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Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions

  • Ahmed Alsaedi , Bashir Ahmad EMAIL logo , Hana Al-Hutami and Boshra Alharbi
Published/Copyright: May 2, 2023
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this article, we introduce and study a new class of hybrid fractional q -integro-difference equations involving Riemann-Liouville q -derivatives, supplemented with nonlocal boundary conditions containing Riemann-Liouville q -integrals of different orders. The existence of a unique solution to the given problem is shown by applying Banach’s fixed point theorem. We also present the existing criteria for solutions to the problem at hand by applying Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Illustrative examples are given to demonstrate the application of the obtained results. Some new results follow as special cases of this work.

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 hybrid fractional q -integro-difference equation involving Riemann-Liouville q -derivatives and nonlocal q -integral boundary conditions. In precise terms, we study the following nonlocal hybrid q -fractional integral boundary value problem:

(1) υ D q α [ u ( x ) − f ( x , u ( x ) ) ] + ( 1 − υ ) D q β u ( x ) = a g ( x , u ( x ) ) + b I q ε h ( x , u ( x ) ) , 0 < x < 1 ,

(2) u ( 0 ) = 0 , u ( 1 ) = ξ ∫ 0 η ( η − q s ) ( γ 1 − 1 ) Γ q ( γ 1 ) u ( s ) d q s + ( 1 − ξ ) ∫ 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 , 1 < α , β ≤ 2 , 0 < ε < 1 , 0 < υ ≤ 1 , 0 ≤ ξ ≤ 1 , α − β > 0 , 0 < η , σ < 1 , D q α , and D q β denote the Riemann-Liouville fractional q -derivatives of order α and β , respectively, I q ε denotes the Riemann-Liouville fractional q -integral of order ε , and f , g , h : [ 0 , 1 ] × R → R are continuous functions, a , b ∈ R .

Let us now dwell on some recent works on fractional q -difference equations. In 2013, Zhou and Liu [1] applied Mönch’s fixed point theorem together with the technique of measure of weak noncompactness to investigate the existence of solutions for the following fractional q -difference equation with boundary conditions:

D q α c u ( t ) + f ( t , u ( t ) ) = 0 , 0 ≤ t ≤ 1 , 0 < q < 1 , u ( 0 ) = ( D q 2 u ) ( 0 ) = 0 , γ ( D q u ) ( 1 ) + β ( D q 2 u ) ( 1 ) = 0 ,

where 2 < α ≤ 3 , γ , β ≥ 0 , and f : [ 0 , 1 ] × R → R is a continuous function.

In [2], the authors studied the following nonlinear boundary value problem of fractional q -integro-difference equation:

( λ D q α + ( 1 − λ ) D q β ) u ( t ) = a f ( t , u ( t ) ) + b I q δ g ( t , u ( t ) ) , t ∈ [ 0 , 1 ] , a , b ∈ R + , u ( 0 ) = 0 , μ ∫ 0 1 ( 1 − q s ) ( γ 1 − 1 ) Γ q ( γ 1 ) u ( s ) d q s + ( 1 − μ ) ∫ 0 1 ( 1 − q s ) ( γ 2 − 1 ) Γ q ( γ 2 ) u ( s ) d q s = 0 , γ 1 , γ 2 > 0 ,

where 0 < q < 1 , 1 < α , β < 2 , 0 < δ < 1 , 0 < λ ≤ 1 , 0 ≤ μ ≤ 1 , α − β > 1 , and D q α denotes the Riemann-Liouville fractional q -derivative of order α , and f , g : [ 0 , 1 ] × R → R are continuous functions. For some more results on boundary value problems involving fractional q -difference operators, we refer the reader to previous studies [3–11].

Recently, in [12], a coupled system of nonlinear fractional q -integro-difference equations equipped with coupled q -integral boundary conditions was studied. In [13], the authors proved some existence results for a Langevin-type q -variant system of nonlinear fractional integro-difference equations with nonlocal boundary conditions. On the other hand, for some recent works on fractional differential equations, for example, see [14–19].

The objective of this present work is to investigate the criteria ensuring the existence and uniqueness of solutions to problems (1) and (2). We make use of the Banach fixed point theorem [20] to obtain a uniqueness result, while two existence results are established by means of Krasnoselskii’s fixed point theorem [21] and Leray-Schauder’s nonlinear alternative [22]. Moreover, our results generalize the ones presented in [2] in the sense that we consider a hybrid Riemann-Liouville-type fractional q -integro-difference equation subject to nonlocal Riemann-Liouville q -integral boundary conditions. The second condition in (2) can be interpreted as the value of the unknown function u ( x ) at x = 1 is proportional to the sum of its two-strip contributions formulated in terms of Riemann-Liouville integral on the segments ( 0 , η ) and ( 0 , σ ) .

We arrange the remainder of this article as follows. In Section 2, we recall some basic concepts related to this study. Section 3 contains the main results, while examples illustrating the obtained results are presented in Section 4.

2 Auxiliary material

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

For every a ∈ R , the q -number [ a ] q is defined by [ a ] q = 1 − q a 1 − q , where q ∈ ( 0 , 1 ) is an arbitrary real number. Also, the q -shifted factorial of real number a is defined by ( a ; q ) 0 = 1 and ( a ; q ) n = ∏ j = 0 n − 1 ( 1 − a q j ) for 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 a real number, then ( a − b ) ( ϱ ) = a ϱ ∏ j = 0 ∞ a − b q j a − b q ϱ + j and a ( ϱ ) = a ϱ when b = 0 . 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 ) = [ α ] q Γ q ( ϱ ) [23].

Definition 2.1

[23] 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 = t ϱ ( 1 − q ) ϱ ∑ k = 0 ∞ q k ( q ϱ ; q ) k ( q ; q ) k u ( t q k ) , ϱ > 0 , t ∈ ( 0 , ∞ ) .

Definition 2.2

[24] The fractional q -derivative of the Riemann-Liouville type of order ϱ ≥ 0 is defined by ( D q 0 u ) ( t ) = u ( t ) and

( D q ϱ u ) ( t ) = ( D q m I q m − ϱ u ) ( t ) , ϱ > 0 ,

where m is the smallest integer greater than or equal to ϱ , I q ( ⋅ ) is the Riemann-Liouville fractional q -integral of order ( ⋅ ) , and D q m is the q -derivative of integer order m .

Alternatively, the Riemann-Liouville fractional q -derivative of order ϱ > 0 for a function u : ( 0 , ∞ ) → R is defined by [23]:

D q ϱ u ( t ) = 1 Γ q ( n − ϱ ) ∫ 0 t u ( s ) ( t − q s ) ϱ − n + 1 d q s , n − 1 < ϱ < n .

Recall that I q β I q ϱ u ( t ) = I q β + ϱ u ( t ) for ϱ , β ∈ R + [23]. Further, according to Lemma 2.8 in [13],

I q ϱ ( ( x − a ) ( σ ) ) = Γ q ( σ + 1 ) Γ q ( ϱ + σ + 1 ) ( x − a ) ( ϱ + σ ) , 0 < a < x < b , ϱ ∈ R + , σ ∈ ( − 1 , ∞ ) .

In particular, for σ = 0 and a = 0 , using q -integration by parts, we have

( I q ϱ 1 ) ( x ) = 1 Γ q ( ϱ ) ∫ 0 x ( x − q t ) ( ϱ − 1 ) d q t = 1 Γ q ( ϱ ) ∫ 0 x D q ( ( x − t ) ( ϱ ) ) − [ ϱ ] q d q t = − 1 Γ q ( ϱ + 1 ) ∫ 0 x D q ( ( x − t ) ( ϱ ) ) d q t = 1 Γ q ( ϱ + 1 ) x ( ϱ ) .

3 Main results

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

Lemma 3.1

Let y ∈ C ( [ 0 , 1 ] , R ) and

(3) Δ ≔ 1 − ξ Γ q ( α ) η α + γ 1 − 1 Γ q ( α + γ 1 ) − ( 1 − ξ ) Γ q ( α ) σ α + γ 2 − 1 Γ q ( α + γ 2 ) ≠ 0 .

The function u is a solution for the fractional q -difference boundary value problem

(4) υ D q α [ u ( x ) − f ( x , u ( x ) ) ] + ( 1 − υ ) D q β u ( x ) = y ( x ) , 0 < x < 1 , u ( 0 ) = 0 , u ( 1 ) = ξ ∫ 0 η ( η − q s ) ( γ 1 − 1 ) Γ q ( γ 1 ) u ( s ) d q s + ( 1 − ξ ) ∫ 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 for the fractional q -integral equation

(5) u ( x ) = f ( x , u ( x ) ) + ( υ − 1 ) υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) u ( s ) d q s + 1 υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) y ( s ) d q s + x α − 1 Δ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) f ( s , u ( s ) ) d q s + ξ ( υ − 1 ) υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) u ( s ) d q s + ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) y ( s ) d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) f ( s , u ( s ) ) d q s + ( 1 − ξ ) ( υ − 1 ) υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) u ( s ) d q s + ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) y ( s ) d q s − ( υ − 1 ) υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) u ( s ) d q s − 1 υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) y ( s ) d q s − f ( 1 , u ( 1 ) ) .

Proof

Let u be a solution of the q -fractional boundary value problem (4). Then, we have

D q α [ u ( x ) − f ( x , u ( x ) ) ] = υ − 1 υ D q β u ( x ) + 1 υ y ( x ) .

Taking the Riemann-Liouville fractional q -integral of order α to both sides of the above equation, we obtain

u ( x ) − f ( x , u ( x ) ) = υ − 1 υ I q α D q β u ( x ) + 1 υ I q α y ( x ) + c 1 x α − 1 + c 2 x α − 2 ,

where c 1 and c 2 ∈ R are the arbitrary constants. Since 1 < α < 2 , it follows from the first boundary condition that c 2 = 0 . Thus,

(6) u ( x ) = f ( x , u ( x ) ) + υ − 1 υ I q α − β u ( x ) + 1 υ I q α y ( x ) + c 1 x α − 1 .

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 ) f ( s , u ( s ) ) d q s + υ − 1 υ Γ q ( α − β + Θ ) ∫ 0 x ( x − q s ) ( α − β + Θ − 1 ) u ( s ) d q s + 1 υ 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 .

Now, by using the second boundary condition and substituting the values Θ ∈ { γ 1 , γ 2 } into the aforementioned expression, we find that

c 1 = 1 Δ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) f ( s , u ( s ) ) d q s + ξ ( υ − 1 ) υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) u ( s ) d q s + ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) y ( s ) d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) f ( s , u ( s ) ) d q s + ( 1 − ξ ) ( υ − 1 ) υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) u ( s ) d q s + ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) y ( s ) d q s − ( υ − 1 ) υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) u ( s ) d q s − 1 υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) y ( s ) d q s − f ( 1 , u ( 1 ) ) ,

where Δ is defined in (3). Substituting the value of c 1 in (6), we obtain solution (5). Conversely, it is clear that u is a solution for the fractional q -difference equation (4) whenever u is a solution for the fractional q -integral equation (5). This completes the proof.□

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

(7) ( T u ) ( x ) = f ( x , u ( x ) ) + ( υ − 1 ) υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) u ( s ) d q s + a υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) g ( s , u ( s ) ) d q s + b υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) h ( s , u ( s ) ) d q s + x α − 1 Δ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) f ( s , u ( s ) ) d q s + ξ ( υ − 1 ) υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) u ( s ) d q s + a ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) g ( s , u ( s ) ) d q s + b ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) h ( s , u ( s ) ) d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) f ( s , u ( s ) ) d q s + ( 1 − ξ ) ( υ − 1 ) υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) u ( s ) d q s + a ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) g ( s , u ( s ) ) d q s + b ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) h ( s , u ( s ) ) d q s − ( υ − 1 ) υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) u ( s ) d q s − a υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) g ( s , u ( s ) ) d q s − b υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) h ( s , u ( s ) ) d q s − f ( 1 , u ( 1 ) ) ,

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) and (2).

In the sequel, we set

(8) Ω 1 = 1 + ξ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − ξ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣ , Ω 2 = ∣ a ∣ υ 1 Γ q ( α + 1 ) + ξ η α + γ 1 ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − ξ ) σ α + γ 2 ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) , Ω 3 = ∣ b ∣ υ 1 Γ q ( α + ε + 1 ) + ξ η α + ε + γ 1 ∣ Δ ∣ Γ q ( α + ε + γ 1 + 1 ) + ( 1 − ξ ) σ α + ε + γ 2 ∣ Δ ∣ Γ q ( α + ε + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + ε + 1 ) , Ω 4 = ∣ υ − 1 ∣ υ 1 Γ q ( α − β + 1 ) + ξ η α − β + γ 1 ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − ξ ) σ α − β + γ 2 ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) .

In our first result, we prove the uniqueness of solutions to problems (1) and (2) with the aid of the Banach contraction mapping principle [20].

Theorem 3.2

Let f , g , h ∈ ( [ 0 , 1 ] × R , R ) satisfy the following conditions:

  1. there exists a positive constant L 1 such that, for each pair of elements u , v ∈ R ,

    ∣ f ( x , u ) − f ( x , v ) ∣ ≤ L 1 ∣ u − v ∣ , x ∈ [ 0 , 1 ] ;

  2. there exists a positive constant L 2 such that, for each pair of elements u , v ∈ R ,

    ∣ g ( x , u ) − g ( x , v ) ∣ ≤ L 2 ∣ u − v ∣ , x ∈ [ 0 , 1 ] ;

  3. there exists a positive constant L 3 such that, for each pair of elements u , v ∈ R ,

    ∣ h ( x , u ) − h ( x , v ) ∣ ≤ L 3 ∣ u − v ∣ , x ∈ [ 0 , 1 ] .

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

(9) Ω ≔ L 1 Ω 1 + L 2 Ω 2 + L 3 Ω 3 + Ω 4 < 1 ,

where Ω 1 , Ω 2 , Ω 3 , and Ω 4 are defined by (8).

Proof

Let us verify that the operator T : E → E defined by (7) satisfies the hypothesis of the Banach contraction mapping principle [20]. Setting sup x ∈ [ 0 , 1 ] ∣ f ( x , 0 ) ∣ = K 1 < + ∞ , sup x ∈ [ 0 , 1 ] ∣ g ( x , 0 ) ∣ = K 2 < + ∞ , and sup x ∈ [ 0 , 1 ] ∣ h ( x , 0 ) ∣ = K 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 assumptions ( H 1 ) , ( H 2 ) , and ( H 3 ) that

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

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

‖ T u ‖ = sup x ∈ [ 0 , 1 ] ∣ f ( x , u ( x ) ) ∣ + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + x ( α − 1 ) ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s

+ ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ∣ f ( 1 , u ( 1 ) ) ∣

≤ ( L 1 r + K 1 ) sup x ∈ [ 0 , 1 ] 1 + x ( α − 1 ) ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) d q s + 1 + ( L 2 r + K 2 ) sup x ∈ [ 0 , 1 ] ∣ a ∣ υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) d q s + ( L 3 r + K 3 ) sup x ∈ [ 0 , 1 ] ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) d q s + r sup x ∈ [ 0 , 1 ] ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) d q s

≤ ( L 1 r + K 1 ) 1 + ξ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − ξ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣ + ( L 2 r + K 2 ) ∣ a ∣ υ 1 Γ q ( α + 1 ) + ξ η ( α + γ 1 ) ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − ξ ) σ ( α + γ 2 ) ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) + ( L 3 r + K 3 ) ∣ b ∣ υ 1 Γ q ( α + ε + 1 ) + ξ η ( α + ε + γ 1 ) ∣ Δ ∣ Γ q ( α + ε + γ 1 + 1 ) + ( 1 − ξ ) σ ( α + ε + γ 2 ) ∣ Δ ∣ Γ q ( α + ε + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + ε + 1 ) + r ∣ υ − 1 ∣ υ 1 Γ q ( α − β + 1 ) + ξ η ( α − β + γ 1 ) ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − ξ ) σ ( α − β + γ 2 ) ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) ,

which, on using (9) 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 ( x , u ( x ) ) − f ( x , v ( x ) ) ∣ + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) − g ( s , v ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) − h ( s , v ( s ) ) ∣ d q s + x α − 1 ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f ( s , u ( s ) ) − f ( s , v ( s ) ) ∣ d q s + ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ g ( s , u ( s ) ) − g ( s , v ( s ) ) ∣ d q s + ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) ∣ h ( s , u ( s ) ) − h ( s , v ( s ) ) ∣ d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f ( s , u ( s ) ) − f ( s , v ( s ) ) ∣ d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ g ( s , u ( s ) ) − g ( s , v ( s ) ) ∣ d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) ∣ h ( s , u ( s ) ) − h ( s , v ( s ) ) ∣ d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) − g ( s , v ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) − h ( s , v ( s ) ) ∣ d q s + ∣ f ( 1 , u ( 1 ) ) − f ( 1 , v ( 1 ) ) ∣ ≤ L 1 1 + ξ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − ξ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣ ‖ u − v ‖ + L 2 ∣ a ∣ υ 1 Γ q ( α + 1 ) + ξ η ( α + γ 1 ) ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − ξ ) σ ( α + γ 2 ) ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) ‖ u − v ‖

+ L 3 ∣ b ∣ υ 1 Γ q ( α + ε + 1 ) + ξ η ( α + ε + γ 1 ) ∣ Δ ∣ Γ q ( α + ε + γ 1 + 1 ) + ( 1 − ξ ) σ ( α + ε + γ 2 ) ∣ Δ ∣ Γ q ( α + ε + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + ε + 1 ) ‖ u − v ‖ + ∣ υ − 1 ∣ υ 1 Γ q ( α − β + 1 ) + ξ η ( α − β + γ 1 ) ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − ξ ) σ ( α − β + γ 2 ) ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) ‖ u − v ‖ ≤ Ω ‖ u − v ‖ ,

which, by condition (9), implies that T is a contraction. In consequence, it follows by the conclusion of the Banach contraction mapping principle [20] that the operator T has a unique fixed point, which is indeed the unique solution of problems (1) and (2). The proof is completed.□

In the next two results, we present the existence criteria for solutions to problems (1) and (2). The first result is based on Krasnoselskii’s fixed point theorem [21], while the second one relies on Leray-Schauder nonlinear alternative [22].

Theorem 3.3

Assume that

( H 4 ) there exist ψ 1 , ψ 2 , and ψ 3 ∈ C ( [ 0 , 1 ] , R + ) with

∣ f ( x , u ) ∣ ≤ ψ 1 ( x ) , ∣ g ( x , u ) ∣ ≤ ψ 2 ( x ) , ∣ h ( x , u ) ∣ ≤ ψ 3 ( x ) ∀ ( x , u ) ∈ [ 0 , 1 ] × R ,

and ‖ ψ i ‖ = sup x ∈ [ 0 , 1 ] ∣ ψ i ( x ) ∣ , i = 1 , 2 , 3 .

If Ω 4 < 1 , where Ω 4 is given in (8), then the fractional hybrid q-difference equation (1) with q-integral nonlocal boundary conditions (2) has at least one solution on [ 0 , 1 ] .

Proof

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

(10) ρ ≥ ‖ ψ 1 ‖ Ω 1 + ‖ ψ 2 ‖ Ω 2 + ‖ ψ 3 ‖ Ω 3 1 − Ω 4 , Ω 4 < 1 ,

where Ω 1 , Ω 2 , Ω 3 , and Ω 4 are given in (8). Clearly B ρ is a closed, bounded, convex, and nonempty subset of Banach space E . Now we verify that the operator T : E → E defined by (7) satisfies the hypothesis of Krasnoselskii’s fixed point theorem [21]. For each x ∈ [ 0 , 1 ] , we define two operators from B ρ to E as follows:

(11) ( T 1 u ) ( x ) = ( υ − 1 ) υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) u ( s ) d q s + x α − 1 Δ ξ ( υ − 1 ) υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) u ( s ) d q s + ( 1 − ξ ) ( υ − 1 ) υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) u ( s ) d q s − ( υ − 1 ) υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) u ( s ) d q s ,

(12) ( T 2 u ) ( x ) = f ( x , u ( x ) ) + a υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) g ( s , u ( s ) ) d q s + b υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) h ( s , u ( s ) ) d q s + x α − 1 Δ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) f ( s , u ( s ) ) d q s + a ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) g ( s , u ( s ) ) d q s + b ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) h ( s , u ( s ) ) d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) f ( s , u ( s ) ) d q s + a ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) g ( s , u ( s ) ) d q s + b ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) h ( s , u ( s ) ) d q s − a υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) g ( s , u ( s ) ) d q s − b υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) h ( s , u ( s ) ) d q s − f ( 1 , u ( 1 ) ) .

For any u , v ∈ B ρ , we have

∣ T 1 u ( x ) + T 2 v ( x ) ∣ ≤ sup x ∈ [ 0 , 1 ] ∣ f ( x , v ( x ) ) ∣ + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ g ( s , v ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) ∣ h ( s , v ( s ) ) ∣ d q s + x α − 1 ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f ( s , v ( s ) ) ∣ d q s + ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ g ( s , v ( s ) ) ∣ d q s + ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) ∣ h ( s , v ( s ) ) ∣ d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f ( s , v ( s ) ) ∣ d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ g ( s , v ( s ) ) ∣ d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) ∣ h ( s , v ( s ) ) ∣ d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ g ( s , v ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) ∣ h ( s , v ( s ) ) ∣ d q s + ∣ f ( 1 , v ( 1 ) ) ∣

≤ ‖ ψ 1 ‖ sup x ∈ [ 0 , 1 ] 1 + x ( α − 1 ) ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) d q s + 1 + ‖ ψ 2 ‖ sup x ∈ [ 0 , 1 ] ∣ a ∣ υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) d q s + ‖ ψ 3 ‖ sup x ∈ [ 0 , 1 ] ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) d q s

+ ρ sup x ∈ [ 0 , 1 ] ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) d q s ≤ ‖ ψ 1 ‖ 1 + ξ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − ξ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣

+ ‖ ψ 2 ‖ × ∣ a ∣ υ 1 Γ q ( α + 1 ) + ξ η ( α + γ 1 ) ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − ξ ) σ ( α + γ 2 ) ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) + ‖ ψ 3 ‖ ∣ b ∣ υ 1 Γ q ( α + ε + 1 ) + ξ η ( α + ε + γ 1 ) ∣ Δ ∣ Γ q ( α + ε + γ 1 + 1 ) + ( 1 − ξ ) σ ( α + ε + γ 2 ) ∣ Δ ∣ Γ q ( α + ε + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + ε + 1 ) + ρ ∣ υ − 1 ∣ υ 1 Γ q ( α − β + 1 ) + ξ η ( α − β + γ 1 ) ∣ Δ ∣ Γ q ( α − β + γ 1 + 1 ) + ( 1 − ξ ) σ ( α − β + γ 2 ) ∣ Δ ∣ Γ q ( α − β + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α − β + 1 ) ≤ ‖ ψ 1 ‖ Ω 1 + ‖ ψ 2 ‖ Ω 2 + ‖ ψ 3 ‖ Ω 3 + ρ Ω 4 ,

which implies that ‖ T 1 u + T 2 v ‖ ≤ ρ by condition (10) 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 ( x , u ( x ) ) ∣ + ∣ a ∣ υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + x α − 1 ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ∣ f ( 1 , u ( 1 ) ) ∣ ≤ ‖ ψ 1 ‖ 1 + ξ η γ 1 ∣ Δ ∣ Γ q ( γ 1 + 1 ) + ( 1 − ξ ) σ γ 2 ∣ Δ ∣ Γ q ( γ 2 + 1 ) + 1 ∣ Δ ∣ + ‖ ψ 2 ‖ × ∣ a ∣ υ 1 Γ q ( α + 1 ) + ξ η ( α + γ 1 ) ∣ Δ ∣ Γ q ( α + γ 1 + 1 ) + ( 1 − ξ ) σ ( α + γ 2 ) ∣ Δ ∣ Γ q ( α + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + 1 ) + ‖ ψ 3 ‖ ∣ b ∣ υ 1 Γ q ( α + ε + 1 ) + ξ η ( α + ε + γ 1 ) ∣ Δ ∣ Γ q ( α + ε + γ 1 + 1 ) + ( 1 − ξ ) σ ( α + ε + γ 2 ) ∣ Δ ∣ Γ q ( α + ε + γ 2 + 1 ) + 1 ∣ Δ ∣ Γ q ( α + ε + 1 ) = ‖ ψ 1 ‖ Ω 1 + ‖ ψ 2 ‖ Ω 2 + ‖ ψ 3 ‖ Ω 3 .

In order to establish the equicontinuity of the operator T 2 , we assume that x 1 , x 2 ∈ [ 0 , 1 ] such that x 2 > x 1 . We will show that T 2 maps bounded sets into equicontinuous sets. For each u ∈ B ρ , we have

∣ T 2 u ( x 2 ) − T 2 u ( x 1 ) ∣ ≤ sup x ∈ [ 0 , 1 ] ∣ f ( x 2 , u ( x 2 ) ) − f ( x 1 , u ( x 1 ) ) ∣ + ∣ a ∣ υ Γ q ( α ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − 1 ) − ( x 1 − q s ) ( α − 1 ) ] ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x 1 [ ( x 2 − q s ) ( α + ε − 1 ) − ( x 1 − q s ) ( α + ε − 1 ) ] ∣ h ( s , u ( s ) ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ x 1 x 2 ( x 2 − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ x 1 x 2 ( x 2 − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + x 2 ( α − 1 ) − x 1 ( α − 1 ) ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s ≤ ∣ f ( x 2 , u ( x 2 ) ) − f ( x 1 , u ( x 1 ) ) ∣ + ∣ a ∣ ‖ ψ 2 ‖ υ Γ q ( α ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − 1 ) − ( x 1 − q s ) ( α − 1 ) ] d q s + ∣ b ∣ ‖ ψ 3 ‖ υ Γ q ( α + ε ) ∫ 0 x 1 [ ( x 2 − q s ) ( α + ε − 1 ) − ( x 1 − q s ) ( α + ε − 1 ) ] d q s + ∣ a ∣ ‖ ψ 2 ‖ υ Γ q ( α ) ∫ x 1 x 2 ( x 2 − q s ) ( α − 1 ) d q s + ∣ b ∣ ‖ ψ 3 ‖ υ Γ q ( α + ε ) ∫ x 1 x 2 ( x 2 − q s ) ( α + ε − 1 ) d q s + ( x 2 ( α − 1 ) − x 1 ( α − 1 ) ) ∣ Δ ∣ ‖ ψ 1 ‖ ξ η γ 1 Γ q ( γ 1 + 1 ) + ∣ a ∣ ξ ‖ ψ 2 ‖ η ( α + γ 1 ) υ Γ q ( α + γ 1 + 1 ) + ∣ b ∣ ξ ‖ ψ 3 ‖ η ( α + ε + γ 1 ) υ Γ q ( α + ε + γ 1 + 1 ) + ‖ ψ 1 ‖ ( 1 − ξ ) σ γ 2 Γ q ( γ 2 + 1 ) + ∣ a ∣ ‖ ψ 2 ‖ ( 1 − ξ ) σ ( α + γ 2 ) υ Γ q ( α + γ 2 + 1 ) + ∣ b ∣ ‖ ψ 3 ‖ ( 1 − ξ ) σ ( α + ε + γ 2 ) υ Γ q ( α + ε + γ 2 + 1 ) + ∣ a ∣ ‖ ψ 2 ‖ υ Γ q ( α ) + ∣ b ∣ ‖ ψ 3 ‖ υ Γ q ( α + ε ) .

Observe that the right-hand side of the above 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, the Arzelá-Ascoli theorem implies that T 2 is completely continuous and so 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 ] , one can write

‖ T 1 u − T 1 v ‖ ≤ sup x ∈ [ 0 , 1 ] ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + x α − 1 ∣ Δ ∣ ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) − v ( s ) ∣ d q s ≤ Ω 4 ‖ u − v ‖ .

Since Ω 4 < 1 , so T 1 is a contraction. Thus, all the assumptions of Krasnoselskii’s fixed point theorem [21] are satisfied. Therefore, the fractional hybrid q -difference equation (1) with q -integral nonlocal boundary conditions (2) has at least one solution on [ 0 , 1 ] and the proof is completed.□

Theorem 3.4

Assume that:

  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 ( x , u ) ∣ ≤ ϕ 1 ( x ) χ 1 ( ∣ u ∣ ) , ∣ g ( x , u ) ∣ ≤ ϕ 2 ( x ) χ 2 ( ∣ u ∣ ) , and ∣ h ( x , u ) ∣ ≤ ϕ 3 ( x ) χ 3 ( ∣ u ∣ ) for each ( x , u ) ∈ [ 0 , 1 ] × R ;

  2. there exists a constant G > 0 such that

    ( 1 − Ω 4 ) G ‖ ϕ 1 ‖ χ 1 ( G ) Ω 1 + ‖ ϕ 2 ‖ χ 2 ( G ) Ω 2 + ‖ ϕ 3 ‖ χ 3 ( G ) Ω 3 > 1 , Ω 4 < 1 ,

    where Ω 1 , Ω 2 , Ω 3 , and Ω 4 are defined by (8).

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

Proof

We verify the hypothesis of Leray-Schauder’s nonlinear alternative [22] in several steps. Let us first show that the operator T , defined by (7), 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, for x ∈ [ 0 , 1 ] , we have

‖ T u ‖ ≤ sup x ∈ [ 0 , 1 ] ∣ f ( x , u ( x ) ) ∣ + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + x α − 1 ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) ∣ d q s

+ ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ∣ f ( 1 , u ( 1 ) ) ∣

≤ ‖ ϕ 1 ‖ χ 1 ( ω ) sup x ∈ [ 0 , 1 ] 1 + x ( α − 1 ) ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) d q s + 1 + ‖ ϕ 2 ‖ χ 2 ( ω ) sup x ∈ [ 0 , 1 ] ∣ a ∣ υ Γ q ( α ) ∫ 0 x ( x − q s ) ( α − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) d q s + ‖ ϕ 3 ‖ χ 3 ( ω ) sup x ∈ [ 0 , 1 ] ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x ( x − q s ) ( α + ε − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) d q s + ω sup x ∈ [ 0 , 1 ] ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x ( x − q s ) ( α − β − 1 ) d q s + x ( α − 1 ) ∣ Δ ∣ ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 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 4 .

Second, we show that T 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 ) ∣ ≤ sup x ∈ [ 0 , 1 ] { ∣ f ( x 2 , u ( x 2 ) ) − f ( x 1 , u ( x 1 ) ) ∣ + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − β − 1 ) − ( x 1 − q s ) ( α − β − 1 ) ] ∣ u ( s ) ∣ d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ x 1 x 2 ( x 2 − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − 1 ) − ( x 1 − q s ) ( α − 1 ) ] ∣ g ( s , u ( s ) ) ∣ d q s

+ ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 x 1 [ ( x 2 − q s ) ( α + ε − 1 ) − ( x 1 − q s ) ( α + ε − 1 ) ] ∣ h ( s , u ( s ) ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ x 1 x 2 ( x 2 − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ x 1 x 2 ( x 2 − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + x 2 ( α − 1 ) − x 1 ( α − 1 ) ∣ Δ ∣ ξ Γ q ( γ 1 ) ∫ 0 η ( η − q s ) ( γ 1 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ξ ∣ υ − 1 ∣ υ Γ q ( α − β + γ 1 ) ∫ 0 η ( η − q s ) ( α − β + γ 1 − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ ξ υ Γ q ( α + γ 1 ) ∫ 0 η ( η − q s ) ( α + γ 1 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ξ υ Γ q ( α + ε + γ 1 ) ∫ 0 η ( η − q s ) ( α + ε + γ 1 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ( 1 − ξ ) Γ q ( γ 2 ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) ∣ f ( s , u ( s ) ) ∣ d q s + ( 1 − ξ ) ∣ υ − 1 ∣ υ Γ q ( α − β + γ 2 ) ∫ 0 σ ( σ − q s ) ( α − β + γ 2 − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ ( 1 − ξ ) υ Γ q ( α + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + γ 2 − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ ( 1 − ξ ) υ Γ q ( α + ε + γ 2 ) ∫ 0 σ ( σ − q s ) ( α + ε + γ 2 − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s + ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ 0 1 ( 1 − q s ) ( α − β − 1 ) ∣ u ( s ) ∣ d q s + ∣ a ∣ υ Γ q ( α ) ∫ 0 1 ( 1 − q s ) ( α − 1 ) ∣ g ( s , u ( s ) ) ∣ d q s + ∣ b ∣ υ Γ q ( α + ε ) ∫ 0 1 ( 1 − q s ) ( α + ε − 1 ) ∣ h ( s , u ( s ) ) ∣ d q s

≤ ∣ f ( x 2 , u ( x 2 ) ) − f ( x 1 , u ( x 1 ) ) ∣ + ω ∣ υ − 1 ∣ υ 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 ∣ υ − 1 ∣ υ Γ q ( α − β ) ∫ x 1 x 2 ( x 2 − q s ) ( α − β − 1 ) d q s + ∣ a ∣ ‖ ϕ 2 ‖ χ 2 ( ω ) υ Γ q ( α ) ∫ 0 x 1 [ ( x 2 − q s ) ( α − 1 ) − ( x 1 − q s ) ( α − 1 ) ] d q s + ∣ b ∣ ‖ ϕ 3 ‖ χ 3 ( ω ) υ Γ q ( α + ε ) ∫ 0 x 1 [ ( x 2 − q s ) ( α + ε − 1 ) − ( x 1 − q s ) ( α + ε − 1 ) ] d q s + ∣ a ∣ ‖ ϕ 2 ‖ χ 2 ( ω ) υ Γ q ( α ) ∫ x 1 x 2 ( x 2 − q s ) ( α − 1 ) d q s + ∣ b ∣ ‖ ϕ 3 ‖ χ 3 ( ω ) υ Γ q ( α + ε ) ∫ x 1 x 2 ( x 2 − q s ) ( α + ε − 1 ) d q s

+ ( x 2 ( α − 1 ) − x 1 ( α − 1 ) ) ∣ Δ ∣ ‖ ϕ 1 ‖ χ 1 ( ω ) ξ η γ 1 Γ q ( γ 1 + 1 ) + ξ ω ∣ υ − 1 ∣ η α − β + γ 1 υ Γ q ( α − β + γ 1 + 1 ) + ∣ a ∣ ξ ‖ ϕ 2 ‖ χ 2 ( ω ) η ( α + γ 1 ) υ Γ q ( α + γ 1 + 1 ) + ∣ b ∣ ξ ‖ ϕ 3 ‖ χ 3 ( ω ) η ( α + ε + γ 1 ) υ Γ q ( α + ε + γ 1 + 1 ) + ‖ ϕ 1 ‖ χ 1 ( ω ) ( 1 − ξ ) σ γ 2 Γ q ( γ 2 + 1 ) + ω ( 1 − ξ ) ∣ υ − 1 ∣ σ ( α − β + γ 2 ) υ Γ q ( α − β + γ 2 + 1 ) + ∣ a ∣ ‖ ϕ 2 ‖ χ 2 ( ω ) ( 1 − ξ ) σ ( α + γ 2 ) υ Γ q ( α + γ 2 + 1 ) + ∣ b ∣ ‖ ϕ 3 ‖ χ 3 ( ω ) ( 1 − ξ ) σ ( α + ε + γ 2 ) υ Γ q ( α + ε + γ 2 + 1 ) + ω ∣ υ − 1 ∣ υ Γ q ( α − β + 1 ) + ∣ a ∣ ‖ ϕ 2 ‖ χ 2 ( ω ) υ Γ q ( α ) + ∣ b ∣ ‖ ϕ 3 ‖ χ 3 ( ω ) υ Γ q ( α + ε ) .

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 the Leray-Schauder nonlinear alternative [22], 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 4 ‖ u ‖ .

Consequently, we have

( 1 − Ω 4 ) ‖ u ‖ ‖ ϕ 1 ‖ χ 1 ( ‖ u ‖ ) Ω 1 + ‖ ϕ 2 ‖ χ 2 ( ‖ u ‖ ) Ω 2 + Ω 3 ‖ ϕ 3 ‖ χ 3 ( ‖ u ‖ ) ≤ 1 .

By the condition ( H 6 ) , we can find a positive number G such that ‖ u ‖ ≠ G . Introduce a set

(13) 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 the nonlinear alternative of Leray-Schauder type [22] that the operator T has a fixed point in U ¯ . Thus, there exists a solution of problems (1) and (2) on [ 0 , 1 ] . The proof is complete.□

4 Examples

Example 4.1

(Illustration of Theorem 3.2)

Let us consider the fractional hybrid q -difference equation

(14) 0.99 D 0.5 1.5 [ u ( x ) − f ( x , u ( x ) ) ] + ( 1 − 0.99 ) D 0.5 1.01 u ( x ) = 0.25 g ( x , u ( x ) ) + 0.25 I 0.5 1 ∕ 5 h ( x , u ( x ) ) ,

with q -integral nonlocal boundary conditions

(15) u ( 0 ) = 0 , u ( 1 ) = ∫ 0 0.25 ( 0.25 − q s ) ( 0.1 − 1 ) Γ q ( 0.1 ) u ( s ) d q s + ( 1 − 0.1 ) ∫ 0 0.35 ( 0.35 − q s ) ( 0.1 − 1 ) Γ q ( 0.1 ) u ( s ) d q s ,

where α = 1.5 , q = 0.5 , β = 1.01 , η = 0.25 , σ = 0.35 , ε = 1 ∕ 5 , υ = 0.99 , ξ = 0.1 , γ 1 = γ 2 = 0.1 , a = b = 0.25 , x ∈ [ 0 , 1 ] , and

f ( x , u ( x ) ) = 0.07 ∣ cos ( π x ) ∣ ∣ u ( x ) ∣ 3 + ∣ u ( x ) ∣ , g ( x , u ( x ) ) = 90 ∣ u ( x ) ∣ 1,000 ( 1 + ∣ u ( x ) ∣ ) , h ( x , u ( x ) ) = 8 ∣ u ( x ) ∣ 100 ( 1 + ∣ u ( x ) ∣ ) .

Then, L 1 = 7 ∕ 100 , L 2 = 9 ∕ 100 , L 3 = 8 ∕ 100 as

∣ f ( x , u ( x ) ) − f ( x , v ( x ) ) ∣ ≤ 7 100 ( ∣ u ( x ) − v ( x ) ∣ ) , ∣ g ( x , u ( x ) ) − g ( x , v ( x ) ) ∣ ≤ 9 100 ( ∣ u ( x ) − v ( x ) ∣ ) ,

∣ h ( x , u ( x ) ) − h ( x , v ( x ) ) ∣ ≤ 8 100 ( ∣ u ( x ) − v ( x ) ∣ ) .

With the given data, it is found that Δ ≃ 0.480477 , Ω 1 ≃ 5.01705 , Ω 2 ≃ 0.729128 , Ω 3 ≃ 0.654832 , Ω 4 ≃ 0.0458087 , and Ω ≃ 0.51501 < 1 . Clearly, the assumptions of Theorem 3.2 hold, and hence, problems (14) and (15) have a unique solution on [ 0 , 1 ] by the conclusion of Theorem 3.2.

Example 4.2

(Illustration of Theorem 3.3)

Consider the fractional hybrid q -difference equation

(16) 0.99 D 0.5 1.5 [ u ( x ) − f ( x , u ( x ) ) ] + ( 1 − 0.99 ) D 0.5 1.01 u ( x ) = 0.25 g ( x , u ( x ) ) + 0.25 I 0.5 1 ∕ 5 h ( x , u ( x ) ) ,

subject to q -integral nonlocal boundary conditions

(17) u ( 0 ) = 0 , u ( 1 ) = ∫ 0 0.25 ( 0.25 − q s ) ( 0.1 − 1 ) Γ q ( 0.1 ) u ( s ) d q s + ( 1 − 0.1 ) ∫ 0 0.35 ( 0.35 − q s ) ( 0.1 − 1 ) Γ q ( 0.1 ) u ( s ) d q s .

Here, α = 1.5 , q = 0.5 , β = 1.01 , η = 0.25 , σ = 0.35 , ε = 1 ∕ 5 , υ = 0.99 , ξ = 0.1 , γ 1 = γ 2 = 0.1 , a = b = 0.25 , x ∈ [ 0 , 1 ] , and

f ( x , u ) = 1 9 x 2 ∣ u ( x ) ∣ 1 + ∣ u ( x ) ∣ + 4 , g ( x , u ) = 1 2 ∣ sin u ∣ x 1 + ∣ sin u ∣ + 1 4 , h ( x , u ) = 1 5 x ∣ u ( x ) ∣ 1 + ∣ u ( x ) ∣ + 7 .

On the other hand, there exists a continuous function ψ 1 ( x ) = ( x + 3 ) ∕ 5 , ψ 2 ( x ) = ( 4 x + 1 ) ∕ 8 , and ψ 3 ( x ) = ( x + 7 ) ∕ 5 on [ 0 , 1 ] . Also, we have ‖ ψ 1 ‖ = sup x ∈ [ 0 , 1 ] ψ 1 ( x ) ≃ 0.055556 , ‖ ψ 2 ‖ = sup x ∈ [ 0 , 1 ] ψ 2 ( x ) ≃ 0.3125 , and ‖ ψ 3 ‖ = sup x ∈ [ 0 , 1 ] ψ 3 ( x ) ≃ 1.6 . Using the given values, we have that Ω 4 ≃ 0.0458087 < 1 . Clearly, all the assumptions of Theorem 3.3 are satisfied. Therefore, the conclusion of Theorem 3.3 applies, and hence, the fractional q -integro-difference equation (16) with q -integral nonlocal boundary conditions (17) has at least one solution on [ 0 , 1 ] .

Example 4.3

(Illustration of Theorem 3.4) Consider the fractional hybrid q -difference equation

(18) 0.99 D 0.5 1.5 [ u ( x ) − f ( x , u ( x ) ) ] + ( 1 − 0.99 ) D 0.5 1.01 u ( x ) = 0.25 g ( x , u ( x ) ) + 0.25 I 0.5 1 ∕ 5 h ( x , u ( x ) ) ,

subject to q -integral nonlocal boundary conditions

(19) u ( 0 ) = 0 , u ( 1 ) = ∫ 0 0.25 ( 0.25 − q s ) ( 0.1 − 1 ) Γ q ( 0.1 ) u ( s ) d q s + ( 1 − 0.1 ) ∫ 0 0.35 ( 0.35 − q s ) ( 0.1 − 1 ) Γ q ( 0.1 ) u ( s ) d q s .

Here, α = 1.5 , q = 0.5 , β = 1.01 , η = 0.25 , σ = 0.35 , ε = 1 / 5 , υ = 0.99 , ξ = 0.1 , γ 1 = γ 2 = 0.1 , a = b = 0.25 , x ∈ [ 0 , 1 ] , and

f ( x , u ) = 3 cos ( u ) ( 3 + x ) 2 , g ( x , u ) = ( x + 1 ) u 2 u 2 + 1 , h ( x , u ) = 4 sin ( u ) ( x + 4 ) 2 .

With the given data, we obtain Ω 1 ≃ 5.01705 , Ω 2 ≃ 0.729128 , Ω 3 ≃ 0.654832 , Ω 4 ≃ 0.0458087 , and

∣ f ( x , u ) ∣ ≤ 3 cos ( ∣ u ∣ ) ( 3 + x ) 2 = ϕ 1 ( x ) χ 1 ( ∣ u ∣ ) , ∣ g ( x , u ) ∣ ≤ ( x + 1 ) ∣ u ∣ 2 ∣ u ∣ 2 + 1 = ϕ 2 ( x ) χ 2 ( ∣ u ∣ ) , ∣ h ( x , u ) ∣ ≤ 4 sin ( ∣ u ∣ ) ( x + 4 ) 2 = ϕ 3 ( x ) χ 3 ( ∣ u ∣ ) .

Clearly, ϕ 1 ( x ) = 3 ( 3 + x ) 2 , χ 1 ( ∣ u ∣ ) = cos ( ∣ u ∣ ) , ϕ 2 ( x ) = x + 1 , χ 2 ( ∣ u ∣ ) = ∣ u ∣ 2 ∣ u ∣ 2 + 1 , ϕ 3 ( x ) = 4 ( x + 4 ) 2 , and χ 3 ( ∣ u ∣ ) = sin ( ∣ u ∣ ) . Moreover, the condition ( H 6 ) implies that there exists G > 2.28404 . Thus, the hypotheses of Theorem 3.4 are satisfied. Therefore, the conclusion of Theorem 3.4 implies that the hybrid q -difference equation (18) with nonlocal q -integral boundary conditions (19) has at least one solution on [ 0 , 1 ] .

5 Conclusion

We have developed the existence theory for hybrid fractional q -integro-difference equations involving Riemann-Liouville q -derivatives of different orders, complemented with nonlocal Riemann-Liouville q -integral boundary conditions. Our results are new and contribute significantly to the known literature on nonlocal hybrid q -fractional integral boundary value problems. In particular, our work generalizes the results proved in [2]. Several special cases also follow from the results of this article, for instance, see the following examples.

  1. For ξ = 1 , we deduce the results for the nonlinear hybrid Riemann-Liouville fractional q -integro-difference equation (1) subject to the nonlocal boundary conditions of the form:

    u ( 0 ) = 0 , u ( 1 ) = ∫ 0 η ( η − q s ) ( γ 1 − 1 ) Γ q ( γ 1 ) u ( s ) d q s , γ 1 > 0 .

  2. Our results correspond to the following problem for υ = ξ = 1 , a = 1 , and b = 0 :

    D q α [ u ( x ) − f ( x , u ( x ) ) ] = g ( 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 , γ 1 > 0 .

  3. We obtain the results for the following problem if we fix υ = 1 ∕ 2 , a = 0 , b = 1 ∕ 2 in the present results:

    D q α [ u ( x ) − f ( x , u ( x ) ) ] + D q β u ( x ) = I q ε h ( x , u ( x ) ) , 0 < x < 1 ,

    u ( 0 ) = 0 , u ( 1 ) = ξ ∫ 0 η ( η − q s ) ( γ 1 − 1 ) Γ q ( γ 1 ) u ( s ) d q s + ( 1 − ξ ) ∫ 0 σ ( σ − q s ) ( γ 2 − 1 ) Γ q ( γ 2 ) u ( s ) d q s , γ 1 , γ 2 > 0 .

The aforementioned results are indeed new.

Acknowledgments

The Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU), Jeddah, Saudi Arabia, has funded this project, under Grant No. (KEP-PhD: 41-130-1443). The authors, therefore, acknowledge with thanks DSR technical and financial support.

  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: 41-130-1443).

  2. Author contributions: All authors accepted responsibility for the entire content of this manuscript and approved its submission.

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

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

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Received: 2022-11-01
Revised: 2023-02-18
Accepted: 2023-03-18
Published Online: 2023-05-02

© 2023 the author(s), published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
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  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
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  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
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  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
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  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
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https://doi.org/10.1515/dema-2022-0222
Keywords for this article
q-difference equation; Riemann-Liouville q-derivative; q-integral; nonlocal boundary conditions; existence; fixed point
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. A novel class of bipolar soft separation axioms concerning crisp points
  3. Duality for convolution on subclasses of analytic functions and weighted integral operators
  4. Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness
  5. On the existence of nonnegative radial solutions for Dirichlet exterior problems on the Heisenberg group
  6. Hyers-Ulam stability of isometries on bounded domains-II
  7. Asymptotic study of Leray solution of 3D-Navier-Stokes equations with exponential damping
  8. Semi-Hyers-Ulam-Rassias stability for an integro-differential equation of order 𝓃
  9. Jordan triple (α,β)-higher ∗-derivations on semiprime rings
  10. The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping
  11. Approximation of the image of the Lp ball under Hilbert-Schmidt integral operator
  12. Best proximity points in ℱ-metric spaces with applications
  13. Approximation spaces inspired by subset rough neighborhoods with applications
  14. A numerical Haar wavelet-finite difference hybrid method and its convergence for nonlinear hyperbolic partial differential equation
  15. A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation
  16. Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination
  17. On local fractional integral inequalities via generalized ( h ˜ 1 , h ˜ 2 ) -preinvexity involving local fractional integral operators with Mittag-Leffler kernel
  18. On some geometric results for generalized k-Bessel functions
  19. Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems
  20. Some results of homogeneous expansions for a class of biholomorphic mappings defined on a Reinhardt domain in ℂn
  21. Graded weakly 1-absorbing primary ideals
  22. The existence and uniqueness of solutions to a functional equation arising in psychological learning theory
  23. Some aspects of the n-ary orthogonal and b(αn,βn)-best approximations of b(αn,βn)-hypermetric spaces over Banach algebras
  24. Numerical solution of a malignant invasion model using some finite difference methods
  25. Increasing property and logarithmic convexity of functions involving Dirichlet lambda function
  26. Feature fusion-based text information mining method for natural scenes
  27. Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach
  28. The study of solutions for several systems of PDDEs with two complex variables
  29. Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces
  30. Generalized Stević-Sharma operators from the minimal Möbius invariant space into Bloch-type spaces
  31. On initial value problem for elliptic equation on the plane under Caputo derivative
  32. A dimension expanded preconditioning technique for block two-by-two linear equations
  33. Asymptotic behavior of Fréchet functional equation and some characterizations of inner product spaces
  34. Small perturbations of critical nonlocal equations with variable exponents
  35. Dynamical property of hyperspace on uniform space
  36. Some notes on graded weakly 1-absorbing primary ideals
  37. On the problem of detecting source points acting on a fluid
  38. Integral transforms involving a generalized k-Bessel function
  39. Ruled real hypersurfaces in the complex hyperbolic quadric
  40. On the monotonic properties and oscillatory behavior of solutions of neutral differential equations
  41. Approximate multi-variable bi-Jensen-type mappings
  42. Mixed-type SP-iteration for asymptotically nonexpansive mappings in hyperbolic spaces
  43. On the equation fn + (f″)m ≡ 1
  44. Results on the modified degenerate Laplace-type integral associated with applications involving fractional kinetic equations
  45. Characterizations of entire solutions for the system of Fermat-type binomial and trinomial shift equations in ℂn#
  46. Commentary
  47. On I. Meghea and C. S. Stamin review article “Remarks on some variants of minimal point theorem and Ekeland variational principle with applications,” Demonstratio Mathematica 2022; 55: 354–379
  48. Special Issue on Fixed Point Theory and Applications to Various Differential/Integral Equations - Part II
  49. On Cauchy problem for pseudo-parabolic equation with Caputo-Fabrizio operator
  50. Fixed-point results for convex orbital operators
  51. Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators
  52. Asymptotic behavior of resolvents of equilibrium problems on complete geodesic spaces
  53. A system of additive functional equations in complex Banach algebras
  54. New inertial forward–backward algorithm for convex minimization with applications
  55. Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator
  56. Analysis of Cauchy problem with fractal-fractional differential operators
  57. Common best proximity points for a pair of mappings with certain dominating property
  58. Investigation of hybrid fractional q-integro-difference equations supplemented with nonlocal q-integral boundary conditions
  59. The structure of fuzzy fractals generated by an orbital fuzzy iterated function system
  60. On the structure of self-affine Jordan arcs in ℝ2
  61. Solvability for a system of Hadamard-type hybrid fractional differential inclusions
  62. Three solutions for discrete anisotropic Kirchhoff-type problems
  63. On split generalized equilibrium problem with multiple output sets and common fixed points problem
  64. Special Issue on Computational and Numerical Methods for Special Functions - Part II
  65. Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function
  66. Certain aspects of Nörlund ℐ-statistical convergence of sequences in neutrosophic normed spaces
  67. On completeness of weak eigenfunctions for multi-interval Sturm-Liouville equations with boundary-interface conditions
  68. Some identities on generalized harmonic numbers and generalized harmonic functions
  69. Study of degenerate derangement polynomials by λ-umbral calculus
  70. Normal ordering associated with λ-Stirling numbers in λ-shift algebra
  71. Analytical and numerical analysis of damped harmonic oscillator model with nonlocal operators
  72. Compositions of positive integers with 2s and 3s
  73. Kinematic-geometry of a line trajectory and the invariants of the axodes
  74. Hahn Laplace transform and its applications
  75. Discrete complementary exponential and sine integral functions
  76. Special Issue on Recent Methods in Approximation Theory - Part II
  77. On the order of approximation by modified summation-integral-type operators based on two parameters
  78. Bernstein-type operators on elliptic domain and their interpolation properties
  79. A class of strongly convergent subgradient extragradient methods for solving quasimonotone variational inequalities
  80. Special Issue on Recent Advances in Fractional Calculus and Nonlinear Fractional Evaluation Equations - Part II
  81. Application of fractional quantum calculus on coupled hybrid differential systems within the sequential Caputo fractional q-derivatives
  82. On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative
  83. A certain class of fractional difference equations with damping: Oscillatory properties
  84. Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals
  85. Special Issue on Recent Advances for Computational and Mathematical Methods in Scientific Problems - Part II
  86. The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method
  87. Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves
  88. Petri net analysis of a queueing inventory system with orbital search by the server
  89. Asymptotic stability of an epidemiological fractional reaction-diffusion model
  90. On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces
  91. Special Issue on Application of Fractional Calculus: Mathematical Modeling and Control - Part I
  92. New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization
  93. Pell-Lucas polynomials for numerical treatment of the nonlinear fractional-order Duffing equation
  94. Impacts of Brownian motion and fractional derivative on the solutions of the stochastic fractional Davey-Stewartson equations
  95. Some results on fractional Hahn difference boundary value problems
  96. Properties of a subclass of analytic functions defined by Riemann-Liouville fractional integral applied to convolution product of multiplier transformation and Ruscheweyh derivative
  97. Special Issue on Development of Fuzzy Sets and Their Extensions - Part I
  98. The cross-border e-commerce platform selection based on the probabilistic dual hesitant fuzzy generalized dice similarity measures
  99. Comparison of fuzzy and crisp decision matrices: An evaluation on PROBID and sPROBID multi-criteria decision-making methods
  100. Rejection and symmetric difference of bipolar picture fuzzy graph
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