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Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms

  • Ahmed Alsaedi , Boshra Alharbi and Bashir Ahmad EMAIL logo
Published/Copyright: September 19, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

In this article, we introduce and study a new class of higher-order fractional q-difference equations involving Riemann-Liouville q-derivatives with dual hybrid terms, supplemented with nonlocal multipoint q-integral boundary conditions. The existence of a unique solution to the given problem is shown by applying Banach’s fixed point theorem. We also present existing criteria for solutions to the problem at hand with the aid of Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Illustrative examples are given to demonstrate the application of the obtained results.

Keywords: q-difference equation; Riemann-Liouville q-derivative; q-integral; nonlocal boundary conditions; existence; fixed point
MSC 2010: 34A08; 39A13; 34B10; 34B15

1 Introduction

In this article, we explore the existence criteria for solutions of a new boundary value problem consisting of a nonlinear higher-order fractional q-difference equation with dual hybrid terms involving Riemann-Liouville q-derivatives and nonlocal multipoint q-integral boundary conditions. In precise terms, we study the following nonlocal nonlinear higher-order hybrid q-fractional integral boundary value problem:

(1) λ D q α [ u ( x ) − f 1 ( x , u ( x ) ) ] + ( 1 − λ ) D q β [ u ( x ) − f 2 ( x , u ( x ) ) ] = f 3 ( x , u ( x ) ) , 0 < x < 1 ,

(2) u ( 0 ) = 0 , ∑ i = 1 n a i u ( ξ i ) = 0 , u ( 1 ) = μ ∫ 0 η ( η − q s ) ( γ 1 − 1 ) Γ q ( γ 1 ) u ( s ) d q s + ( 1 − μ ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) Γ q ( γ 2 ) u ( s ) d q s , γ 1 , γ 2 > 0 ,

where 0 < q < 1 , 2 < α < 3 , 1 < β ≤ 2 , 0 < λ ≤ 1 , 0 ≤ μ ≤ 1 , 0 < η , σ , ξ i < 1 , a i ∈ R + , i = 1 , … , n , D q α , and D q β denote the Riemann-Liouville fractional q-derivative operators of order α and β , respectively, and f 1 , f 2 , f 3 : [ 0 , 1 ] × R → R are continuous functions.

Let us now review some recent works on boundary value problems of fractional q-difference equations. Agarwal et al. [1] proved some existence results for a Langevin-type q-variant system of nonlinear fractional integro-difference equations with nonlocal boundary conditions. In [2] a coupled system of nonlinear fractional q-integro-difference equations equipped with coupled q-integral boundary conditions was studied. Alsaedi et al. [3] proved some existence results for a hybrid fractional q-integro-difference equation with nonlocal q-integral boundary conditions. A nonlinear boundary value problem for a fractional q-integro-difference equation was studied in [4]. For some more results on boundary value problems involving fractional q-difference operators, we refer the reader to articles [5–12] and references cited therein.

The objective of the present work is to investigate the criteria ensuring the existence and uniqueness of solutions to problem (1)–(2). We make use of the Banach’s fixed point theorem [13] to obtain a uniqueness result, while two existence results are established by means of Krasnoselskii’s fixed point theorem [14] and Leray-Schauder’s nonlinear alternative [15].

We arrange the remainder of this article as follows. In Section 2, we recall some basic concepts related to our present study. In Section 3, we first prove a subsidiary result, which lays the foundation to convert the given nonlinear problem into a fixed point problem. Then, we use the tools of fixed point theory to establish the existence and uniqueness results for the problem at hand. Examples illustrating these results are also presented in this section. Some interesting observations are discussed in Section 4.

2 Auxiliary material

Let us first recall some necessary concepts and definitions about q-fractional calculus.

The q -number [ a ] q is defined by [ a ] q = 1 − q a 1 − q for every a ∈ R and q ∈ ( 0 , 1 ) . Also, the q-shifted factorial of real number a is defined by

( a ; q ) 0 = 1 , ( a ; q ) n = ∏ j = 0 n − 1 ( 1 − a q j ) , n ∈ N ∪ { ∞ } .

For a , b ∈ R , the q-analogue of the power function ( a − b ) n with n ∈ N 0 ≔ { 0 , 1 , 2 , … } is given by

( a − b ) ( 0 ) = 1 , ( a − b ) ( n ) = ∏ j = 0 n − 1 ( a − b q j ) .

In general, if ϱ is the real number, then

( a − b ) ( ϱ ) = a ϱ ∏ j = 0 ∞ 1 − b a q j 1 − b a q ϱ + j .

If ϱ > 0 and 0 ≤ a ≤ b ≤ t , then ( t − b ) ( ϱ ) ≤ ( t − a ) ( ϱ ) . The q-Gamma function Γ q ( ϱ ) is defined as

Γ q ( ϱ ) = ( 1 − q ) ( ϱ − 1 ) ( 1 − q ) ϱ − 1 , ϱ ∈ R \ { 0 , − 1 , − 2 , … } ,

which satisfies the relation Γ q ( ϱ + 1 ) = [ ϱ ] q Γ q ( ϱ ) [16].

Definition 2.1

[16] Let ϱ ≥ 0 and u : ( 0 , ∞ ) → R be a continuous function. The Riemann-Liouville fractional q-integral for the function u of order ϱ is defined by ( I q 0 u ) ( t ) = u ( t ) and

( I q ϱ u ) ( t ) = 1 Γ q ( ϱ ) ∫ 0 t ( t − q s ) ( ϱ − 1 ) u ( s ) d q s , ϱ > 0 ,

for t ∈ ( 0 , ∞ ) , provided that the right-hand side is point-wise defined on ( 0 , ∞ ) .

Recall that I q β I q ϱ u ( t ) = I q β + ϱ u ( t ) for ϱ , β ∈ R + [16] and

I q ϱ t β = Γ q ( β + 1 ) Γ q ( ϱ + β + 1 ) t ϱ + β , β ∈ ( − 1 , ∞ ) , ϱ ≥ 0 , t > 0 .

If u ≡ 1 , then I q ϱ 1 ( t ) = 1 Γ q ( ϱ + 1 ) t ϱ for all t > 0 .

Definition 2.2

[16] The Riemann-Liouville fractional q-derivative of order ϱ > 0 for a function u : ( 0 , ∞ ) → R is defined by ( D q 0 u ) ( t ) = u ( t ) and

( D q ϱ u ) ( t ) = ( D q p I q p − ϱ u ) ( t ) ,

where p is the smallest integer greater than or equal to ϱ .

Now, we state the following fixed point theorems that will be applied to prove the existence of solutions for the given problem.

Theorem 2.1

(Krasnoselskii’s fixed point theorem [14]) Let Y be a closed, convex, bounded, and nonempty subset of a Banach space X. Let ℋ 1 and ℋ 2 be the operators defined on Y onto X such that

  1. ℋ 1 u + ℋ 2 v ∈ Y whenever u , v ∈ Y ;

  2. ℋ 1 is compact and continuous;

  3. ℋ 2 is a contraction mapping.

Then, there exists w ∈ Y such that w = ℋ 1 w + ℋ 2 w .

Theorem 2.2

(Leray-Schauder’s nonlinear alternative [15]) Let T : S → S be a completely continuous operator (i.e., a map restricted to any bounded set in S is compact). Let Φ ( T ) = { u ∈ S : u = ζ T ( u ) for some ζ ∈ ( 0 , 1 ) } . Then, either the set Φ ( T ) is unbounded or T has at least one fixed point.

3 Main results

We begin this section with an auxiliary lemma that characterizes the structure of solutions for problems (1)–(2).

Lemma 3.1

Let ρ 1 , ρ 2 , y ∈ C ( [ 0 , 1 ] , R ) . Then, the function u is a solution to the linear hybrid fractional q-difference boundary value problem

(3) λ D q α [ u ( x ) − ρ 1 ( x ) ] + ( 1 − λ ) D q β [ u ( x ) − ρ 2 ( x ) ] = y ( x ) , 0 < x < 1 , u ( 0 ) = 0 , ∑ i = 1 n a i u ( ξ i ) = 0 , u ( 1 ) = μ ∫ 0 η ( η − q s ) ( γ 1 − 1 ) Γ q ( γ 1 ) u ( s ) d q s + ( 1 − μ ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) Γ q ( γ 2 ) u ( s ) d q s ,

if and only if u is a solution to the fractional q-integral equation

(4) u ( x ) = ρ 1 ( x ) + ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) y ( s ) d q s + 1 Δ x α − 1 ∑ i = 1 n a i ξ i α − 2 − x α − 2 ∑ i = 1 n a i ξ i α − 1 μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ρ 1 ( s ) d q s + μ ( λ − 1 ) λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) y ( s ) d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ρ 1 ( s ) d q s + ( 1 − μ ) ( λ − 1 ) λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) y ( s ) d q s − ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s − 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) y ( s ) d q s − ρ 1 ( 1 ) + Δ 2 x α − 1 − Δ 1 x α − 2 Δ ( λ − 1 ) λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) y ( s ) d q s + ∑ i = 1 n a i ρ 1 ( ξ i ) ,

where

(5) Δ ≔ Δ 1 ∑ i = 1 n a i ξ i α − 2 − Δ 2 ∑ i = 1 n a i ξ i α − 1 ≠ 0 , Δ 1 ≔ 1 − μ Γ q ( α ) η α + γ 1 − 1 Γ q ( α + γ 1 ) − ( 1 − μ ) Γ q ( α ) σ α + γ 2 − 1 Γ q ( α + γ 2 ) , Δ 2 ≔ 1 − μ Γ q ( α − 1 ) η α + γ 1 − 2 Γ q ( α + γ 1 − 1 ) − ( 1 − μ ) Γ q ( α − 1 ) σ α + γ 2 − 2 Γ q ( α + γ 2 − 1 ) .

Proof

Let u be a solution of the q-fractional boundary value problem (3). Dividing the q-fractional differential equation in (3) by λ and applying the Riemann-Liouville fractional q-integral operator of order α to both sides of the resulting equation, we obtain

(6) u ( x ) − ρ 1 ( x ) = λ − 1 λ I q α D q β [ u ( x ) − ρ 2 ( x ) ] + 1 λ I q α y ( x ) + c 1 x α − 1 + c 2 x α − 2 + c 3 x α − 3 ,

where c 1 , c 2 , c 3 ∈ R are the arbitrary constants. Since 2 < α < 3 , it follows by using the first condition ( u ( 0 ) = 0 ) in (6) that c 3 = 0 . Thus, we have

(7) u ( x ) = ρ 1 ( x ) + λ − 1 λ I q α − β [ u ( x ) − ρ 2 ( x ) ] + 1 λ I q α y ( x ) + c 1 x α − 1 + c 2 x α − 2 .

On the other hand, if Θ ∈ { γ 1 , γ 2 } , then we have

I q Θ u ( x ) = 1 Γ q ( Θ ) ∫ 0 x ( x − q s ) ( Θ − 1 ) ρ 1 ( s ) d q s + λ − 1 λ Γ q ( α − β + Θ ) ∫ 0 x ( x − q s ) ( α − β + Θ − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + 1 λ Γ q ( α + Θ ) ∫ 0 x ( x − q s ) ( α + Θ − 1 ) y ( s ) d q s + c 1 Γ q ( α ) Γ q ( α + Θ ) x α + Θ − 1 + c 2 Γ q ( α − 1 ) Γ q ( α + Θ − 1 ) x α + Θ − 2 .

Now, using the remaining boundary conditions in (7) together with the aforementioned expression, we find that

c 1 = 1 Δ ∑ i = 1 n a i ξ i α − 2 μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ρ 1 ( s ) d q s + μ ( λ − 1 ) λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) y ( s ) d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ρ 1 ( s ) d q s + ( 1 − μ ) ( λ − 1 ) λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) y ( s ) d q s − ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s − 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) y ( s ) d q s − ρ 1 ( 1 ) + Δ 2 Δ ( λ − 1 ) λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) y ( s ) d q s + ∑ i = 1 n a i ρ 1 ( ξ i ) , c 2 = − 1 Δ ∑ i = 1 n a i ξ i α − 1 μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ρ 1 ( s ) d q s + μ ( λ − 1 ) λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) y ( s ) d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ρ 1 ( s ) d q s + ( 1 − μ ) ( λ − 1 ) λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) y ( s ) d q s − ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s − 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) y ( s ) d q s − ρ 1 ( 1 ) + Δ 1 Δ ( λ − 1 ) λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) [ u ( s ) − ρ 2 ( s ) ] d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) y ( s ) d q s + ∑ i = 1 n a i ρ 1 ( ξ i ) ,

where Δ is defined in (5). Substituting the value of c 1 and c 2 in (7), we obtain the solution (4). The converse of the lemma follows by substituting (4) directly into problem (3). This completes the proof.□

In relation to problems (1)–(2), we introduce an operator T : E → E by

(8) ( T u ) ( x ) = f 1 ( x , u ( x ) ) + ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) [ u ( s ) − f 2 ( s , u ( s ) ) ] d q s + 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) f 3 ( s , u ( s ) ) d q s + 1 Δ x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) × μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) f 1 ( s , u ( s ) ) d q s + μ ( λ − 1 ) λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) [ u ( s ) − f 2 ( s , u ( s ) ) ] d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) f 3 ( s , u ( s ) ) d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) f 1 ( s , u ( s ) ) d q s + ( 1 − μ ) ( λ − 1 ) λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) [ u ( s ) − f 2 ( s , u ( s ) ) ] d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) f 3 ( s , u ( s ) ) d q s − ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) [ u ( s ) − f 2 ( s , u ( s ) ) ] d q s − 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) f 3 ( s , u ( s ) ) d q s − f 1 ( 1 , u ( 1 ) ) + Δ 2 x α − 1 − Δ 1 x α − 2 Δ ( λ − 1 ) λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) [ u ( s ) − f 2 ( s , u ( s ) ) ] d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) f 3 ( s , u ( s ) ) d q s + ∑ i = 1 n a i f 1 ( ξ i , u ( ξ i ) ) ,

where u ∈ E and x ∈ [ 0 , 1 ] . Here, E = C ( [ 0 , 1 ] , R ) is the Banach space of all continuous real-valued functions defined on [ 0 , 1 ] equipped with the norm ‖ u ‖ = sup x ∈ [ 0 , 1 ] ∣ u ( x ) ∣ , u ∈ E . Here, one can note that the fixed points of the operator T are solutions to problems (1)–(2).

In the sequel, we set the notation

(9) Ω 1 = 1 + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − μ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣ + ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ∣ Δ ∣ ∑ i = 1 n a i , Ω 2 = ∣ λ − 1 ∣ λ 1 Γ q ( α − β + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η α − β + γ 1 ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − μ ) σ α − β + γ 2 ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) + ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ∣ Δ ∣ Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β , Ω 3 = 1 λ 1 Γ q ( α + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η α + γ 1 ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − μ ) σ α + γ 2 ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) + ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ∣ Δ ∣ Γ q ( α + 1 ) ∑ i = 1 n a i ξ i α .

In our first result, we prove the uniqueness of solutions to the problems (1)–(2) with the aid of the Banach contraction mapping principle [13].

Theorem 3.2

Let f 1 , f 2 , f 3 ∈ ( [ 0 , 1 ] × R , R ) satisfy the following condition:

  1. There exist positive constants L 1 , L 2 , and L 3 such that, for each pair of elements u , v ∈ R ,

    ∣ f i ( x , u ) − f i ( x , v ) ∣ ≤ L i ∣ u − v ∣ , i = 1 , 2 , 3 , x ∈ [ 0 , 1 ] .

Then, the fractional hybrid q-difference equation (1) supplemented with nonlocal multipoint q-integral boundary conditions (2) has a unique solution on [ 0 , 1 ] , provided that

(10) Ω ≔ L 1 Ω 1 + L 3 Ω 3 + ( L 2 + 1 ) Ω 2 < 1 ,

where Ω 1 , Ω 2 , and Ω 3 are defined in (9).

Proof

Let us verify that the operator T : E → E defined by (8) satisfies hypothesis of the Banach contraction mapping principle [13]. Setting sup x ∈ [ 0 , 1 ] ∣ f i ( x , 0 ) ∣ = K i < + ∞ , i = 1 , 2 , 3 , and fixing r ≥ ( K 1 Ω 1 + K 2 Ω 2 + K 3 Ω 3 ) ⁄ ( 1 − Ω ) , we show that T B r ⊂ B r , where B r = { u ∈ E : ‖ u ‖ ≤ r } . For any u ∈ B r , x ∈ [ 0 , 1 ] , it follows by the assumption ( H 1 ) that

∣ f i ( x , u ( x ) ) ∣ ≤ ∣ f i ( x , u ( x ) ) − f i ( x , 0 ) ∣ + ∣ f i ( x , 0 ) ∣ ≤ L i r + K i , i = 1 , 2 , 3 .

Then, for any u ∈ B r , x ∈ [ 0 , 1 ] , we have

‖ T u ‖ = sup x ∈ [ 0 , 1 ] ∣ f 1 ( x , u ( x ) ) ∣ + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) − f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ × μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f 1 ( s , u ( s ) ) ∣ d q s + μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) − f 2 ( s , u ( s ) ) ∣ d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f 1 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) − f 2 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) − f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ∣ f 1 ( 1 , u ( 1 ) ) ∣ + ∣ Δ 2 x α − 1 − Δ 1 x α − 2 ∣ ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) ∣ u ( s ) − f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ∑ i = 1 n a i ∣ f 1 ( ξ i , u ( ξ i ) ) ∣ ≤ ( L 1 r + K 1 ) sup x ∈ [ 0 , 1 ] 1 + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) d q s + 1 + ∣ Δ 2 x α − 1 − Δ 1 x α − 2 ∣ ∣ Δ ∣ ∑ i = 1 n a i + ( ( 1 + L 2 ) r + K 2 ) sup x ∈ [ 0 , 1 ] ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) × ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) d q s + ∣ Δ 2 x α − 1 − Δ 1 x α − 2 ∣ ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) d q s + ( L 3 r + K 3 ) sup x ∈ [ 0 , 1 ] 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ × μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) d q s + 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) d q s + ∣ Δ 2 x α − 1 − Δ 1 x α − 2 ∣ λ ∣ Δ ∣ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) d q s ≤ ( L 1 r + K 1 ) 1 + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − μ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣ + ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ∣ Δ ∣ ∑ i = 1 n a i + ( ( 1 + L 2 ) r + K 2 ) ∣ λ − 1 ∣ λ 1 Γ q ( α − β + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) × μ η ( α − β + γ 1 ) ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − μ ) σ ( α − β + γ 2 ) ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) + ∣ Δ 2 ∣ + ∣ Δ 1 ∣ ∣ Δ ∣ Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β + ( L 3 r + K 3 ) 1 λ 1 Γ q ( α + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η ( α + γ 1 ) ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − μ ) σ ( α + γ 2 ) ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) + ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ∣ Δ ∣ Γ q ( α + 1 ) ∑ i = 1 n a i ξ i α ,

which, on using (10) and definition of r , implies that ‖ T u ‖ ≤ r . Thus, T B r ⊂ B r as u ∈ B r is an arbitrary element. For any x ∈ [ 0 , 1 ] and a pair of elements u , v ∈ R , we obtain

‖ T u − T v ‖ ≤ sup x ∈ [ 0 , 1 ] ∣ f 1 ( x , u ( x ) ) − f 1 ( x , v ( x ) ) ∣ + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) − f 2 ( s , v ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) − f 3 ( s , v ( s ) ) ∣ d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f 1 ( s , u ( s ) ) − f 1 ( s , v ( s ) ) ∣ d q s + μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ f 2 ( s , u ( s ) ) − f 2 ( s , v ( s ) ) ∣ d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ f 3 ( s , u ( s ) ) − f 3 ( s , v ( s ) ) ∣ d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f 1 ( s , u ( s ) ) − f 1 ( s , v ( s ) ) ∣ d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ f 2 ( s , u ( s ) ) − f 2 ( s , v ( s ) ) ∣ d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ f 3 ( s , u ( s ) ) − f 3 ( s , v ( s ) ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) − f 2 ( s , v ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) − f 3 ( s , v ( s ) ) ∣ d q s + ∣ f 1 ( 1 , u ( 1 ) ) − f 1 ( 1 , v ( 1 ) ) ∣ + ∣ Δ 2 x α − 1 − Δ 1 x α − 2 ∣ ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) − f 2 ( s , v ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) − f 3 ( s , v ( s ) ) ∣ d q s + ∑ i = 1 n a i ∣ f 1 ( ξ i , u ( ξ i ) ) − f 1 ( ξ i , v ( ξ i ) ) ∣ ≤ L 1 1 + 1 ∣ Δ ∣ ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η γ 1 Γ q ( γ 1 + 1 ) + ( 1 − μ ) σ γ 2 Γ q ( γ 2 + 1 ) + 1 + ∣ Δ 2 ∣ + ∣ Δ 1 ∣ ∣ Δ ∣ ∑ i = 1 n a i ‖ u − v ‖ + L 2 ∣ λ − 1 ∣ λ 1 Γ q ( α − β + 1 ) + 1 ∣ Δ ∣ ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) × μ η ( α − β + γ 1 ) Γ q ( α − β + γ 1 + 1 ) + ( 1 − μ ) σ ( α − β + γ 2 ) Γ q ( α − β + γ 2 + 1 ) + 1 Γ q ( α − β + 1 ) + ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ∣ Δ ∣ Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β ‖ u − v ‖ + L 3 1 λ 1 Γ q ( α + 1 ) + 1 ∣ Δ ∣ ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η ( α + γ 1 ) Γ q ( α + γ 1 + 1 ) + ( 1 − μ ) σ ( α + γ 2 ) Γ q ( α + γ 2 + 1 ) + 1 Γ q ( α + 1 ) + ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ∣ Δ ∣ Γ q ( α + 1 ) ∑ i = 1 n a i ξ i α ‖ u − v ‖ + ∣ λ − 1 ∣ λ 1 Γ q ( α − β + 1 ) + 1 ∣ Δ ∣ ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η ( α − β + γ 1 ) Γ q ( α − β + γ 1 + 1 ) + ( 1 − μ ) σ ( α − β + γ 2 ) Γ q ( α − β + γ 2 + 1 ) + 1 Γ q ( α − β + 1 ) + ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ∣ Δ ∣ Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β ‖ u − v ‖ ≤ Ω ‖ u − v ‖ ,

which, by condition (10), implies that T is a contraction. In consequence, we deduce by the conclusion of the Banach contraction mapping principle [13] that the operator T has a unique fixed point, which is indeed a unique solution to problem (1)–(2). The proof is complete.□

Example 3.3

Let us consider the fractional hybrid q-difference equation

(11) 8 9 D 3 ⁄ 5 11 ⁄ 5 u ( x ) − 1 1,000 tan − 1 u ( x ) − 1 3 + 1 9 D 3 ⁄ 5 9 ⁄ 8 u ( x ) − 1 50 sin u ( x ) − 1 ∣ x ∣ + 1 = 1 250 ∣ cos π x ∣ ∣ u ( x ) ∣ 1 + ∣ u ( x ) ∣ + 1 x 2 + 1 , x ∈ [ 0 , 1 ] ,

with multipoint q-integral boundary conditions

(12) u ( 0 ) = 0 , ∑ i = 1 3 a i u ( ξ i ) = 0 , u ( 1 ) = 7 8 ∫ 0 1 ⁄ 4 ( 1 ⁄ 4 − q s ) 1 9 − 1 Γ q 1 9 u ( s ) d q s + 1 8 ∫ 0 1 ⁄ 2 ( 1 ⁄ 2 − q s ) 1 8 − 1 Γ q ( 1 8 ) u ( s ) d q s ,

where α = 11 5 , q = 3 5 , β = 9 8 , η = 1 4 , σ = 1 2 , λ = 8 9 , μ = 7 8 , γ 1 = 1 9 , γ 2 = 1 8 , n = 3 , a 1 = 1 4 , a 2 = 1 2 , a 3 = 3 4 , ξ 1 = 5 8 , ξ 2 = 3 4 , ξ 3 = 7 8 , and

f 1 ( x , u ( x ) ) = 1 1,000 tan − 1 u ( x ) + 1 3 , f 2 ( x , u ( x ) ) = sin u ( x ) 50 + 1 ∣ x ∣ + 1 ,

f 3 ( x , u ( x ) ) = ∣ cos π x ∣ ∣ u ∣ 250 ( 1 + ∣ u ∣ ) + 1 x 2 + 1 .

Note that L 1 = 0.001 , L 2 = 0.02 , and L 3 = 0.004 as

∣ f 1 ( x , u ( x ) ) − f 1 ( x , v ( x ) ) ∣ ≤ 1 1,000 ( ∣ u ( x ) − v ( x ) ∣ ) , ∣ f 2 ( x , u ( x ) ) − f 2 ( x , v ( x ) ) ∣ ≤ 1 50 ( ∣ u ( x ) − v ( x ) ∣ ) ,

∣ f 3 ( x , u ( x ) ) − f 3 ( x , v ( x ) ) ∣ ≤ 1 250 ( ∣ u ( x ) − v ( x ) ∣ ) .

With the given data, it is found that Δ 1 ≃ 0.816702 , Δ 2 ≃ 0.319267 , Δ ≃ 0.805323 , Ω 1 ≃ 9.17736 , Ω 2 ≃ 0.794863 , Ω 3 ≃ 3.5155 , and Ω ≃ 0.834 < 1 . Clearly, the assumptions of Theorem 3.2 hold true. Therefore, problem (11)–(12) have a unique solution on [ 0 , 1 ] by the conclusion of Theorem 3.2.

In the next two results, we present the existence criteria for solutions to problem (1)–(2). The first result is based on Theorem 2.1 (Krasnoselskii’s fixed point theorem), while the second one relies on Theorem 2.2 (Leray-Schauder’s nonlinear alternative).

Theorem 3.4

Assume that

  1. there exist ψ 1 , ψ 2 , ψ 3 ∈ C ( [ 0 , 1 ] , R + ) such that ∣ f i ( x , u ) ∣ ≤ ψ i ( x ) , ∀ ( x , u ) ∈ [ 0 , 1 ] × R , and ‖ ψ i ‖ = sup x ∈ [ 0 , 1 ] ∣ ψ i ( x ) ∣ , i = 1 , 2 , 3 .

If Ω 2 < 1 , where Ω 2 is given in (9), then the fractional hybrid q-difference equation (1) supplemented with nonlocal multipoint q-integral boundary conditions (2) has at least one solution on [ 0 , 1 ] .

Proof

Let us define B ρ ≔ { u ∈ E : ‖ u ‖ ≤ ρ } with

(13) ρ ≥ ‖ ψ 1 ‖ Ω 1 + ‖ ψ 2 ‖ Ω 2 + ‖ ψ 3 ‖ Ω 3 1 − Ω 2 , Ω 2 < 1 ,

where Ω 1 , Ω 2 , and Ω 3 are given in (9). Clearly, B ρ is a closed, bounded, convex, and nonempty subset of the Banach space E . Now, we verify that the operator T : E → E defined by (8) satisfies the hypothesis of Krasnoselskii’s fixed point theorem (Theorem 2.1). For each x ∈ [ 0 , 1 ] , we define two operators from B ρ to E as follows:

( T 1 u ) ( x ) = ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) u ( s ) d q s + 1 Δ x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) μ ( λ − 1 ) λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) u ( s ) d q s + ( 1 − μ ) ( λ − 1 ) λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) u ( s ) d q s − ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) u ( s ) d q s + [ Δ 2 x α − 1 − Δ 1 x α − 2 ] Δ ( λ − 1 ) λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) u ( s ) d q s , ( T 2 u ) ( x ) = f 1 ( x , u ( x ) ) − ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) f 2 ( s , u ( s ) ) d q s + 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) f 3 ( s , u ( s ) ) d q s + 1 Δ x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) f 1 ( s , u ( s ) ) d q s − μ ( λ − 1 ) λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) f 2 ( s , u ( s ) ) d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) f 3 ( s , u ( s ) ) d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) f 1 ( s , u ( s ) ) d q s − ( 1 − μ ) ( λ − 1 ) λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) f 2 ( s , u ( s ) ) d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) f 3 ( s , u ( s ) ) d q s + ( λ − 1 ) λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) f 2 ( s , u ( s ) ) d q s − 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) f 3 ( s , u ( s ) ) d q s − f 1 ( 1 , u ( 1 ) ) + [ Δ 2 x α − 1 − Δ 1 x α − 2 ] Δ − ( λ − 1 ) λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) f 2 ( s , u ( s ) ) d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) f 3 ( s , u ( s ) ) d q s + ∑ i = 1 n a i f 1 ( ξ i , u ( ξ i ) ) .

For any u , v ∈ B ρ , we have

∣ T 1 u ( x ) + T 2 v ( x ) ∣ ≤ sup x ∈ [ 0 , 1 ] ∣ f 1 ( x , v ( x ) ) ∣ + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ f 2 ( s , v ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ f 3 ( s , v ( s ) ) ∣ d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f 1 ( s , v ( s ) ) ∣ d q s + μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) × ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) ∣ d q s + μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ f 2 ( s , v ( s ) ) ∣ d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ f 3 ( s , v ( s ) ) ∣ d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f 1 ( s , v ( s ) ) ∣ d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) ∣ d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) × ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ f 2 ( s , v ( s ) ) ∣ d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ f 3 ( s , v ( s ) ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ f 2 ( s , v ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ f 3 ( s , v ( s ) ) ∣ d q s + ∣ f 1 ( 1 , u ( 1 ) ) ∣ + [ ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ] ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) ∣ f 2 ( s , v ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) ∣ f 3 ( s , v ( s ) ) ∣ d q s + ∑ i = 1 n a i ∣ f 1 ( ξ i , v ( ξ i ) ) ∣ ≤ ‖ ψ 1 ‖ sup x ∈ [ 0 , 1 ] 1 + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) d q s + 1 + [ ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ] ∣ Δ ∣ ∑ i = 1 n a i + ‖ ψ 2 ‖ sup x ∈ [ 0 , 1 ] ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) d q s + [ ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ] ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) d q s + ‖ ψ 3 ‖ sup x ∈ [ 0 , 1 ] 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ × μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) d q s + 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) d q s + [ ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ] ∣ Δ ∣ 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) d q s + ρ sup x ∈ [ 0 , 1 ] ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) d q s + [ ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ] ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) d q s ≤ ‖ ψ 1 ‖ 1 + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − μ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣ + [ ∣ Δ 2 ∣ + ∣ Δ 1 ∣ ] ∣ Δ ∣ ∑ i = 1 n a i + ‖ ψ 2 ‖ ∣ λ − 1 ∣ λ 1 Γ q ( α − β + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) × μ η ( α − β + γ 1 ) ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − μ ) σ ( α − β + γ 2 ) ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) + [ ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ] ∣ Δ ∣ Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β + ‖ ψ 3 ‖ 1 λ 1 Γ q ( α + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) × μ η ( α + γ 1 ) ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − μ ) σ ( α + γ 2 ) ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) + [ ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ] ∣ Δ ∣ Γ q ( α + 1 ) ∑ i = 1 n a i ξ i α + ρ ∣ λ − 1 ∣ λ 1 Γ q ( α − β + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η ( α − β + γ 1 ) ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − μ ) σ ( α − β + γ 2 ) ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) + [ ∣ Δ 1 ∣ + ∣ Δ 2 ∣ ] ∣ Δ ∣ Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β ≤ ‖ ψ 1 ‖ Ω 1 + ‖ ψ 2 ‖ Ω 2 + ‖ ψ 3 ‖ Ω 3 + ρ Ω 2 ,

which implies that ‖ T 1 u + T 2 v ‖ ≤ ρ by condition (13) and so T 1 u + T 2 v ∈ B ρ for all u , v ∈ B ρ . From the continuity of f and g , it follows that the operator T 2 is continuous on B ρ .

In the next step, we show that the operator T 2 is compact. Let us first show that T 2 is uniformly bounded. For each u ∈ B ρ and x ∈ [ 0 , 1 ] , we have

‖ T 2 u ‖ ≤ sup x ∈ [ 0 , 1 ] ∣ f 1 ( x , u ( x ) ) ∣ + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ × μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f 1 ( s , u ( s ) ) ∣ d q s + μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f 1 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ∣ f 1 ( 1 , u ( 1 ) ) ∣ + ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ∑ i = 1 n a i ∣ f 1 ( ξ i , u ( ξ i ) ) ∣ ≤ ‖ ψ 1 ‖ 1 + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − μ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣ + ∣ Δ 2 ∣ + ∣ Δ 1 ∣ ∣ Δ ∣ ∑ i = 1 n a i + ‖ ψ 2 ‖ ∣ λ − 1 ∣ λ 1 Γ q ( α − β + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η ( α − β + γ 1 ) ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − μ ) σ ( α − β + γ 2 ) ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) + ∣ Δ 2 ∣ + ∣ Δ 1 ∣ ∣ Δ ∣ 1 Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β + ‖ ψ 3 ‖ 1 λ 1 Γ q ( α + 1 ) + ∑ i = 1 n a i ξ i α − 2 ( 1 + ξ i ) μ η ( α + γ 1 ) ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − μ ) σ ( α + γ 2 ) ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) + [ ∣ Δ 2 ∣ + ∣ Δ 1 ∣ ] ∣ Δ ∣ 1 Γ q ( α + 1 ) ∑ i = 1 n a i ξ i α = ‖ ψ 1 ‖ Ω 1 + ‖ ψ 2 ‖ Ω 2 + ‖ ψ 3 ‖ Ω 3 ,

where Ω 1 , Ω 2 , and Ω 3 are given in (9). In order to establish the equicontinuity of the operator T 2 , we assume that x 1 , x 2 ∈ [ 0 , 1 ] with x 2 > x 1 . Then, for each u ∈ B ρ , we have

∣ T 2 u ( x 2 ) − T 2 u ( x 1 ) ∣ ≤ sup x ∈ [ 0 , 1 ] ∣ f 1 ( x 2 , u ( x 2 ) ) − f 1 ( x 1 , u ( x 1 ) ) ∣ + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − β − 1 ) − ( x 1 − q s ) ( α − β − 1 ) ] ∣ f 2 ( s , u ( s ) ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ x 1 x 2 ( x 2 − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − 1 ) − ( x 1 − q s ) ( α − 1 ) ] ∣ f 3 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ x 1 x 2 ( x 2 − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + 1 ∣ Δ ∣ ∣ x 2 α − 1 − x 1 α − 1 ∣ ∑ i = 1 n a i ξ i α − 2 + ∣ x 2 α − 2 − x 1 α − 2 ∣ ∑ i = 1 n a i ξ i α − 1 × μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f 1 ( s , u ( s ) ) ∣ d q s + μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f 1 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + [ ∣ Δ 2 ∣ ∣ x 2 α − 1 − x 1 α − 1 ∣ + ∣ Δ 1 ∣ ∣ x 2 α − 2 − x 1 α − 2 ∣ ] ∣ Δ ∣ × ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) ∣ f 2 ( s , u ( s ) ) ∣ d q s + 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) ∣ f 3 ( s , u ( s ) ) ∣ d q s + ∑ i = 1 n a i ∣ f 1 ( ξ i , u ( ξ i ) ) ≤ ∣ f 1 ( x 2 , u ( x 2 ) ) − f 1 ( x 1 , u ( x 1 ) ) ∣ + ∣ λ − 1 ∣ ‖ ψ 2 ‖ λ Γ q ( α − β ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − β − 1 ) − ( x 1 − q s ) ( α − β − 1 ) ] d q s + ∣ λ − 1 ∣ ‖ ψ 2 ‖ λ Γ q ( α − β ) ∫ x 1 x 2 ( x 2 − q s ) ( α − β − 1 ) d q s + ‖ ψ 3 ‖ λ Γ q ( α ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − 1 ) − ( x 1 − q s ) ( α − 1 ) ] d q s + ‖ ψ 3 ‖ λ Γ q ( α ) ∫ x 1 x 2 ( x 2 − q s ) ( α − 1 ) d q s + ∣ x 2 ( α − 1 ) − x 1 ( α − 1 ) ∣ ∑ i = 1 n a i ξ i α − 2 + ∣ x 2 ( α − 2 ) − x 1 ( α − 2 ) ∣ ∑ i = 1 n a i ξ i α − 1 ∣ Δ ∣ ‖ ψ 1 ‖ μ η γ 1 Γ q ( γ 1 + 1 ) + ‖ ψ 2 ‖ μ ∣ λ − 1 ∣ η ( α − β + γ 1 ) λ Γ q ( α − β + γ 1 + 1 ) + ‖ ψ 3 ‖ μ η ( α + γ 1 ) λ Γ q ( α + γ 1 + 1 ) + ‖ ψ 1 ‖ ( 1 − μ ) σ γ 2 Γ q ( γ 2 + 1 ) + ‖ ψ 2 ‖ ( 1 − μ ) ∣ λ − 1 ∣ σ ( α − β + γ 2 ) λ Γ q ( α − β + γ 2 + 1 ) + ‖ ψ 3 ‖ ( 1 − μ ) σ ( α + γ 2 ) λ Γ q ( α + γ 2 + 1 ) + ‖ ψ 2 ‖ ∣ λ − 1 ∣ λ Γ q ( α − β + 1 ) + ‖ ψ 3 ‖ λ Γ q ( α + 1 ) + ∣ Δ 2 ∣ ∣ x 2 α − 1 − x 1 α − 1 ∣ + ∣ Δ 1 ∣ ∣ x 2 α − 2 − x 1 α − 2 ∣ ∣ Δ ∣ × ‖ ψ 1 ‖ ∑ i = 1 n a i + ‖ ψ 2 ‖ ∣ λ − 1 ∣ λ Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β + ‖ ψ 3 ‖ λ Γ q ( α + 1 ) ∑ i = 1 n a i ξ i α .

Observe that the right-hand side of the aforementioned inequality is independent of u ∈ B ρ and tends to zero as x 1 → x 2 . This shows that T 2 is equicontinuous. Therefore, the operator T 2 is relatively compact on B ρ , and hence, by the Arzelá-Ascoli theorem, T 2 is completely continuous. In consequence, T 2 is the compact operator on B ρ .

Finally, we prove that the operator T 1 is a contraction. For any u , v ∈ B ρ and x ∈ [ 0 , 1 ] , we obtain

‖ T 1 u − T 1 v ‖ ≤ sup x ∈ [ 0 , 1 ] ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ × μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) × ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + [ ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ] ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s ≤ Ω 2 ‖ u − v ‖ .

Since Ω 2 < 1 , it follows from the aforementioned inequality that T 1 is a contraction. Thus, all the assumptions of Theorem 2.1 (Krasnoselskii’s fixed point theorem) are satisfied. Therefore, the fractional hybrid q-difference equation (1) with nonlocal multipoint q-integral boundary conditions (2) has at least one solution on [ 0 , 1 ] . This completes the proof.□

Example 3.5

Consider the fractional hybrid q-difference equation

(14) 8 9 D 3 ⁄ 5 11 ⁄ 5 u ( x ) − 1 9 x 2 ∣ u ( x ) ∣ 1 + ∣ u ( x ) ∣ + 4 + 1 9 D 3 ⁄ 5 9 ⁄ 8 u ( x ) − 1 2 x ∣ cos u ( x ) ∣ + 1 4 = 1 5 [ x ∣ tan − 1 u ( x ) ∣ + 7 ] ,

subject to multipoint q-integral boundary conditions (12) with

f 1 ( x , u ) = 1 9 x 2 ∣ u ∣ 1 + ∣ u ∣ + 4 , f 2 ( x , u ) = 1 2 x ∣ cos u ∣ + 1 4 , f 3 ( x , u ) = 1 5 ( x ∣ tan − 1 u ∣ + 7 ) .

It is easy to check that there exist continuous functions

ψ 1 ( x ) = x 2 + 4 9 , ψ 2 ( x ) = 4 x + 1 8 , ψ 3 ( x ) = π x + 14 10 ,

on [ 0 , 1 ] . Also, we have ‖ ψ 1 ‖ = sup x ∈ [ 0 , 1 ] ψ 1 ( x ) ≃ 0.555556 , ‖ ψ 2 ‖ = sup x ∈ [ 0 , 1 ] ψ 2 ( x ) ≃ 0.625 and ‖ ψ 3 ‖ = sup x ∈ [ 0 , 1 ] ψ 3 ( x ) ≃ 1.714159 . As in Example 11, Ω 2 ≃ 0.794863 < 1 . Clearly, all the assumptions of Theorem 3.4 are satisfied. Therefore, by the conclusion of Theorem 3.4, the fractional q-integro-difference equation (14) with multipoint q-integral boundary conditions (12) has at least one solution on [ 0 , 1 ] .

Theorem 3.6

Assume that the following conditions hold:

  1. There exist continuous nondecreasing functions χ 1 , χ 2 , χ 3 : [ 0 , ∞ ) → ( 0 , ∞ ) and functions ϕ 1 , ϕ 2 , ϕ 3 ∈ C ( [ 0 , 1 ] , R + ) such that ∣ f i ( x , u ) ∣ ≤ ϕ i ( x ) χ i ( ∣ u ∣ ) , i = 1 , 2 , 3 , for each ( x , u ) ∈ [ 0 , 1 ] × R ;

  2. There exists a constant G > 0 such that

    ( 1 − Ω 2 ) G ‖ ϕ 1 ‖ χ 1 ( G ) Ω 1 + ‖ ϕ 2 ‖ χ 2 ( G ) Ω 2 + ‖ ϕ 3 ‖ χ 3 ( G ) Ω 3 > 1 , Ω 2 < 1 ,

    where Ω 1 , Ω 2 , and Ω 3 are defined in (9).

Then, the fractional hybrid q-difference equation (1) with nonlocal multipoint q-integral boundary conditions (2)has at least one solution on [ 0 , 1 ] .

Proof

Let us verify the hypotheses of the Leray-Schauder’s nonlinear alternative (Theorem 2.2) in several steps. We first show that the operator T : E → E defined by (8) maps bounded sets (balls) into bounded sets in E . For a positive number ω , let B ω = { u ∈ E : ‖ u ‖ ≤ ω } be a bounded ball in E . Then, as argued in the proof of the last result, we obtain

‖ T u ‖ ≤ ‖ ϕ 1 ‖ χ 1 ( ω ) sup x ∈ [ 0 , 1 ] 1 + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) d q s + ( 1 − μ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) d q s + 1 + [ ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ] ∣ Δ ∣ ∑ i = 1 n a i + ( ω + ‖ ϕ 2 ‖ χ 2 ( ω ) ) sup x ∈ [ 0 , 1 ] ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ ∣ λ − 1 ∣ λ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) d q s + ( 1 − μ ) ∣ λ − 1 ∣ λ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) d q s + ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) d q s + ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ∣ Δ ∣ ∣ λ − 1 ∣ λ Γ q ( α − β ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − β − 1 ) d q s + ‖ ϕ 3 ‖ χ 3 ( ω ) sup x ∈ [ 0 , 1 ] 1 λ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) d q s + x α − 2 ∑ i = 1 n a i ξ i α − 2 ( x − ξ i ) ∣ Δ ∣ μ λ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) d q s + ( 1 − μ ) λ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) d q s + 1 λ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) d q s + [ ∣ Δ 2 ∣ x α − 1 + ∣ Δ 1 ∣ x α − 2 ] ∣ Δ ∣ 1 λ Γ q ( α ) ∑ i = 1 n a i ∫ 0 ξ i ( ξ i − q s ) ( α − 1 ) d q s ≤ ‖ ϕ 1 ‖ χ 1 ( ω ) Ω 1 + ‖ ϕ 2 ‖ χ 2 ( ω ) Ω 2 + ‖ ϕ 3 ‖ χ 3 ( ω ) Ω 3 + ω Ω 2 .

Secondly, we show that the operator T : E → E defined by (8) maps bounded sets into equicontinuous sets of E . Let x 1 , x 2 ∈ [ 0 , 1 ] with x 1 < x 2 and u ∈ B ω . Then, we have

∣ T u ( x 2 ) − T u ( x 1 ) ≤ ∣ f 1 ( x 2 , u ( x 2 ) ) − f 1 ( x 1 , u ( x 1 ) ) ∣ + ( ω + ‖ ϕ 2 ‖ χ 2 ( ω ) ) ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − β − 1 ) − ( x 1 − q s ) ( α − β − 1 ) ] d q s

+ ( ω + ‖ ϕ 2 ‖ χ 2 ( ω ) ) ∣ λ − 1 ∣ λ Γ q ( α − β ) ∫ x 1 x 2 ( x 2 − q s ) ( α − β − 1 ) d q s + ‖ ϕ 3 ‖ χ 3 ( ω ) λ Γ q ( α ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − 1 ) − ( x 1 − q s ) ( α − 1 ) ] d q s + ‖ ϕ 3 ‖ χ 3 ( ω ) λ Γ q ( α ) ∫ x 1 x 2 ( x 2 − q s ) ( α − 1 ) d q s + 1 ∣ Δ ∣ ( x 2 α − 1 − x 1 α − 1 ) ∑ i = 1 n a i ξ i α − 2 + ( x 2 α − 2 − x 1 α − 2 ) ∑ i = 1 n a i ξ i α − 1 ‖ ϕ 1 ‖ χ 1 ( ω ) μ η γ 1 Γ q ( γ 1 + 1 ) + μ ( ω + ‖ ϕ 2 ‖ χ 2 ( ω ) ) ∣ λ − 1 ∣ η α − β + γ 1 λ Γ q ( α − β + γ 1 + 1 ) + μ ‖ ϕ 3 ‖ χ 3 ( ω ) η ( α + γ 1 ) λ Γ q ( α + γ 1 + 1 ) + ‖ ϕ 1 ‖ χ 1 ( ω ) ( 1 − μ ) σ γ 2 Γ q ( γ 2 + 1 ) + ( ω + ‖ ϕ 2 ‖ χ 2 ( ω ) ) ( 1 − μ ) ∣ λ − 1 ∣ σ ( α − β + γ 2 ) λ Γ q ( α − β + γ 2 + 1 ) + ‖ ϕ 3 ‖ χ 3 ( ω ) ( 1 − μ ) σ ( α + γ 2 ) λ Γ q ( α + γ 2 + 1 ) + ( ω + ‖ ϕ 2 ‖ χ 2 ( ω ) ) ∣ λ − 1 ∣ λ Γ q ( α − β + 1 ) + ‖ ϕ 3 ‖ χ 3 ( ω ) λ Γ q ( α + 1 ) + ∣ Δ 2 ∣ ( x 2 α − 1 − x 1 α − 1 ) + ∣ Δ 1 ∣ ( x 2 α − 1 − x 1 α − 1 ) ∣ Δ ∣ ‖ ϕ 1 ‖ χ 1 ( ω ) ∑ i = 1 n a i + ( ω + ‖ ϕ 2 ‖ χ 2 ( ω ) ) ∣ λ − 1 ∣ λ Γ q ( α − β + 1 ) ∑ i = 1 n a i ξ i α − β + ‖ ϕ 3 ‖ χ 3 ( ω ) λ Γ q ( α + 1 ) ∑ i = 1 n a i ξ i α .

Obviously, the right-hand side of the aforementioned inequality tends to zero independently of u ∈ B ω as x 2 → x 1 . Therefore, it follows by the Arzelá-Ascoli theorem that T : E → E is completely continuous.

In order to complete the hypothesis of Theorem 2.2 (Leray-Schauder’s nonlinear alternative), it will be shown that the set of all solutions to the equation u = θ T u is bounded for θ ∈ [ 0 , 1 ] . For that, let u be a solution of u = θ T u for θ ∈ [ 0 , 1 ] . Then, for x ∈ [ 0 , 1 ] , we apply the strategy used in the first step to obtain

‖ u ‖ ≤ Ω 1 ‖ ϕ 1 ‖ χ 1 ( ‖ u ‖ ) + Ω 2 ‖ ϕ 2 ‖ χ 2 ( ‖ u ‖ ) + Ω 3 ‖ ϕ 3 ‖ χ 3 ( ‖ u ‖ ) + Ω 2 ‖ u ‖ .

Consequently, we have

( 1 − Ω 2 ) ‖ u ‖ ‖ ϕ 1 ‖ χ 1 ( ‖ u ‖ ) Ω 1 + ‖ ϕ 2 ‖ χ 2 ( ‖ u ‖ ) Ω 2 + ‖ ϕ 3 ‖ χ 3 ( ‖ u ‖ ) Ω 3 ≤ 1 .

By the condition ( H 5 ) , we can find a positive number G such that ‖ u ‖ ≠ G . Introduce a set

(15) U = { u ∈ E : ‖ u ‖ < G } ,

and observe that the operator T : U ¯ → E is continuous and completely continuous ( U ¯ is closure of U ). With this choice of U , we cannot find u ∈ ∂ U ( ∂ U is boundary of U ) satisfying u = θ T u . Therefore, it follows by Theorem 2.2 that the operator T has a fixed point in U ¯ . Thus, there exists a solution of problems (1)–(2) on [ 0 , 1 ] . The proof is complete.□

Example 3.7

Consider the fractional hybrid q-difference equation

(16) 8 9 D 3 ⁄ 5 11 ⁄ 5 u ( x ) − ∣ sin u ( x ) ∣ ( 10 + x ) 2 + 1 9 D 3 ⁄ 5 9 ⁄ 8 u ( x ) − ∣ u ( x ) ∣ ( 8 + x ) 2 ( 1 + ∣ u ( x ) ∣ ) = ∣ u 2 ( x ) ∣ ( 9 + x ) 2 ( 1 + ∣ u 2 ( x ) ∣ ) , x ∈ [ 0 , 1 ] ,

subject to multipoint q-integral boundary conditions (12). Here,

f 1 ( x , u ) = ∣ sin u ( x ) ∣ ( 10 + x ) 2 , f 2 ( x , u ) = ∣ u ( x ) ∣ ( 8 + x ) 2 ( 1 + ∣ u ( x ) ∣ ) , f 3 ( x , u ) = ∣ u 2 ( x ) ∣ ( 9 + x ) 2 ( 1 + ∣ u 2 ( x ) ∣ ) .

With the given data, we obtain Ω 1 ≃ 9.17736 , Ω 2 ≃ 0.794863 , Ω 3 ≃ 3.5155 , and

∣ f 1 ( x , u ) ∣ ≤ ∣ sin u ( x ) ∣ ( 10 + x ) 2 = ϕ 1 ( x ) χ 1 ( ∣ u ∣ ) , ∣ f 2 ( x , u ) ∣ ≤ 1 ( 8 + x ) 2 ∣ u ( x ) ∣ ( 1 + ∣ u ( x ) ∣ ) = ϕ 2 ( x ) χ 2 ( ∣ u ∣ ) , ∣ f 3 ( x , u ) ∣ ≤ 1 ( 9 + x ) 2 ∣ u 2 ( x ) ∣ ( 1 + ∣ u 2 ( x ) ∣ ) = ϕ 3 ( x ) χ 3 ( ∣ u ∣ ) .

Clearly, ϕ 1 ( x ) = 1 ( 10 + x ) 2 , ϕ 2 ( x ) = 1 ( 8 + x ) 2 , ϕ 3 ( x ) = 1 ( 9 + x ) 2 , χ 1 ( ∣ ∣ u ∣ ∣ ) = ∣ ∣ u ∣ ∣ , χ 2 ( ∣ ∣ u ∣ ∣ ) = χ 3 ( ∣ ∣ u ∣ ∣ ) = 1 . Moreover, condition ( H 5 ) implies that there exists G > 0.498942 . Thus, the hypotheses of Theorem 3.6 are satisfied. Therefore, the conclusion of Theorem 3.6 implies that the hybrid q-difference equation (16) with multipoint q-integral boundary conditions (12) has at least one solution on [ 0 , 1 ] .

4 Conclusions

We have developed the existence theory for a higher-order fractional q-difference equation involving Riemann-Liouville q-derivative operators with dual hybrid terms, supplemented with nonlocal multipoint q-integral boundary conditions. The standard tools of the fixed point theory are applied to derive the desired results. Our work is a valuable contribution to the literature on nonlocal hybrid q-fractional integral boundary value problems. Our results correspond to some new ones as special cases by fixing the parameters involved in the problem appropriately. Here are two examples.

  1. For μ = 1 , our results correspond to the ones for the following problem:

    λ D q α [ u ( x ) − f 1 ( x , u ( x ) ) ] + ( 1 − λ ) D q β [ u ( x ) − f 2 ( x , u ( x ) ) ] = f 3 ( x , u ( x ) ) , 0 < x < 1 , u ( 0 ) = 0 , u ( 1 ) = ∫ 0 η ( η − q s ) ( γ 1 − 1 ) Γ q ( γ 1 ) u ( s ) d q s , ∑ i = 1 n a i u ( ξ i ) = 0 .

  2. For λ = 1 , we obtain the existence results for the following equation:

    D q α [ u ( x ) − f 1 ( x , u ( x ) ) ] = f 3 ( x , u ( x ) ) , 0 < x < 1 ,

    subject to the boundary conditions (2).

Acknowledgements

The authors are grateful for the reviewers’ valuable comments that improved the original manuscript.

  1. Funding information: The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia, has funded this project, under Grant No (KEP-PhD: 35-130-1443).

  2. Author contributions: All authors accept responsibility for the content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. All authors contributed equally to the preparation of this manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2024-02-07
Revised: 2024-07-12
Accepted: 2024-07-27
Published Online: 2024-09-19

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/math-2024-0051
Keywords for this article
q-difference equation; Riemann-Liouville q-derivative; q-integral; nonlocal boundary conditions; existence; fixed point
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  59. On Laguerre-Sobolev matrix orthogonal polynomials
  60. Pullback attractors for fractional lattice systems with delays in weighted space
  61. Singularities of spherical surface in R4
  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
  64. On the energy decay of a coupled nonlinear suspension bridge problem with nonlinear feedback
  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
  66. Hardy-type inequalities for a class of iterated operators and their application to Morrey-type spaces
  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
  90. Integral mean estimates of Turán-type inequalities for the polar derivative of a polynomial with restricted zeros
  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
  113. On k-prime graphs
  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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