Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator

Ahmed Alsaedi; Madeaha Alghanmi; Bashir Ahmad; Boshra Alharbi

doi:10.1515/dema-2022-0195

Article Open Access

Uniqueness of solutions for a ψ-Hilfer fractional integral boundary value problem with the p-Laplacian operator

Ahmed Alsaedi , Madeaha Alghanmi , Bashir Ahmad and Boshra Alharbi

Published/Copyright: February 15, 2023

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From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this article, we discuss the existence of a unique solution to a ψ -Hilfer fractional differential equation involving the p -Laplacian operator subject to nonlocal ψ -Riemann-Liouville fractional integral boundary conditions. Banach’s fixed point theorem is the main tool of our study. Examples are given for illustrating the obtained results.

Keywords: ψ-Hilfer fractional derivative; the p-Laplacian operator; existence; fixed point

MSC 2010: 34A08; 34B10; 34B15

1 Introduction

Fractional differential operators are found to be of great utility in the mathematical modeling of natural and engineering phenomena, for example, see [1–6] and the references cited therein. In contrast to the integer-order differential operators, these operators are nonlocal in nature and account for the history of the physical phenomena under consideration. In the literature, there do exist several kinds of fractional integrals and derivatives, for instance, see [7–10]. Hilfer in [11] proposed a generalized fractional derivative of order α and type β , which is known as the Hilfer fractional derivative, and it interpolates between the Riemann-Liouville and Caputo derivatives. This two-parameter fractional derivative operator appeared in the modeling of diffusion models, dielectric relaxation in glass-forming materials, etc. [12,13]. For some recent works on Hilfer fractional differential equations, we refer the reader to the articles [14–18]. In [19], Sousa and Capelas de Oliveira generalized the Hilfer fractional derivative by introducing the concept of ψ -Hilfer fractional derivative. One of the advantages of this derivative is that it covers a wide class of fractional derivatives, which can be fixed by choosing the function ψ appropriately. Later, this derivative gained much attention, and many researchers turned to investigate it, for example, see [20–26].

On the other hand, differential equations with the p -Laplacian operator appeared for the first time when Leibenson [27] was attempting to derive an accurate formula to model turbulent flow in the porous medium. Keeping in mind the application of differential equations involving the p -Laplacian operator in the areas of mechanics, nonlinear dynamics, glaciology, nonlinear elasticity, flow through porous media, and so on, many authors focused on this topic. For details and examples, one can see the article [28–32]. As far as we know, there is no work dealing with ψ -Hilfer fractional differential equations with the p -Laplacian operator.

The objective this study is to investigate a ψ -Hilfer fractional boundary value problem involving the p -Laplacian operator ϕ p ( ⋅ ) and ψ -Riemann-Liouville fractional integral boundary conditions given as follows:

(1) ( ϕ p ( H D 0 + α , β , ψ y ( t ) ) ) ′ + f ( t , y ( t ) ) = 0 , 0 < t < 1 , y ( 0 ) = 0 , D 0 + α , β H y ( 0 ) = 0 , y ( 1 ) = ∑ i = 1 m a i I 0 + σ i , ψ y ( μ i ) , 0 < μ i < 1 , a i ∈ R ,

where ϕ p ( t ) = t ∣ t ∣ p − 2 , 1 p + 1 q = 1 , p , q > 1 , D 0 + α , β , ψ H denotes the ψ -Hilfer fractional derivative operator of order α ∈ ( 1 , 2 ] and type β ∈ [ 0 , 1 ] , I 0 + σ i , ψ is the ψ -Riemann-Liouville fractional integral operator of order σ i > 0 , and f : [ 0 , 1 ] × R → R is a continuous function.

The rest of the article is organized as follows: Section 2 presents the background material related to our work, Section 3 contains the main results for problem (1), and Section 4 presents examples to illustrate the obtained results.

2 Preliminaries

Let C ( [ a , b ] , R ) denote the Banach space of all continuous functions from [ a , b ] into R endowed with the supremum norm ‖ y ‖ = sup t ∈ [ a , b ] ∣ y ( t ) ∣ . Let C n ( [ a , b ] , R ) be the class all n -times continuously differential functions from [ a , b ] into R .

In the forthcoming analysis, it is assumed that ψ is an increasing and positive monotone function on ( 0 , 1 ] possessing a continuous derivative ψ ′ ( t ) ≠ 0 on ( 0 , 1 ) .

Definition 1

[7] For α > 0 and − ∞ ≤ a < b ≤ ∞ , the left-sided ψ -Riemann-Liouville fractional integral of an integrable function f on [ a , b ] is defined as follows:

(2) ( I a + α , ψ f ) ( t ) = 1 Γ ( α ) ∫ a t ψ ′ ( s ) ( ψ ( t ) − ψ ( s ) ) α − 1 f ( s ) d s .

Definition 2

[7] Let n − 1 ≤ α < n ( n = [ α ] + 1 ) , the left-sided ψ -Riemann-Liouville fractional derivative of a function f ∈ C ( [ a , b ] , R ) is given as follows:

(3) ( D a + α , ψ f ) ( t ) = 1 Γ ( n − α ) 1 ψ ′ ( t ) d d t n ∫ a t ψ ′ ( s ) ( ψ ( t ) − ψ ( s ) ) n − α − 1 f ( s ) d s .

Definition 3

[33] Let n − 1 ≤ α < n ( n = [ α ] + 1 ) , the left-sided ψ -Caputo fractional derivative of a function f ∈ C n ( [ a , b ] , R ) is defined as follows:

(4) ( D a + α , ψ C f ) ( t ) = 1 Γ ( n − α ) ∫ a t ψ ′ ( s ) [ ψ ( t ) − ψ ( s ) ] 1 − n + α 1 ψ ′ ( s ) d d s n f ( s ) d s .

Definition 4

[19] For α > 0 , n = [ α ] + 1 , β ∈ [ 0 , 1 ] , and − ∞ ≤ a < x < b ≤ ∞ , the left-sided ψ -Hilfer fractional derivative for a function f ∈ C n ( [ a , b ] , R ) is defined as follows:

(5) ( D a + α , β , ψ H f ) ( t ) = I a + β ( n − α ) , ψ 1 ψ ′ ( t ) d d t n I a + ( 1 − β ) ( n − α ) , ψ f ( t ) ,

which can alternatively be written as

(6) ( D a + α , β , ψ H f ) ( t ) = I a + γ − α , ψ D a + γ , ψ f ( t ) ,

where γ = α + β ( n − α ) .

Lemma 1

[19] Let n − 1 < α ≤ n , β ∈ [ 0 , 1 ] and γ = α + β ( n − α ) .

If f ∈ C n ( [ a , b ] , R ) , then
(7) I a + α , ψ ( D a + α , β , ψ H f ) ( t ) = f ( t ) − ∑ i = 1 n ( ψ ( t ) − ψ ( a ) ) γ − i Γ ( γ − i + 1 ) d ψ [ n − i ] I a + ( 1 − β ) ( n − α ) , ψ f ( a ) ,
where d ψ [ n − i ] f ( t ) = 1 ψ ′ ( t ) d d t n − i .
If f ∈ C ( [ a , b ] , R ) , then
(8) D a + α , β , ψ H I a + α , ψ f ( t ) = f ( t ) .

In the following lemma, we solve the linear variant of problem (1).

Lemma 2

For 1 < α < 2 , β ∈ [ 0 , 1 ] , γ = α + β ( 2 − α ) , and h ∈ C ( [ 0 , 1 ] , R ) , the integral representation of the solution for the following linear ψ -Hilfer p -Laplacian fractional integral boundary value problem:

(9) ( ϕ p ( D 0 + α , β , ψ H y ( t ) ) ) ′ + h ( t ) = 0 , y ( 0 ) = 0 , D 0 + α , β ψ H y ( 0 ) = 0 , y ( 1 ) = ∑ i = 1 m a i I 0 + σ i , ψ y ( μ i ) ,

is as follows:

(10) y ( t ) = − 1 Γ ( α ) ∫ 0 t ψ ′ ( τ ) ( ψ ( t ) − ψ ( τ ) ) α − 1 ϕ q ∫ 0 τ h ( s ) d s d τ + ( ψ ( t ) − ψ ( 0 ) ) γ − 1 Λ Γ ( γ ) × 1 Γ ( α ) ∫ 0 1 ψ ′ ( τ ) ( ψ ( 1 ) − ψ ( τ ) ) α − 1 ϕ q ∫ 0 τ h ( s ) d s d τ − ∑ i = 1 m a i Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( τ ) ( ψ ( μ i ) − ψ ( τ ) ) α + σ i − 1 ϕ q ∫ 0 τ h ( s ) d s d τ ,

where it is assumed that

Λ = ( ψ ( 1 ) − ψ ( 0 ) ) γ − 1 Γ ( γ ) − ∑ i = 1 m a i ( ψ ( μ i ) − ψ ( 0 ) ) γ + σ i − 1 Γ ( γ + σ i ) ≠ 0 .

Proof

Setting ϕ p ( D 0 + α , β , ψ H y ( t ) ) = θ ( t ) , problem (9) can be split into two problems:

(11) θ ( t ) ′ + h ( t ) = 0 , θ ( 0 ) = 0

and

(12) ϕ p ( D 0 + α , β , ψ H y ( t ) ) = θ ( t ) , y ( 0 ) = 0 , y ( 1 ) = ∑ i = 1 n a i I 0 + σ i , ψ y ( μ i ) .

Solving (11), we obtain θ ( t ) = − ∫ 0 t h ( s ) d s . Applying I 0 + α , ψ to the ψ -Hilfer p -Laplacian fractional differential equation in (12), we obtain

(13) y ( t ) = 1 Γ ( α ) ∫ 0 t ψ ′ ( s ) ( ψ ( t ) − ψ ( s ) ) α − 1 ϕ q ( θ ( s ) ) d s + c 0 ( ψ ( t ) − ψ ( 0 ) ) γ − 2 + c 1 ( ψ ( t ) − ψ ( 0 ) ) γ − 1 ,

where c 0 and c 1 are arbitrary constants. Using the condition y ( 0 ) = 0 in (13) yields c 0 = 0 . Then, from (13), we have

(14) y ( 1 ) = 1 Γ ( α ) ∫ 0 1 ψ ′ ( s ) ( ψ ( t ) − ψ ( s ) ) α − 1 ϕ q ( θ ( s ) ) d s + c 1 ( ψ ( 1 ) − ψ ( 0 ) ) γ − 1

and

(15) I 0 + σ i , ψ y ( μ i ) = 1 Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( s ) ( ψ ( t ) − ψ ( s ) ) α + σ i − 1 ϕ q ( θ ( s ) ) d s + c 1 Γ ( γ ) Γ ( γ + σ i ) ( ψ ( μ i ) − ψ ( 0 ) ) γ + σ i − 1 .

Inserting (14) and (15) in the condition: y ( 1 ) = ∑ i = 1 m a i I 0 + σ i , ψ y ( μ i ) , we find that

(16) c 1 = 1 Λ Γ ( γ ) ∑ i = 1 m a i Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( s ) ( ψ ( t ) − ψ ( s ) ) α + σ i − 1 ϕ q ( θ ( s ) ) d s − 1 Γ ( α ) ∫ 0 1 ψ ′ ( s ) ( ψ ( t ) − ψ ( s ) ) α − 1 ϕ q ( θ ( s ) ) d s .

Substituting the values of c 0 , c 1 , and θ ( t ) = − ∫ 0 t h ( s ) d s in (13), we obtain the solution (10). This completes the proof.□

The following lemma provides bounds for the p -Laplace operator, which can easily be proved by using the mean value theorem when the function ϕ p ( k ) = ∣ k ∣ p − 2 k is differentiable for all values of k except k = 0 . Moreover, ∂ ϕ ( k ) ∂ k is bounded by ( p − 1 ) max ∣ k ∣ p − 2 in the case of p > 2 and bounded by ( p − 1 ) min ∣ k ∣ p − 2 when 1 < p < 2 .

Lemma 3

(See (2.1) and (2.2) on page 3268 in [28])

For 1 < p ≤ 2 , ∣ k 1 ∣ , ∣ k 2 ∣ ≥ Δ 1 > 0 , and k 1 k 2 > 0 , we have
∣ ϕ p ( k 2 ) − ϕ p ( k 1 ) ∣ ≤ ( p − 1 ) Δ 1 p − 2 ∣ k 2 − k 1 ∣ ;
For p > 2 , ∣ k 1 ∣ , ∣ k 2 ∣ ≤ Δ 2 , and k 1 k 2 > 0 , we have
∣ ϕ p ( k 2 ) − ϕ p ( k 1 ) ∣ ≤ ( p − 1 ) Δ 2 p − 2 ∣ k 2 − k 1 ∣ .

3 Main results

This section is devoted to the uniqueness results for problem (1) for different values of p . Consider the following composition operator:

(17) G = G 2 ∘ G 1 ,

where G 1 : C ( [ 0 , 1 ] , R ) → C ( [ 0 , 1 ] , R ) is defined as follows:

(18) ( G 1 y ) ( t ) = ϕ q ∫ 0 t f ( s , y ( s ) ) d s ,

and G 2 : C ( [ 0 , 1 ] , R ) → C ( [ 0 , 1 ] , R ) is given as follows:

(19) ( G 2 y ) ( t ) = − 1 Γ ( α ) ∫ 0 t ψ ′ ( τ ) ( ψ ( t ) − ψ ( τ ) ) α − 1 y ( τ ) d τ + ( ψ ( t ) − ψ ( 0 ) ) γ − 1 Λ Γ ( γ ) × 1 Γ ( α ) ∫ 0 1 ψ ′ ( τ ) ( ψ ( 1 ) − ψ ( τ ) ) α − 1 y ( τ ) d τ − ∑ i = 1 m − 1 a i 1 Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( τ ) ( ψ ( μ i ) − ψ ( τ ) ) α + σ i − 1 y ( τ ) d τ .

Clearly G : C ( [ 0 , 1 ] , R ) → C ( [ 0 , 1 ] , R ) is a continuous operator. For computational convenience, we set

(20) Ω = ( ψ ( 1 ) − ψ ( 0 ) ) α Γ ( α + 1 ) + ( ψ ( 1 ) − ψ ( 0 ) ) γ + α − 1 Λ Γ ( γ ) Γ ( α + 1 ) + ∑ i = 1 m a i ( ψ ( 1 ) − ψ ( 0 ) ) γ − 1 ( ψ ( μ i ) − ψ ( 0 ) ) α + σ i Λ Γ ( γ ) Γ ( α + σ i + 1 ) .

Now, we present our main results, which will be proved with the aid of the Banach contraction mapping principle and Lemma 2.

Theorem 1

Assume that 1 < p ≤ 2 and the following conditions hold:

There exist a nonnegative integrable function g on [ 0 , 1 ] and a positive constant M such that
0 < ∫ 0 1 g ( t ) d t ≤ M and ∣ f ( t , y ) ∣ ≤ g ( t ) for ( t , y ) ∈ [ 0 , 1 ] × R ;
There exists a positive constant k such that
∣ f ( t , y 1 ) − f ( t , y 2 ) ∣ ≤ k ∣ y 2 − y 1 ∣ , ( t , y i ) ∈ [ 0 , 1 ] × R , i = 1 , 2 .

Then, the boundary value problem (1) has a unique solution provided that

(21) 0 < k < 1 ( q − 1 ) M q − 2 Ω ,

where Ω is given by (20).

Proof

By the assumption ( A 1 ), we have

∫ 0 t f ( s , y ( s ) ) d s = ∫ 0 t ∣ f ( t , y ( s ) ) ∣ d s ≤ ∫ 0 t g ( s ) d s ≤ M .

Setting r ≥ M q − 1 Ω , we define B r = { y ∈ C ( [ 0 , 1 ] , R ) : ‖ y ‖ ≤ r } and show that G B r ⊂ B r , where the operator G is defined by (17). For y ∈ B r , it follows by the definition of ϕ p ( ⋅ ) that

∣ ( G 1 y ) ( t ) ∣ = ϕ q ∫ 0 t f ( s , y ( s ) ) d s = ϕ q ∫ 0 t f ( s , y ( s ) ) d s ≤ M q − 1 .

Consequently, we obtain

∣ ( G y ) ( t ) ∣ = ∣ ( G 2 ∘ G 1 y ) ( t ) ∣ = 1 Γ ( α ) ∫ 0 t ψ ′ ( τ ) ( ψ ( t ) − ψ ( τ ) ) α − 1 ( ( G 1 y ) ( τ ) ) d τ + ( ψ ( t ) − ψ ( 0 ) ) γ − 1 Λ Γ ( γ ) × 1 Γ ( α ) ∫ 0 1 ψ ′ ( τ ) ( ψ ( 1 ) − ψ ( τ ) ) α − 1 ( ( G 1 y ) ( τ ) ) d τ − ∑ i = 1 m − 1 a i 1 Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( τ ) ( ψ ( μ i ) − ψ ( τ ) ) α + σ i − 1 ( ( G 1 y ) ( τ ) ) d τ ≤ M q − 1 1 Γ ( α ) ∫ 0 t ψ ′ ( τ ) ( ψ ( t ) − ψ ( τ ) ) α − 1 d τ + ( ψ ( t ) − ψ ( 0 ) ) γ − 1 Λ Γ ( γ ) × 1 Γ ( α ) ∫ 0 1 ψ ′ ( τ ) ( ψ ( 1 ) − ψ ( τ ) ) α − 1 d τ + ∑ i = 1 m − 1 a i 1 Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( τ ) ( ψ ( μ i ) − ψ ( τ ) ) α + σ i − 1 d τ ≤ M q − 1 ( ψ ( 1 ) − ψ ( 0 ) ) α Γ ( α + 1 ) + ( ψ ( 1 ) − ψ ( 0 ) ) γ + α − 1 Λ Γ ( γ ) Γ ( α + 1 ) + ∑ i = 1 m a i ( ψ ( 1 ) − ψ ( 0 ) ) − 1 ( ψ ( μ i ) − ψ ( 0 ) ) α + σ i Λ Γ ( γ ) Γ ( α + σ i + 1 ) = M q − 1 Ω ,

which, on taking the norm for t ∈ [ 0 , 1 ] , yields ‖ G y ‖ ≤ r . Because y ∈ B r is an arbitrary element, we obtain G B r ⊂ B r .

In order to show that G is a contraction, let y , z ∈ C ( [ 0 , 1 ] ) . Then, using Lemma 3 and the fact that 1 < p ≤ 2 ⇒ q > 2 , we obtain

∣ ( G 1 y ) ( t ) − ( G 1 z ) ( t ) ∣ = ϕ q ∫ 0 t f ( s , y ( s ) ) d s − ϕ q ∫ 0 t f ( s , z ( s ) ) d s ≤ ( q − 1 ) M q − 2 ∫ 0 t f ( s , y ( s ) ) d s − ∫ 0 t f ( s , z ( s ) ) d s ≤ k ( q − 1 ) M q − 2 t ‖ y − z ‖ .

In consequence, we obtain

∣ ( G y ) ( t ) − ( G z ) ( t ) ∣ = ∣ ( G 2 ∘ G 1 y ) ( t ) − ( G 2 ∘ G 1 z ) ( t ) ∣ = 1 Γ ( α ) ∫ 0 t ψ ′ ( τ ) ( ψ ( t ) − ψ ( τ ) ) α − 1 ( ( G 1 y ) ( τ ) − ( G 1 z ) ( τ ) ) d τ + ( ψ ( t ) − ψ ( 0 ) ) γ − 1 Λ Γ ( γ ) × 1 Γ ( α ) ∫ 0 1 ψ ′ ( τ ) ( ψ ( 1 ) − ψ ( τ ) ) α − 1 ( ( G 1 y ) ( τ ) − ( G 1 z ) ( τ ) ) d τ − ∑ i = 1 m − 1 a i 1 Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( τ ) ( ψ ( μ i ) − ψ ( τ ) ) α + σ i − 1 ( ( G 1 y ) ( τ ) − ( G 1 z ) ( τ ) ) d τ ≤ k ( q − 1 ) M q − 2 ‖ y − z ‖ 1 Γ ( α ) ∫ 0 t ψ ′ ( τ ) ( ψ ( t ) − ψ ( τ ) ) α − 1 τ d τ + ( ψ ( t ) − ψ ( 0 ) ) γ − 1 Λ Γ ( γ ) × 1 Γ ( α ) ∫ 0 1 ψ ′ ( τ ) ( ψ ( 1 ) − ψ ( τ ) ) α − 1 τ d τ + ∑ i = 1 m − 1 a i 1 Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( τ ) ( ψ ( μ i ) − ψ ( τ ) ) α + σ i − 1 τ d τ ≤ k ( q − 1 ) M q − 2 ( ψ ( 1 ) − ψ ( 0 ) ) α Γ ( α + 1 ) + ( ψ ( 1 ) − ψ ( 0 ) ) γ + α − 1 Λ Γ ( γ ) Γ ( α + 1 ) + ∑ i = 1 m a i ( ψ ( 1 ) − ψ ( 0 ) ) γ − 1 ( ψ ( μ i ) − ψ ( 0 ) ) α + σ i Λ Γ ( γ ) Γ ( α + σ i + 1 ) ‖ y − z ‖ = k ( q − 1 ) M q − 2 Ω ‖ y − z ‖ .

Taking the norm of the above inequality for t ∈ [ 0 , 1 ] , we obtain

‖ ( G y ) ( t ) − ( G z ) ( t ) ‖ ≤ k ( q − 1 ) M q − 2 Ω ,

which together with the condition (21) implies that the operator G is a contraction. Hence, by the Banach’s contraction mapping principle, the operator G has a unique fixed point. Hence, problem (1) has a unique solution on [ 0 , 1 ] . This completes the proof.□

Theorem 2

Assume that p > 2 and the following conditions hold:

There exist constants m > 0 and δ with 0 < δ ≤ 1 2 − q such that
f ( t , y ) ≥ m δ t δ − 1 , f o r ( t , y ) ∈ [ 0 , 1 ] × R ;
There exists a constant k > 0 such that
∣ f ( t , y ) − f ( t , z ) ∣ ≤ k ∣ y − z ∣ , f o r ( t , y i ) ∈ [ 0 , 1 ] × R , i = 1 , 2 .

(22) 0 < k < 1 ( q − 1 ) m q − 2 Ω ,

where Ω is given by (20), then there exists a unique solution to the boundary value problem (1) on [ 0 , 1 ] .

Proof

By ( A 3 ) , we have

∫ 0 t f ( s , y ( s ) ) d s ≥ ∫ 0 t m δ s δ − 1 d s = m t δ .

Choosing r ¯ ≥ max 1 − f 0 k , f 0 Ω 1 − k Ω , where f 0 = max t ∈ [ 0 , 1 ] ∣ f ( t , 0 ) ∣ , we define B r ¯ = { y ∈ C ( [ 0 , 1 ] , R ) : ‖ y ‖ ≤ r ¯ } and show that G B r ¯ ⊂ B r ¯ , where the operator G is defined by (17). Using the definition of ϕ q ( ⋅ ) and the values of r ¯ and q , we have for y ∈ B r ¯ that

∣ ( G 1 y ) ( t ) ∣ = ϕ q ∫ 0 t f ( s , y ( s ) ) d s = ϕ q ∫ 0 t f ( s , y ( s ) ) d s ≤ ϕ q ∫ 0 t [ ∣ f ( s , y ( s ) ) − f ( s , 0 ) ∣ + ∣ f ( s , 0 ) ∣ ] d s ≤ ϕ q ∫ 0 1 [ k ‖ y ‖ + f 0 ] d s ≤ ( k r ¯ + f 0 ) q − 1 ≤ ( k r ¯ + f 0 ) .

As in the previous theorem, one can obtain

‖ ( G y ) ( t ) ‖ ≤ ( k r ¯ + f 0 ) Ω ≤ r ¯ ,

which means that ‖ G y ‖ ≤ r ¯ . Thus, G y ∈ B r ¯ . Because y ∈ B r ¯ is an arbitrary element, we obtain G B r ¯ ⊂ B r ¯ .

Next, it will be shown that G is a contraction. Let y , z ∈ C ( [ 0 , 1 ] ) . Because p > 2 implies 1 < q < 2 , by Lemma (3), we obtain

∣ ( G 1 y ) ( t ) − ( G 1 z ) ( t ) ∣ = ϕ q ∫ 0 t f ( s , y ( s ) ) d s − ϕ q ∫ 0 t f ( s , z ( s ) ) d s ≤ ( q − 1 ) ( m t δ ) q − 2 ∫ 0 t f ( s , y ( s ) ) d s − ∫ 0 t f ( s , z ( s ) ) d s ≤ k ( q − 1 ) m q − 2 t δ ( q − 2 ) + 1 ‖ y − z ‖ ≤ k ( q − 1 ) m q − 2 ‖ y − z ‖ .

Consequently, we have

∣ ( G y ) ( t ) − ( G z ) ( t ) ∣ = ∣ ( G 2 ∘ G 1 y ) ( t ) − ( G 2 ∘ G 1 z ) ( t ) ∣ = 1 Γ ( α ) ∫ 0 t ψ ′ ( τ ) ( ψ ( t ) − ψ ( τ ) ) α − 1 ( ( G 1 y ) ( τ ) − ( G 1 z ) ( τ ) ) d τ + ( ψ ( t ) − ψ ( 0 ) ) γ − 1 Λ Γ ( γ ) × 1 Γ ( α ) ∫ 0 1 ψ ′ ( τ ) ( ψ ( 1 ) − ψ ( τ ) ) α − 1 ( ( G 1 y ) ( τ ) − ( G 1 z ) ( τ ) ) d τ − ∑ i = 1 m − 1 a i 1 Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( τ ) ( ψ ( μ i ) − ψ ( τ ) ) α + σ i − 1 ( ( G 1 y ) ( τ ) − ( G 1 z ) ( τ ) ) d τ ≤ k ( q − 1 ) m q − 2 ‖ y − z ‖ 1 Γ ( α ) ∫ 0 t ψ ′ ( τ ) ( ψ ( t ) − ψ ( τ ) ) α − 1 τ δ ( q − 2 ) + 1 d τ + ( ψ ( t ) − ψ ( 0 ) ) γ − 1 Λ Γ ( γ ) × 1 Γ ( α ) ∫ 0 1 ψ ′ ( τ ) ( ψ ( 1 ) − ψ ( τ ) ) α − 1 τ δ ( q − 2 ) + 1 d τ + ∑ i = 1 m − 1 a i 1 Γ ( α + σ i ) ∫ 0 μ i ψ ′ ( τ ) ( ψ ( μ i ) − ψ ( τ ) ) α + σ i − 1 τ δ ( q − 2 ) + 1 d τ ≤ k ( q − 1 ) m q − 2 ( ψ ( 1 ) − ψ ( 0 ) ) α Γ ( α + 1 ) + ( ψ ( 1 ) − ψ ( 0 ) ) γ + α − 1 Λ Γ ( γ ) Γ ( α + 1 ) + ∑ i = 1 m a i ( ψ ( 1 ) − ψ ( 0 ) ) γ − 1 ( ψ ( μ i ) − ψ ( 0 ) ) α + σ Λ Γ ( γ ) Γ ( α + σ i + 1 ) ‖ y − z ‖ = k ( q − 1 ) m q − 2 Ω ‖ y − z ‖ ,

which, on taking norm for t ∈ [ 0 , 1 ] , yields ‖ ( G 1 y ) − ( G 1 z ) ‖ ≤ k ( q − 1 ) m q − 2 Ω ‖ y − z ‖ . Thus, it follows from condition (22) that the operator G is a contraction. Hence, by Banach’s contraction principle, the operator G has a unique fixed point, which is indeed a unique solution to problem (1) on [ 0 , 1 ] . This finishes the proof.□

Theorem 3

Let p > 2 . Assume that ( A 4 ) and the following condition are satisfied:

There exist a number m > 0 and 0 < δ ≤ 1 2 − q such that
f ( t , y ) ≤ − m δ t δ − 1 , f o r ( t , y ) ∈ [ 0 , 1 ] × R .

Then, the boundary value problem (1) has a unique solution on [ 0 , 1 ] if k satisfies

0 < k < 1 ( q − 1 ) m q − 2 Ω .

Proof

Since the proof is similar to that of the last theorem, we omit it.□

4 Examples

Consider the following problem:

(23) ( ϕ p ( D 0 + 3 ∕ 2 , 1 ∕ 2 , ψ H y ( x ) ) ) ′ + f ( t , y ( t ) ) = 0 , y ( 0 ) = D 0 + 3 ∕ 2 , 1 ∕ 2 , ψ H y ( 0 ) = 0 , y ( 1 ) = 1 ∕ 4 I 0 + 5 ∕ 3 y ( 1 ∕ 2 ) + 3 I 0 + 4 ∕ 5 y ( 3 ∕ 4 ) ,

where α = 3 ∕ 2 , β = 1 ∕ 2 , σ 1 = 5 ∕ 3 , σ 2 = 4 ∕ 5 , μ 1 = 1 ∕ 2 , μ 2 = 3 ∕ 4 , a 1 = 1 ∕ 4 , a 2 = 3 , and ψ ( t ) = t 2 + t . We will fix p and f ( t , y ( t ) ) later.

For illustrating Theorem 3.1, let us take

(24) f ( t , y ) = 1 400 + t e t sin y + ∣ y ∣ ∣ y ∣ + 1 + cos t ,

and p = 4 ∕ 3 , which implies that q = 4 . From the given data, we find that γ = 1.75 and Ω = 7.222698841 . Condition ( A 1 ) is satisfied with g ( t ) = 2 e t + cos t t + 400 , and so M = 0.2137530673 . In addition, ( A 2 ) and (21) are satisfied with k = 0.2718281828 . Thus, all the conditions of Theorem 3.1 are satisfied, and hence, its conclusion implies that problem (23) with p = 4 ∕ 3 and f ( t , y ) given by (24) has a unique solution on [ 0 , 1 ] .

Next, we illustrate Theorem 3.2 by choosing

(25) f ( t , y ) = 2 e t 3 ∣ y ∣ ∣ y ∣ + 12 + 1 ,

and p = 7 ∕ 2 , which means that q = 7 ∕ 5 . Let us take δ = 4 ∕ 3 < 1 ∕ ( 2 − q ) and m = 1 ∕ 2 . Then, f ( t , y ) ≥ 2 ∕ 3 t 1 ∕ 3 = δ m t δ − 1 , that is, ( A 3 ) is satisfied. In addition, ( A 4 ) and (22) hold true with k = 0.1510156571 . Thus, it follows by the conclusion of Theorem 3.2 that problem (23) with p = 7 ∕ 2 and f ( t , y ) given by (25) has a unique solution on [ 0 , 1 ] .

5 Conclusion

We have presented the uniqueness results for a ψ -Hilfer fractional differential equation involving the p -Laplacian operator complemented with nonlocal ψ -Riemann-Liouville fractional integral boundary conditions with the aid of Banach’s contraction mapping principle. Our first result deals with the case 1 < p ≤ 2 , while the second and third results are obtained under different conditions on the nonlinear function f ( t , y ) involved in the given problem when p > 2 . Our results are new in the given configuration and enrich the literature on the p -Laplacian fractional order boundary value problems.

Acknowledgments

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. The authors thank the reviewers for their constructive remarks on their work.

Funding information: The DSR at KAU, Jeddah, Saudi Arabia, has funded this project, under grant no. KEP-PhD: 35-130-1443.
Author contributions: All authors accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state that there is no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, Oxford, 2005. 10.1093/oso/9780198526049.001.0001Search in Google Scholar

[2] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Danbury, 2006. Search in Google Scholar

[3] M. Javidi and B. Ahmad, Dynamic analysis of time fractional order phytoplankton-toxic phytoplankton-zooplankton system, Ecological Modelling 318 (2015), no. 3, 8–18, DOI: https://doi.org/10.1016/j.ecolmodel.2015.06.016. 10.1016/j.ecolmodel.2015.06.016Search in Google Scholar

[4] H. A. Fallahgoul, S. M. Focardi, and F. J. Fabozzi, Fractional Calculus and Fractional Processes with Applications to Financial Economics. Theory and Application, Elsevier/Academic Press, London, 2017. 10.1016/B978-0-12-804248-9.50002-4Search in Google Scholar

[5] A. N. Chatterjee and B. Ahmad, A fractional-order differential equation model of COVID-19 infection of epithelial cells, Chaos Solitons Fractals 147 (2021), 110952, DOI: https://doi.org/10.1016/j.chaos.2021.110952.10.1016/j.chaos.2021.110952Search in Google Scholar PubMed PubMed Central

[6] H. Yan, Y. Qiao, L. Duan, and J. Miao, Synchronization of fractional-order gene regulatory networks mediated by miRNA with time delays and unknown parameters, J. Franklin Inst. 359 (2022), no. 5, 2176–2191, DOI: https://doi.org/10.1016/j.jfranklin.2022.01.028. 10.1016/j.jfranklin.2022.01.028Search in Google Scholar

[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam, 2006. 10.1016/S0304-0208(06)80001-0Search in Google Scholar

[8] K. Diethelm, The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2010. 10.1007/978-3-642-14574-2Search in Google Scholar

[9] B. Ahmad, A. Alsaedi, S. K. Ntouyas, and J. Tariboon, Hadamard-type Fractional Differential Equations, Inclusions and Inequalities, Springer, Cham, 2017. 10.1007/978-3-319-52141-1Search in Google Scholar

[10] B. Ahmad, M. Alghanmi, S. K. Ntouyas, and A. Alsaedi, Fractional differential equations involving generalized derivative with Stieltjes and fractional integral boundary conditions, Appl. Math. Lett. 84 (2018), 111–117, DOI: https://doi.org/10.1016/j.aml.2018.04.024. 10.1016/j.aml.2018.04.024Search in Google Scholar

[11] R. Hilfer, (Ed.), Applications of Fractional Calculus in Physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. 10.1142/3779Search in Google Scholar

[12] R. Hilfer, Experimental evidence for fractional time evolution in glass forming materials, J. Chem. Phys. 284 (2002), no. 1–2, 399–408, DOI: https://doi.org/10.1016/S0301-0104(02)00670-5. 10.1016/S0301-0104(02)00670-5Search in Google Scholar

[13] I. Ali and N. A. Malik, Hilfer fractional advection-diffusion equations with power-law initial condition; a numerical study using variational iteration method, Comput. Math. Appl. 68 (2014), no. 10, 1161–1179, DOI: https://doi.org/10.1016/j.camwa.2014.08.021. 10.1016/j.camwa.2014.08.021Search in Google Scholar

[14] A. Wongchareon, B. Ahmad, S. K. Ntouyas, and J. Tariboon, Three-point boundary value problem for the Langevin equation with the Hilfer fractional derivative, Adv. Math. Phys. 2020 (2020), Article ID 9606428, 11 pages, DOI: https://doi.org/10.1155/2020/9606428. 10.1155/2020/9606428Search in Google Scholar

[15] J. E. Restrepo and D. Suragan, Hilfer-type fractional differential equations with variable coefficients, Chaos Solitons Fractals 150 (2021), 111146, DOI: https://doi.org/10.1016/j.chaos.2021.111146. 10.1016/j.chaos.2021.111146Search in Google Scholar

[16] C. Nuchpong, S. K. Ntouyas, A. Samadi, and J. Tariboon, Boundary value problems for Hilfer type sequential fractional differential equations and inclusions involving Riemann-Stieltjes integral multi-strip boundary conditions, Adv. Difference Equ. 2021 (2021), no. 268, 19, DOI: https://doi.org/10.1186/s13662-021-03424-7.10.1186/s13662-021-03424-7Search in Google Scholar

[17] Y. Zhou and J. W. He, A Cauchy problem for fractional evolution equations with Hilfer’s fractional derivative on semi-infinite interval, Fract. Calc. Appl. Anal. 25 (2022), 924–961, DOI: https://doi.org/10.1007/s13540-022-00057-9.10.1007/s13540-022-00057-9Search in Google Scholar

[18] A. Alsaedi, B. Ahmad, A. Assolami, and S. K. Ntouyas, On a nonlinear coupled system of differential equations involving Hilfer fractional derivative and Riemann-Liouville mixed operators with nonlocal integro-multi-point boundary conditions, AIMS Math. 7 (2022), no. 7, 12718–12741, DOI: https://doi.org/10.3934/math.2022704. 10.3934/math.2022704Search in Google Scholar

[19] J. V. C. Sousa and E. Capelas de Oliveira, On the Ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 60 (2018), 72–91, DOI: https://doi.org/10.1016/j.cnsns.2018.01.005. 10.1016/j.cnsns.2018.01.005Search in Google Scholar

[20] C. Bai, Existence and uniqueness of solutions for fractional boundary value problems with p-Laplacian operator, Adv. Difference Equ. 2018 (2018), no. 4, 12, DOI: https://doi.org/10.1186/s13662-017-1460-3.10.1186/s13662-017-1460-3Search in Google Scholar

[21] J. V. C. Sousa, K. D. Kucche, and E. Capelas de Oliveira, Stability of ψ-Hilfer impulsive fractional differential equations, Appl. Math. Lett. 88 (2019), 73–80, DOI: https://doi.org/10.1016/j.aml.2018.08.013.10.1016/j.aml.2018.08.013Search in Google Scholar

[22] M. S. Abdo, S. T. M. Thabet, and B. Ahmad, The existence and Ulam-Hyers stability results for ψ-Hilfer fractional integrodifferential equations, J. Pseudo-Differ. Oper. Appl. 11 (2020), 1757–1780, DOI: https://doi.org/10.1007/s11868-020-00355-x.10.1007/s11868-020-00355-xSearch in Google Scholar

[23] W. Abdelhedi, Fractional differential equations with a ψ-Hilfer fractional derivative, Comput. Appl. Math. 40 (2021), no. 53, 19, DOI: https://doi.org/10.1007/s40314-021-01447-0.10.1007/s40314-021-01447-0Search in Google Scholar

[24] A. Wongcharoen, S. K. Ntouyas, P. Wongsantisuk, and J. Tariboon, Existence results for a nonlocal coupled system of sequential fractional differential equations involving ψ-Hilfer fractional derivatives, Adv. Math. Phys. 2021 (2021), Art. ID 5554619, 9, DOI: https://doi.org/10.1155/2021/5554619. 10.1155/2021/5554619Search in Google Scholar

[25] N. Vieira, M. M. Rodrigues, and M. Ferreira, Time-fractional diffusion equation with ψ-Hilfer derivative, Comput. Appl. Math. 41 (2022), no. 230, 26, DOI: https://doi.org/10.1007/s40314-022-01911-5.10.1007/s40314-022-01911-5Search in Google Scholar

[26] M. Vellappandi, V. Govindaraj, and C. Sousa, Fractional optimal reachability problems with ψ-Hilfer fractional derivative, Math. Methods Appl. Sci. 45 (2022), no. 10, 6255–6267, DOI: https://doi.org/10.1002/mma.8168. 10.1002/mma.8168Search in Google Scholar

[27] L. S. Leibenson, General problem of the movement of a compressible fluid in a porous medium (Russian), Bull. Acad. Sci. URSS. Ser. Geograph. Geophys. [Izvestia Akad. Nauk SSSR] 9 (1945), 7–10. Search in Google Scholar

[28] X. Liu, M. Jia, and X. Xiang, On the solvability of a fractional differential equation model involving the p-Laplacian operator, Comput. Math. Appl. 64 (2012), no. 10, 3267–3275, DOI: https://doi.org/10.1016/j.camwa.2012.03.001.10.1016/j.camwa.2012.03.001Search in Google Scholar

[29] X. Liu, M. Jia, and W. Ge, The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator, Appl. Math. Lett. 65 (2017), 56–62, DOI: https://doi.org/10.1016/j.aml.2016.10.001.10.1016/j.aml.2016.10.001Search in Google Scholar

[30] J. Tan and M. Li, Solutions of fractional differential equations with p-Laplacian operator in Banach spaces, Bound. Value Probl. 2018 (2018), no. 15, 13, DOI: https://doi.org/10.1186/s13661-018-0930-1. 10.1186/s13661-018-0930-1Search in Google Scholar

[31] S. Wang and Z. Bai, Existence and uniqueness of solutions for a mixed p-Laplace boundary value problem involving fractional derivatives, Adv. Difference Equ. 2020 (2020), no. 694, 9, DOI: https://doi.org/10.1186/s13662-020-03154-2. 10.1186/s13662-020-03154-2Search in Google Scholar

[32] J. V. C. Sousa, Existence and uniqueness of solutions for the fractional differential equations with P-Laplacian in HPν,η;ψ, J. Appl. Anal. Comput. 12 (2022), no. 2, 622–661, DOI: https://doi.org/10.11948/20210258. 10.11948/20210258Search in Google Scholar

[33] R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul. 44 (2017), 460–481, DOI: https://doi.org/10.1016/j.cnsns.2016.09.006.10.1016/j.cnsns.2016.09.006Search in Google Scholar

Received: 2022-07-26

Revised: 2022-10-18

Accepted: 2023-01-05

Published Online: 2023-02-15

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

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0195

Keywords for this article

ψ-Hilfer fractional derivative; the p-Laplacian operator; existence; fixed point

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