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Hilfer proportional nonlocal fractional integro-multipoint boundary value problems

  • Ayub Samadi , Sotiris K. Ntouyas , Asawathep Cuntavepanit and Jessada Tariboon EMAIL logo
Published/Copyright: October 27, 2023
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Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we introduce and study a boundary value problem for ( k , χ ¯ * ) -Hilfer generalized proportional fractional differential equation of order in an interval (1, 2], equipped with integro-multipoint nonlocal boundary conditions. In the scalar case setting, the existence results are proved via Leray-Schauder nonlinear alternative and Krasnosel’skiĭ’s fixed point theorem, while the existence of a unique solution is established by applying Banach’s contraction mapping principle. In Banach’s space setting, an existence result is proved via Mönch’s fixed point theorem and the measure of noncompactness. Finally, the obtained theoretical results are well illustrated by constructed examples.

Keywords: Hilfer proportional fractional derivative; proportional fractional integral; fixed point theorems; existence results
MSC 2010: 26A33; 34A08; 34B15

1 Introduction

Fractional differential equations (FDEs) has been applied as a noteworthy tools in various fields of science and engineering. Thus, numerous applications have been applied in variant fields of science, see [1–5] and therein cited references. For a systematic development of fractional calculus and FDEs, see [6–14]. The topic of boundary value problems (BVPs) for different kinds of fractional derivatives constitute a very interesting area in mathematics. A variety of fractional derivative operators have been appeared in the literature, such as Caputo, Riemann-Liouville, Erdélyi-Kober, Hadamard, and Hilfer. Hilfer [3] proposed a new fractional derivative, which include Riemann-Liouville and Caputo fractional derivatives as special cases. Katugampola in [15,16] introduced a new type of fractional operators, which include Riemann-Liouville and Hadamard fractional operators. Jarad et al. [17] introduced the generalized proportional fractional derivative, while in [18,19], introduced the concept of proportional derivatives of a function with respect to another function. Moreover, in a study by Ahmed et al. [20] were introduced the Hilfer generalized proportional fractional derivative. Recently, in a study by Mallah et al. [21], the χ ¯ * -Hilfer generalized proportional fractional derivative was introduced. In a study by Dorrego [22], the k -Riemann-Liouville fractional derivative was introduced, and recently, in a study by Kucche and Mali [23], the ( k , χ ¯ * ) -Hilfer fractional derivative was introduced and studied.

Recently, in a study by Tariboon et al. [24], we studied nonlocal BVPs for Hilfer generalized fractional proportional differential equations of order ( 1 , 2 ] , supplemented with multipoint boundary conditions. In a series of articles [25–30], we initiated the study of BVPs for ( k , χ ¯ * ) -Hilfer FDEs of order ( 1 , 2 ] . In a study by Ntouyas et al. [31], a new problem was investigated consisting of a χ ¯ * -Hilfer fractional proportional differential equation and nonlocal integro-multistrip-multipoint boundary conditions given by

(1) D x 1 + α ¯ , β * , σ , χ ¯ * H ϖ ( w ) = Ψ ( w , ϖ ( w ) ) , w ∈ [ x 1 , x 2 ] , ϖ ( x 1 ) = 0 , ∫ x 1 x 2 χ ¯ * ′ ( s ) ϖ ( s ) d s = ∑ i = 1 n φ i ∫ ξ i η i χ ¯ * ′ ( s ) ϖ ( s ) d s + ∑ j = 1 m θ j ϖ ( ζ j ) ,

where D x 1 + α ¯ , β * , σ , χ ¯ * H denotes the χ ¯ * -Hilfer fractional proportional derivative operator of order α ¯ ∈ ( 1 , 2 ] and type β * ∈ [ 0 , 1 ] , σ ∈ ( 0 , 1 ] , x 1 < ζ j < ξ i < η i < x 2 , φ i , θ j ∈ R , i = 1 , 2 , … , n , j = 1 , 2 , … , m , χ ¯ * : [ x 1 , x 2 ] → R is an increasing function with χ ¯ * ′ ( w ) ≠ 0 for all w ∈ [ x 1 , x 2 ] , and Ψ : [ x 1 , x 2 ] × R → R is a continuous function. Existence and uniqueness results are established via fixed point theorems due to Banach, Krasnosel’skiĭ, Schaefer, and Leray-Schauder alternative.

Motivated by the aforementioned articles, in the present article we combine both ( k , ψ ¯ * ) -Hilfer FDEs and Hilfer generalized proportional FDEs. In precise terms, in this article, we study the following BVP consisting of ( k , χ ¯ * ) -Hilfer generalized proportional fractional derivative equipped with integro-multipoint nonlocal boundary conditions, given by

(2) D a + α ¯ , β * , σ , χ ¯ * k , H ϖ ( w ) = Π ( w , ϖ ( w ) ) , w ∈ [ x 1 , x 2 ] , ϖ ( x 1 ) = 0 , ϖ ( x 2 ) = ∑ i = 1 m λ i ϖ ( ξ i ) + λ I a + δ , σ , χ ¯ * k ϖ ( η ) ,

where D a + α ¯ , β * , σ , χ ¯ * k , H indicates the ( k , χ ¯ * ) -Hilfer generalized proportional fractional derivative of order α ¯ , 1 < α ¯ ≤ 2 and parameter β * , 0 < β * ≤ 1 , λ , λ i ∈ R , i = 1 , 2 , … , m , Π : [ x 1 , x 2 ] × R ⟶ R is a continuous function, χ ¯ * : [ x 1 , x 2 ] ⟶ R is an increasing function with χ ¯ * ′ ( w ) ≠ 0 for all w ∈ [ x 1 , x 2 ] , I a + δ , σ , χ ¯ * is a generalized proportional fractional integral (GPFI) of order δ > 0 , ξ i , η ∈ ( x 1 , x 2 ) , and σ ∈ ( 0 , 1 ] .

Comparing problems (1) and (2), we note that problems (1) concerns χ ¯ * -Hilfer fractional proportional differential equations, while equation (2) concerns ( k , χ ¯ * ) -Hilfer generalized proportional FDEs. Moreover, it is obvious that, in both problems, the boundary conditions are entirely different, nonlocal integro-multistrip-multipoint boundary conditions in the first, and nonlocal integro-multipoint boundary conditions in the second. In addition, in problem (1), we studied only the scalar case for differential equations and inclusions, whereas in problem (2), we investigate both cases, scalar case and Banach space case.

We establish the results with the help of classical fixed point theorems. First, in the scalar case setting, the existence results are established via Leray-Schauder nonlinear alternative and Krasnosel’skiĭ’s fixed point theorem, while the existence of a unique solution is established by applying Banach contraction mapping principle. Next, we prove an existence result in the Banach space setting via the measure of noncompactness and a fixed point theorem due to Mönch.

The novelty of the present study lies in the fact that we consider a very general BVP concerning ( k , χ ¯ * ) -Hilfer generalized proportional fractional derivative, supplemented with integro-multipoint nonlocal boundary conditions, which is a new topic of research. We emphasize that the ( k , χ ¯ * ) -Hilfer generalized proportional FDE considered in equation (2) is of more general form and include many special cases by fixing the values of χ ¯ * and β . For instance, the ( k , χ ¯ * ) -Hilfer generalized proportional FDE in equation (2) corresponds:

  1. for β * = 0 to ( k , χ ¯ * ) -Riemann-Liouville;

  2. for χ ¯ * ( t ) = t to k -Riemann-Liouville;

  3. for χ ¯ * ( t ) = t ρ to k -Hilfer-Katugampola;

  4. χ ¯ * ( t ) = t ρ and β * = 0 to k -Katugampola;

  5. when χ ¯ * ( t ) = t ρ and β * = 1 to k -Caputo-Katugampola;

  6. for χ ¯ * ( t ) = log t to k -Hilfer-Hadamard;

  7. for χ ¯ * ( t ) = log t and β * = 0 to k -Hadamard; and

  8. for χ ¯ * ( t ) = log t and β * = 1 to k -Caputo-Hadamard.

Our results are new and contribute significantly to the new research topic of ( k , χ ¯ * ) -Hilfer generalized proportional FDEs, for which as a new subject the literature is very limited. The used method is standard, but its configuration in the problem (2) is new.

The relict part of this article has been vocalized as follows: Section 2 consists of some preparatory definitions and results to construct the main results. In section 3, an subsidiary result is presented to transmute the problem (2) into a fixed point problem. In Section 4, the existence and uniqueness results for the problem (2) are established, while in Section 5, an existence result is manufactured based on Mönch’s theorem and the technique of the measure of noncompactness. In Section 6, some numerical examples are constructed to illustrate the obtained theoretical results.

2 Preliminaries

Here, some concepts are introduced from the fractional calculus.

By C ( [ x 1 , x 2 ] , E ) , we denote the Banach space of all continuous functions ϖ : [ x 1 , x 2 ] → E with norm defined as follows:

‖ ϖ ‖ ∞ = sup { ‖ ϖ ( w ) ‖ , w ∈ [ x 1 , x 2 ] } .

The notation

‖ ϖ ‖ = sup { ∣ ϖ ( w ) ∣ , w ∈ [ x 1 , x 2 ] }

is used when E = R .

Now, we recall the following definitions.

Definition 2.1

[17] For σ ∈ ( 0 , 1 ] and α ¯ ∈ C with ℜ ( α ¯ ) > 0 , the left-sided GPFI of order α ¯ > 0 is presented as follows:

I x 1 + α ¯ , σ p ( w ) = 1 σ α ¯ Γ ( α ¯ ) ∫ x 1 w e σ − 1 σ ( w − s ) ( w − s ) α ¯ − 1 p ( s ) d s , w > x 1 .

Definition 2.2

[17] Let n − 1 < α ¯ < n , n ∈ N , σ ∈ ( 0 , 1 ] , and 0 ≤ β ≤ 1 . Then, the fractional proportional derivative of Hilfer type with order α ¯ , parameter β , and proportional number σ is presented as follows:

(3) ( D x 1 + α ¯ , β , σ p ) ( w ) = I x 1 + β ( n − α ¯ ) , σ [ D σ ( I x 1 + ( 1 − β ) ( n − 1 ) , σ p ) ] ( w ) ,

in which D σ p ( w ) = ( 1 − σ ) p ( w ) + σ p ′ ( w ) and I ( ⋅ ) , σ is the GPFI defined in Definition 2.1.

Definition 2.3

[17] Let σ ∈ ( 0 , 1 ] , α ¯ ∈ C with ℜ ( α ¯ ) > 0 , k > 0 , χ ¯ * ∈ C n ( [ x 1 , x 2 ] , R ) , χ ¯ * ′ ( w ) ≠ 0 , and w ∈ [ x 1 , x 2 ] . Then, the fractional operator

(4) I x 1 + α ¯ , σ , χ ¯ * k p ( w ) = 1 k σ α ¯ ⁄ k Γ k ( α ¯ ) ∫ x 1 w χ ¯ * ′ ( s ) e σ − 1 σ ( χ ¯ * ( w ) − χ ¯ * ( s ) ) ( χ ¯ * ( w ) − χ ¯ * ( s ) ) α ¯ k − 1 p ( s ) d s

indicates the ( k , χ ¯ * ) -left-side GPFI of order α ¯ > 0 with respect to χ ¯ * of function p .

Definition 2.4

[17] Let n − 1 < α ¯ < n , n ∈ N , σ ∈ ( 0 , 1 ] , k > 0 , 0 ≤ β ≤ 1 , χ ¯ * ∈ C n ( [ x 1 , x 2 ] , R ) , χ ¯ * ′ ( w ) ≠ 0 , and w ∈ [ x 1 , x 2 ] . Then, the ( k , χ ¯ * ) -Hilfer generalized proportional fractional derivative is defined as follows:

(5) D x 1 + α ¯ , β , σ , χ ¯ * k , H p ( w ) = I x 1 + β ( k n − α ¯ ) , σ , χ ¯ * k [ D n , σ ( I x 1 + ( 1 − β ) ( k n − α ¯ ) , σ , χ ¯ * k p ) ( w ) ] ,

where D n , σ = D σ ⋯ D σ ︸ n -times .

Remark 2.1

If k ⟶ 1 and χ ¯ * ( w ) = r in equation (5), then equation (5) reduces to equation (3).

Lemma 2.1

[32] Let b 1 , b 2 ∈ C so that ℜ ( b 1 ) ≥ 0 and ℜ ( α ¯ ) > 0 . Then, we have, for σ ∈ ( 0 , 1 ] ,

I a + α ¯ , σ , χ ¯ * k e σ − 1 σ [ χ ¯ * ( s ) − χ ¯ * ( a ) ] ( χ ¯ * ( s ) − χ ¯ * ( a ) ) δ ¯ k − 1 ( w ) = Γ k ( δ ¯ ) σ α ¯ k Γ k ( b 2 + α ¯ ) e σ − 1 σ [ χ ¯ * ( w ) − χ ¯ * ( a ) ] ( χ ¯ * ( w ) − χ ¯ * ( a ) ) b 2 + α ¯ k − 1 , D a + α ¯ , σ , χ ¯ * k , H e σ − 1 σ [ χ ¯ * ( s ) − χ ¯ * ( a ) ] ( χ ¯ * ( s ) − χ ¯ * ( a ) ) b 2 k − 1 ( w ) = Γ k ( b 2 ) σ α ¯ k Γ k ( b 2 − α ¯ ) e σ − 1 σ [ χ ¯ * ( w ) − χ ¯ * ( a ) ] ( χ ¯ * ( w ) − χ ¯ * ( a ) ) b 2 − α ¯ k − 1 .

3 A secondary result

The following lemma will be applied to prove the next lemma of this section. It is a modification of Theorem 5.5 in the study by Kucche and Mali [23].

Lemma 3.1

Let x 1 < x 2 , k > 0 , n − 1 < α ¯ < n , 0 ≤ β ≤ 1 , σ ∈ ( 0 , 1 ] , γ = α ¯ + β ( k n − α ¯ ) ∈ [ α ¯ , k n ] , ϖ ∈ L 1 ( x 1 , x 2 ) , and I x 1 + n k − γ , σ , χ ¯ * k ϖ ∈ C n ( [ x 1 , x 2 ] , R ) , then

I x 1 + α ¯ , σ , χ ¯ * k ( D x 1 + α ¯ , β , σ , χ ¯ * k , H ) ϖ ( w ) = ϖ ( w ) − ∑ j = 1 n e σ − 1 σ ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − j k σ γ k − j Γ k ( γ − j k + k ) k χ ¯ * ′ ( w ) d d r n − j I x 1 + n k − γ , σ , χ ¯ * k ϖ ( w ) w = x 1 .

The next lemma concerns a linear variant of equation (2) and used in converting equation (2) into a fixed point problem.

Lemma 3.2

Let x 1 < x 2 , k > 0 , 1 < α ¯ ≤ 2 , β ∈ [ 0 , 1 ] , σ ∈ ( 0 , 1 ] , k > 0 , η ∈ [ x 1 , x 2 ] , δ > 0 , λ , λ i ∈ R , i = 1 , 2 , … , m , γ = α ¯ + β ( 2 k − α ¯ ) , h ∈ C 2 ( [ x 1 , x 2 ] , R ) and

Λ ≔ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 Γ k ( γ ) e σ − 1 σ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) − ∑ i = 1 m λ i e σ − 1 σ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) γ k − 1 Γ k ( γ ) − λ Γ k ( γ ) σ α ¯ k Γ k ( γ + α ¯ ) e σ − 1 σ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + γ k − 1 ≠ 0 .

Then, ϖ ∈ C ( [ x 1 , x 2 ] , R ) is a solution to the ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint nonlocal BVP

(6) D x 1 + α ¯ , β , σ , χ ¯ * k , H ϖ ( w ) = h ( w ) , w ∈ [ x 1 , x 2 ] , ϖ ( x 1 ) = 0 , ϖ ( x 2 ) = ∑ i = 1 m λ i ϖ ( ξ i ) + λ I x 1 + δ , σ , χ ¯ * k ϖ ( η ) ,

if and only if

(7) ϖ ( w ) = I x 1 + α ¯ , σ , χ ¯ * k h ( w ) + e σ − 1 σ ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 Λ Γ k ( γ ) λ I x 1 + α ¯ + δ , σ , χ ¯ * k h ( η ) + ∑ i = 1 m λ i I x 1 + α ¯ , σ , χ ¯ * k h ( ξ i ) − I x 1 + α ¯ , σ , χ ¯ * k h ( x 2 ) .

Proof

By using Lemma 3.1, we have

I x 1 + α ¯ , σ , χ ¯ * k ( D x 1 + α ¯ , β , σ , χ ¯ * k , H ϖ ( w ) ) = ϖ ( z ) − ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 k Γ k ( γ ) σ γ k − 1 e σ − 1 σ ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) [ D x 1 + 1 , σ , χ ¯ * ( I x 1 + 2 k − γ , σ , χ ¯ * k ϖ ( w ) ) ] w = x 1 − ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 2 k Γ k ( γ − k ) σ γ k − 2 e σ − 1 σ ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) [ D x 1 + 0 , σ , χ ¯ * ( I x 1 + 2 k − γ , σ , χ ¯ * k ϖ ( w ) ) ] w = x 1 .

Consequently,

(8) ϖ ( w ) = I x 1 + α ¯ , σ , χ ¯ * k h ( w ) + c 0 e σ − 1 σ ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 Γ k ( γ ) + c 1 e σ − 1 σ ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 2 Γ k ( γ − k ) ,

where c 0 = [ D x 1 + 1 , σ , χ ¯ * ( I x 1 + 2 k − γ , σ , χ ¯ * k ϖ ( w ) ) ] w = x 1 and c 1 = [ D x 1 + 0 , σ , χ ¯ * ( I x 1 + 2 k − γ , σ , χ ¯ * k ϖ ( w ) ) ] w = x 1 . Now applying ϖ ( x 1 ) = 0 in equation (8), we conclude that c 1 = 0 , since γ k − 2 < 0 . On the other hand, using the second boundary condition in equation (6) and Lemma 2.1, we conclude that

I x 1 + α ¯ , σ , χ ¯ * k h ( x 2 ) + c 0 e σ − 1 σ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 Γ k ( γ ) = ∑ i = 1 m λ i I x 1 + α ¯ , σ , χ ¯ * k h ( ξ i ) + c 0 ∑ i = 1 m λ i e ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) σ − 1 σ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 Γ k ( γ ) + λ I x 1 + α ¯ + δ , σ , χ ¯ * k h ( η ) + λ c 0 Γ k ( γ ) σ α ¯ k Γ k ( γ + α ¯ ) e σ − 1 σ ( χ ¯ * ( η ) − χ ¯ * ( a ) ) ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + γ k − 1 ,

from which we obtain

c 0 = 1 Λ λ I x 1 + α ¯ + δ , σ , χ ¯ * k h ( η ) + ∑ i = 1 m λ i I x 1 + α ¯ , σ , χ ¯ * k h ( ξ i ) − I x 1 + α ¯ , σ , χ ¯ * k h ( x 2 ) .

By substituting c 0 in equation (8), we obtain equation (7). The converse is obtained by direct computation.□

4 Existence and uniqueness results

In view of Lemma 3.2, we define an operator T : C ( [ x 1 , x 2 ] , R ) ⟶ C ( [ x 1 , x 2 ] , R ) by

(9) ( T ϖ ) ( w ) = I x 1 + α ¯ , σ , χ ¯ * k Π ( w , ϖ ( w ) ) + e σ − 1 σ ( χ ¯ * ( z ) − χ ¯ * ( x 1 ) ) ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 Λ Γ k ( γ ) × λ I x 1 + α ¯ + δ , σ , χ ¯ * k Π ( η , ϖ ( η ) ) + ∑ i = 1 m λ i I x 1 + α ¯ , σ , χ ¯ * k Π ( ξ i , ϖ ( ξ i ) ) − I x 1 + α ¯ , σ , χ ¯ * k Π ( x 2 , ϖ ( x 2 ) ) , w ∈ [ x 1 , x 2 ] .

Obviously, the fixed point of T are the solutions of equation (2). For computational convenience, we set

(10) Ψ = ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) .

Now we will apply some classical fixed point theorems to establish existence and uniqueness results about equation (2).

4.1 Uniqueness result

We establish the uniqueness result by applying Banach fixed point theorem [33].

Theorem 4.1

Suppose that:

  1. there exists N > 0 such that:

    ∣ Π ( w , ϖ 1 ) − Π ( w , ϖ 2 ) ∣ ≤ N ∣ ϖ 1 − ϖ 2 ∣ ∀ w ∈ [ x 1 , x 2 ] , ϖ 1 , ϖ 2 ∈ R .

Then, the ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint nonlocal BVP (2) has a unique solution on [ x 1 , x 2 ] , provided that

(11) N Ψ < 1 ,

where Ψ is given by equation (10).

Proof

Let M = sup w ∈ [ x 1 , x 2 ] ∣ Π ( w , 0 ) ∣ and B τ = { ϖ ∈ C ( [ x 1 , x 2 ] , R ) : ‖ ϖ ‖ ≤ τ } with τ ≥ M Ψ 1 − N Ψ . First we show that T B τ ⊆ B τ . Note that

∣ Π ( w , ϖ ( w ) ) ∣ ≤ ∣ Π ( w , ϖ ( w ) ) − Π ( w , 0 ) ∣ + ∣ Π ( w , 0 ) ∣ ≤ N ∣ ϖ ( w ) ∣ + M ≤ N ‖ ϖ ‖ + M ≤ N τ + M .

For any ϖ ∈ B τ and using 0 < e σ − 1 σ [ χ ¯ * ( w ) − χ ¯ * ( x 1 ) ] ≤ 1 , we have

∣ ( T ϖ ) ( w ) ∣ ≤ I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( w , ϖ ( w ) ) ∣ + ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) [ ∣ λ ∣ I x 1 + α ¯ + δ , σ , χ ¯ * k ∣ Π ( η , ϖ ( η ) ) ∣ + ∑ i = 1 m ∣ λ i ∣ I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( ξ i , ϖ ( ξ i ) ) ∣ + I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( x 2 , ϖ ( x 2 ) ) ∣ ≤ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) ( N ‖ ϖ ‖ + M ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) ( N ‖ ϖ ‖ + M ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) ( N ‖ ϖ ‖ + M ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) ( N ‖ ϖ ‖ + M ) ≤ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( a ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) ( N τ + M ) = ( N τ + M ) Ψ ≤ τ .

Hence, we have ‖ T ϖ ‖ ≤ τ and obtain T B τ ⊂ B τ .

For w ∈ [ x 1 , x 2 ] and ϖ 1 , ϖ 2 ∈ C ( [ x 1 , x 2 ] , R ) , we obtain

∣ ( T ϖ 2 ) ( w ) − ( T ϖ 1 ) ( w ) ∣ ≤ I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( w , ϖ 2 ( w ) ) − Π ( w , ϖ 1 ( w ) ) ∣ + ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) × ∣ λ ∣ I x 1 + α ¯ + δ , σ , χ ¯ * k ∣ Π ( η , ϖ 2 ( η ) ) − Π ( η , ϖ 1 ( η ) ) ∣ + ∑ i = 1 m ∣ λ i ∣ I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( ξ i , ϖ 2 ( ξ i ) ) − Π ( ξ i , ϖ 1 ( ξ i ) ) ∣ + I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( x 2 , ϖ 2 ( x 2 ) ) − Π ( x 2 , ϖ 1 ( x 2 ) ) ∣ ≤ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) N ‖ ϖ 2 − ϖ 1 ‖ = N Ψ ‖ ϖ 2 − ϖ 1 ‖ .

Thus, we conclude that ‖ T ϖ 2 − T ϖ 1 ‖ ≤ N Ψ ‖ ϖ 2 − ϖ 1 ‖ . Since N Ψ < 1 , we infer that T is a contraction. Now, by Banach’s contraction principle, the operator T has a unique fixed point, which is the unique solution of the nonlocal ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint BVP (2), and this completes the proof.□

4.2 Existence results

Now, to establish the existence results for equation (2), we apply a fixed point theorem due to Krasnosel’skiĭ [34], as well as the Leray-Schauder nonlinear alternative [35].

Theorem 4.2

Let the continuous function Π : [ x 1 , x 2 ] × R ⟶ R satisfies the assumption ( M 1 ) . Moreover, suppose that:

  1. ∣ Π ( w , ϖ ) ∣ ≤ ω ( w ) , for all ( w , ϖ ) ∈ [ x 1 , x 2 ] × R and ω ∈ C ( [ x 1 , x 2 ] , R ) .

If N Ψ 1 < 1 , then the nonlocal ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint BVP (2) has at least one solution on [ x 1 , x 2 ] , where

(12) Ψ 1 ≔ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) .

Proof

Let sup w ∈ [ x 1 , x 2 ] ω ( w ) = ‖ ω ‖ and B ρ = { ϖ ∈ C ( [ x 1 , x 2 ] , R ) : ‖ ϖ ‖ ≤ ρ } with ρ ≥ ‖ ω ‖ Ψ . We split the operator T into the operators D 1 and D 2 defined on B ρ as follows:

( D 1 ϖ ) ( w ) = I a + α ¯ , σ , χ ¯ * k Π ( w , ϖ ( w ) ) , w ∈ [ x 1 , x 2 ] ,

and

( D 2 ϖ ) ( w ) = ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 Λ Γ k ( γ ) λ I x 1 + α ¯ + δ , σ , χ ¯ * k Π ( η , ϖ ( η ) ) + ∑ i = 1 m λ i I x 1 + α ¯ , σ , χ ¯ * k Π ( ξ i , ϖ ( ξ i ) ) − I x 1 + α ¯ , σ , χ ¯ * k Π ( x 2 , ϖ ( x 2 ) ) , w ∈ [ x 1 , x 2 ] .

For ϖ 1 , ϖ 2 ∈ B ρ , we have

∣ ( D 1 ϖ 1 ) ( w ) + ( D 2 ϖ 2 ) ( w ) ∣ ≤ sup w ∈ [ x 1 , x 2 ] I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( w , ϖ 1 ( w ) ) ∣ + ( χ ¯ * ( s ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) × ∣ λ ∣ I x 1 + α ¯ + δ , σ , χ ¯ * k ∣ Π ( η , ϖ 2 ( η ) ) ∣ + ∑ i = 1 m ∣ λ i ∣ I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( ξ i , ϖ 2 ( ξ i ) ) ∣ − I x 1 + α ¯ , σ , χ ¯ * k Π ( x 2 , ϖ 2 ( x 2 ) ) ≤ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) ‖ ω ‖ = Ψ ‖ ω ‖ ≤ ρ .

Hence, we have ‖ ( D 1 ϖ 1 ) + ( D 2 ϖ 2 ) ‖ ≤ ρ , which implies that ( D 1 ϖ 1 ) + ( D 2 ϖ 2 ) ∈ B ρ .

As in Theorem 4.1, by using the condition N Ψ 1 < 1 , we can prove that D 2 is a contraction.

The operator D 1 is continuous, since Π is continuous. Besides, since ‖ D 1 ϖ ‖ ≤ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) ‖ ω ‖ , the operator D 1 is uniformly bounded on B ρ . Now the compactness property of operator D 1 is verified. For w 1 , w 2 ∈ [ x 1 , x 2 ] , w 1 < w 2 , we obtain

∣ ( D 1 ϖ ) ( w 2 ) − ( D 1 ϖ ) ( w 1 ) ∣ ≤ 1 k σ α ¯ k Γ k ( α ¯ ) ∫ a w 2 χ ¯ * ′ ( s ) [ ( χ ¯ * ( w 2 ) − χ ¯ * ( s ) ) α ¯ k − 1 − ( χ ¯ * ( w 1 ) − χ ¯ * ( s ) ) α ¯ k − 1 ] Π ( s , ϖ ( s ) ) d s + ∫ w 1 w 2 χ ¯ * ′ ( s ) ( χ ¯ * ( w 2 ) − χ ¯ * ( s ) ) α ¯ k − 1 Π ( s , ϖ ( s ) ) d s ≤ ‖ ω ‖ σ α ¯ k ( α ¯ + k ) [ 2 ( χ ¯ * ( w 2 ) − χ ¯ * ( w 1 ) ) α ¯ k + ∣ ( χ ¯ * ( w 2 ) − χ ¯ * ( x 1 ) ) α ¯ k − ( χ ¯ * ( w 1 ) − χ ¯ * ( x 1 ) ) α ¯ k ∣ ] ,

which tends to zero as w 2 − w 1 ⟶ 0 independently of ϖ . Thus, D 1 is equicontinuous. Now, by applying Arzelá-Ascoli theorem, we conclude that D 1 is completely continuous. By Krasnosel’skiĭ’s fixed point theorem, the nonlocal ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint BVP (2) has at least one solution on [ x 1 , x 2 ] .□

Theorem 4.3

Suppose that the continuous function Π : [ x 1 , x 2 ] × R → R satisfies the following assumptions:

  1. There exist continuous and nondecreasing function β : [ 0 , ∞ ) → ( 0 , ∞ ) and a continuous and positive function ϑ such that for all ( w , ϖ ) ∈ [ x 1 , x 2 ] × R , we have

    ∣ Π ( w , ϖ ) ∣ ≤ ϑ ( w ) β ( ∣ ϖ ∣ ) .

  2. There exists a constant L > 0 such that

    L β ( L ) ‖ ϑ ‖ Ψ > 1 .

    Then, the nonlocal ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint BVP (2) has at least one solution on [ x 1 , x 2 ] .

Proof

First, we show that bounded sets are mapped into bounded sets by T in C ( [ x 1 , x 2 ] , R ) . Let B ζ = { ϖ ∈ C ( [ x 1 , x 2 ] , R ) : ‖ ϖ ‖ ≤ ζ } . Then, for w ∈ [ x 1 , x 2 ] , we have

∣ ( T ϖ ) ( w ) ∣ ≤ I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( w , ϖ ( w ) ) ∣ + ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) [ ∣ λ ∣ I x 1 + α ¯ + δ , σ , χ ¯ * k ∣ Π ( η , ϖ ( η ) ) ∣ + ∑ i = 1 m ∣ λ i ∣ I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( ξ i , ϖ ( ξ i ) ) ∣ + I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( x 2 , ϖ ( x 2 ) ) ∣ ≤ ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) λ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) + ∑ i = 1 m λ i ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) ‖ ϑ ‖ β ( ‖ ϖ ‖ ) ,

and consequently,

‖ T ϖ ‖ ≤ β ( ζ ) ‖ ϑ ‖ Ψ .

Next, it is shown that bounded sets of C ( [ x 1 , x 2 ] , R ) are mapped by T into equicontinuous sets. For w 1 , w 2 ∈ [ x 1 , x 2 ] , w 1 < w 2 and ϖ ∈ B ζ , we have

∣ ( T ϖ ) ( w 2 ) − ( T ϖ ) ( w 1 ) ∣ ≤ 1 σ α ¯ k Γ k ( α ¯ ) ∫ a w 1 χ ¯ * ′ ( s ) [ χ ¯ * ( w 2 ) − χ ¯ * ( s ) α ¯ k − 1 − ( χ ¯ * ( w 1 ) − χ ¯ * ( s ) ) α ¯ k − 1 ] Π ( s , ϖ ( s ) ) d s + ∫ w 1 w 2 χ ¯ * ′ ( s ) ( χ ¯ * ( w 2 ) − χ ¯ * ( s ) ) α ¯ k − 1 Π ( s , ϖ ( s ) ) d s + ( χ ¯ * ( w 2 ) − χ ¯ * ( x 1 ) ) α ¯ k − 1 − ( χ ¯ * ( w 1 ) − χ ¯ * ( x 1 ) ) α ¯ k − 1 ∣ Λ ∣ Γ k ( α ¯ ) ∣ [ ∣ λ ∣ I x 1 + α ¯ + δ , σ , χ ¯ * k ∣ Π ( η , ϖ ( η ) ) ∣ + ∑ i = 1 m ∣ λ i ∣ I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( ξ i , ϖ ( ξ i ) ) ∣ + I x 1 + α ¯ , σ , χ ¯ * k ∣ Π ( x 2 , ϖ ( x 2 ) ) ∣ ≤ ‖ ϑ ‖ β ( ζ ) σ α ¯ k Γ k ( α ¯ + k ) [ 2 ( χ ¯ * ( w 2 ) − χ ¯ * ( w 1 ) ) α ¯ k + ∣ ( χ ¯ * ( w 2 ) − χ ¯ * ( x 1 ) ) α ¯ k − ( χ ¯ * ( w 1 ) − χ ¯ * ( x 1 ) ) α ¯ k ∣ ] + ( χ ¯ * ( w 2 ) − χ ¯ * ( x 1 ) ) α ¯ k − 1 − ( χ ¯ * ( w 1 ) − χ ¯ * ( x 1 ) ) α ¯ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) ‖ ϑ ‖ β ( ζ ) .

As w 2 − w 1 ⟶ 0 , we conclude that ∣ ( T ϖ ) ( w 2 ) − ( T ϖ ) ( w 1 ) ∣ tends to zero independently of ϖ ∈ B ζ . By Arzelá-Ascoli theorem, the operator T : C ( [ x 1 , x 2 ] , R ) ⟶ C ( [ x 1 , x 2 ] , R ) is completely continuous.

Finally, it remains to show that the set ϖ = ν T ϖ for ν ∈ ( 0 , 1 ) of all solutions is bounded. Let ϖ be a solution. Thus, for w ∈ [ x 1 , x 2 ] and in view of the computations in the first step, we have

∣ ϖ ( w ) ∣ ≤ β ( ‖ ϖ ‖ ) ‖ ϑ ‖ Ψ ,

or

‖ ϖ ‖ β ( ‖ ϖ ‖ ) ‖ ϑ ‖ Ψ ≤ 1 .

Due to ( M 4 ) , there exists L > 0 such that ‖ ϖ ‖ ≠ L . Let

U = { ϖ ∈ C ( [ x 1 , x 2 ] , R ) : ‖ ϖ ‖ < L } .

Note that T : U ¯ → C ( [ x 1 , x 2 ] , R ) is continuous and also completely continuous. Besides, in view of the choice of U , we cannot find ϖ ∈ ∂ U such that ϖ = ν T ϖ for some ν ∈ ( 0 , 1 ) . By Leray-Schauder nonlinear alternative, the operator T has a fixed point ϖ ∈ U ¯ , which is a solution to equation (2). The proof is finished.□

5 An existence result via the measure of noncompactness

In this section, we consider the case where Π : [ x 1 , x 2 ] × E ⟶ E and E is a Banach space. An existence result is proved for the nonlocal ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint BVP (2) via the measure of noncompactness and a fixed point theorem due to Mönch. The principle of the measure of noncompactness was first studied and presented by Kuratowski [36]. We will discuss the definitions and related theorems as follows.

Definition 5.1

[37] Let E be a Banach space and M E indicates the set of all bounded subsets of E . The Kuratowski measure of noncompactness is the mapping ϒ : M E ⟶ [ 0 , ∞ ) defined as follows:

ϒ ( P ) = inf { ε > 0 : P ⊆ ∪ i = 1 m P i , diam ( P i ) ≤ ε } .

The key properties of the measure of noncompactness ϒ are as follows:

  1. ϒ ( P ) = 0 ⇔ P ¯ is compact;

  2. ϒ ( P ) = ϒ ( P ¯ ) ;

  3. P 1 ⊂ P 2 ⇒ ϒ ( P 1 ) ≤ ϒ ( P 2 ) ;

  4. ϒ ( P 1 + P 2 ) ≤ ϒ ( P 1 ) + ϒ ( P 2 ) ;

  5. ϒ ( λ P ) = ∣ λ ∣ ϒ ( P ) , λ ∈ R ; and

  6. ϒ ( conv P ) = ϒ ( P ) .

Lemma 5.1

[35] Let G ⊆ C ( [ x 1 , x 2 ] , E ) be a bounded and equicontinuous subset. Then, the function t ⟶ ϒ ( G ( t ) ) is continuous on [ x 1 , x 2 ] , i.e.,

ϒ C ( G ) = max t ∈ [ x 1 , x 2 ] ϒ ( G ( t ) ) ,

and

ϒ ∫ x 1 x 2 θ ( s ) d s ≤ ∫ x 1 x 2 ϒ ( G ( s ) ) d s ,

where G ( s ) = { θ ( s ) : θ ∈ G } , s ∈ [ x 1 , x 2 ] .

In the next theorem, we state the Mönch’s fixed point theorem.

Theorem 5.1

[38] Let E be a Banach space and D ⊂ E be a bounded, closed, and convex subset such that 0 ∈ D . In addition, let T : D ⟶ D be a continuous mapping. If

(13) W = conv ¯ T ( W ) , o r W = T ( W ) ∪ { 0 } ⇒ ϒ ( W ) = 0 ,

for all subset W of D, then T has a fixed point.

Theorem 5.2

Suppose that:

  1. the function Π : [ x 1 , x 2 ] × E ⟶ E satisfies the Carathéodory conditions, (i.e., (i) Π ( w , ϖ ) is measurable in w for ϖ ∈ E and (ii) Π ( z , ϖ ) is continuous in ϖ for w ∈ [ x 1 , x 2 ] );

  2. there exist Φ Π ∈ C ( [ x 1 , x 2 ] , E ) and nondecreasing function ϕ ∈ C ( R + , R + ) such that:

    ∣ Π ( w , ϖ ) ∣ ≤ Φ Π ( w ) ϕ ( ‖ ϖ ‖ ∞ ) , w ∈ [ x 1 , x 2 ] , ϖ ∈ E

  3. it holds

    Φ ( Π ( w , Q ) ) ≤ Φ Π ( w ) Φ ( Q )

    for all w ∈ [ x 1 , x 2 ] and each bounded set Q ⊆ E .

If

(14) Φ Π * χ ¯ * < 1 ,

where Φ Π * = sup w ∈ [ x 1 , x 2 ] Φ Π ( z ) , then the nonlocal ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint BVP (2)has at least one solution on [ x 1 , x 2 ] .

Proof

Let T : C ( [ x 1 , x 2 ] , E ) ⟶ C ( [ x 1 , x 2 ] , E ) given by equation (9). Define

B τ = { ϖ ∈ C ( [ x 1 , x 2 ] , E ) ; ‖ ϖ ‖ ∞ ≤ τ } ,

where

Φ Π * ϕ ( τ ) Ψ ≤ τ .

Step 1. The ball B τ maps T into itself.

For all ϖ ∈ B τ and w ∈ [ x 1 , x 2 ] , we have

‖ ( T ϖ ) ( w ) ‖ ≤ I x 1 + α ¯ , σ , χ ¯ * k ‖ Π ( w , ϖ ( w ) ) ‖ + ( χ ¯ * ( w ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) [ ∣ λ ∣ I x 1 + α ¯ + δ , σ , χ ¯ * k ‖ Π ( η , ϖ ( η ) ) ‖ + ∑ i = 1 m ∣ λ i ∣ I x 1 + α ¯ , σ , χ ¯ * k ‖ Π ( ξ i , ϖ ( ξ i ) ) ‖ + I x 1 + α ¯ , σ , χ ¯ * k ‖ Π ( x 2 , ϖ ( x 2 ) ) ‖ ≤ Φ Π * ϕ ( τ ) ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ Φ Π * ϕ ( τ ) ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ k ( α ¯ + δ + k ) + Φ Π * ϕ ( τ ) ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( 0 ) ) α ¯ k σ σ Γ k ( α ¯ + k ) + Φ Π * ϕ ( τ ) ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k σ σ Γ k ( α ¯ + k ) = Φ Π * ϕ ( τ ) Ψ ≤ τ .

Thus, T : B τ → B τ .

Step 2. T is continuous.

Let { ϖ n } ⊆ B τ and ϖ n ⟶ q as n ⟶ ∞ . Since the function Π satisfies the Carathéodory conditions, we deduce that Π ( s , ϖ n ( s ) ) ⟶ Π ( s , ϖ ( s ) ) as n → ∞ . Thus, due to ( C 2 ) and the Lebesgue dominated convergence theorem, we deduce that ‖ T ϖ n − T ϖ ‖ ∞ → 0 as n → ∞ . Consequently, T is continuous on B τ .

Step 3. T is equicontinuous with respect to z.

For w 1 , w 2 ∈ [ x 1 , x 2 ] , with w 1 < w 2 and ϖ ∈ B τ , we have

‖ ( T ϖ ) ( w 2 ) − ( T ϖ ) ( w 1 ) ‖ ≤ 1 k σ α ¯ Γ k ( α ¯ ) ∫ x 1 w 2 χ ¯ * ′ ( s ) [ ( χ ¯ * ( w 2 ) − χ ¯ * ( s ) ) α ¯ k − 1 − ( χ ¯ * ( w 1 ) − χ ¯ * ( s ) ) α ¯ k − 1 ] ‖ Π ( s , ϖ ( s ) ) ‖ d s + 1 k σ α ¯ Γ k ( α ¯ ) ∫ w 1 w 2 χ ¯ * ′ ( s ) ( χ ¯ * ( w 2 ) − χ ¯ * ( s ) ) α ¯ k − 1 ‖ Π ( s , ϖ ( s ) ) ‖ d s + ( χ ¯ * ( w 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 − ( χ ¯ * ( w 1 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) Φ Π * ϕ ( z ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ ( α ¯ + δ + k ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k Γ ( α ¯ + k ) ≤ Φ Π * ϕ ( τ ) σ α ¯ k Γ k ( α ¯ + k ) ∣ ( χ ¯ * ( w 2 ) − χ ¯ * ( x 1 ) ) α ¯ k − ( χ ¯ * ( w 1 ) − χ ¯ * ( x 1 ) ) α ¯ k ∣ + 2 ( χ ¯ * ( w 2 ) − χ ¯ * ( w 1 ) ) α ¯ k + ( χ ¯ * ( w 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 − ( χ ¯ * ( w 1 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) Φ Π * ϕ ( τ ) ∣ λ ∣ ( χ ¯ * ( η ) − χ ¯ * ( x 1 ) ) α ¯ + δ k σ α ¯ + δ k Γ ( α ¯ + δ + k ) + ∑ i = 1 m ∣ λ i ∣ ( χ ¯ * ( ξ i ) − χ ¯ * ( x 1 ) ) α ¯ k σ α ¯ k Γ k ( α ¯ + k ) + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) α ¯ k Γ ( α ¯ + k ) .

Due to the above inequality, we conclude that ‖ ( T ϖ ) ( z 2 ) − ( T ϖ ) ( z 1 ) ‖ → 0 , independent of ϖ ∈ B τ . Thus, T is equicontinuous on B τ .

Step 4. The condition (13) of Theorem 5.1  is satisfied.

Assume that Q ⊆ conv ¯ ( T ( Q ) ) ∪ { 0 } is a bounded and equicontinuous subset. Thus, the function T ( w ) = Φ ( Q ( w ) ) is a continuous function. Now, in view of Lemma 5.1 and ( C 3 ) , we obtain

T ( w ) = Φ ( Q ( w ) ) ≤ Φ ( c o n v ¯ ( T ( Q ) ∪ { 0 } ) ) ≤ Φ ( T ( Q ) ( w ) ) ≤ Φ 1 k σ α ¯ Γ k ( α ¯ ) ∫ x 1 r e σ − 1 σ ( w − s ) χ ¯ * ′ ( s ) ( χ ¯ * ( w ) − χ ¯ * ( s ) ) α ¯ k Π ( s , ϖ ( s ) ) d s : ϖ ∈ Q + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) × ∣ λ ∣ Φ 1 k σ α ¯ Γ k ( α ¯ ) ∫ x 1 η e σ − 1 σ ( η − s ) χ ¯ * ′ ( s ) ( χ ¯ * ( η ) − χ ¯ * ( s ) ) α ¯ k Π ( s , ϖ ( s ) ) d s : ϖ ∈ C + ∑ i = 1 m ∣ λ i ∣ Φ { I x 1 + α ¯ , σ , χ ¯ * k Π ( ξ i , ϖ ( ξ i ) ) : ϖ ∈ Q } + Φ { I x 1 + α ¯ , σ , χ ¯ * k Π ( x 2 , ϖ ( x 2 ) ) : ϖ ∈ Q } ] ≤ 1 k σ α ¯ Γ k ( α ¯ ) ∫ x 1 r e σ − 1 σ ( w − s ) χ ¯ * ′ ( s ) ( χ ¯ * ( w ) − χ ¯ * ( s ) ) α ¯ k Φ ( Π ( s , Q ( s ) ) ) d s + ( χ ¯ * ( x 2 ) − χ ¯ * ( x 1 ) ) γ k − 1 ∣ Λ ∣ Γ k ( γ ) ∣ λ ∣ 1 k σ α ¯ Γ k ( α ¯ ) ∫ x 1 η e σ − 1 σ ( η − s ) χ ¯ * ′ ( s ) ( χ ¯ * ( η ) − χ ¯ * ( s ) ) α ¯ k Φ ( Π ( s , Q ( s ) ) ) d s + ∑ i = 1 m ∣ λ i ∣ I x 1 + α ¯ , σ , χ ¯ * k Φ ( Π ( ξ i , Q ( ξ i ) ) ) d s + I x 1 + α ¯ , σ , χ ¯ * k Φ ( Π ( x 2 , Q ( x 2 ) ) ) ≤ ‖ T ‖ ∞ Φ Π * Ψ .

Hence, we have

‖ T ‖ ∞ ≤ Ψ Φ Π * ‖ T ‖ ∞ .

By equation (14), ‖ T ‖ ∞ = 0 . Thus, for all w ∈ [ x 1 , x 2 ] , we obtain T ( w ) = 0 , and therefore, Φ ( Q ( w ) ) = 0 . Consequently, Q ( z ) is relatively compact in E , and by Arzelá-Ascoli theorem, Q is relatively compact in B τ . By Theorem 5.1, T has a fixed point. Thus, the nonlocal BVP (2) has at least one solution. The proof is completed.□

6 Example

In this section, some examples will be presented to illustrate our theoretical results obtained in previous sections.

Example 6.1

Consider the following Hilfer fractional proportional boundary value problem, subject to nonlocal integro-multipoint boundary conditions, given by

(15) D 1 4 + 7 4 , 2 3 , 1 3 , r + 1 r + 2 5 2 , H ϖ ( w ) = Π ( w , ϖ ( w ) ) , w ∈ 1 4 , 7 4 , q 1 4 = 0 , q 7 4 = 1 11 q 1 2 + 3 22 q 5 4 + 5 33 q 3 2 + 7 44 I 1 4 + 3 2 , 1 3 , r + 1 r + 2 5 2 q 3 4 .

Here, k = 5 ⁄ 2 , α ¯ = 7 ⁄ 4 , β = 2 ⁄ 3 , σ = 1 ⁄ 3 , and χ ¯ * ( w ) = ( w + 1 ) ⁄ ( w + 2 ) with χ ¯ * ′ ( w ) = 1 ⁄ ( w + 2 ) 2 > 0 , x 1 = 1 ⁄ 4 , x 2 = 7 ⁄ 4 , m = 3 , λ 1 = 1 ⁄ 11 , λ 2 = 3 ⁄ 22 , λ 3 = 5 ⁄ 33 , ξ 1 = 1 ⁄ 2 , ξ 2 = 5 ⁄ 4 , ξ 3 = 3 ⁄ 2 , λ = 7 ⁄ 44 , and δ = 3 ⁄ 2 , η = 3 ⁄ 4 .   We can compute that γ = 47 ⁄ 12 , ∣ Λ ∣ ≈ 0.0416439 , Ψ ≈ 3.50653297 , and Ψ 1 ≈ 3.1333259 .

( i ) Consider the function Π ∈ C ( [ 1 ⁄ 4 , 7 ⁄ 4 ] , R ) presented by

(16) Π ( w , ϖ ) = 5 4 r + 37 ϖ 2 + 2 ∣ ϖ ∣ 1 + ∣ ϖ ∣ + 1 2 w 3 + 1 4 .

Observe that Π satisfies the Lipschitz condition as ∣ Π ( z , ϖ 1 ) − Π ( z , ϖ 2 ) ∣ ≤ ( 5 ⁄ 19 ) ∣ ϖ 1 − ϖ 2 ∣ for all z ∈ [ 1 ⁄ 4 , 7 ⁄ 4 ] and ϖ i ∈ R , i = 1 , 2 , with Lipschitz constant N = 5 ⁄ 19 . Hence, we can conclude that the BVP (15) with function Π given in equation (16) has a unique solution on the interval [ 1 ⁄ 4 , 7 ⁄ 4 ] by Theorem 4.1, since N Ψ ≈ 0.9227718342 < 1 .

( i i ) Let us modify the function Π in equation (16) as follows:

(17) Π ( w , ϖ ) = 5 4 z + 15 ∣ ϖ ∣ 1 + ∣ ϖ ∣ + 1 2 w 3 + 1 4 .

Then, we see that ∣ Π ( w , ϖ 1 ) − Π ( w , ϖ 2 ) ∣ ≤ ( 5 ⁄ 16 ) ∣ ϖ 1 − ϖ 2 ∣ , and thus Π satisfies the Lipschitz condition with constant N = 5 ⁄ 16 . By Theorem 4.1, the BVP (15) with function Π given in equation (17) does not have a unique solution since N Ψ ≈ 1.095791553 > 1 . However, the function Π in equation (17) is bounded by

∣ Π ( w , ϖ ) ∣ ≤ 5 4 z + 15 + 1 2 w 3 + 1 4 ≔ ω ( w )

for all w ∈ [ 1 ⁄ 4 , 7 ⁄ 4 ] and ϖ ∈ R . By Theorem 4.2, there exists at least one solution of ( k , χ ¯ * ) -Hilfer proportional FDEs, subject to nonlocal integro-multipoint boundary conditions (15) with the function Π given in equation (17), since N Ψ 1 ≈ 0.9791643438 < 1 .

( i i i ) Let the function Π be defined as follows:

(18) Π ( w , ϖ ) = 1 4 w + 2 ϖ 2022 5 ( 1 + ϖ 2020 ) + ϖ 2023 4 ( 1 + ϖ 2022 ) + 1 7 e − ϖ 2 .

We have

∣ Π ( w , ϖ ) ∣ ≤ 1 4 w + 2 1 5 ϖ 2 + 1 4 ∣ ϖ ∣ + 1 7 .

Then, we choose ϑ ( w ) = 1 ⁄ ( 4 w + 2 ) and β ( ∣ ϖ ∣ ) = ( 1 ⁄ 5 ) ϖ 2 + ( 1 ⁄ 4 ) ∣ ϖ ∣ + ( 1 ⁄ 7 ) . Then, we have ‖ ϑ ‖ = 1 ⁄ 3 , and there exists a constant L ∈ ( 0.257878762 , 2.769850872 ) satisfying the inequality in ( M 4 ) of Theorem 4.3. The conclusion of Theorem 4.3 implies that the nonlocal integro-multipoint BVP (15) with the function Π given in equation (18) has at least one solution on [ 1 ⁄ 4 , 7 ⁄ 4 ] .

( i v ) Let ϖ = ( ϖ 1 , ϖ 2 , ϖ 3 , … , ϖ n , … ) and B = { ϖ : ϖ n → 0 } be the Banach space of real sequences converging to zero, with the norm ‖ ϖ ‖ ∞ = sup n ≥ 1 ∣ ϖ n ∣ . If Π : [ 1 ⁄ 4 , 7 ⁄ 4 ] × B → B is given by

(19) Π ( w , ϖ ) = 5 20 w + 13 1 2 n + ∣ ϖ n ∣ 3 1 + n n ≥ 1 , ϖ = ϖ n ∈ B ,

then the condition ( C 1 ) of Theorem 5.2 is satisfied. Next, we have

‖ Π ( w , ϖ ) ‖ ∞ ≤ 5 20 w + 13 1 2 n + ∣ ϖ n ∣ 3 1 + n ∞ ≤ 5 20 z + 13 ( 1 + ‖ ϖ ‖ ∞ 3 ) ≔ Φ Π ( w ) ϕ ( ‖ ϖ ‖ ∞ ) .

Therefore, the condition ( C 2 ) is fulfilled with Φ Π ( w ) = 5 ⁄ ( 20 z + 13 ) and ϕ ( ‖ ϖ ‖ ∞ ) = 1 + ‖ ϖ ‖ ∞ 3 . Furthermore, if K ⊆ B is a bounded set, then we have Φ ( Π ( w , K ) ) ≤ Φ Π ( w ) Φ ( K ) . Setting Φ Π * = 5 ⁄ 18 , we obtain Φ Π * χ ¯ * ≈ 0.9740369361 < 1 . Thus, by applying the result in Theorem 5.2, the nonlocal BVP (15) with the nonlinear function h given in equation (19) has at least one solution on [ 1 ⁄ 4 , 7 ⁄ 4 ] .

7 Conclusion

In this article, we presented the criteria ensuring the existence and uniqueness of solutions of a ( k , χ ¯ * ) -Hilfer generalized proportional FDEs of order ( 1 , 2 ] supplemented with nonlocal integro-multipoint boundary conditions. We used standard fixed point theorems to establish the desired results, which are well illustrated by constructing numerical examples. We examined both cases, the scalar cases and the Banach space case. Our results are new and contribute to enrich the literature on nonlocal ( k , χ ¯ * )-Hilfer proportional fractional integro-multipoint BVPs.

  1. Funding information: This research was funded by the National Science, Research and Innovation Fund and King Mongkut’s University of Technology North Bangkok with contract no. KMUTNB-FF-66-11.

  2. Conflict of interest: The authors state that there is no conflict of interest.

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Received: 2023-04-20
Revised: 2023-09-27
Accepted: 2023-09-27
Published Online: 2023-10-27

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

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

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https://doi.org/10.1515/math-2023-0137
Keywords for this article
Hilfer proportional fractional derivative; proportional fractional integral; fixed point theorems; existence results
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  1. Special Issue on Future Directions of Further Developments in Mathematics
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  101. Construction of analytical solutions to systems of two stochastic differential equations
  102. Two-distance vertex-distinguishing index of sparse graphs
  103. Regularity and abundance on semigroups of partial transformations with invariant set
  104. Liouville theorems for Kirchhoff-type parabolic equations and system on the Heisenberg group
  105. Spin(8,C)-Higgs pairs over a compact Riemann surface
  106. Properties of locally semi-compact Ir-topological groups
  107. Transcendental entire solutions of several complex product-type nonlinear partial differential equations in ℂ2
  108. Ordering stability of Nash equilibria for a class of differential games
  109. A new reverse half-discrete Hilbert-type inequality with one partial sum involving one derivative function of higher order
  110. About a dubious proof of a correct result about closed Newton Cotes error formulas
  111. Ricci ϕ-invariance on almost cosymplectic three-manifolds
  112. Schur-power convexity of integral mean for convex functions on the coordinates
  113. A characterization of a ∼ admissible congruence on a weakly type B semigroup
  114. On Bohr's inequality for special subclasses of stable starlike harmonic mappings
  115. Properties of meromorphic solutions of first-order differential-difference equations
  116. A double-phase eigenvalue problem with large exponents
  117. On the number of perfect matchings in random polygonal chains
  118. Evolutoids and pedaloids of frontals on timelike surfaces
  119. A series expansion of a logarithmic expression and a decreasing property of the ratio of two logarithmic expressions containing cosine
  120. The 𝔪-WG° inverse in the Minkowski space
  121. Stability result for Lord Shulman swelling porous thermo-elastic soils with distributed delay term
  122. Approximate solvability method for nonlocal impulsive evolution equation
  123. Construction of a functional by a given second-order Ito stochastic equation
  124. Global well-posedness of initial-boundary value problem of fifth-order KdV equation posed on finite interval
  125. On pomonoid of partial transformations of a poset
  126. New fractional integral inequalities via Euler's beta function
  127. An efficient Legendre-Galerkin approximation for the fourth-order equation with singular potential and SSP boundary condition
  128. Eigenfunctions in Finsler Gaussian solitons
  129. On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces
  130. Some estimates for commutators of sharp maximal function on the p-adic Lebesgue spaces
  131. A preconditioned iterative method for coupled fractional partial differential equation in European option pricing
  132. A digital Jordan surface theorem with respect to a graph connectedness
  133. A quasi-boundary value regularization method for the spherically symmetric backward heat conduction problem
  134. The structure fault tolerance of burnt pancake networks
  135. Average value of the divisor class numbers of real cubic function fields
  136. Uniqueness of exponential polynomials
  137. An application of Hayashi's inequality in numerical integration
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