

Γ𝑞(𝛼 + 1)
+
‖𝐴‖ℒ(𝑋)𝜂

∗

Γ𝑞(𝛼 + 1)
≔ ℓ. 

Step 3: The operator 𝐹 maps bounded sets into equicontinuous sets of 𝐶([0,1], 𝑋). 

For 𝑡1, 𝑡2 ∈ [0,1]  with 𝑡1 < 𝑡2, and for 𝑢 ∈ 𝐵𝜂∗ , where 𝐵𝜂∗  is the bounded subset of 𝐶([0,1], 𝑋) 

defined in step 2, we have 

|𝐹(𝑢)(𝑡2) − 𝐹(𝑢)(𝑡1)|

= |
1

Γ𝑞(𝛼)
∫ [(𝑡2 − 𝑞𝑠)

(𝛼−1) − (𝑡1 − 𝑞𝑠)
(𝛼−1)]𝑓(𝑠, 𝑢(𝑠))𝑑𝑞𝑠

𝑡1

0

+
1

Γ𝑞(𝛼)
∫ [(𝑡2 − 𝑞𝑠)

(𝛼−1) − (𝑡1 − 𝑞𝑠)
(𝛼−1)]𝐴𝑢(𝑠)𝑑𝑞𝑠

𝑡1

0

+
1

Γ𝑞(𝛼)
∫ (𝑡2 − 𝑞𝑠)

(𝛼−1)𝑓(𝑠, 𝑢(𝑠))𝑑𝑞𝑠 +
1

Γ𝑞(𝛼)
∫ (𝑡2 − 𝑞𝑠)

(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠
𝑡2

𝑡1

𝑡2

𝑡1

|

≤
𝑀

Γ𝑞(𝛼)
∫ [(𝑡1 − 𝑞𝑠)

(𝛼−1) − (𝑡2 − 𝑞𝑠)
(𝛼−1)]𝑓(𝑠, 𝑢(𝑠))𝑑𝑞𝑠

𝑡1

0

+
𝜂∗‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼)
∫ [(𝑡1 − 𝑞𝑠)

(𝛼−1) − (𝑡2 − 𝑞𝑠)
(𝛼−1)]𝑑𝑞𝑠

𝑡1

0

+
𝑀

Γ𝑞(𝛼)
∫ (𝑡2 − 𝑞𝑠)

(𝛼−1)𝑑𝑞𝑠 +
𝜂∗‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼)
∫ (𝑡2 − 𝑞𝑠)

(𝛼−1)𝑑𝑞𝑠
𝑡2

𝑡1

𝑡2

𝑡1

≤
𝑀

Γ𝑞(𝛼 + 1)
[(𝑡2 − 𝑡1)

(𝛼) + 𝑡1
(𝛼) − 𝑡2

(𝛼)] +
𝜂∗‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
[(𝑡2 − 𝑡1)

(𝛼) + 𝑡1
(𝛼) − 𝑡2

(𝛼)]

+
𝑀

Γ𝑞(𝛼 + 1)
(𝑡2 − 𝑡1)

(𝛼) +
𝜂∗‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
(𝑡2 − 𝑡1)

(𝛼)

≤ [
2𝑀

Γ𝑞(𝛼 + 1)
+
2𝜂∗‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
] (𝑡2 − 𝑡1)

(𝛼) + [
𝑀

Γ𝑞(𝛼 + 1)
+
𝜂∗‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
] (𝑡2 − 𝑡1)

(𝛼). 

As 𝑡1 → 𝑡2, the right-hand side of the preceding inequality tends to zero. As a consequence of 

step 1 to 3 tougher with the Arzelà-Ascoli theorem, we can conclude that the operator  

𝐹: 𝐶([0,1], 𝑋) → 𝐶([0,1], 𝑋) is continuous and completely continuous. 

 



Communications on Applied Nonlinear Analysis 

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Step 4: A priori bounds. 

By Schauder’s fixed point theorem, it is sufficient to show that the set: 

ℰ = {𝑢 ∈ 𝐶([0,1], 𝑋) |𝑢 = 𝜆𝐹(𝑢),   𝜆 ∈]0,1[} 

is bounded. 

Let 𝑢 ∈ ℰ. Then there exists 𝜆 ∈]0,1[ such that 𝑢 = 𝜆𝐹(𝑢). Hence, for every 𝑡 ∈ [0,1], 
we have 

𝑢(𝑡) = 𝜆 [𝑎∫ 𝑢(𝑠)𝑑𝑞𝑠 + 𝑏 +
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

𝑓(𝑠, 𝑢(𝑠))𝑑𝑞𝑠
1

0

+
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)𝐴𝑢(𝑠)𝑑𝑞𝑠
𝑡

0

]. 

Under hypothesis (𝐻3), it follows that for every 𝑡 ∈ [0,1]: 

|𝐹(𝑢)(𝑡)| ≤ |𝑎|∫ |𝑢(𝑠)|𝑑𝑞𝑠
1

0

+ |𝑏| +
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

|𝑓(𝑠, 𝑢(𝑠))|𝑑𝑞𝑠

+
1

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)
𝑡

0

‖𝐴‖ℒ(𝑋)|𝑢(𝑠)|𝑑𝑞𝑠

≤ |𝑎| 𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)| + |𝑏| +
𝑀

Γ𝑞(𝛼)
∫ (𝑡 − 𝑞𝑠)(𝛼−1)𝑑𝑞𝑠
𝑡

0

+
‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼)
 𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|∫ (𝑡 − 𝑞𝑠)

(𝛼−1)
𝑡

0

𝑑𝑞𝑠

≤ |𝑎| 𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)| + |𝑏| +
𝑀

Γ𝑞(𝛼 + 1)
+
‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|. 

Then, for every 𝑡 ∈ [0,1], we have 

‖𝐹(𝑢)‖∞ ≤ |𝑎| 𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)| + |𝑏| +
𝑀

Γ𝑞(𝛼 + 1)
+
‖𝐴‖ℒ(𝑋)

Γ𝑞(𝛼 + 1)
𝑠𝑢𝑝𝑠∈[0,1]|𝑢(𝑠)|  ≔ 𝑅. 

This shows that the set ℰ is bounded. 

Consequently, by Schaefer fixed point theorem, we conclude that 𝐹 admits a fixed point 

which is a solution of the problem (1.1)-(1.2). 

4.   Application 

In this section, we provide an illustrative example to demonstrate the applicability of our result. 

Consider the following fractional q-difference problem with an integral boundary condition: 

𝐷1
3

1

2𝑐 𝑢(𝑡) =
𝑡

100
𝑢(𝑡) +

𝑢(𝑡)

(1+𝑐𝑜𝑠(𝑢(𝑡)))
,      𝑡 ∈ [0,1],                                             (4.1) 

𝑢(0) =
2

3
∫ 𝑢(𝑠)𝑑1

3

𝑠 +
1

2
.

1

0
                                                              (4.2)  



Communications on Applied Nonlinear Analysis 

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Let  

𝑓(𝑡, 𝑢) =
𝑢

(1 + 𝑐𝑜𝑠(𝑢))
,   (𝑡, 𝑢) ∈ [0,1] ×]0,+∞[    

and 

𝐴(𝑡) =
𝑡

100
. 

For all 𝑢, 𝑣 ∈]0,+∞[ and 𝑡 ∈ [0,1]. We have 

|𝑓(𝑡, 𝑢) − 𝑓(𝑡, 𝑣)| ≤ |
𝑢 − 𝑣

(1 + cos(𝑢))(1 + cos(𝑣))
| ≤

1

4
|𝑢 − 𝑣|. 

We see that the condition |𝑎| +
𝑘+‖𝐴‖

Γ𝑞(𝛼+1)
≈ 0.93 < 1 holds with 𝛼 =

1

2
, 𝑘 =

1

4
, 𝑞 =

1

3
, ‖𝐴‖ =

1

100
  

and Γ1
3

(
3

2
) ≈ 0.9376. Consequently, by Theorem 14, the fractional q-difference problem (4.1)-

(4.2) admits a unique solution on the interval  [0,1]. 

5.   Conclusion 

In this study, we establish the existence of solutions to Cauchy problems for fractional q-

difference equations with integral conditions in Banach spaces. The analysis is conducted 

employing the Banach fixed point theorem and the Schaefer fixed point theorem. Furthermore, 

an illustrative example is provided to demonstrate the applicability and effectiveness of the 

obtained results. 

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