Home Mathematics Analysis of solutions for the fractional differential equation with Hadamard-type
Article Open Access

Analysis of solutions for the fractional differential equation with Hadamard-type

  • Huijuan Zhu , Yuanfang Ru EMAIL logo and Fanglei Wang
Published/Copyright: October 20, 2023
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 21 Issue 1

Abstract

We mainly consider the existence and stability results of the positive solutions for the fractional differential equation with Hadamard-type by applying fixed point theorems, if the nonlinearity may be continuous or singular. We also construct some examples to show the applicability of the results.

Keywords: Hadamard fractional boundary value problem; existence and uniqueness; positive solutions; fixed point theorem; Ulam-Hyers stability
MSC 2010: 34C25; 34B15

1 Introduction

During the past two decades, many real-world processes and phenomena in signal processing, fluid flow, electrical systems, control theory, etc., are characterized by fractional differential equations (FDEs), which are considered much better and improved over ones that are only dependent on integer-order operators (see [1–3]). Therefore, many researchers focused on the study of the FDEs using various methods, such as fixed points [4–13], variational methods [14–16], and upper and lower solution methods [17,18]. Especially, in [19–22], combining some fixed point theorems with the monotonic iteration technique, the authors studied the existence of solutions of fractional equations.

In 1892, the well-known Hadamard-type fractional derivative was defined in [23] with salient features, which contains a logarithmic function of any exponent in the integral kernel. Since then, it is important to study the property of the Hadamard-type fractional derivative and applications. For example, Ahmad and Ntouyas [24] applied the fixed point theorem to study the solutions of the FDEs:

D 1 + α z ( t ) = g ( z ( t ) ) , 1 < t < e , z ( 1 ) = 0 , z ( e ) = α z ( ρ ) ,

where D 1 + α denotes the Hadamard-type fractional derivative, 1 < γ < 2 , and ρ ∈ ( 1 , e ) . Butzer et al. [25,26] studied semigroups and bounded properties of the Hadamard-type fractional operators that are used in [27]. Also, many researchers applied the fixed point theorem with the monotonic iteration technique or numerical techniques to study the solutions of the Hadamard-type FDEs (see [28–31]). In [32], the authors studied the general Hadamard-type singular FDEs. Using the double monotone iterative technique, the uniqueness of positive solutions is obtained. Especially, the authors [31] have studied the existence of positive solutions for a class of weakly singular Hadamard-type FDE with various kinds of boundary conditions.

Inspired by these references, we are interested in studying the problem:

(1.1) ( D 1 + α x ( t ) ) ″ + λ 2 D 1 + α x ( t ) + f ( t , x ( t ) , − D 1 + α x ( t ) ) = 0 , t ∈ ( 1 , e ) , x ( 1 ) = x ′ ( 1 ) = D 1 + α + 1 x ( 1 ) = 0 , x ′ ( e ) = D 1 + α + 1 x ( e ) = 0 ,

where 2 < α ≤ 3 and 0 < λ ≤ 1 e ( e − 1 ) . When the function f may be continuous or singular, we will investigate the properties of positive solutions of this equation, such as existence, uniqueness, and stability. The discussions are based on Banach’s contraction mapping and some fixed point theorems. Compared to the results in the references, the main results obtained by us have some new features. First, the problem satisfies the Neumann boundary condition, which is obviously different to the periodic boundary value condition. Second, we give some sufficient conditions for the uniqueness result, which are based on the iterative method. Finally, we further study the stability.

The organization of this article is as follows. In Section 2, we present some definitions and preliminary lemmas. In Section 3, we study the regular or singular Problem (1.1) and give the proof of the related theorems. In Section 4, we further show the stability results. In Section 5, some examples are given to illustrate the main results.

2 Preliminaries

For convenience, the basic definition for the Hadamard-type fractional derivatives is given as follows.

Definition 2.1

[10] The Hadamard derivative of fractional order α > 0 for a function x : [ a , + ∞ ) → R is defined by:

D a + α x ( t ) = 1 Γ ( n − α ) t d d t n ∫ a t ln t s n − α − 1 x ( s ) d s s ,

where n = [ α ] + 1 , [ α ] stands for the largest integer, which is less than or equal to α .

Now, let − D 1 + α x ( t ) = u ( t ) . Then, combining with the boundary condition, it follows from [31] that

x ( t ) = ∫ 1 e G 1 ( t , s ) u ( s ) d s s ,

where

G 1 ( t , s ) = 1 Γ ( α ) ( ln t ) α − 1 ( 1 − ln s ) α − 2 − ( ln t − ln s ) α − 1 , 1 ≤ s ≤ t ≤ e , ( ln t ) α − 1 ( 1 − ln s ) α − 2 , 1 ≤ t ≤ s ≤ e .

Thus, Problem (1.1) can be reduced to the Neumann boundary value problem:

(2.1) u ″ ( t ) + λ 2 u ( t ) = f t , ∫ 1 e G 1 ( t , s ) u ( s ) d s s , u ( t ) , t ∈ ( 1 , e ) , u ′ ( 1 ) = u ′ ( e ) = 0 .

Meanwhile, the solution of the nonhomogeneous problem

(2.2) u ″ ( t ) + λ 2 u ( t ) = h ( t ) , u ′ ( 1 ) = u ′ ( e ) = 0 ,

can be expressed by u ( t ) = ∫ 1 e G 2 ( t , s ) h ( s ) d s , where

G 2 ( t , s ) = 1 λ sin λ ( e − 1 ) cos λ ( 1 − t ) cos λ ( s − e ) + sin λ ( e − 1 ) sin λ ( t − s ) , 1 ≤ s ≤ t ≤ e , cos λ ( 1 − t ) cos λ ( s − e ) , 1 ≤ t ≤ s ≤ e .

Next, we give the positive estimates of G i ( t , s ) ( i = 1 , 2 ) .

Lemma 2.2

G i ( t , s ) ( i = 1 , 2 ) satisfy the properties:

  1. G i ( t , s ) ≥ 0 , i = 1 , 2 ;

  2. for all t , s ∈ [ 1 , e ] ,

    1. ( ln t ) α − 1 G 1 ( e , s ) ≤ G 1 ( t , s ) ≤ ( ln t ) α − 1 Γ ( α ) ;

    2. G 2 ( s , s ) ≤ G 2 ( t , s ) ≤ cos λ ( e − t ) λ sin λ ( e − 1 ) .

Proof

It is obvious that Property (1) is satisfied. Now, we assert Property (2).

(i) For 1 ≤ s ≤ t ≤ e ,

G 1 ( t , s ) = ( ln t ) α − 1 Γ ( α ) ( 1 − ln s ) α − 2 − 1 − ln s ln t α − 1 ≥ ( ln t ) α − 1 Γ ( α ) ln s ( 1 − ln s ) α − 2 = ( ln t ) α − 1 G 1 ( e , s ) .

Similarly, we have G 1 ( t , s ) ≥ ( ln t ) α − 1 G 1 ( e , s ) , with 1 ≤ t ≤ s ≤ e for t , s ∈ [ 1 , e ] .

Moreover, for 1 ≤ s ≤ t ≤ e ,

G 1 ( t , s ) ≤ 1 Γ ( α ) ( ln t ) α − 1 ( 1 − ln s ) α − 2 ≤ ( ln t ) α − 1 Γ ( α ) .

Hence, for 1 ≤ t ≤ s ≤ e , G 1 ( t , s ) ≤ ( ln t ) α − 1 Γ ( α ) is easily visible. Then, (i) is demonstrated.

(ii) Since G 2 ( t , s ) is continuous, taking the derivative of G 2 ( t , s ) with respect to t , we obtain the following:

For 1 ≤ s ≤ t ≤ e ,

∂ G 2 ( t , s ) ∂ t = 1 sin λ ( e − 1 ) [ cos λ ( e − s ) sin λ ( 1 − t ) + sin λ ( e − 1 ) cos λ ( t − s ) ]

and

∂ 2 G 2 ( t , s ) ∂ t 2 = λ sin λ ( e − 1 ) [ − cos λ ( e − s ) cos λ ( 1 − t ) − sin λ ( e − 1 ) sin λ ( t − s ) ] ≤ 0 ,

so ∂ G 2 ( t , s ) ∂ t ≥ ∂ G 2 ( e , s ) ∂ t = sin λ ( s − 1 ) sin λ ( e − 1 ) ≥ 0 .

For 1 ≤ t ≤ s ≤ e ,

∂ G 2 ( t , s ) ∂ t = 1 sin λ ( e − 1 ) cos λ ( e − s ) sin λ ( 1 − t ) ≤ 0 .

Therefore, G 2 ( t , s ) is decreasing with respect to t on [ 1 , s ] and increasing with respect to t on [ s , e ] , which means that G 2 ( t , s ) ≥ G 2 ( s , s ) for t , s ∈ [ 1 , e ] .

Furthermore, for 1 ≤ t ≤ s ≤ e , from 0 < λ ≤ 1 e ( e − 1 ) < π 2 ( e − 1 ) , we have 0 < cos λ ( s − e ) ≤ 1 and

G 2 ( t , s ) = 1 λ sin λ ( e − 1 ) cos λ ( 1 − t ) cos λ ( s − e ) ≤ cos λ ( 1 − t ) λ sin λ ( e − 1 ) .

For 1 ≤ s ≤ t ≤ e , taking the derivative of G 2 ( t , s ) with respect to s , using the similar discussion earlier, ∂ G 2 ( t , s ) ∂ s is monotonically decreasing with respect to s on [ 1 , t ] . Since ∂ G 2 ( t , 1 ) ∂ s = 0 , G 2 ( t , s ) is decreasing with respect to s on [ 1 , t ] , namely,

G 2 ( t , s ) ≤ G 2 ( t , 1 ) = cos λ ( e − t ) λ sin λ ( e − 1 ) .

Hence,

G 2 ( s , s ) ≤ G 2 ( t , s ) ≤ cos λ ( e − t ) λ sin λ ( e − 1 ) ,

which completes the proof.□

3 Existence

For convenience, in this part, let E = C [ 1 , e ] be a Banach space equipped with the norm ‖ x ‖ ≔ max t ∈ [ 1 , e ] ∣ x ( t ) ∣ . Let B r = { u ∈ E : ‖ u ‖ < r } , where r > sup t ∈ [ 1 , e ] f ( t , 0 , 0 ) Γ ( α ) ( e − 1 ) λ Γ ( α ) sin λ ( e − 1 ) − ( ℒ 1 + ℒ 2 Γ ( α ) ) ( e − 1 ) .

Theorem 3.1

Let f : [ 1 , e ] × R × R → R + be continuous. If the following assumption holds:

(H1) there exist ℒ 1 , ℒ 2 > 0 such that

∣ f ( t , x , u ) − f ( t , y , v ) ∣ ≤ ℒ 1 ∣ x − y ∣ + ℒ 2 ∣ u − v ∣ , ∀ x , y , u , v ∈ E , and t ∈ [ 1 , e ] .

Then, (1.1) has a unique solution if ℒ 1 Γ ( α ) + ℒ 2 ( e − 1 ) λ sin λ ( e − 1 ) < 1 .

Proof

Define the operator T by:

T u ( t ) = ∫ 1 e G 2 ( t , s ) f s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ , u ( s ) d s .

Then, for any u ∈ B r , by virtue of Lemma 2.2 and (H1), we have

‖ T u ( t ) ‖ < ∫ 1 e G 2 ( t , s ) f s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ , u ( s ) − f ( s , 0 , 0 ) + sup t ∈ [ 1 , e ] f ( t , 0 , 0 ) d s < ℒ 1 Γ ( α ) + ℒ 2 ‖ u ‖ + sup t ∈ [ 1 , e ] f ( t , 0 , 0 ) ∫ 1 e G 2 ( t , s ) d s < ℒ 1 Γ ( α ) + ℒ 2 ‖ u ‖ + sup t ∈ [ 1 , e ] f ( t , 0 , 0 ) e − 1 λ sin λ ( e − 1 ) < r ,

which follows that T ( B r ) ⊂ B r .

In addition, for any u , v ∈ E , we obtain

‖ T u ( t ) − T v ( t ) ‖ < ∫ 1 e G 2 ( t , s ) ℒ 1 ∫ 1 e G 1 ( s , τ ) ∣ u ( τ ) − v ( τ ) ∣ d τ τ + ℒ 2 ∣ u ( s ) − v ( s ) ∣ d s < ℒ 1 Γ ( α ) + ℒ 2 ‖ u − v ‖ ∫ 1 e G 2 ( t , s ) d s < ℒ 1 Γ ( α ) + ℒ 2 e − 1 λ sin λ ( e − 1 ) ‖ u − v ‖ ,

namely, T is contractional.

At last, we conclude that T has a unique fixed point by Banach’s contraction mapping principle, which means that (2.1) has a unique solution.□

Theorem 3.2

Let f : [ 1 , e ] × R + × R + → R + be continuous, and the following assumption holds:

(H2) There exist a constant γ ∈ ( 0 , 1 ) and two functions h 1 , h 2 ∈ P with h 1 , h 2 ≠ 0 on any subinterval of ( 1 , e ) , such that

h 1 ( t ) ( u + v ) γ ≤ f ( t , u , v ) ≤ h 2 ( t ) ( u + v ) γ , ( u , v ) ∈ [ 0 , ∞ ) × [ 0 , ∞ ) , and t ∈ [ 1 , e ] .

Then, Problem (1.1) has at least one positive solution.

Proof

Let

P = { u ∈ C [ 1 , e ] : u ( t ) ≥ 0 }

and

P 0 = { u ∈ P : l ≤ u ( t ) ≤ L } .

Obviously, P 0 is a subset of P and P ⊆ E . Now, we show that T has a fixed point in P 0 .

By Lemma 2.2, for any u ∈ P 0 , we obtain

(3.1) l ( ln s ) α − 1 α ( α − 1 ) Γ ( α ) ≤ ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ ≤ L ( ln s ) α − 1 Γ ( α ) ≤ L Γ ( α ) .

From (3.1) and (H2), we have

h 1 ( s ) l γ ≤ f s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ , u ( s ) ≤ h 2 ( s ) L γ 1 + 1 Γ ( α ) γ .

Let

M 1 = ∫ 1 e G 2 ( s , s ) h 1 ( s ) d s , M 2 = ( 1 + Γ ( α ) ) γ ( Γ ( α ) ) γ λ sin λ ( e − 1 ) ∫ 1 e h 2 ( s ) d s .

Then, we have

T u ( t ) ≥ l γ ∫ 1 e G 2 ( s , s ) h 1 ( s ) d s = M 1 l γ

and

T u ( t ) ≤ L γ 1 + 1 Γ ( α ) γ λ sin λ ( e − 1 ) ∫ 1 e h 2 ( s ) d s = M 2 L γ .

Take l = 1 L . Since M i > 0 and 0 < γ < 1 , actually there exists a L > 1 such that M 1 L 1 − γ ≥ 1 and M 2 L γ − 1 ≤ 1 , which implies that T : P 0 → P 0 . Thus, T has a fixed point u ∈ P 0 by Schauder’s fixed point theorem. Therefore, Theorem 3.2 is proved.□

If f ( t , u , v ) is nonnegative and continuous on [ 1 , e ] × ( 0 , ∞ ) 2 and may be singular at u = 0 , v = 0 . Now, we study the problem:

(3.2) u ″ ( t ) + λ 2 u ( t ) = f t , ∫ 1 e G 1 ( t , s ) u ( s ) d s s + 1 n , u ( t ) + 1 n , u ′ ( 1 ) = u ′ ( e ) = 0 ,

where n ∈ N + with n ≥ 2 . Similar to the previous discussion, u is a solution of equation of (3.2) if and if only u satisfies the following equation:

u ( t ) = ∫ 1 e G 2 ( t , s ) f s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ + 1 n , u ( s ) + 1 n d s .

Theorem 3.3

Suppose that the following conditions hold:

(H3) f ( t , u , v ) = φ ( t , u , v ) + ψ ( t , u , v ) , where φ : [ 1 , e ] × [ 0 , + ∞ ) × [ 0 , + ∞ ) → [ 0 , + ∞ ) , ψ : [ 1 , e ] × ( 0 , + ∞ ) × ( 0 , + ∞ ) → [ 0 , + ∞ ) are continuous, φ ( t , u , v ) is nondecreasing, and ψ ( t , u , v ) is nonincreasing in u , v > 0 .

(H4) For any u , v > 0 , c ∈ ( 0 , 1 ) , there exists 0 < σ < 1 2 such that

c σ φ ( t , u , v ) ≤ φ ( t , c u , c v ) , c σ ψ ( t , u , v ) ≤ ψ ( t , c − 1 u , c − 1 v ) .

Then, there exists a unique positive solution x * for Problem (1.1) satisfying

M cos λ ( e − t ) ≤ x * ( t ) ≤ 1 M cos λ ( e − t ) ,

where 0 < M < 1 is a constant.

Furthermore, for any given initial u 0 , v 0 ∈ P 0 ˜ , construct two sequences as follows:

u n = ∫ 1 e G 2 ( t , s ) φ s , ∫ 1 e G 1 ( s , τ ) u n − 1 ( τ ) d τ τ , u n − 1 ( s ) + ψ s , ∫ 1 e G 1 ( s , τ ) v n − 1 ( τ ) d τ τ , v n − 1 ( s ) d s , n ∈ N + , v n = ∫ 1 e G 2 ( t , s ) φ s , ∫ 1 e G 1 ( s , τ ) v n − 1 ( τ ) d τ τ , v n − 1 ( s ) + ψ s , ∫ 1 e G 1 ( s , τ ) u n − 1 ( τ ) d τ τ , u n − 1 ( s ) d s , n ∈ N + ,

which converge uniformly to − D 1 + α x * ( t ) on [ 1 , e ] as n → ∞ , i.e.,

u n → − D 1 + α x * a n d v n → − D 1 + α x * as n → ∞ .

Remark 3.4

By (H4), for c ≥ 1 and u , v > 0 , one has

c σ φ ( t , u , v ) ≥ φ ( t , c u , c v ) , c σ ψ ( t , u , v ) ≥ ψ ( t , c − 1 u , c − 1 v ) .

We first give the following necessary definition and lemma, before proving Theorem 3.3.

Definition 3.5

[22] Assume P to be a normal cone of a Banach space E , T : P × P → P is said to be mixed monotone if T ( y , z ) is non-decreasing in y and non-increasing in z , i.e., y 1 ≤ y 2 ( y 1 , y 2 ∈ P ) implies T ( y 1 , z ) ≤ T ( y 2 , z ) for any z ∈ P , and z 1 ≤ z 2 ( z 1 , z 2 ∈ P ) implies T ( y , z 1 ) ≥ T ( y , z 2 ) for any y ∈ P . The element y * ∈ P is called a fixed point of T if T ( y * , y * ) = y * .

Lemma 3.6

[33] Let P be a normal, solid cone of Banach space E and T : P 0 ˜ → P 0 ˜ be a mixed monotone operator. Assume that there exists 0 < σ < 1 such that

(3.3) T ( c y , c − 1 z ) ≥ c σ T ( y , z ) , y , z ∈ P 0 ˜ , 0 < c < 1 .

Then, the operator T has a unique fixed point y * ∈ P 0 ˜ . Moreover, for any initial y 0 , z 0 ∈ P 0 ˜ , by constructing successively the sequences

y n = T ( y n − 1 , z n − 1 ) , z n = T ( y n − 1 , z n − 1 ) , n = 1 , 2 , … ,

we have ‖ y n − y * ‖ → 0 , ‖ z n − z * ‖ → 0 as n → + ∞ .

Proof of Theorem 3.3

Let P 0 ˜ be a normal and solid cone in E , which is defined by:

P 0 ˜ = u ∈ E : M cos λ ( e − t ) ≤ u ( t ) ≤ 1 M cos λ ( e − t ) ,

where M ∈ ( 0 , 1 ) with M < min { 1 , L 1 , L 2 } ,

L 1 = ∫ 1 e ( [ a ( cos λ ( e − s ) + 1 ) ] σ M − σ φ ( s , 1 , 1 ) + ( M b ) − σ ( ln s ) − 2 σ ψ ( s , 1 , 1 ) ) d s λ sin λ ( e − 1 ) − 1 , L 2 = ∫ 1 e M σ [ a ( cos λ ( e − s ) + 1 ) ] − σ G 2 ( s , s ) ψ ( s , 1 , 1 ) d s ,

in which

a = max N 2 Γ ( α ) , 1 , b = min N 1 Γ ( α ) , 1

and

N 1 = ∫ 1 e ln τ ( 1 − ln τ ) α − 2 cos λ ( e − τ ) d τ τ , N 2 = ∫ 1 e cos λ ( e − τ ) d τ τ .

Now, we define an operator T by:

T ( u , v ) ( t ) = ∫ 1 e G 2 ( t , s ) φ s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ + 1 n , u ( s ) + 1 n + ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ + 1 n , v ( s ) + 1 n d s .

First, for u ∈ P 0 ˜ , we have

(3.4) M N 1 ( ln s ) α − 1 Γ ( α ) ≤ ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ ≤ N 2 M Γ ( α ) .

Let f ( s ) = ln s − cos λ ( e − s ) , s ∈ [ 1 , e ] , and then,

f ′ ( s ) = 1 s − λ sin λ ( e − s ) = 1 s g ( s ) ,

which g ( s ) = 1 − λ s sin λ ( e − s ) . Since 0 < λ ≤ 1 e ( e − 1 ) , we have g ( s ) ≥ 0 , which implies that f ′ ( s ) ≥ 0 . Therefore, f ( s ) is increasing on [ 1 , e ] , which follows that f ( s ) ≤ f ( e ) = 0 . This means that ln s ≤ cos λ ( e − s ) , i.e., ( ln s ) 2 ≤ cos 2 λ ( e − s ) for s ∈ [ 1 , e ] .

For any u , v ∈ P 0 ˜ , from (H3)–(H4) and Remark 3.4, it follows that

φ s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ + 1 n , u ( s ) + 1 n ≤ a σ M − σ φ ( s , ( ln s ) α − 1 + 1 , cos λ ( e − s ) + 1 ) ≤ [ a ( cos λ ( e − s ) + 1 ) ] σ M − σ φ ( s , 1 , 1 )

and

ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ + 1 n , v ( s ) + 1 n ≤ ψ s , M N 1 ( ln s ) α − 1 Γ ( α ) + 1 n , M cos λ ( e − s ) + 1 n ≤ M − σ ψ s , N 1 ( ln s ) α − 1 Γ ( α ) , cos λ ( e − s ) ≤ ( M b ) − σ ψ ( s , ( ln s ) α − 1 , cos λ ( e − s ) ) ≤ ( M b ) − σ ψ ( s , ( ln s ) 2 , cos 2 λ ( e − s ) ) ≤ ( M b ) − σ ( ln s ) − 2 σ ψ ( s , 1 , 1 ) .

Consequently, applying Lemma 2.2, we have

T ( u , v ) = ∫ 1 e G 2 ( t , s ) φ s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ + 1 n , u ( s ) + 1 n + ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ + 1 n , v ( s ) + 1 n d s ≤ cos λ ( e − t ) λ sin λ ( e − 1 ) ∫ 1 e ( [ a ( cos λ ( e − s ) + 1 ) ] σ M − σ φ ( s , 1 , 1 ) + ( M b ) − σ ( ln s ) − 2 σ ψ ( s , 1 , 1 ) ) d s ≤ 1 M cos λ ( e − t ) .

Moreover,

ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ + 1 n , v ( s ) + 1 n ≥ ψ s , N 2 ( ln s ) α − 1 M Γ ( α ) + 1 n , 1 M cos λ ( e − s ) + 1 n ≥ M σ a − σ ψ ( s , ( ln s ) α − 1 + 1 , cos λ ( e − s ) + 1 ) ≥ M σ a − σ ψ ( s , ln s + 1 , cos λ ( e − s ) + 1 ) ≥ M σ [ a ( cos λ ( e − s ) + 1 ) ] − σ ψ ( s , 1 , 1 ) .

Meanwhile, Lemma 2.2 yields that

T ( u , v ) = ∫ 1 e G 2 ( t , s ) φ s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ , u ( s ) + ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ , v ( s ) d s ≥ ∫ 1 e G 2 ( t , s ) ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ , v ( s ) d s ≥ cos λ ( e − t ) ∫ 1 e M σ [ a ( cos λ ( e − s ) + 1 ) ] − σ G 2 ( s , s ) ψ ( s , 1 , 1 ) d s ≥ M cos λ ( e − t ) .

From the aforementioned discussions, it concludes that T : P 0 ˜ × P 0 ˜ → P 0 ˜ .

Next, for any u 1 , u 2 ∈ P 0 ˜ with u 1 ≤ u 2 , from the monotonicity of ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ and φ , we have

T ( u 1 , v ) = ∫ 1 e G 2 ( t , s ) φ s , ∫ 1 e G 1 ( s , τ ) u 1 ( τ ) d τ τ + 1 n , u 1 ( s ) + 1 n + ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ + 1 n , v ( s ) + 1 n d s ≤ ∫ 1 e G 2 ( t , s ) φ s , ∫ 1 e G 1 ( s , τ ) u 2 ( τ ) d τ τ + 1 n , u 2 ( s ) + 1 n + ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ + 1 n , v ( s ) + 1 n d s = T ( u 2 , v ) ,

i.e., T ( u , v ) is non-decreasing with respect to u , for any v ∈ P 0 ˜ . In an analogous manner, if v 1 , v 2 ∈ P 0 ˜ with v 1 ≥ v 2 , we obtain

T ( u , v 1 ) ( t ) ≥ T ( u , v 2 ) ( t ) , u ∈ P 0 ˜ .

Hence, T is a mixed monotone operator on P 0 ˜ × P 0 ˜ → P 0 ˜ .

Now, for every u , v ∈ P 0 ˜ , t ∈ [ 1 , e ] and c ∈ ( 0 , 1 ) , by (H4), we have

T ( c u , c − 1 v ) ( t ) ≥ ∫ 1 e G 2 ( t , s ) c σ φ s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ + 1 n , u ( s ) + 1 n + ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ + 1 n , v ( s ) + 1 n d s = c σ ∫ 1 e G 2 ( t , s ) φ s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ + 1 n , u ( s ) + 1 n + ψ s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ + 1 n , v ( s ) + 1 n d s = c σ T ( u , v ) ( t ) ,

which shows that (3.3) of Lemma 3.6 holds.

By virtue of Lemma 3.6, one can show that Problem (2.1) has a unique positive solution u * satisfying T ( u * , u * ) = u * . Hence, Problem (1.1) has a unique positive solution x * ∈ P 0 ˜ , where x * ( t ) = ∫ 1 e G 1 ( t , s ) u ( s ) d s s and

M cos λ ( e − t ) ≤ x * ( t ) ≤ 1 M cos λ ( e − t ) .

Finally, for any given u 0 , v 0 ∈ P 0 ˜ , construct two iterative sequences { u n } and { v n } by:

u n = T ( u n − 1 , v n − 1 ) , v n = T ( v n − 1 , u n − 1 ) , n = 1 , 2 , … ,

which converges uniformly to u * on [ 1 , e ] , as n → ∞ , i.e.,

u n → − D 1 + α x * ( t ) , v n → − D 1 + α x * ( t ) as n → ∞ .

Therefore, Theorem 3.3 is proved.□

4 Stability

Theorem 4.1

Suppose that (H1) holds and f is nonnegative and continuous on [ 1 , e ] × R × R , then Problem (1.1) is Ulam-Hyers-Rassias stable if ℒ 1 Γ ( α ) + ℒ 2 e − 1 λ sin λ ( e − 1 ) < 1 .

Proof

For Ψ ( t ) ∈ C [ 1 , e ] , let y ∈ C [ 1 , e ] , which satisfies the following inequality:

∣ ( D 1 + α y ( t ) ) ″ + λ 2 D 1 + α y ( t ) + f ( t , y ( t ) , − D 1 + α y ( t ) ) ∣ ≤ ε Ψ ( t ) .

Let − D 1 + α y ( t ) = v ( t ) , then we obtain that, for ε > 0 ,

v ( t ) − ∫ 1 e G 2 ( t , s ) f s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ , v ( s ) d s ≤ ε Ψ ( t ) , t ∈ [ 1 , e ] .

Since Problem (1.1) can be equivalently rewritten to Problem (2.1), it is obvious that we only need to discuss the stability of Problem (2.1).

By Theorem 3.1, the solution u ( t ) of Problem (2.1) can be rewritten as:

u ( t ) = ∫ 1 e G 2 ( t , s ) f s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ , u ( s ) d s .

Then, for t ∈ [ 1 , e ] , according to Lemma 2.2, we have

∣ u ( t ) − v ( t ) ∣ ≤ v ( t ) − ∫ 1 e G 2 ( t , s ) f s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ , v ( s ) d s + ∫ 1 e G 2 ( t , s ) f s , ∫ 1 e G 1 ( s , τ ) u ( τ ) d τ τ , u ( s ) − f s , ∫ 1 e G 1 ( s , τ ) v ( τ ) d τ τ , v ( s ) d s ≤ ε Ψ ( t ) + ℒ 1 Γ ( α ) + ℒ 2 ‖ u − v ‖ ∫ 1 e G 2 ( t , s ) d s < ε Ψ ( t ) + ℒ 1 Γ ( α ) + ℒ 2 e − 1 λ sin λ ( e − 1 ) ‖ u − v ‖ ,

which yields

∣ v ( t ) − u ( t ) ∣ ≤ ε 1 − ℒ 1 Γ ( α ) + ℒ 2 e − 1 λ sin λ ( e − 1 ) Ψ ( t ) = Λ ε Ψ ( t ) , t ∈ [ 1 , e ] .

Consequently, the solution of Problem (2.1) is Ulam-Hyers-Rassias stable with the Ψ ( t ) .□

5 Examples

Example 5.1

Considering the following equation:

− D 1 + 5 2 y ( t ) ″ − π 2 4 e 2 D 1 + 5 2 y ( t ) = t y + ∣ − D 5 2 y ( t ) ∣ 1 2 , t ∈ [ 1 , e ] , y ( 1 ) = y ′ ( 1 ) = y ′ ( e ) = 0 , D 1 + 7 2 y ( 1 ) = D 1 + 7 2 y ( e ) = 0 .

Denote f ( t , y ( t ) , − D α y ( t ) ) = t y + ∣ − D 5 2 y ( t ) ∣ 1 2 and γ = 1 2 . Taking h 1 ( t ) ≡ 1 , h 2 ( t ) ≡ e , then

( u + v ) 1 2 ≤ f ( t , u , v ) ≤ e ( u + v ) 1 2 ,

which means (H2) holds. Let α = 5 2 and λ = π 2 e , then

G 1 ( t , s ) = 1 Γ 5 2 ( ln t ) 3 2 ( 1 − ln s ) 1 2 − ( ln t − ln s ) 3 2 , 1 ≤ s ≤ t ≤ e , ( ln t ) 3 2 ( 1 − ln s ) 1 2 , 1 ≤ t ≤ s ≤ e ,

and

G 2 ( t , s ) = 2 e π sin π 2 e ( e − 1 ) cos π 2 e ( 1 − t ) cos π 2 e ( s − e ) + sin π 2 e ( e − 1 ) sin π 2 e ( t − s ) , 1 ≤ s ≤ t ≤ e , cos π 2 e ( 1 − t ) cos π 2 e ( s − e ) , 1 ≤ t ≤ s ≤ e .

Moreover,

M 1 = 2 e π sin π 2 e ( e − 1 ) ∫ 1 e cos π 2 e ( 1 − s ) cos π 2 e ( s − e ) d s ≈ 2.46679963

and

M 2 = 1 + 1 Γ 5 2 1 2 e ( e − 1 ) π 2 e sin π 2 e ( e − 1 ) − 1 ≈ 10.5179295 .

Let L = 38,000, then M 1 L 1 2 > 1 and M 2 L − 1 2 > 1 hold. Therefore, this equation satisfies the conditions of Theorem 3.2; then, the equation has at least one positive solution.

Example 5.2

Considering the equation:

− D 1 + 5 2 y ( t ) ″ − π 2 4 e 2 D 1 + 5 2 y ( t ) = t y 1 5 ( t ) − D 1 + 5 2 y ( t ) 1 4 + t − 1 2 y − 1 8 ( t ) − D 1 + 5 2 y ( t ) − 1 6 , t ∈ [ 1 , e ] , y ( 1 ) = y ′ ( 1 ) = y ′ ( e ) = 0 , D 1 + 7 2 y ( 1 ) = D 1 + 7 2 y ( e ) = 0 .

Let φ ( t , u , v ) = t ( u 1 5 + v 1 4 ) , ψ ( t , u , v ) = t − 1 2 ( u − 1 8 + v − 1 6 ) and σ = 1 4 , then

φ ( t , c u , c v ) = t ( c 1 5 u 1 5 + c 1 4 v 1 4 ) ≥ c 1 4 t ( u 1 5 + v 1 4 ) = c 1 4 φ ( t , u , v )

and

ψ ( t , c − 1 u , c − 1 v ) = t − 1 2 c 1 8 u − 1 8 + c 1 6 v − 1 6 ≥ c 1 4 t − 1 2 u − 1 8 + v − 1 6 = c 1 4 ψ ( t , u , v ) ,

for every u , v > 0 and c ∈ ( 0 , 1 ) . Obviously, ψ : [ 1 , e ] × ( 0 , + ∞ ) 2 → [ 0 , + ∞ ) and φ : [ 1 , e ] × [ 0 , + ∞ ) 2 → [ 0 , + ∞ ) are continuous. Moreover, φ ( t , u , v ) is non-decreasing and ψ ( t , u , v ) is nonincreasing in u , v > 0 . Then, from Theorem 3.3, the equation has a unique positive solution x * satisfying

M cos π 2 e ( e − t ) ≤ x * ( t ) ≤ 1 M cos π 2 e ( e − t ) ,

where 0 < M < 1 .

6 Conclusion

In this article, we first consider the existence and stability of positive solutions to a class of FDEs with Hadamard-type under continuous nonlinearities using the immovable point theorem. Then, we consider the existence of positive solutions to this model under singular nonlinearities using the mixed immovable point theorem. Finally, we construct some examples to illustrate the applicability of the results.

Acknowledgement

The authors are grateful to the anonymous referee whose careful reading of this manuscript and valuable comments enhanced the presentation of this manuscript.

  1. Funding information: The work was supported by the Scientific Research Plan of Jiangsu Drug Administration (No. 202102) and the Fundamental Research Funds for the Central Universities (2013/B220202082).

  2. Author contributions: H. Zhu was mainly responsible for the calculation of this article, F. Wang was responsible for the typesetting of this article, and Y. Ru was responsible for the revisions and touch-ups.

  3. Conflict of interest: The authors declare that they have no conflict of interest.

References

[1] M. Eslami, F. S. Khodadad, F. Nazari, and H. Rezazadeh, The first integral method applied to the Bogoyavlenskii equations by means of conformable fractional derivative, Opt. Quant. Electron. 49 (2017), 391. 10.1007/s11082-017-1224-zSearch in Google Scholar

[2] M. Eslami, H. Rezazadeh, M. Rezazadeh, and S. S. Mosavi, Exact solutions to the space-time fractional Schrodinger-Hirota equation and the space-time modified KDV-Zakharov-Kuznetsov equation, Opt. Quant. Electron. 49 (2017), 279. 10.1007/s11082-017-1112-6Search in Google Scholar

[3] J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus, Springer, Dordrecht, 2007. 10.1007/978-1-4020-6042-7Search in Google Scholar

[4] R. Agarwal, S. Hristova, and D. O’Regan, Existence and Ulam type stability for nonlinear Riemann-Liouville fractional differential equations with constant delay, Electron. J. Qual. Theory Differ. Equ. 67 (2020), 1–18. 10.14232/ejqtde.2020.1.67Search in Google Scholar

[5] R. Agarwal, S. Hristova, and D. O’Regan, Ulam type stability for non-instantaneous impulsive Caputo fractional differential equations with finite state dependent delay, Georgian Math. J. 28 (2021), no. 4, 499–517. 10.1515/gmj-2020-2061Search in Google Scholar

[6] R. Agarwal, D. O’Regan, and S. Hristova, Stability of Caputo fractional differential equations by Lyapunov functions, Appl. Math. 60 (2015), 653–676. 10.1007/s10492-015-0116-4Search in Google Scholar

[7] M. Benchohra, S. Bouriah, and J. Henderson, Existence and stability results for nonlinear implicit neutral fractional differential equations with finite delay and impulses, Comm. Appl. Nonlinear Anal. 22 (2015), no. 1, 46–67. 10.7603/s40956-015-0002-9Search in Google Scholar

[8] X. Dong, Z. Bai, and S. Zhang, Positive solutions to boundary value problems of p-Laplacian with fractional derivative, Bound. Value Probl. 5 (2017), 1–15. 10.1186/s13661-016-0735-zSearch in Google Scholar

[9] M. Jleli, D. O’Regan, and B. Samet, Lyapunov-type inequalities for coupled systems of nonlinear fractional differential equations via a fixed point approach, J. Fixed Point Theory Appl. 21 (2019), 45. 10.1007/s11784-019-0683-1Search in Google Scholar

[10] K. Lan and W. Lin, Positive solutions of systems of Caputo fractional differential equations, Comm. Appl. Anal. 17 (2013), no. 1, 61–86. Search in Google Scholar

[11] F. Wang, L. Zhang, Y. Ru, and R. Liu, On positive solutions of fractional differential equations with change of sign, Math. Methods Appl. Sci. 42 (2019), no. 11, 3900–3911. 10.1002/mma.5619Search in Google Scholar

[12] W. Yukunthorn, B. Ahmad, S. K. Ntouyas, and J. Tariboon, On Caputo-Hadamard type fractional impulsive hybrid systems with nonlinear fractional integral conditions, Nonlinear Anal. Hybrid Syst. 19 (2016), 77–92. 10.1016/j.nahs.2015.08.001Search in Google Scholar

[13] H. Zhu, B. Han, and J. Shen, Some results on fractional m-point boundary value problems, J. Funct. Spaces 2021 (2021), 3152688. 10.1155/2021/3152688Search in Google Scholar

[14] K. Perera, M. Squassina, and Y. Yang, A note on the Dancer-Fucík spectra of the fractional p-Laplacian and Laplacian operators, Adv. Nonlinear Anal. 4 (2015), 13–23. 10.1515/anona-2014-0038Search in Google Scholar

[15] K. Perera, M. Squassina, and Y. Yang, Bifurcation and multiplicity results for critical fractional p-Laplacian problems, Math. Nachr. 289 (2016), no. 2–3, 332–342. 10.1002/mana.201400259Search in Google Scholar

[16] P. Pucci, M. Xiang, and B. Zhang, Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations, Adv. Nonlinear Anal. 5 (2016), 27–55. 10.1515/anona-2015-0102Search in Google Scholar

[17] K. Dhawan, R. K. Vats, and V. Vijayakumar, Analysis of neutral fractional differential equation via the method of upper and lower solution, Qual. Theory Dyn. Syst. 22 (2023), 93. 10.1007/s12346-023-00795-ySearch in Google Scholar

[18] X. Zhang, D. Kong, H. Tian, Y. Wu, and B. Wiwatanapataphee, An upper-lower solution method for the eigenvalue problem of Hadamard-type singular fractional differential equation, Nonlinear Anal. Model. Control 27 (2022), no. 4, 789–802. 10.15388/namc.2022.27.27491Search in Google Scholar

[19] Z. Liu, J. Sun, and I. Szanto, Monotone iterative technique for Riemann-Liouville fractional integro-differential equations with advanced arguments, Results Math. 63 (2013), 1277–1287. 10.1007/s00025-012-0268-4Search in Google Scholar

[20] X. M. Zhang, A new method for searching the integral solution of system of Riemann-Liouville fractional differential equations with non-instantaneous impulses, J. Comput. Appl. Math. 388 (2021), 113307. 10.1016/j.cam.2020.113307Search in Google Scholar

[21] L. Zhang, B. Ahmad, and G. Wang, Analysis and application for a diffusion equations with a new fractional derivative without singular kernel, Electron. J. Differential Equations 2017 (2017), no. 289, 1–6. 10.1186/s13662-017-1356-2Search in Google Scholar

[22] X. Zhang, L. Liu, and Y. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett. 37 (2014), 26–33. 10.1016/j.aml.2014.05.002Search in Google Scholar

[23] J. Hadamard, Essai sur letude des fonctions donnees par leur developpement de Taylor, J. Math. Pures Appl. 8 (1892), 101–186. Search in Google Scholar

[24] B. Ahmad and S. K. Ntouyas, On three-point Hadamard-type fractional boundary value problems, Int. Electron. J. Pure. Appl. Math. 8 (2014), no. 4, 31–42. 10.1186/1029-242X-2014-454Search in Google Scholar

[25] P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Compositions of Hadamard-type fractional integration operators and the semigroup property, J. Math. Anal. Appl. 269 (2002), 387–400. 10.1016/S0022-247X(02)00049-5Search in Google Scholar

[26] P. L. Butzer, A. A. Kilbas, and J. J. Trujillo, Fractional calculus in the Mellin setting and Hadamard-type fractional integrals, J. Math. Anal. Appl. 269 (2002), 1–27. 10.1016/S0022-247X(02)00001-XSearch in Google Scholar

[27] Y. Bai and H. Kong, Existence of solutions for nonlinear Caputo-Hadamard fractional differential equations via the method of upper and lower solutions, J. Nonlinear Sci. Appl. 10 (2017), 5744–5752. 10.22436/jnsa.010.11.12Search in Google Scholar

[28] M. Benchohra, S. Bouriah, and J. R. Graef, Boundary value problems for nonlinear implicit Caputo-Hadamard type fractional differential equations with impulses, Mediterr. J. Math. 14 (2017), 206. 10.1007/s00009-017-1012-9Search in Google Scholar

[29] 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. Search in Google Scholar

[30] G. Wang, K. Pei, R. P. Agrawal, L. Zhang, and B. Ahmad, Nonlocal Hadamard fractional boundary value problem with Hadamard integral and discrete boundary conditions on a half-line, J. Comput. Appl. Math. 343 (2018), 230–239. 10.1016/j.cam.2018.04.062Search in Google Scholar

[31] X. Zhang, L. Yu, J. Jiang, Y. Wu, and Y. Cui, Positive solutions for a weakly singular Hadamard-type fractional differential equation with changing-sign nonlinearity, J. Funct. Spaces 2020 (2020), 5623589. 10.1155/2020/5623589Search in Google Scholar

[32] X. Zhang, P. Xu, Y. Wu, and B. Wiwatanapataphee, The uniqueness and iterative properties of solutions for a general Hadamard-type singular fractional turbulent flow model, Nonlinear Anal. Model. Control 27 (2022), no. 3, 428–444. 10.15388/namc.2022.27.25473Search in Google Scholar

[33] D. Guo, Fixed points of mixed monotone operators with applications, Appl. Anal. 31 (1988), no. 3, 215–224. 10.1080/00036818808839825Search in Google Scholar
Received: 2023-02-20
Revised: 2023-09-19
Accepted: 2023-09-20
Published Online: 2023-10-20

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

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

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  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
https://doi.org/10.1515/math-2023-0131
Keywords for this article
Hadamard fractional boundary value problem; existence and uniqueness; positive solutions; fixed point theorem; Ulam-Hyers stability
Creative Commons
BY 4.0

Articles in the same Issue

  1. Special Issue on Future Directions of Further Developments in Mathematics
  2. What will the mathematics of tomorrow look like?
  3. On H 2-solutions for a Camassa-Holm type equation
  4. Classical solutions to Cauchy problems for parabolic–elliptic systems of Keller-Segel type
  5. Control of multi-agent systems: Results, open problems, and applications
  6. Logical perspectives on the foundations of probability
  7. Subharmonic solutions for a class of predator-prey models with degenerate weights in periodic environments
  8. A non-smooth Brezis-Oswald uniqueness result
  9. Luenberger compensator theory for heat-Kelvin-Voigt-damped-structure interaction models with interface/boundary feedback controls
  10. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part II)
  11. Positive solution for a nonlocal problem with strong singular nonlinearity
  12. Analysis of solutions for the fractional differential equation with Hadamard-type
  13. Hilfer proportional nonlocal fractional integro-multipoint boundary value problems
  14. A comprehensive review on fractional-order optimal control problem and its solution
  15. The θ-derivative as unifying framework of a class of derivatives
  16. Review Articles
  17. On the use of L-functionals in regression models
  18. Minimal-time problems for linear control systems on homogeneous spaces of low-dimensional solvable nonnilpotent Lie groups
  19. Regular Articles
  20. Existence and multiplicity of solutions for a new p(x)-Kirchhoff problem with variable exponents
  21. An extension of the Hermite-Hadamard inequality for a power of a convex function
  22. Existence and multiplicity of solutions for a fourth-order differential system with instantaneous and non-instantaneous impulses
  23. Relay fusion frames in Banach spaces
  24. Refined ratio monotonicity of the coordinator polynomials of the root lattice of type Bn
  25. On the uniqueness of limit cycles for generalized Liénard systems
  26. A derivative-Hilbert operator acting on Dirichlet spaces
  27. Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines
  28. Solutions to a modified gauged Schrödinger equation with Choquard type nonlinearity
  29. A symbolic approach to multiple Hurwitz zeta values at non-positive integers
  30. Some results on the value distribution of differential polynomials
  31. Lucas non-Wieferich primes in arithmetic progressions and the abc conjecture
  32. Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point
  33. Some results for a p(x)-Kirchhoff type variation-inequality problems in non-divergence form
  34. Homotopy cartesian squares in extriangulated categories
  35. A unified perspective on some autocorrelation measures in different fields: A note
  36. Total Roman domination on the digraphs
  37. Well-posedness for bilevel vector equilibrium problems with variable domination structures
  38. Binet's second formula, Hermite's generalization, and two related identities
  39. Non-solid cone b-metric spaces over Banach algebras and fixed point results of contractions with vector-valued coefficients
  40. Multidimensional sampling-Kantorovich operators in BV-spaces
  41. A self-adaptive inertial extragradient method for a class of split pseudomonotone variational inequality problems
  42. Convergence properties for coordinatewise asymptotically negatively associated random vectors in Hilbert space
  43. Relating the super domination and 2-domination numbers in cactus graphs
  44. Compatibility of the method of brackets with classical integration rules
  45. On the inverse Collatz-Sinogowitz irregularity problem
  46. Positive solutions for boundary value problems of a class of second-order differential equation system
  47. Global analysis and control for a vector-borne epidemic model with multi-edge infection on complex networks
  48. Nonexistence of global solutions to Klein-Gordon equations with variable coefficients power-type nonlinearities
  49. On 2r-ideals in commutative rings with zero-divisors
  50. A comparison of some confidence intervals for a binomial proportion based on a shrinkage estimator
  51. The construction of nuclei for normal constituents of Bπ-characters
  52. Weak solution of non-Newtonian polytropic variational inequality in fresh agricultural product supply chain problem
  53. Mean square exponential stability of stochastic function differential equations in the G-framework
  54. Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents
  55. Solitons for the coupled matrix nonlinear Schrödinger-type equations and the related Schrödinger flow
  56. The dual index and dual core generalized inverse
  57. Study on Birkhoff orthogonality and symmetry of matrix operators
  58. Uniqueness theorems of the Hahn difference operator of entire function with a Picard exceptional value
  59. Estimates for certain class of rough generalized Marcinkiewicz functions along submanifolds
  60. On semigroups of transformations that preserve a double direction equivalence
  61. Positive solutions for discrete Minkowski curvature systems of the Lane-Emden type
  62. A multigrid discretization scheme based on the shifted inverse iteration for the Steklov eigenvalue problem in inverse scattering
  63. Existence and nonexistence of solutions for elliptic problems with multiple critical exponents
  64. Interpolation inequalities in generalized Orlicz-Sobolev spaces and applications
  65. General Randić indices of a graph and its line graph
  66. On functional reproducing kernels
  67. On the Waring-Goldbach problem for two squares and four cubes
  68. Singular moduli of rth Roots of modular functions
  69. Classification of self-adjoint domains of odd-order differential operators with matrix theory
  70. On the convergence, stability and data dependence results of the JK iteration process in Banach spaces
  71. Hardy spaces associated with some anisotropic mixed-norm Herz spaces and their applications
  72. Remarks on hyponormal Toeplitz operators with nonharmonic symbols
  73. Complete decomposition of the generalized quaternion groups
  74. Injective and coherent endomorphism rings relative to some matrices
  75. Finite spectrum of fourth-order boundary value problems with boundary and transmission conditions dependent on the spectral parameter
  76. Continued fractions related to a group of linear fractional transformations
  77. Multiplicity of solutions for a class of critical Schrödinger-Poisson systems on the Heisenberg group
  78. Approximate controllability for a stochastic elastic system with structural damping and infinite delay
  79. On extremal cacti with respect to the first degree-based entropy
  80. Compression with wildcards: All exact or all minimal hitting sets
  81. Existence and multiplicity of solutions for a class of p-Kirchhoff-type equation RN
  82. Geometric classifications of k-almost Ricci solitons admitting paracontact metrices
  83. Positive periodic solutions for discrete time-delay hematopoiesis model with impulses
  84. On Hermite-Hadamard-type inequalities for systems of partial differential inequalities in the plane
  85. Existence of solutions for semilinear retarded equations with non-instantaneous impulses, non-local conditions, and infinite delay
  86. On the quadratic residues and their distribution properties
  87. On average theta functions of certain quadratic forms as sums of Eisenstein series
  88. Connected component of positive solutions for one-dimensional p-Laplacian problem with a singular weight
  89. Some identities of degenerate harmonic and degenerate hyperharmonic numbers arising from umbral calculus
  90. Mean ergodic theorems for a sequence of nonexpansive mappings in complete CAT(0) spaces and its applications
  91. On some spaces via topological ideals
  92. Linear maps preserving equivalence or asymptotic equivalence on Banach space
  93. Well-posedness and stability analysis for Timoshenko beam system with Coleman-Gurtin's and Gurtin-Pipkin's thermal laws
  94. On a class of stochastic differential equations driven by the generalized stochastic mixed variational inequalities
  95. Entire solutions of two certain Fermat-type ordinary differential equations
  96. Generalized Lie n-derivations on arbitrary triangular algebras
  97. Markov decision processes approximation with coupled dynamics via Markov deterministic control systems
  98. Notes on pseudodifferential operators commutators and Lipschitz functions
  99. On Graham partitions twisted by the Legendre symbol
  100. Strong limit of processes constructed from a renewal process
  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
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 19.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/math-2023-0131/html
Scroll to top button