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Article

Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations

1
Department of Social and Applied Science, College of Industrial Technology, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
2
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
3
Intelligent and Nonlinear Dynamic Innovations Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand
*
Author to whom correspondence should be addressed.
Mathematics 2023, 11(4), 867; https://doi.org/10.3390/math11040867
Submission received: 24 January 2023 / Revised: 4 February 2023 / Accepted: 5 February 2023 / Published: 8 February 2023

Abstract

:
In this paper, we introduce and study a new class of boundary value problems, consisting of a mixed-type ψ 1 -Hilfer and ψ 2 -Caputo fractional order differential equation supplemented with integro-differential nonlocal boundary conditions. The uniqueness of solutions is achieved via the Banach contraction principle, while the existence of results is established by using the Leray–Schauder nonlinear alternative. Numerical examples are constructed illustrating the obtained results.

1. Introduction

Fractional order differential equations have recently proved to be valuable tools in the modeling of many phenomena in various fields of mathematics, physics, viscoelasticity, electrochemistry, engineering, control, porous media, electromagnetic, etc., see [1,2,3,4,5] and references cited therein. For a theoretical approach of fractional calculus, see the monographs [6,7,8,9,10,11]. Many processes in physics and engineering can be described accurately by using differential equations containing different types of fractional derivatives such as Riemann–Liouville, Caputo, Hadamard, Erdeyl–Kober, Hilfer, Caputo–Hadamard, etc. Hilfer proposed in [12] a fractional derivative operator generalizing both Riemann–Liouville and Caputo fractional derivative operators. For the advantages of the Hilfer derivative, see [13]. In [14], the ψ -Hilfer fractional derivative operator was introduced. Initial and boundary value problems including the ψ -Hilfer fractional derivative operator have been studied by many researchers, see [15,16,17,18,19,20] and references therein.
In the present paper, we investigate a new class of boundary value problems, consisting of mixed-type fractional differential equations including ψ 1 -Hilfer and ψ 2 -Caputo fractional derivative operators supplemented with nonlocal integro-differential boundary conditions. More precisely, we consider the following sequential ψ 1 -Hilfer and ψ 2 -Caputo fractional differential equation with nonlocal integro-differential boundary conditions
H D α , β ; ψ 1 ( C D γ ; ψ 2 π ) ( t ) = Π ( t , π ( t ) ) , 0 < α , β , γ < 1 , t [ 0 , x 1 ] , C D γ ; ψ 2 π ( 0 ) = 0 , π ( T ) = i = 1 m λ i C D γ ; ψ 2 π ( η i ) + j = 1 n δ j I μ j ; ψ 2 π ( ξ j ) ,
where H D α , β ; ψ 1 and C D γ ; ψ 2 are the ψ 1 -Hilfer and ψ 2 -Caputo fractional derivatives with respect to functions ψ 1 and ψ 2 , respectively, when ψ 1 ( t ) , ψ 2 ( t ) > 0 for all t [ 0 , x 1 ] . In addition, the given constants λ i , δ j R and some points η i , ξ j ( 0 , x 1 ) , I μ j ; ψ 2 is the Riemann–Liouville fractional integral of order μ j > 0 , with respect to a function ψ 2 , for i = 1 , , m , j = 1 , , n and Π : [ 0 , x 1 ] × R R is a nonlinear continuous function. Existence and uniqueness are established via Banach’s fixed point theorem and the Leray–Schauder nonlinear alternative.
The novelty of this study lies in the fact that we introduce a new class of nonlocal boundary value problems in which we combine ψ 1 -Hilfer and ψ 2 -Caputo fractional derivative operators and as far as we know, this is the only paper dealing with this combination. By fixing the parameters in the nonlocal integro-differential fractional boundary value problem (1), we obtain some new results as special cases. For example, we get to:
(i)
Hilfer and Caputo fractional nonlocal integro-differential boundary value problem if ψ 1 ( t ) = ψ 2 ( t ) = t ;
(ii)
ψ 2 -Hilfer and Caputo-type fractional nonlocal integro-differential boundary value problem if ψ 1 ( t ) = t ;
(iii)
ψ 1 -Hilfer and Caputo-type nonlocal integro-differential boundary value problem if ψ 2 ( t ) = t .
The remaining part of this article is organized as follows: in Section 2, some preliminary definitions and results that will be applied in the next sections are recalled. In addition, an auxiliary result is proved to convert the problem (1) into a fixed point problem. In Section 3, the main results for the nonlocal integro-differential boundary value problem (1) are established, while in Section 4, these results are discussed for some special cases. Section 5 includes some numerical examples illustrating the main results.

2. Preliminaries

Now, some notations and definitions of fractional calculus are recalled. In the following, we assume that ψ C 1 ( [ 0 , x 1 ] , R ) is an increasing function with ψ ( t ) > 0 for all t [ 0 , x 1 ] .
Definition 1
([7]). Given α > 0 and h ^ L 1 ( [ 0 , x 1 ] , R ) , the ψ-Riemann–Liouville fractional integral of order α of a function h ^ with respect to ψ is defined by
I 0 α ; ψ h ^ ( t ) = 1 Γ ( α ) a t ψ ( s ) ( ψ ( t ) ψ ( s ) ) α 1 h ^ ( s ) d s .
To abbreviate, we use I 0 α ; ψ h ^ ( t ) as I α ; ψ h ^ ( t ) throughout this paper.
Definition 2
([14]). Suppose that n 1 < α < n , n N and h ^ , ψ C n ( [ 0 , x 1 ] , R ) . The ψ-Hilfer fractional derivative H D α , β ; ψ ( · ) of order α of a function h ^ with a parameter β [ 0 , 1 ] is defined by
H D α , β ; ψ h ^ ( t ) = I β ( n α ) ; ψ 1 ψ ( t ) d d t n I ( 1 β ) ( n α ) ; ψ h ^ ( t ) ,
provided that the right-hand side exists.
Definition 3
([21]). The ψ-Caputo fractional derivative C D α ; ψ ( · ) of order α of a function h ^ is expressed as
C D α ; ψ h ^ ( t ) = I n α ; ψ 1 ψ ( t ) d d t n h ^ ( t ) ,
where n 1 < α < n , n N and h ^ , ψ C n ( [ 0 , x 1 ] , R ) .
Remark 1
([22]). The following relations hold:
ρ = α + β ( n α ) , n 1 < α , ρ < n , 0 β 1 ,
and
ρ α , ρ > β , n ρ < n β ( n α ) .
Lemma 1
([14]). Let α , μ > 0 and δ > 1 be constants. Then, we have:
(i)
I α ; ψ I μ ; ψ h ^ ( t ) = I α + μ ; ψ h ^ ( t ) ;
(ii)
I α ; ψ ψ ( t ) ψ ( 0 ) δ 1 = Γ ( δ ) Γ ( α + δ ) ( ψ ( t ) ψ ( 0 ) ) α + δ 1 .
Lemma 2.
Let h ^ L ( 0 , x 1 ) , n 1 < α n , n N ,   0 β 1 ,   ρ = α + n β α β ,   I ( n α ) ( 1 β ) ; ψ h ^ A C k [ 0 , x 1 ] .  (Here, by A C k [ 0 , x 1 ] , we denote the space of k times absolutely continuous functions on [ 0 , x 1 ] .) Then, we have
I α ; ψ H D α , β ; ψ h ^ ( t ) = h ^ ( t ) k = 1 n ( ψ ( t ) ψ ( a ) ) ρ k Γ ( ρ k + 1 ) h ^ ψ [ n k ] I ( 1 β ) ( n α ) ; ψ h ^ ( a ) ,
where h ^ ψ [ n k ] = 1 ψ ( t ) d d t n k and
I α ; ψ C D α ; ψ h ^ ( t ) = h ^ ( t ) k = 0 n 1 h ^ ψ [ k ] h ^ ( a ) k ! ( ψ ( t ) ψ ( a ) ) k .
A linear variant of the sequential Hilfer–Caputo fractional integro-differential boundary value problem (1) is investigated in the next lemma.
Lemma 3.
Let h C ( [ 0 , x 1 ] , R ) be a given function and all constants are as in boundary value problem (1). Then, the sequential Hilfer–Caputo fractional integro-differential linear boundary value problem
H D α , β ; ψ 1 ( C D γ ; ψ 2 π ) ( t ) = h ( t ) , t [ 0 , x 1 ] , C D γ ; ψ 2 π ( 0 ) = 0 , π ( x 1 ) = i = 1 m λ i C D γ ; ψ 2 π ( η i ) + j = 1 n δ j I μ j ; ψ 2 π ( ξ j )
is equivalent to the integral equation
π ( t ) = 1 A i = 1 m λ i I α ; ψ 1 h ( η i ) + j = 1 n δ j I μ j + γ ; ψ 2 I α ; ψ 1 h ( ξ j ) I γ ; ψ 2 I α ; ψ 1 h ( x 1 ) + I γ ; ψ 2 I α ; ψ 1 h ( t ) ,
where it is assumed that
A : = 1 j = 1 n δ j ψ 2 ( ξ j ) ψ 2 ( 0 ) μ j Γ ( μ j + 1 ) 0 .
Proof. 
Operating the fractional integral I α ; ψ 1 to both sides of the first equation in (2) and applying Lemma 2, we obtain for t [ 0 , x 1 ] , that
C D γ : ψ 2 π ( t ) = c 0 Γ ( ρ 1 ) ( ψ 1 ( t ) ψ 1 ( 0 ) ) ρ 1 1 + I α ; ψ 1 h ( t ) ,
where ρ 1 = α + ( 1 α ) β and c 0 R . Since ρ 1 ( α , 1 ) , by Remark 1, from C D γ : ψ 2 π ( 0 ) = 0 , we have c 0 = 0 . Therefore, we get
C D γ : ψ 2 π ( t ) = I α ; ψ 1 h ( t ) ,
which leads to
i = 1 m λ i C D γ : ψ 2 π ( η i ) = i = 1 m λ i I α ; ψ 1 h ( η i ) .
Acting I γ ; ψ 2 in (5) yields
π ( t ) = c 1 + I γ ; ψ 2 I α ; ψ 1 h ( t ) .
In addition, we have
j = 1 n δ j I μ j ; ψ 2 π ( ξ j ) = c 1 j = 1 n δ j ψ 2 ( ξ j ) ψ 2 ( 0 ) μ j Γ ( μ j + 1 ) + j = 1 n δ j I μ j + γ ; ψ 2 I α ; ψ 1 h ( ξ j ) .
From the second boundary condition (2) with (6) and (8), we get
c 1 = 1 A i = 1 m λ i I α ; ψ 1 h ( η i ) + j = 1 n δ j I μ j + γ ; ψ 2 I α ; ψ 1 h ( ξ j ) I γ ; ψ 2 I α ; ψ 1 h ( x 1 ) ,
where A is defined in (4). Substituting the value of c 1 in (7), we get the solution (3). On the other hand, by taking the fractional differential operator of ψ 2 -Caputo and ψ 1 -Hilfer of orders γ and α , respectively, we get the first equation in problem (2). By direct computation, it is easy to see that (3) satisfies the two boundary conditions in (2). Therefore, the proof is completed. □

3. Main Results

In this section, we establish existence and uniqueness of solutions to the sequential Hilfer–Caputo fractional integro-differential boundary value problem (1) on an interval J = [ 0 , x 1 ] . At first, we denote the Banach space of all continuous functions from J to R equipped with the norm π = sup { | π ( t ) | : t J } by C = C ( J , R ) . Having in mind Lemma 3, we define an operator W : C C by
( W π ) ( t ) = 1 A [ i = 1 m λ i I α ; ψ 1 Π ( η i , π ( η i ) ) + j = 1 n δ j I μ j + γ ; ψ 2 I α ; ψ 1 Π ( ξ j , π ( ξ j ) ) I γ ; ψ 2 I α ; ψ 1 Π ( x 1 , π ( x 1 ) ) ] + I γ ; ψ 2 I α ; ψ 1 Π ( t , π ( t ) ) ,
where
I α ; ψ 1 Π ( η i , π ( η i ) ) = 1 Γ ( α ) 0 η i ψ 1 ( s ) ( ψ 1 ( η i ) ψ 1 ( s ) ) α 1 Π ( s , π ( s ) ) d s
and
I ϕ ; ψ 2 I α ; ψ 1 Π ( l , π ( l ) ) = 1 Γ ( α ) Γ ( ϕ ) 0 l 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 ( ψ 2 ( l ) ψ 2 ( u ) ) ϕ 1 Π ( s , π ( s ) ) d s d u ,
with ϕ { γ , μ j + γ } and l { t , x 1 , ξ j } . Note that if Π ( t , π ) 1 , we have
I ϕ ; ψ 2 I α ; ψ 1 ( 1 ) ( l ) = 1 Γ ( α + 1 ) Γ ( ϕ ) 0 l ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( 0 ) ) α ( ψ 2 ( l ) ψ 2 ( u ) ) ϕ 1 d u : = A ψ 1 , ψ 2 α , ϕ ( l ) .
For convenience, we put
A 1 = 1 | A | i = 1 m | λ i | ψ 1 ( η i ) ψ 1 ( 0 ) α Γ ( α + 1 ) + j = 1 n | δ j | A ψ 1 , ψ 2 α , μ j + γ ( ξ j ) + | A | + 1 | A | A ψ 1 , ψ 2 α , γ ( x 1 ) .
In the following theorem, we prove the existence and uniqueness of solutions of the fractional integro-differential boundary value problem of sequential Hilfer and Caputo fractional derivatives (1) by applying the Banach contraction mapping principle.
Theorem 1.
Let Π : J × R R such that:
( H 1 )
There exists L > 0 such that
| Π ( t , π 1 ) Π ( t , π 2 ) | L | π 1 π 2 | ,
t J and π 1 , π 2 R .
If
A 1 L < 1 ,
where A 1 is given by (11). Then, the fractional integro-differential boundary value problem of sequential Hilfer and Caputo fractional derivatives (1) has a unique solution on J.
Proof. 
Let M = sup { | Π ( t , 0 ) | : t J } and B r = { π C : π r * } with
r * M A 1 1 A 1 L .
Now, we will show that W B r * B r * . For any π B r * , we obtain
| W π ( t ) | sup t J | W π ( t ) | 1 | A | [ i = 1 m | λ i | I α ; ψ 1 | Π ( η i , π ( η i ) ) Π ( η i , 0 ) | + | Π ( η i , 0 ) | + j = 1 n | δ j | I μ j + γ ; ψ 2 I α ; ψ 1 | Π ( ξ j , π ( ξ j ) ) Π ( ξ j , 0 ) | + | Π ( ξ j , 0 ) | + I γ ; ψ 2 I α ; ψ 1 ( | Π ( x 1 , π ( x 1 ) ) Π ( x 1 , 0 ) | + | Π ( x 1 , 0 ) | ) ] + I γ ; ψ 2 I α ; ψ 1 | Π ( x 1 , π ( x 1 ) ) Π ( x 1 , 0 ) | + | Π ( x 1 , 0 ) | 1 | A | [ i = 1 m | λ i | ( L r * + M ) I α ; ψ 1 ( 1 ) ( η i ) + j = 1 n | δ j | ( L r * + M ) I μ j + γ ; ψ 2 I α ; ψ 1 ( 1 ) ( ξ j ) + ( L r * + M ) I γ ; ψ 2 I α ; ψ 1 ( 1 ) ( x 1 ) ] + ( L r * + M ) I γ ; ψ 2 I α ; ψ 1 ( 1 ) ( x 1 ) = 1 | A | [ ( L r * + M ) i = 1 m | λ i | ψ 1 ( η i ) ψ 1 ( 0 ) α Γ ( α + 1 ) + ( L r * + M ) j = 1 n | δ j | A ψ 1 , ψ 2 α , μ j + γ ( ξ j ) + ( L r * + M ) A ψ 1 , ψ 2 α , γ ( x 1 ) ] + ( L r * + M ) A ψ 1 , ψ 2 α , γ ( x 1 ) = ( L r * + M ) A 1 r * ,
which holds from (14). This shows that W B r * B r * . Next, we let π 1 , π 2 B r * , then we have
| W π 1 ( t ) W π 2 ( t ) | sup t J | W π 1 ( t ) W π 2 ( t ) | 1 | A | [ i = 1 m | λ i | I α ; ψ 1 | Π ( η i , π 1 ( η i ) ) Π ( η i , π 2 ( η i ) ) | + j = 1 n | δ j | I μ j + γ ; ψ 2 I α ; ψ 1 | Π ( ξ j , π 1 ( ξ j ) ) Π ( ξ j , π 2 ( ξ j ) ) | + I γ ; ψ 2 I α ; ψ 1 | Π ( x 1 , π 1 ( x 1 ) ) Π ( x 1 , π 2 ( x 1 ) ) | ] + I γ ; ψ 2 I α ; ψ 1 | Π ( x 1 , π 1 ( x 1 ) ) Π ( x 1 , π 2 ( x 1 ) ) | , L | A | [ π 1 π 2 i = 1 m | λ i | I α ; ψ 1 ( 1 ) ( η i ) + π 1 π 2 j = 1 n | δ j | I μ j + γ ; ψ 2 I α ; ψ 1 ( 1 ) ( ξ j ) + π 1 π 2 I γ ; ψ 2 I α ; ψ 1 ( 1 ) ( x 1 ) ] + L π 1 π 2 I γ ; ψ 2 I α ; ψ 1 ( 1 ) ( x 1 ) = A 1 L π 1 π 2 .
Therefore, the operator W satisfies the inequality W π 1 W π 2 A 1 L π 1 π 2 . Since, A 1 L < 1 , W is a contraction. Therefore, the operator W has a unique fixed point in the ball B r , by Banach’s contraction mapping. Consequently, the sequential Hilfer–Caputo fractional integro-differential boundary value problem (1) has a unique solution on J. □
Next, the nonlinear alternative of the Leray–Schauder-type [23] is used to prove the existence of at least one solution to the sequential Hilfer–Caputo fractional integro-differential boundary value problem (1).
Theorem 2.
Assume that Π : J × R R is a continuous function satisfying the conditions:
( H 2 )
There exists a continuous function Ω : [ 0 , ) ( 0 , ) which is nondecreasing and u 1 , u 2 : J R + two continuous functions such that
| Π ( t , π ) | u 1 ( t ) Ω ( | π | ) + u 2 ( t ) ,
for all t J and π R ;
( H 3 )
There exists a positive constant K such that
K ( u 1 Ω ( K ) + u 2 ) A 1 > 1 .
Then, the sequential Hilfer–Caputo fractional integro-differential boundary value problem (1) has at least one solution on J.
Proof. 
We show that the operator W defined by (10) is compact on a bounded ball B ρ , when B ρ = { π C : π ρ } . For any π B ρ , we have
| W π ( t ) | sup t J | W π ( t ) | 1 | A | [ i = 1 m | λ i | I α ; ψ 1 | Π ( η i , π ( η i ) ) | + j = 1 n | δ j | I μ j + γ ; ψ 2 I α ; ψ 1 | Π ( ξ j , π ( ξ j ) ) | + I γ ; ψ 2 I α ; ψ 1 | Π ( x 1 , π ( x 1 ) ) | ] + I γ ; ψ 2 I α ; ψ 1 | Π ( x 1 , π ( x 1 ) ) | , 1 | A | [ ( u 1 Ω ( ρ ) + u 2 ) i = 1 m | λ i | I α ; ψ 1 ( η i ) + ( u 1 Ω ( ρ ) + u 2 ) j = 1 n | δ j | I μ j + γ ; ψ 2 I α ; ψ 1 ( ξ j ) + ( u 1 Ω ( ρ ) + u 2 ) I γ ; ψ 2 I α ; ψ 1 ( x 1 ) ] + ( u 1 Ω ( ρ ) + u 2 ) I γ ; ψ 2 I α ; ψ 1 ( x 1 ) = ( u 1 Ω ( ρ ) + u 2 ) A 1 : = Φ , a constant ,
which yields W π Φ . Therefore, the set W ( B ρ ) is uniformly bounded. To show that W ( B ρ ) is an equicontinuous set, we let t 1 and t 2 be the two points in J such that t 1 < t 2 . Thus, for any π B ρ , we have
| W π ( t 2 ) W π ( t 1 ) | = | 1 Γ ( α ) Γ ( γ ) 0 t 2 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 Π ( s , π ( s ) ) d s d u 1 Γ ( α ) Γ ( γ ) 0 t 1 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × ( ψ 2 ( t 1 ) ψ 2 ( u ) ) α 1 Π ( s , π ( s ) ) d s d u | = | 1 Γ ( α ) Γ ( γ ) 0 t 1 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 Π ( s , π ( s ) ) d s d u + 1 Γ ( α ) Γ ( γ ) t 1 t 2 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 Π ( s , π ( s ) ) d s d u 1 Γ ( α ) Γ ( γ ) 0 t 1 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × ( ψ 2 ( t 1 ) ψ 2 ( u ) ) α 1 Π ( s , π ( s ) ) d s d u | = | 1 Γ ( α ) Γ ( γ ) 0 t 1 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 { ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 ( ψ 2 ( t 1 ) ψ 2 ( u ) ) α 1 } Π ( s , π ( s ) ) d s d u + 1 Γ ( α ) Γ ( γ ) t 1 t 2 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 Π ( s , π ( s ) ) d s d u | 1 Γ ( α ) Γ ( γ ) 0 t 1 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × | ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 ( ψ 2 ( t 1 ) ψ 2 ( u ) ) α 1 | | Π ( s , π ( s ) ) | d s d u + 1 Γ ( α ) Γ ( γ ) t 1 t 2 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 | Π ( s , π ( s ) ) | d s d u ( u 1 Ω ( ρ ) + u 2 ) Γ ( α ) Γ ( γ ) 0 t 1 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × | ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 ( ψ 2 ( t 1 ) ψ 2 ( u ) ) α 1 | d s d u + ( u 1 Ω ( ρ ) + u 2 ) Γ ( α ) Γ ( γ ) t 1 t 2 0 u ψ 1 ( s ) ψ 2 ( u ) ( ψ 1 ( u ) ψ 1 ( s ) ) α 1 × ( ψ 2 ( t 2 ) ψ 2 ( u ) ) α 1 d s d u .
Observe that if t 1 t 2 , then we have | W π ( t 2 ) W π ( t 1 ) | 0 independently of π . Therefore, the set W ( B ρ ) is an equicontinuous set. Hence, the set W ( B ρ ) is relatively compact. By applying the Arzelá–Ascoli theorem, the operator W is completely continuous.
Finally, we show that the set of all solutions to equations π = λ W π is bounded for λ ( 0 , 1 ) . Let π C and π = λ W π for some λ ( 0 , 1 ) . Then, for any t J , as in the first step, we obtain
| π ( t ) | = λ | W π ( t ) | sup t J | W π ( t ) | ( u 1 Ω ( π ) + u 2 ) A 1 ,
and, consequently,
π ( u 1 Ω ( π ) + u 2 ) A 1 1 .
From ( H 3 ) , π K . After that, we define U = { π B ρ : π < K } . Now, we can see that W : U ¯ C is continuous and completely continuous. Thus, there is no π U such that π = λ W π with 0 < λ < 1 . By the nonlinear alternative of the Leray–Schauder-type, we get that the operator W has a fixed point π U ¯ , which is a solution of the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) on J. The proof is completed. □

4. Some Special Cases

In this section, we present some special cases and some interesting behavior of solutions to the investigated problem (1).
Corollary 1.
Assume that Π : J × R R is a continuous function.
(a)
If | Π ( t , π ) | M , where M is a positive constant, then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solution J.
(b)
If u 1 ( t ) = 1 , Ω ( u ) = B u + C , u 2 ( t ) = D , where B 0 , C , D > 0 , then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solution J if A 1 B < 1 .
(c)
If u 1 ( t ) = 1 , Ω ( u ) = B u 2 + C , u 2 ( t ) = D , where B 0 , C , D > 0 , then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solution J , if 4 A 1 2 B ( C + D ) < 1 .
If we put ψ 1 ( t ) = ψ 2 ( t ) = ψ ( t ) , then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) is reduced to
H D α , β ; ψ ( C D γ ; ψ π ) ( t ) = Π ( t , π ( t ) ) , 0 < α , β , γ < 1 , t [ 0 , x 1 ] , C D γ ; ψ π ( 0 ) = 0 , π ( x 1 ) = i = 1 m λ i C D γ ; ψ π ( η i ) + j = 1 n δ j I μ j ; ψ π ( ξ j ) .
The following constants are used in the next corollaries.
A * = 1 j = 1 n δ j ψ ( ξ j ) ψ ( 0 ) μ j Γ ( μ j + 1 ) , A 1 * = 1 | A * | i = 1 m | λ i | ψ ( η i ) ψ ( 0 ) α Γ ( α + 1 ) + j = 1 n | δ j | ψ ( ξ j ) ψ ( 0 ) α + μ j + γ Γ ( α + μ j + γ + 1 ) + | A * | + 1 | A * | ψ ( x 1 ) ψ ( 0 ) α + γ Γ ( α + γ + 1 ) .
Corollary 2.
If f satisfies the Lipschitz condition in ( H 1 ) and if A 1 * L < 1 , then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has a unique solution on J .
Corollary 3.
If the continuous function f satisfies ( H 2 ) in Theorem 2 and if there exists a positive constant M such that
M ( u 1 Ω ( M ) + u 2 ) A 1 * > 1 ,
then the nonlocal fractional integro-differential sequential Hilfer and Caputo boundary value problem (17) has at least one solutions on J.
If n = p + q , and μ w = 0 for w = 1 , , q , then the problem (17) can be reduced to the following problem with integro-differential multi-point boundary conditions as
H D α , β ; ψ ( C D γ ; ψ π ) ( t ) = Π ( t , π ( t ) ) , 0 < α , β , γ < 1 , t [ 0 , x 1 ] , C D γ ; ψ π ( 0 ) = 0 , π ( x 1 ) = i = 1 m λ i C D γ ; ψ π ( η i ) + j = 1 p δ j I μ j ; ψ π ( ξ j ) + w = p + 1 q ζ w π ( θ w ) .
In addition, we put
A ^ = 1 j = 1 p δ j ψ ( ξ j ) ψ ( 0 ) μ j Γ ( μ j + 1 ) w = p + 1 q ζ w , A ^ 1 = 1 | A ^ | { i = 1 m | λ i | ψ ( η i ) ψ ( 0 ) α Γ ( α + 1 ) + j = 1 p | δ j | ψ ( ξ j ) ψ ( 0 ) α + μ j + γ Γ ( α + μ j + γ + 1 ) + w = p + 1 q | ζ w | ψ ( θ w ) ψ ( 0 ) α + γ Γ ( α + γ + 1 ) } + | A ^ | + 1 | A ^ | ψ ( x 1 ) ψ ( 0 ) α + γ Γ ( α + γ + 1 ) .
The existence and uniqueness results for the integro-differential multi-point boundary value problem (18) are similar to the Corollaries 2 and 3 by replacing A ^ 1 with A 1 * .

5. Illustrative Examples

Example 1.
Let us consider the following integro-differential boundary conditions to the sequential ψ 1 -Hilfer and ψ 2 -Caputo fractional differential equation of the form
H D 1 4 , 3 4 ; e 1 10 t C D 1 2 ; t 2 + t π ( t ) = Π ( t , π ( t ) ) , 0 < t < 9 8 , C D 1 2 ; t 2 + t π ( 0 ) = 0 , π 9 8 = 3 67 C D 1 2 ; t 2 + t π 3 8 + 5 77 C D 1 2 ; t 2 + t π 5 8
+ 7 87 C D 1 2 ; t 2 + t π 7 8 + 2 39 I 4 5 ; t 2 + t π 1 2 + 4 59 I 7 5 ; t 2 + t π 3 4 .
From the boundary value problem (19), we set constants as α = 1 / 4 , β = 3 / 4 , γ = 1 / 2 , x 1 = 9 / 8 , λ 1 = 3 / 67 , λ 2 = 5 / 77 , λ 3 = 7 / 87 , η 1 = 3 / 8 , η 2 = 5 / 8 , η 3 = 7 / 8 , δ 1 = 2 / 39 , δ 2 = 4 / 59 , μ 1 = 4 / 5 , μ 2 = 7 / 5 , ξ 1 = 1 / 2 , ξ 2 = 3 / 4 and functions ψ 1 ( t ) = e ( 1 / 10 ) t and ψ 2 ( t ) = t 2 + t . From above information, we can compute that A 0.8763925133 and A 1 2.374946616 . Observe that the two functions satisfy  ψ 1 , ψ 2 > 0 .
(i)
If the function Π is defined by
Π ( t , π ( t ) ) = 1 2 t + 5 ( 2 | π | + π 2 1 + | π | ) + 1 7 t 2 + 8 t + 1 9 .
From the given nonlinear unbounded Lipschitzian function in (21), we get | Π ( t , π ) Π ( t , z ) | ( 2 / 5 ) | π z | for t [ 0 , 9 / 8 ] , π , z R . Setting L = 2 / 5 , we have A 1 L 0.9499786464 < 1 which fulfills the condition in (13). The result in Theorem 1 can be used to conclude that the boundary value problem (19) and (20) with the given function in (21) has a unique solution on [ 0 , 9 / 8 ]
(ii)
Let the function Π be defined as
Π ( t , π ( t ) ) = 1 t + 4 π 2024 5 ( 1 + π 2022 ) + 1 3 t + 6 + 1 2 t + 7 .
We have
| Π ( t , π ) | 1 t + 4 1 5 π 2 + 1 6 + 1 2 t + 7 .
Choosing u 1 ( t ) = 1 / ( t + 4 ) , u 2 ( t ) = 1 / ( 2 t + 7 ) and Ω ( π ) = ( 1 / 5 ) π 2 + ( 1 / 6 ) , we get u 1 = 1 / 4 , u 2 = 1 / 7 and then, there exists a K ( 0.463775263 , 7.957466657 ) satisfying the inequality in (16). Therefore, all assumptions in Theorem 2 agree with function Π in (22). Then, using integro-differential boundary conditions to the sequential ψ 1 -Hilfer and ψ 2 -Caputo fractional differential Equations (19), (20) and (22) have at least one solution on [ 0 , 9 / 8 ] .
(iii)
If ψ 1 ( t ) = ψ 2 ( t ) = t 2 + t , then (19) is expressed as
H D 1 4 , 3 4 ; t 2 + t C D 1 2 ; t 2 + t π ( t ) = Π ( t , π ( t ) ) , 0 < t < 9 8 ,
and we can find that A * 0.8763925133 , A 1 * 4.810643110 . If
Π ( t , p i ) = 1 2 t + 10 2 | π | + π 2 1 + | x | + 1 7 t 2 + 8 t + 1 9 .
Then, L = 1 / 5 and we have A 1 * L 0.9621286220 . This means that boundary value problem (23), (20) and (24) has a unique solution on [ 0 , 9 / 8 ] .
In addition, if function Π in (23) is given in (22), then there exists a constant K ( 0.235228817 , 3.922219479 ) which satisfies the Corollary 3. So, the boundary value problem (23), (20) and (22) has at least one solution on [ 0 , 9 / 8 ] .
(iv)
If the boundary conditions in (20) is replaced by
C D 1 2 ; t 2 + t π ( 0 ) = 0 , π 9 8 = 3 67 C D 1 2 ; t 2 + t π 3 8 + 5 77 C D 1 2 ; t 2 + t π 5 8 + 7 87 C D 1 2 ; t 2 + t π 7 8 + 2 39 I 4 5 ; t 2 + t π 1 2 + 4 59 π 3 4 .
Then, we get A ^ 0.8884626894 , A ^ 1 4.816166032 . If
Π ( t , π ) = W π 2 + Z ,
where constants W , Z > 0 and W Z < 1 / ( 4 A ^ 1 2 ) 0.01077797341 . Then, there exists a positive constant M satisfying the Corollary 3 when replacing A 1 * by A ^ 1 . Hence, the boundary value problem (23), (25) and (26) has at least one solution on [ 0 , 9 / 8 ] .

6. Conclusions

In this paper, we have studied a new kind of boundary value problem consisting of a combination of two fractional derivative operators, one ψ 1 -Hilfer and one ψ 2 -Caputo, supplemented with nonlocal integro-differential boundary conditions. This combination, as far as we know, is new in the literature. Our uniqueness result is derived via Banach’s contraction principle, while the Leray–Schauder nonlinear alternative is used to derive the existence result. The main results are well illustrated by constructing numerical examples.

Author Contributions

Conceptualization, S.K.N. and J.T.; methodology, S.S., S.K.N., C.S. and J.T.; formal analysis, S.S., S.K.N., C.S. and J.T.; writing—original draft preparation, S.S. and C.S.; writing—review and editing, S.K.N. and J.T.; supervision, S.K.N.; funding acquisition, J.T. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-62-KNOW-41.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Gaul, L.; Klein, P.; Kempfle, S. Damping description involving fractional operators. Mech. Syst. Signal Process. 1991, 5, 81–88. [Google Scholar] [CrossRef]
  2. Glockle, W.G.; Nonnenmacher, T.F. A fractional calculus approach of self-similar protein dynamics. Biophys. J. 1995, 68, 46–53. [Google Scholar] [CrossRef] [PubMed]
  3. Hilfer, R. Experimental evidence for fractional time evolution in glass forming materials. J. Chem. Phys. 2002, 284, 399–408. [Google Scholar] [CrossRef]
  4. Mainardi, F. Fractional calculus: Some basic problems in continuum and statistical mechanics. In Fractals and Fractional Calculus in Continuum Mechanics; Carpinteri, A., Mainardi, F., Eds.; Springer: Vienna, Austria, 1997; pp. 291–348. [Google Scholar]
  5. Metzler, F.; Schick, W.; Kilian, H.G.; Nonnenmacher, T.F. Relaxation in filled polymers: A fractional calculus approach. J. Chem. Phys. 1995, 103, 7180–7186. [Google Scholar] [CrossRef]
  6. Diethelm, K. The Analysis of Fractional Differential Equations; Lecture Notes in Mathematics; Springer: New York, NY, USA, 2010. [Google Scholar]
  7. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of the Fractional Differential Equations; North-Holland Mathematics Studies; Elsevier: Amsterdam, The Netherlands, 2006; Volume 204. [Google Scholar]
  8. Miller, K.S.; Ross, B. An Introduction to the Fractional Calculus and Differential Equations; John Wiley: NewYork, NY, USA, 1993. [Google Scholar]
  9. Podlubny, I. Fractional Differential Equations; Academic Press: New York, NY, USA, 1999. [Google Scholar]
  10. Zhou, Y. Basic Theory of Fractional Differential Equations; World Scientific: Singapore, 2014. [Google Scholar]
  11. Ahmad, B.; Ntouyas, S.K. Nonlocal Nonlinear Fractional-Order Boundary Value Problems; World Scientific: Singapore, 2021. [Google Scholar]
  12. Hilfer, R. (Ed.) Applications of Fractional Calculus in Physics; World Scientific: Singapore, 2000. [Google Scholar]
  13. Kamocki, R. A new representation formula for the Hilfer fractional derivative and its application. J. Comput. Appl. Math. 2016, 308, 39–45. [Google Scholar] [CrossRef]
  14. Vanterler da Sousa, J.; de Oliveira, E.C. On the ψ-Hilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 2018, 60, 72–91. [Google Scholar] [CrossRef]
  15. Vanterler da Sousa, J.; Kucche, K.D.; de Oliveira, E.C. On the Ulam-Hyers stabilities of the solutions of ψ-Hilfer fractional differential equation with abstract Volterra operator. Math. Methods Appl. Sci. 2019, 42, 3021–3032. [Google Scholar] [CrossRef]
  16. Vanterler da Sousa, J.; de Oliveira, E.C. On the Ulam–Hyers–Rassias stability for nonlinear fractional differential equations using the ψ–Hilfer operator. J. Fixed Point Theory Appl. 2018, 20, 96. [Google Scholar] [CrossRef]
  17. Ahmad, B.; Ntouyas, S.K.; Alsaedi, A.; Alotaibi, F.M. Existence results for a ψ-Hilfer type nonlocal fractional boundary value problem via topological degree theory. Dyn. Syst. Appl. 2021, 30, 1091–1103. [Google Scholar] [CrossRef]
  18. Sitho, S.; Ntouyas, S.K.; Samadi, A.; Tariboon, J. Boundary value problems for ψ-Hilfer type sequential fractional differential equations and inclusions with integral multi-point boundary conditions. Mathematics 2021, 9, 1001. [Google Scholar] [CrossRef]
  19. Kiataramkul, C.; Ntouyas, S.K.; Tariboon, J. Existence results for ψ-Hilfer fractional integro-differential hybrid boundary value problems for differential equations and inclusions. Adv. Math. Phys. 2021, 2021, 9044313. [Google Scholar] [CrossRef]
  20. Kiataramkul, C.; Ntouyas, S.K.; Tariboon, J. An existence result for ψ-Hilfer fractional integro-differential hybrid three-point boundary value problems. Fractal. Fract. 2021, 5, 136. [Google Scholar] [CrossRef]
  21. Almeida, R. A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 2017, 44, 460–481. [Google Scholar] [CrossRef]
  22. Mallah, I.; Ahmed, I.; Akgul, A.; Jarad, F.; Alha, S. On ψ-Hilfer generalized proportional fractional operators. AIMS Math. 2021, 7, 82–103. [Google Scholar] [CrossRef]
  23. Granas, A.; Dugundji, J. Fixed Point Theory; Springer: New York, NY, USA, 2003. [Google Scholar]
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Sitho, S.; Ntouyas, S.K.; Sudprasert, C.; Tariboon, J. Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations. Mathematics 2023, 11, 867. https://doi.org/10.3390/math11040867

AMA Style

Sitho S, Ntouyas SK, Sudprasert C, Tariboon J. Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations. Mathematics. 2023; 11(4):867. https://doi.org/10.3390/math11040867

Chicago/Turabian Style

Sitho, Surang, Sotiris K. Ntouyas, Chayapat Sudprasert, and Jessada Tariboon. 2023. "Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations" Mathematics 11, no. 4: 867. https://doi.org/10.3390/math11040867

APA Style

Sitho, S., Ntouyas, S. K., Sudprasert, C., & Tariboon, J. (2023). Integro-Differential Boundary Conditions to the Sequential ψ1-Hilfer and ψ2-Caputo Fractional Differential Equations. Mathematics, 11(4), 867. https://doi.org/10.3390/math11040867

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