Home On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
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On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions

  • Ahmed Alsaedi , Bashir Ahmad EMAIL logo , Badrah Alghamdi and Sotiris K. Ntouyas
Published/Copyright: August 5, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

We study a nonlinear system of Riemann-Liouville fractional differential equations equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We make use of the tools of the fixed-point theory to obtain the desired results, which are well-supported with numerical examples.

Keywords: Riemann-Liouville fractional derivative; integro-differential inclusions; nonlocal multi-point boundary conditions; existence; fixed point theorems
MSC 2010: 34A08; 34B15

1 Introduction

Nonlinear boundary value problems involving Riemann-Liouville fractional derivatives have been studied by many researchers, for example, see [1,2,3, 4,5,6]. In a recent article [7], the authors discussed the existence and Ulam-type stability for nonlinear Riemann-Liouville fractional differential equations with constant delay.

The nonlocal nature of fractional derivative operators significantly contributed to the popularity of fractional calculus. Nowadays, one can find extensive application of fractional-order operators in the mathematical models of several real-world phenomena occurring in physical and applied sciences, such as continuum mechanics [8], bioengineering [9], financial economics [10], fractals [11], etc.

The topic of fractional differential systems also received considerable attention in view of their applications in diverse fields such as anomalous diffusion [12], disease models [13] biological models [14], hybrid systems [15], rheological models [16], diffusion systems [17], ecological models [18], etc. For theoretical details of such systems, see [19,20,21, 22,23,24, 25,26,27, 28,29].

In this paper, we investigate the existence of solutions for a system of nonlinear Riemann-Liouville fractional differential equations

(1) D α u ( t ) = F ( t , u ( t ) , v ( t ) ) , 1 < α ≤ 2 , t ∈ [ 0 , T ] , D β v ( t ) = G ( t , u ( t ) , v ( t ) ) , 1 < β ≤ 2 , t ∈ [ 0 , T ] ,

equipped with nonlocal semi-coupled fractional integro-multipoint boundary conditions of the form:

(2) D α − 2 u ( 0 + ) + a 0 D α − 2 u ( T − ) = λ 1 , D α − 1 u ( 0 + ) + a 1 D α − 1 u ( T − ) = ν I α − 1 v ( η 1 ) + ∑ i = 1 m μ i v ( ξ i ) , D β − 2 v ( 0 + ) + b 0 D β − 2 v ( T − ) = λ 2 , D β − 1 v ( 0 + ) + b 1 D β − 1 v ( T − ) = μ I β − 1 u ( η 2 ) + ∑ j = 1 n σ j u ( ζ j ) ,

where D χ is the Riemann-Liouville fractional derivative of order χ ∈ { α , β } , 0 < η 1 < η 2 < ξ 1 < ξ 2 < … < ξ m < ζ 1 < ζ 2 < … < ζ m < T , a 0 , a 1 , b 0 , b 1 , ν , μ , λ 1 , λ 2 , μ i ( i = 1 , 2 , … , m ) , σ j ( j = 1 , 2 , … , n ) are real constants and a 0 ≠ − 1 , a 1 ≠ − 1 , b 0 ≠ − 1 , and F , G : [ 0 , T ] × R × R ⟶ R are continuous functions.

Here we emphasize that our results are new with respect to the semi-coupled boundary conditions (2) and enrich the related literature on the topic.

In Section 2, we prove an auxiliary lemma dealing with the linear variant of systems (1)–(2), which plays a fundamental role in establishing the existence and uniqueness results for the given nonlinear problem, presented in Section 3. Illustrative examples demonstrating the application of the obtained results are given in Section 4.

2 Preliminaries

Let us begin this section with some related definitions [30,31].

Definition 2.1

The (left) Riemann-Liouville fractional integral of order χ > 0 for ϱ ∈ L 1 [ a , b ] , − ∞ < a ≤ t ≤ b < + ∞ , existing almost everywhere on [ a , b ] , is defined as

( I a + χ ϱ ) ( t ) = 1 Γ ( χ ) ∫ a t ( t − s ) χ − 1 ϱ ( s ) d s , χ > 0 ,

where Γ ( ⋅ ) is the gamma function and ( I a + 0 ϱ ) ( x ) = ϱ ( x ) .

Definition 2.2

For ϱ , ϱ ( m ) ∈ L 1 [ a , b ] , the (left) Riemann-Liouville fractional derivative D a + χ ϱ of order χ ∈ ( m − 1 , m ] , m ∈ N is defined as

D a + χ ϱ ( t ) = 1 Γ ( m − χ ) d m d t m ∫ a t ( t − s ) m − 1 − χ ϱ ( s ) d s , − ∞ ≤ a < t < b ≤ + ∞ .

In the sequel, we will write Riemann-Liouville fractional integral and derivative operators as I χ and D χ instead of I a + χ and D a + χ , respectively.

Lemma 2.1

Let ψ 1 , ψ 2 ∈ C [ 0 , T ] ∩ L [ 0 , T ] and Λ ≠ 0 . Then the unique solution of the following linear system

(3) D α u ( t ) = ψ 1 ( t ) , 1 < α ≤ 2 , t ∈ [ 0 , T ] , D β v ( t ) = ψ 2 ( t ) , 1 < β ≤ 2 , t ∈ [ 0 , T ] ,

subject to the boundary conditions (2) is given by

(4) u ( t ) = λ 1 ρ 1 ( t ) + λ 2 ω 1 ρ 2 ( t ) − a 1 ρ 2 ( t ) I 1 ψ 1 ( T ) − b 1 ρ 3 ( t ) I 1 ψ 2 ( T ) − a 0 ρ 1 ( t ) I 2 ψ 1 ( T ) − b 0 ω 1 ρ 2 ( t ) I 2 ψ 2 ( T ) + μ ρ 3 ( t ) I α + β − 1 ψ 1 ( η 2 ) + ν ρ 2 ( t ) I α + β − 1 ψ 2 ( η 1 ) + ρ 3 ( t ) ∑ j = 1 n σ j I α ψ 1 ( ζ j ) + ρ 2 ( t ) ∑ i = 1 m μ i I β ψ 2 ( ξ i ) + I α ψ 1 ( t ) ,

(5) v ( t ) = λ 2 ρ 1 ∗ ( t ) + λ 1 ν 1 ρ 2 ∗ ( t ) − a 1 ρ 3 ∗ ( t ) I 1 ψ 1 ( T ) − b 1 ρ 2 ∗ ( t ) I 1 ψ 2 ( T ) − a 0 ν 1 ρ 2 ∗ ( t ) I 2 ψ 1 ( T ) − b 0 ρ 1 ∗ ( t ) I 2 ψ 2 ( T ) + μ ρ 2 ∗ ( t ) I α + β − 1 ψ 1 ( η 2 ) + ν ρ 3 ∗ ( t ) I α + β − 1 ψ 2 ( η 1 ) + ρ 2 ∗ ( t ) ∑ j = 1 n σ j I α ψ 1 ( ζ j ) + ρ 3 ∗ ( t ) ∑ i = 1 m μ i I β ψ 2 ( ξ i ) + I β ψ 2 ( t ) ,

where

ρ 1 ( t ) = t α − 2 1 ( 1 + a 0 ) Γ ( α − 1 ) + ν 1 Λ a 0 T Γ ( α ) ( 1 + a 0 ) Γ ( α − 1 ) ω 2 − b 0 T Γ ( β ) ( 1 + a 1 ) Γ ( α ) ω 1 − t α − 1 ν 1 Λ ω 2 − b 0 T Γ ( β ) ( 1 + a 1 ) Γ ( α ) ω 1 , ρ 2 ( t ) = 1 Λ t α − 2 a 0 T ( b 1 + 1 ) Γ ( β ) ( 1 + a 1 ) ( 1 + a 0 ) Γ ( α − 1 ) − t α − 1 ( 1 + b 1 ) Γ ( β ) ( 1 + a 1 ) Γ ( α ) , ρ 3 ( t ) = 1 Λ ω 2 − b 0 T Γ ( β ) ( 1 + a 1 ) Γ ( α ) ω 1 t α − 2 a 0 T Γ ( α ) ( 1 + a 0 ) Γ ( α − 1 ) − t α − 1 , ρ 1 ∗ ( t ) = t β − 2 1 ( 1 + b 0 ) Γ ( β − 1 ) + ω 1 Λ b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) ν 2 − a 0 T ( 1 + a 1 ) ν 1 − t β − 1 ω 1 Λ ν 2 − a 0 T ( 1 + a 1 ) ν 1 , ρ 2 ∗ ( t ) = 1 Λ t β − 2 b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) − t β − 1 , ρ 3 ∗ ( t ) = 1 Λ ν 2 − a 0 T ( 1 + a 1 ) ν 1 t β − 2 b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) − t β − 1 , Λ = − Γ ( β ) ( 1 + b 1 ) + ν 2 − a 0 T ( 1 + a 1 ) ν 1 [ Γ ( α ) ( 1 + a 1 ) ω 2 − b 0 T Γ ( β ) ω 1 ] , ν 1 = 1 ( 1 + a 0 ) Γ ( α − 1 ) μ η 2 α + β − 3 Γ ( α − 1 ) Γ ( α + β − 2 ) + ∑ j = 1 n σ j ζ j α − 2 , ν 2 = 1 ( 1 + a 1 ) Γ ( α ) μ η 2 α + β − 2 Γ ( α ) Γ ( α + β − 1 ) + ∑ j = 1 n σ j ζ j α − 1 , ω 1 = 1 ( 1 + b 0 ) Γ ( β − 1 ) ν η 1 α + β − 3 Γ ( β − 1 ) Γ ( α + β − 2 ) + ∑ i = 1 m μ i ξ i β − 2 , ω 2 = 1 ( 1 + a 1 ) Γ ( α ) ν η 1 α + β − 2 Γ ( β ) Γ ( α + β − 1 ) + ∑ i = 1 m μ i ξ i β − 1 .

Proof

It is well known that the solutions of fractional differential equations in (3) can be written as

(7) u ( t ) = I α ψ 1 ( t ) + c 0 t α − 2 + c 1 t α − 1 ,

(8) v ( t ) = I β ψ 2 ( t ) + c 2 t β − 2 + c 3 t β − 1 ,

where c i ∈ R , i = 0 , 1 , 2 , 3 are unknown arbitrary constants. From (7) and (8), we have

(9) D α − 2 u ( t ) = c 0 Γ ( α − 1 ) + c 1 Γ ( α ) t + I 2 ψ 1 ( t ) ,

(10) D β − 2 v ( t ) = c 2 Γ ( β − 1 ) + c 3 Γ ( β ) t + I 2 ψ 2 ( t ) ,

(11) D α − 1 u ( t ) = c 1 Γ ( α ) + I 1 ψ 1 ( t ) ,

(12) D β − 1 v ( t ) = c 3 Γ ( β ) + I 1 ψ 2 ( t ) .

Using (9) and (10) in (2), we get

(13) ( 1 + a 0 ) Γ ( α − 1 ) c 0 + a 0 T Γ ( α ) c 1 = λ 1 − a 0 I 2 ψ 1 ( T ) ,

(14) ( 1 + b 0 ) Γ ( β − 1 ) c 2 + b 0 T Γ ( β ) c 3 = λ 2 − b 0 I 2 ψ 2 ( T ) .

Combining (11) and (12) with (2), we obtain

(15) Γ ( α ) ( 1 + a 1 ) c 1 − ν η 1 α + β − 2 Γ ( β ) Γ ( α + β − 1 ) + ∑ i = 1 m μ i ξ i β − 1 − b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) ν η 1 α + β − 3 Γ ( β − 1 ) Γ ( α + β − 2 ) + ∑ i = 1 m μ i ξ i β − 2 c 3 = − b 0 ( 1 + b 0 ) Γ ( β − 1 ) ν η 1 α + β − 3 Γ ( β − 1 ) Γ ( α + β − 2 ) + ∑ i = 1 m μ i ξ i β − 2 I 2 ψ 2 ( t ) + ∑ i = 1 m μ i I β ψ 2 ( ξ i ) − a 1 I 1 ψ 1 ( T ) + λ 2 ( 1 + b 0 ) Γ ( β − 1 ) ν η 1 α + β − 3 Γ ( β − 1 ) Γ ( α + β − 2 ) + ∑ i = 1 m μ i ξ i β − 2 + ν I α + β − 1 ψ 2 ( η 1 ) ,

(16) − μ η 2 α + β − 2 Γ ( α ) Γ ( α + β − 1 ) + ∑ j = 1 n σ j ζ j α − 1 c 1 + Γ ( β ) ( 1 + b 1 ) c 3 = ∑ j = 1 n σ j I α ψ 1 ( ζ j ) − b 1 I 1 ψ 2 ( T ) + 1 ( 1 + a 0 ) Γ ( α − 1 ) μ η 2 α + β − 3 Γ ( α − 1 ) Γ ( α + β − 2 ) + ∑ j = 1 n σ j ζ j α − 2 × [ λ 1 − a 0 c 1 T Γ ( α ) − a 0 I 2 ψ 1 ( t ) ] + μ I α + β − 1 ψ 1 ( η 2 ) .

Solving (15) and (16) for c 1 and c 3 , we find that

(17) c 1 = 1 Λ ( b 1 + 1 ) Γ ( β ) ( 1 + a 1 ) Γ ( α ) − ω 1 λ 2 + b 0 ω 1 I 2 ψ 2 ( T ) − ν I α + β − 1 ψ 2 ( η 1 ) − ∑ i = 1 m μ i I β ψ 2 ( ξ i ) + a 1 I 1 ψ 1 ( T ) + 1 Λ ω 2 − b 0 T Γ ( β ) ( 1 + a 1 ) Γ ( α ) ω 1 − λ 1 ν 1 − μ I α + β − 1 ψ 1 ( η 2 ) − ∑ j = 1 n σ j I α ψ 1 ( ζ j ) + b 1 I 1 ψ 2 ( T ) + a 0 ν 1 I 2 ψ 1 ( T ) ,

(18) c 3 = 1 Λ − λ 1 ν 1 − λ 2 ω 1 ν 2 − a 0 T ( 1 + a 1 ) ν 1 + a 0 ν 1 I 2 ψ 1 ( t ) + b 0 ω 1 ν 2 − a 0 T ( 1 + a 1 ) ν 1 I 2 ψ 2 ( T ) − ν ν 2 − a 0 T ( 1 + a 1 ) ν 1 I α + β − 1 ψ 2 ( η 1 ) − μ I α + β − 1 ψ 1 ( η 2 ) − ν 2 − a 0 T ( 1 + a 1 ) ν 1 ∑ i = 1 m μ i I β ψ 2 ( ξ i ) − ∑ j = 1 n σ j I α ψ 1 ( ζ j ) + a 1 ν 2 − a 0 T ( 1 + a 1 ) ν 1 I 1 ψ 1 ( T ) + b 1 I 1 ψ 2 ( T ) .

Substituting the values of c 1 and c 3 in (13) and (14), respectively, we get

c 0 = [ λ 1 − a 0 I 2 ψ 1 ( T ) ] 1 ( 1 + a 0 ) Γ ( α − 1 ) + ν 1 Λ a 0 T Γ ( α ) ( 1 + a 0 ) Γ ( α − 1 ) ω 2 − b 0 T Γ ( β ) ( 1 + a 1 ) Γ ( α ) ω 1 + λ 2 ω 1 Λ a 0 ( 1 + b 1 ) T ∣ Γ ( β ) ( 1 + a 0 ) ( 1 + a 1 ) Γ ( α − 1 ) − I 2 ψ 2 ( T ) ω 1 Λ a 0 b 0 ( 1 + b 1 ) T Γ ( β ) ( 1 + a 0 ) ( 1 + a 1 ) Γ ( α − 1 ) + I α + β − 1 ψ 2 ( η 1 ) ν Λ a 0 ( 1 + b 1 ) T Γ ( β ) ( 1 + a 0 ) ( 1 + a 1 ) Γ ( α − 1 ) + ∑ i = 1 m μ i I β ψ 2 ( ξ i ) 1 Λ a 0 ( 1 + b 1 ) T Γ ( β ) ( 1 + a 0 ) ( 1 + a 1 ) Γ ( α − 1 ) − I 1 ψ 1 ( T ) 1 Λ a 0 a 1 ( 1 + b 1 ) T Γ ( β ) ( 1 + a 0 ) ( 1 + a 1 ) Γ ( α − 1 ) + I α + β − 1 ψ 1 ( η 2 ) μ Λ a 0 T Γ ( α ) ( 1 + a 0 ) Γ ( α − 1 ) ω 2 − b 0 T Γ ( β ) ( 1 + a 1 ) Γ ( α ) ω 1 + ∑ j = 1 n σ j I α ψ 1 ( ζ j ) 1 Λ a 0 T Γ ( α ) ( 1 + a 0 ) Γ ( α − 1 ) ω 2 − b 0 T Γ ( β ) ( 1 + a 1 ) Γ ( α ) ω 1 + I 1 ψ 2 ( T ) − b 1 Λ a 0 T Γ ( α ) ( 1 + a 0 ) Γ ( α − 1 ) ω 2 − b 0 T Γ ( β ) ( 1 + a 1 ) Γ ( α ) ω 1 ,

c 2 = [ λ 2 − b 0 I 2 ψ 2 ( T ) ] 1 ( 1 + b 0 ) Γ ( β − 1 ) + ω 1 Λ b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) ν 2 − a 0 T ( 1 + b 1 ) ν 1 + λ 1 ν 1 Λ b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) − I 2 ψ 1 ( T ) ν 1 Λ a 0 b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) + I α + β − 1 ψ 2 ( η 1 ) ν Λ b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) ν 2 − a 0 T ( 1 + a 1 ) ν 1 + I α + β − 1 ψ 1 ( η 2 ) μ Λ b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) + ∑ i = 1 m μ i I β ψ 2 ( ξ i ) 1 Λ b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) ν 2 − a 0 T ( 1 + a 1 ) ν 1 − I 1 ψ 1 ( T ) 1 Λ b 0 a 1 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) ν 2 − a 0 T ( 1 + a 1 ) ν 1 + ∑ j = 1 n σ j I α ψ 1 ( ζ j ) 1 Λ b 0 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) − I 1 ψ 2 ( T ) 1 Λ b 0 b 1 T Γ ( β ) ( 1 + b 0 ) Γ ( β − 1 ) .

Inserting the value of c 0 , c 1 , c 2 , and c 3 in (7) and (8) together with the notations (6) leads to the solutions (4) and (5). The converse of this lemma follows by direct computation. The proof is completed.□

3 Main results

Let C ( [ 0 , T ] , R ) denote the Banach space of all continuous real-valued functions defined on [ 0 , T ] with norm ‖ u ‖ = sup t ∈ [ 0 , T ] ∣ u ( t ) ∣ . For t ∈ [ 0 , T ] , let C r ( [ 0 , T ] , R ) denote the space of all functions u r such that u r ∈ C ( [ 0 , T ] , R ) , which is a Banach space endowed with norm ‖ u ‖ r = sup t ∈ [ 0 , T ] { t r ∣ u ( t ) ∣ } . Let X = { u : u ∈ C 2 − α ( [ 0 , T ] , R ) } and Y = { v : v ∈ C 2 − β ( [ 0 , T ] , R ) } be equipped with the norm ‖ u ‖ X = sup t ∈ [ 0 , T ] { t 2 − α ∣ u ( t ) ∣ } and ‖ v ‖ Y = sup t ∈ [ 0 , T ] { t 2 − β ∣ v ( t ) ∣ } , respectively. Then the product space ( X × Y , ‖ ⋅ ‖ X × Y ) is a Banach space with the norm

‖ ( u , v ) ‖ X × Y = ‖ u ‖ X + ‖ v ‖ Y .

Next we introduce an operator P : X × Y → X × Y by

P ( u , v ) ( t ) = ( P 1 ( u , v ) ( t ) , P 2 ( u , v ) ( t ) ) ,

where

P 1 ( u , v ) ( t ) = λ 1 ρ 1 ( t ) + λ 2 ω 1 ρ 2 ( t ) − a 1 ρ 2 ( t ) I 1 F ( T , u ( T ) , v ( T ) ) − b 1 ρ 3 ( t ) I 1 G ( T , u ( T ) v ( T ) ) − a 0 ρ 1 ( t ) I 2 F ( T , u ( T ) , v ( T ) ) − b 0 ω 1 ρ 2 ( t ) I 2 G ( T , u ( T ) , v ( T ) ) + μ ρ 3 ( t ) I α + β − 1 F ( η 2 , u ( η 2 ) , v ( η 2 ) ) + ν ρ 2 ( t ) I α + β − 1 G ( η 1 , u ( η 1 ) , v ( η 1 ) ) + ρ 3 ( t ) ∑ j = 1 n σ j I α F ( ζ j , u ( ζ j ) , v ( ζ j ) ) + ρ 2 ( t ) ∑ i = 1 m μ i I β G ( ξ i , u ( ξ i ) , v ( ξ i ) ) + I α F ( t , u ( t ) , v ( t ) ) , t ∈ [ 0 , T ] ,

(19) P 2 ( u , v ) ( t ) = λ 2 ρ 1 ∗ ( t ) + λ 1 ν 1 ρ 2 ∗ ( t ) − a 1 ρ 3 ∗ ( t ) I 1 F ( T , u ( T ) , v ( T ) ) − b 1 ρ 2 ∗ ( t ) I 1 G ( T , u ( T ) , v ( T ) ) − a 0 ν 1 ρ 2 ∗ ( t ) I 2 F ( T , u ( T ) , v ( T ) ) − b 0 ρ 1 ∗ ( t ) I 2 G ( T , u ( T ) , v ( T ) ) + μ ρ 2 ∗ ( t ) I α + β − 1 F ( η 2 , u ( η 2 ) , v ( η 2 ) ) + ν ρ 3 ∗ ( t ) I α + β − 1 G ( η 1 , u ( η 1 ) , v ( η 1 ) ) + ρ 2 ∗ ( t ) ∑ j = 1 n σ j I α F ( ζ j , u ( ζ j ) , v ( ζ j ) ) + ρ 3 ∗ ( t ) ∑ i = 1 m μ i I β G ( ξ i , u ( ξ i ) , v ( ξ i ) ) + I β G ( t , u ( t ) , v ( t ) ) , t ∈ [ 0 , T ] .

For the sake of brevity, we set

(20) N 1 = ∣ μ ∣ δ 3 η 2 2 α + β − 3 Γ ( α − 1 ) Γ ( 2 α + β − 2 ) + η 2 α + 2 β − 3 Γ ( β − 1 ) Γ ( α + 2 β − 2 ) + ∣ a 0 ∣ δ 1 T α Γ ( α − 1 ) Γ ( α + 1 ) + T β Γ ( β − 1 ) Γ ( β + 1 ) + δ 3 ∑ j = 1 n ∣ σ j ∣ ζ j 2 α − 2 Γ ( α − 1 ) Γ ( 2 α − 1 ) + ζ j α + β − 2 Γ ( β − 1 ) Γ ( α + β − 1 ) + ∣ a 1 ∣ δ 2 T α − 1 ( α − 1 ) + T β − 1 ( β − 1 ) + T α Γ ( α − 1 ) Γ ( 2 α − 1 ) + T β Γ ( β − 1 ) Γ ( α + β − 1 ) ,

(21) N 2 = ∣ ν ∣ δ 2 η 1 2 α + β − 3 Γ ( α − 1 ) Γ ( 2 α + β − 2 ) + η 1 α + 2 β − 3 Γ ( β − 1 ) Γ ( α + 2 β − 2 ) + ∣ b 0 ∣ ∣ ω 1 ∣ δ 2 T α Γ ( α − 1 ) Γ ( α + 1 ) + T β Γ ( β − 1 ) Γ ( β + 1 ) + δ 2 ∑ i = 1 m ∣ μ i ∣ ξ i α + β − 2 Γ ( α − 1 ) Γ ( α + β − 1 ) + ξ i 2 β − 2 Γ ( β − 1 ) Γ ( 2 β − 1 ) + ∣ b 1 ∣ δ 3 T α − 1 α − 1 + T β − 1 β − 1 ,

(22) N 1 ∗ = ∣ μ ∣ δ 2 ∗ η 2 2 α + β − 3 Γ ( α − 1 ) Γ ( 2 α + β − 2 ) + η 2 α + 2 β − 3 Γ ( β − 1 ) Γ ( α + 2 β − 2 ) + ∣ a 0 ν 1 ∣ δ 2 ∗ T α Γ ( α − 1 ) Γ ( α + 1 ) + T β Γ ( β − 1 ) Γ ( β + 1 ) + δ 2 ∗ ∑ j = 1 n ∣ σ j ∣ ζ j 2 α − 2 Γ ( α − 1 ) Γ ( 2 α − 1 ) + ζ j α + β − 2 Γ ( β − 1 ) Γ ( α + β − 1 ) + ∣ a 1 ∣ δ 3 ∗ T α − 1 ( α − 1 ) + T β − 1 ( β − 1 ) ,

(23) N 2 ∗ = ∣ ν ∣ δ 3 ∗ η 1 2 α + β − 3 Γ ( α − 1 ) Γ ( 2 α + β − 2 ) + η 1 α + 2 β − 3 Γ ( β − 1 ) Γ ( α + 2 β − 2 ) + ∣ b 0 ∣ δ 1 ∗ T α Γ ( α − 1 ) Γ ( α + 1 ) + T β Γ ( β − 1 ) Γ ( β + 1 ) + δ 3 ∗ ∑ i = 1 m ∣ μ i ∣ ξ i α + β − 2 Γ ( α − 1 ) Γ ( α + β − 1 ) + ξ i 2 β − 2 Γ ( β − 1 ) Γ ( 2 β − 1 ) + ∣ b 1 ∣ δ 2 ∗ T α − 1 α − 1 + T β − 1 β − 1 + T α Γ ( α − 1 ) Γ ( α + β − 1 ) + T β Γ ( β − 1 ) Γ ( 2 β − 1 ) .

Theorem 3.1

Assume that:

  1. F , G : [ 0 , T ] × R × R → R are continuous functions and there exist positive constants L 1 and L 2 such that, for all t ∈ [ 0 , T ] and u i , v i ∈ R , i = 1 , 2 ,

    ∣ F ( t , u 1 , v 1 ) − F ( t , u 2 , v 2 ) ∣ ≤ L 1 ( ∣ u 1 − u 2 ∣ + ∣ v 1 − v 2 ∣ ) ,

    ∣ G ( t , u 1 , v 1 ) − G ( t , u 2 , v 2 ) ∣ ≤ L 2 ( ∣ u 1 − u 2 ∣ + ∣ v 1 − v 2 ∣ ) .

Then the system (1)–(2) has a unique solution on [ 0 , T ] , provided that

(24) L 1 ( N 1 + N 1 ∗ ) + L 2 ( N 2 + N 2 ∗ ) < 1 ,

where, N 1 , N 2 , N 1 ∗ , and N 2 ∗ are, respectively, given by (20), (21), (22), and (23).

Proof

Define sup t ∈ [ 0 , T ] F ( t , 0 , 0 ) = M 1 , sup t ∈ [ 0 , T ] G ( t , 0 , 0 ) = M 2 , and choose r > 0 such that

(25) r ≥ ∣ λ 1 ∣ ( δ 1 + δ 2 ∗ ∣ ν 1 ∣ ) + ∣ λ 2 ∣ ( ∣ ω 1 ∣ δ 2 + δ 1 ∗ ) + M 1 ( e 1 + e 1 ∗ ) + M 2 ( e 2 + e 2 ∗ ) 1 − [ L 1 ( N 1 + N 1 ∗ ) + L 2 ( N 2 + N 2 ∗ ) ] ,

where

(26) e 1 = ∣ a 0 ∣ δ 1 T 2 2 + ∣ a 1 ∣ δ 2 T + ∣ μ ∣ δ 3 η 2 α + β − 1 Γ ( α + β ) + δ 3 ∑ j = 1 n ∣ σ j ∣ ζ j α Γ ( α + 1 ) + T 2 Γ ( α + 1 ) , e 2 = ∣ b 1 ∣ δ 3 T + ∣ ν ∣ δ 2 η 1 α + β − 1 Γ ( α + β ) + δ 2 ∑ i = 1 m ∣ μ i ∣ ξ i β Γ ( β + 1 ) + ∣ b 0 ω 1 ∣ δ 2 T 2 2 ,

e 1 ∗ = ∣ a 0 ν 1 ∣ δ 2 ∗ T 2 2 + ∣ a 1 ∣ δ 3 ∗ T + ∣ μ ∣ δ 2 ∗ η 2 α + β − 1 Γ ( α + β ) + δ 2 ∗ ∑ j = 1 n ∣ σ j ∣ ζ j α Γ ( α + 1 ) , e 2 ∗ = ∣ b 1 ∣ δ 2 ∗ T + ∣ ν ∣ δ 3 ∗ η 1 α + β − 1 Γ ( α + β ) + δ 3 ∗ ∑ i = 1 m ∣ μ i ∣ ξ i β Γ ( β + 1 ) + ∣ b 0 ∣ δ 1 ∗ T 2 2 + T 2 Γ ( β + 1 ) ,

(27) δ m = sup t ∈ [ 0 , T ] { t 2 − α ∣ ρ m ( t ) ∣ } , δ m ∗ = sup t ∈ [ 0 , T ] { t 2 − β ∣ ρ m ∗ ( t ) ∣ } , m = 1 , 2 , 3 .

In the first step, it will be shown that P B r ⊂ B r , where B r = { ( u , v ) ∈ X × Y : ‖ ( u , v ) ‖ X × Y ≤ r } . By the assumption ( A 1 ) , for ( u , v ) ∈ B r , t ∈ [ 0 , T ] , we have

∣ F ( t , u ( t ) , v ( t ) ) ∣ ≤ ∣ F ( t , u ( t ) , v ( t ) ) − F ( t , 0 , 0 ) ∣ + ∣ F ( t , 0 , 0 ) ∣ ≤ L 1 ( ∣ u ∣ + ∣ v ∣ ) + M 1

and

∣ G ( t , u ( t ) , v ( t ) ) ∣ ≤ ∣ G ( t , u ( t ) , v ( t ) ) − G ( t , 0 , 0 ) ∣ + ∣ G ( t , 0 , 0 ) ∣ ≤ L 2 ( ∣ u ∣ + ∣ v ∣ ) + M 2 .

In consequence, by using the relation for beta function B ( ⋅ , ⋅ ) :

B ( a , b ) = ∫ 0 1 u a − 1 ( 1 − u ) b − 1 d u = Γ ( a ) Γ ( b ) Γ ( a + b ) ,

we obtain

‖ P 1 ( u , v ) ‖ X = sup t ∈ [ 0 , T ] { t 2 − α ∣ P 1 ( u , v ) ( t ) ∣ } ≤ sup t ∈ [ 0 , T ] t 2 − α ∣ λ 1 ∣ ∣ ρ 1 ( t ) ∣ + ∣ λ 2 ω 1 ∣ ∣ ρ 2 ( t ) ∣ + ∣ a 0 ∣ ∣ ρ 1 ( t ) ∣ ∫ 0 T ( T − s ) Γ ( 2 ) ( L 1 ( ∣ u ∣ + ∣ v ∣ ) + M 1 ) d s + ∣ a 1 ∣ ∣ ρ 2 ( t ) ∣ ∫ 0 T ( L 1 ( ∣ u ∣ + ∣ v ∣ ) + M 1 ) d s + ∣ b 1 ∣ ∣ ρ 3 ( t ) ∣ ∫ 0 T ( L 2 ( ∣ u ∣ + ∣ v ∣ ) + M 2 ) d s + ∣ ν ∣ ∣ ρ 2 ( t ) ∣ ∫ 0 η 1 ( η 1 − s ) α + β − 2 Γ ( α + β − 1 ) ( L 2 ( ∣ u ∣ + ∣ v ∣ ) + M 2 ) d s + ∣ μ ∣ ∣ ρ 3 ( t ) ∣ ∫ 0 η 2 ( η 2 − s ) α + β − 2 Γ ( α + β − 1 ) ( L 1 ( ∣ u ∣ + ∣ v ∣ ) + M 1 ) d s + ∣ ρ 2 ( t ) ∣ ∑ i = 1 m ∣ μ i ∣ ∫ 0 ξ i ( ξ i − s ) β − 1 Γ ( β ) ( L 2 ( ∣ u ∣ + ∣ v ∣ ) + M 2 ) d s + ∣ ρ 3 ( t ) ∣ ∑ j = 1 n ∣ σ j ∣ ∫ 0 ζ j ( ζ j − s ) α − 1 Γ ( α ) ( L 1 ( ∣ u ∣ + ∣ v ∣ ) + M 1 ) d s + ∣ b 0 ω 1 ∣ ∣ ρ 2 ( t ) ∣ ∫ 0 T ( T − s ) Γ ( 2 ) ( L 2 ( ∣ u ∣ + ∣ v ∣ ) + M 2 ) d s + ∫ 0 t ( t − s ) α − 1 Γ ( α ) ( L 1 ( ∣ u ∣ + ∣ v ∣ ) + M 1 ) d s ‖ P ‖ ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + ∣ a 0 ∣ δ 1 L 1 ∫ 0 T ( T − s ) [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s + M 1 ∣ a 0 ∣ δ 1 T 2 2 + ∣ a 1 ∣ δ 2 L 1 ∫ 0 T [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s + ∣ a 1 ∣ δ 2 M 1 T + ∣ b 1 ∣ δ 3 L 2 ∫ 0 T [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s + M 2 ∣ b 1 ∣ δ 3 T + ∣ ν ∣ δ 2 L 2 ∫ 0 η 1 ( η 1 − s ) α + β − 2 Γ ( α + β − 1 ) [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s + M 2 ∣ ν ∣ δ 2 ( η 1 ) α + β − 1 Γ ( α + β ) + ∣ μ ∣ δ 3 L 1 ∫ 0 η 2 ( η 2 − s ) α + β − 2 Γ ( α + β − 1 ) [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s + M 1 ∣ μ ∣ δ 3 ( η 2 ) α + β − 1 Γ ( α + β ) + δ 2 ∑ i = 1 m ∣ μ i ∣ L 2 ∫ 0 ξ i ( ξ i − s ) β − 1 Γ ( β ) [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s

(28) + M 2 δ 2 ∑ i = 1 m ∣ μ i ∣ ξ i β Γ ( β + 1 ) + δ 3 ∑ j = 1 n ∣ σ j ∣ L 1 ∫ 0 ζ j ( ζ j − s ) α − 1 Γ ( α ) [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s + M 1 δ 3 ∑ j = 1 n ∣ σ j ∣ ζ j α Γ ( α + 1 ) + ∣ b 0 ω 1 ∣ δ 2 L 2 ∫ 0 T ( T − s ) [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s + M 1 ∣ b 0 ω 1 ∣ δ 2 T 2 2 + L 1 t 2 − α ∫ 0 t ( t − s ) α − 1 Γ ( α ) [ s α − 2 ‖ u ‖ X + s β − 2 ‖ v ‖ Y ] d s + M 1 T 2 Γ ( α + 1 ) ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + [ L 1 N 1 + L 2 N 2 ] r + M 1 e 1 + M 2 e 2 .

Similarly, one can get

(29) ∣ ∣ P 2 ( u , v ) ∣ ∣ Y ≤ ∣ λ 2 ∣ δ 1 ∗ + ∣ λ 1 ν 1 ∣ δ 2 ∗ + [ L 1 N 1 ∗ + L 2 N 2 ∗ ] r + M 1 e 1 ∗ + M 2 e 2 ∗ .

In view of (28) and (29) together with (25), we have

‖ P ( u , v ) ‖ X × Y ≤ ∣ λ 1 ∣ [ δ 1 + ∣ ν 1 ∣ δ 2 ∗ ] + ∣ λ 2 ∣ [ δ 2 ∣ ω 1 ∣ + δ 1 ∗ ] + [ L 1 ( N 1 + N 1 ∗ ) + L 2 ( N 2 + N 2 ∗ ) ] r + M 1 ( e 1 + e 1 ∗ ) + M 2 ( e 2 + e 2 ∗ ) ≤ r .

Now, for ( u 1 , v 1 ) , ( u 2 , v 2 ) ∈ X × Y , and for any t ∈ [ 0 , T ] , we get

t 2 − α ∣ P 1 ( u 2 , v 2 ) ( t ) − P 1 ( u 1 , v 1 ) ( t ) ∣ ≤ sup t ∈ [ 0 , T ] t 2 − α ∣ a 0 ρ 1 ( t ) ∣ ∫ 0 T ( T − s ) ∣ F ( s , u 2 , v 2 ) − F ( s , u 1 , v 1 ) ∣ d s + ∣ a 1 ρ 2 ( t ) ∣ ∫ 0 T ∣ F ( s , u 2 , v 2 ) − F ( s , u 1 , v 1 ) ∣ d s + ∣ b 1 ρ 3 ( t ) ∣ ∫ 0 T ∣ G ( s , u 2 , v 2 ) − G ( s , u 1 , v 1 ) ∣ d s + ∣ ν ρ 2 ( t ) ∣ ∫ 0 η 1 ( η 1 − s ) α + β − 2 Γ ( α + β − 1 ) ∣ G ( s , u 2 , v 2 ) − G ( s , u 1 , v 1 ) ∣ d s + ∣ μ ρ 3 ( t ) ∣ ∫ 0 η 2 ( η 2 − s ) α + β − 2 Γ ( α + β − 1 ) ∣ F ( s , u 2 , v 2 ) − F ( s , u 1 , v 1 ) ∣ d s + ∣ ρ 2 ( t ) ∣ ∑ i = 1 m ∣ μ i ∣ ∫ 0 ξ i ( ξ i − s ) β − 1 Γ ( β ) ∣ G ( s , u 2 , v 2 ) − G ( s , u 1 , v 1 ) ∣ d s + ∣ ρ 3 ( t ) ∣ j = 1 n ∣ σ j ∣ ∫ 0 ζ j ( ζ i − s ) α − 1 Γ ( α ) ∣ F ( s , u 2 , v 2 ) − F ( s , u 1 , v 1 ) ∣ d s + ∣ b 0 ω 1 ρ 2 ( t ) ∣ ∫ 0 T ( T − s ) ∣ G ( s , u 2 , v 2 ) − G ( s , u 1 , v 1 ) ∣ d s + ∫ 0 t ( t − s ) α − 1 Γ ( α ) ∣ F ( s , u 2 , v 2 ) − F ( s , u 1 , v 1 ) ∣ d s .

Using ( A 1 ) and the relation ∣ u 2 − u 1 ∣ + ∣ v 2 − v 1 ∣ ≤ t α − 2 ‖ u 2 − u 1 ‖ X + t β − 2 ‖ v 2 − v 1 ‖ Y in (30) yields

(31) ‖ P 1 ( u 2 , v 2 ) − P 1 ( u 1 , v 1 ) ‖ X ≤ ( L 1 N 1 + L 2 N 2 ) [ ‖ u 2 − u 1 ‖ X + ‖ v 2 − v 1 ‖ Y ] .

In a similar manner, one can get

(32) ‖ P 2 ( u 2 , v 2 ) − P 2 ( u 1 , v 1 ) ‖ Y ≤ ( L 1 N 1 ∗ + L 2 N 2 ∗ ) [ ‖ u 2 − u 1 ‖ X + ‖ v 2 − v 1 ‖ Y ] .

Thus, it follows from (31) and (32) that

‖ P ( u 2 , v 2 ) − P ( u 1 , v 1 ) ‖ X × Y ≤ [ L 1 ( N 1 + N 1 ∗ ) + L 2 ( N 2 + N 2 ∗ ) ] [ ‖ u 2 − u 1 ‖ X + ‖ v 2 − v 1 ‖ Y ] ,

which, in view of condition (24), implies that P is a contraction. Hence, by Banach’s fixed point theorem, the operator P has a unique fixed point, which is indeed a unique solution of the problem (1)–(2) on [ 0 , T ] . This completes the proof.□

In the following result, we present the sufficient conditions ensuring the existence of solutions for the problem (1)–(2). We apply Leray-Schauder alternative [32] to prove this result.

Theorem 3.2

Assume that

  1. F , G : [ 0 , T ] × R × R → R are continuous functions and there exist real constants k i , γ i ≥ 0 , ( i = 1 , 2 ) and k 0 > 0 and γ 0 > 0 such that, for all t ∈ [ 0 , T ] and u , v ∈ R ,

    ∣ F ( t , u , v ) ∣ ≤ k 0 + k 1 ∣ u ∣ + k 2 ∣ v ∣ , ∣ G ( t , u , v ) ∣ ≤ γ 0 + γ 1 ∣ u ∣ + γ 2 ∣ v ∣ .

  2. k ( N 1 + N 1 ∗ ) + γ ( N 2 + N 2 ∗ ) < 1 , where k = max { k 1 , k 2 } , γ = max { γ 1 , γ 2 } .

Then the system (1)–(2) has at least one solution on [ 0 , T ] .

Proof

Let us first note that continuity of the operator P follows from that of the functions F and G . Let B ⊂ X × Y be bounded such that ∣ F ( t , u , v ) ∣ ≤ K F , ∣ G ( t , u , v ) ∣ ≤ K G , ∀ ( u , v ) ∈ B , for positive constants K F and K G . Then, for any ( u , v ) ∈ B , we have

t 2 − α ∣ P 1 ( u , v ) ( t ) ∣ ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + K F ∣ a 0 ∣ δ 1 T 2 2 + ∣ a 1 ∣ T δ 2 + ∣ μ ∣ δ 3 η 2 α + β − 1 Γ ( α + β ) + δ 3 ∑ j = 1 n ∣ σ j ∣ ζ j α Γ ( α + 1 ) + T 2 Γ ( α + 1 ) + K G ∣ ν ∣ δ 2 η 1 α + β − 1 Γ ( α + β ) + ∣ b 1 ∣ δ 3 T + δ 2 ∑ i = 1 m ∣ μ i ∣ ξ i β Γ ( β + 1 ) + ∣ b 0 ω 1 ∣ δ 2 T 2 2 = ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + K F e 1 + K G e 2 ,

which implies that

‖ P 1 ( u , v ) ‖ X ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + K F e 1 + K G e 2 .

Similarly, one can show that

‖ P 2 ( u , v ) ‖ Y ≤ ∣ λ 2 ∣ δ 1 ∗ + ∣ λ 1 ν 1 ∣ δ 2 ∗ + K F e 1 ∗ + K G e 2 ∗ .

In consequence, we get

‖ P ( u , v ) ‖ X × Y ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ∣ δ 1 ∗ + ∣ λ 2 ω 1 ∣ δ 2 + ∣ λ 1 ν 1 ∣ δ 2 ∗ + K F ( e 1 + e 1 ∗ ) + K G ( e 2 + e 2 ∗ ) < ∞ ,

which shows that the operator P is uniformly bounded.

Next, we show that P is equicontinuous. Let t 1 , t 2 ∈ [ 0 , T ] with t 1 > t 2 . Then we have

∣ t 1 2 − α P 1 ( u , v ) ( t 1 ) − t 2 2 − α P 1 ( u , v ) ( t 2 ) ∣ ≤ ∣ λ 1 ∣ ∣ t 1 2 − α ρ 1 ( t 1 ) − t 2 2 − α ρ 1 ( t 2 ) ∣ + ∣ λ 2 ω 1 ∣ ∣ t 1 2 − α ρ 2 ( t 1 ) − t 2 2 − α ρ 2 ( t 2 ) ∣ + K F ∣ a 0 ∣ ∣ t 1 2 − α ρ 1 ( t 1 ) − t 2 2 − α ρ 1 ( t 2 ) ∣ T 2 2 + ∣ a 1 ∣ T ∣ ∣ t 1 2 − α ρ 2 ( t 1 ) − t 2 2 − α ρ 2 ( t 2 ) ∣ + ∣ μ ∣ ∣ t 1 2 − α ρ 3 ( t 1 ) − t 2 2 − α ρ 3 ( t 2 ) ∣ ∣ η 2 ∣ α + β − 1 Γ ( α + β ) + ∣ t 1 2 − α ρ 3 ( t 1 ) − t 2 2 − α ρ 3 ( t 2 ) ∣ ∑ j = 1 n ∣ σ j ∣ ζ j α Γ ( α + 1 ) + t 1 2 − α ∫ 0 t 1 ( t 1 − s ) α − 1 Γ ( α ) d s − t 2 2 − α ∫ 0 t 2 ( t 2 − s ) α − 1 Γ ( α ) d s + K G ∣ ν ∣ ∣ η 1 ∣ α + β − 1 Γ ( α + β ) ∣ t 1 2 − α ρ 2 ( t 1 ) − t 2 2 − α ρ 2 ( t 2 ) ∣ + ∣ b 1 ∣ T ∣ t 1 2 − α ρ 3 ( t 1 ) − t 2 2 − α ρ 3 ( t 2 ) ∣

+ ∑ i = 1 m ∣ μ i ∣ ξ i β Γ ( β + 1 ) ∣ t 1 2 − α ρ 2 ( t 1 ) − t 2 2 − α ρ 2 ( t 2 ) ∣ + ∣ b 0 ω 1 ∣ T 2 2 ∣ t 1 2 − α ρ 2 ( t 1 ) − t 2 2 − α ρ 2 ( t 2 ) ∣ ≤ ∣ t 1 − t 2 ∣ ∣ λ 1 ∣ + ∣ a 0 ∣ T 2 2 K F ν 1 Λ ω 2 − b 0 T Γ ( β ) Γ ( α ) ( 1 + a 1 ) ω 1 + ∣ λ 2 ω 1 ∣ + ∣ a 1 ∣ T K F + ∣ ν ∣ K G η 1 α + β − 1 Γ ( α + β ) + K G ∑ i = 1 m ∣ μ i ∣ ξ i β Γ ( β + 1 ) + K G ∣ b 0 ω 1 ∣ T 2 2 Γ ( β ) ( b 1 + 1 ) Λ Γ ( α ) ( a 1 + 1 ) + ∑ j = 1 n ∣ σ j ∣ ζ j α Γ ( α + 1 ) K G + ∣ μ ∣ ∣ η 2 ∣ α + β − 1 Γ ( α + β ) K F + ∣ b 1 ∣ T K G 1 ∣ Λ ∣ ω 2 − b 0 T Γ ( β ) Γ ( α ) ( 1 + a 1 ) ω 1 + 2 K F t 1 2 − α Γ ( α + 1 ) ∣ t 1 − t 2 ∣ α + K F ∣ t 1 2 − t 2 2 ∣ Γ ( α + 1 ) → 0 as t 2 → t 1 ,

independent of ( u , v ) ∈ B . Also, we have

∣ t 1 2 − β P 2 ( u , v ) ( t 1 ) − t 2 2 − β P 2 ( u , v ) ( t 2 ) ∣ ≤ ∣ t 1 − t 2 ∣ ∣ λ 2 ∣ + ∣ b 0 ∣ K G T 2 2 ω 1 Λ ν 2 − a 0 T ( 1 + a 1 ) ν 1 + ∣ λ 1 ν 1 ∣ + ∣ a 0 ν 1 ∣ K F T 2 2 + μ K F η 2 α + β − 1 Γ ( α + β ) + ∣ b 1 ∣ T K G + K F ∑ j = 1 n ∣ σ j ∣ ζ j α Γ ( α ) 1 ∣ Λ ∣ + ∣ a 1 ∣ T K F + ∣ ν ∣ K G η 1 α + β − 1 Γ ( α + β ) + K G ∑ i = 1 m ∣ μ i ∣ ξ i β Γ ( β + 1 ) 1 ∣ Λ ∣ ν 2 − a 0 T ( 1 + a 1 ) ν 1 + 2 K G t 1 2 − β Γ ( β + 1 ) ∣ t 1 − t 2 ∣ β + K G ∣ t 1 2 − t 2 2 ∣ Γ ( β + 1 ) → 0 as t 2 → t 1 ,

independently of ( u , v ) ∈ B . Thus, the operator P is equicontinuous. Thus, we deduce that the operator P is completely continuous.

Finally, we consider the set

V = { ( u , v ) ∈ X × Y ∣ ( u , v ) = m P ( u , v ) , 0 ≤ m ≤ 1 }

and show that it is bounded. Let ( u , v ) ∈ V with ( u , v ) = m P ( u , v ) , u = m P 1 ( u , v ) , and v = m P 2 ( u , v ) . Then we have

‖ u ‖ X ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + ( N 1 k 1 + N 2 γ 1 ) ‖ u ‖ X + ( N 1 k 2 + N 2 γ 2 ) ‖ v ‖ Y + k 0 e 1 + γ 0 e 2 ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + ( N 1 k + N 2 γ ) ‖ u ‖ X + ( N 1 k + N 2 γ ) ‖ v ‖ Y + k 0 e 1 + γ 0 e 2 , ‖ v ‖ Y ≤ ∣ λ 2 ∣ δ 1 ∗ + ∣ λ 1 ν 1 ∣ δ 2 ∗ + ( N 1 ∗ k 1 + N 2 ∗ γ 1 ) ‖ u ‖ X + ( k 2 N 1 ∗ + N 2 ∗ γ 2 ) ‖ v ‖ Y + k 0 e 1 ∗ + γ 0 e 2 ∗ ≤ ∣ λ 2 ∣ δ 1 ∗ + ∣ λ 1 ν 1 ∣ δ 2 ∗ + ( N 1 ∗ k + N 2 ∗ γ ) ‖ u ‖ X + ( k N 1 ∗ + N 2 ∗ γ ) ‖ v ‖ Y + k 0 e 1 ∗ + γ 0 e 2 ∗ ,

which imply that

‖ u ‖ X + ‖ v ‖ Y ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + ∣ λ 2 ∣ δ 1 ∗ + ∣ λ 1 ν 1 ∣ δ 2 ∗ + [ k ( N 1 + N 1 ∗ ) + γ ( N 2 + N 2 ∗ ) ] ‖ u ‖ X + [ k ( N 1 + N 1 ∗ ) + γ ( N 2 + N 2 ∗ ) ] ‖ v ‖ Y + k 0 ( e 1 + e 1 * ) + γ 0 ( e 2 + e 2 ∗ ) .

Thus,

‖ ( u , v ) ‖ X × Y ≤ ∣ λ 1 ∣ δ 1 + ∣ λ 2 ω 1 ∣ δ 2 + ∣ λ 2 ∣ δ 1 ∗ + ∣ λ 1 ν 1 ∣ δ 2 ∗ + k 0 ( e 1 + e 1 * ) + γ 0 ( e 2 + e 2 ∗ ) 1 − [ k ( N 1 + N 1 ∗ ) + γ ( N 2 + N 2 ∗ ) ] .

Hence, the set V is bounded. Thus, by Leray-Schauder alternative, we deduce that the operator P has at least one fixed point, which corresponds to the fact that the problem (1)–(2) has at least one solution on [ 0 , T ] . The proof is completed.□

4 Examples

This section is devoted to the illustration of the results derived in the previous section.

Example 4.1

Consider the system of fractional differential equations consisting of the equations given by

(33) D 6 / 5 u ( t ) = 1 4 1600 + t ( u ( t ) + tan − 1 v ( t ) ) + cos t , t ∈ [ 0 , 1 ] , D 7 / 4 v ( t ) = 1 2500 + t ( sin u ( t ) + v ( t ) ) + t sin t , t ∈ [ 0 , 1 ] ,

supplemented by the following boundary conditions

(34) D − 4 / 5 u ( 0 + ) + 1 4 D − 4 / 5 u ( 1 − ) = − 4 , D 1 / 5 u ( 0 + ) − 3 2 D 1 / 5 u ( 1 − ) = − 2 I 1 / 5 v 1 4 − 6 v 1 2 − 4 v 2 3 , D − 1 / 4 v ( 0 + ) − D − 1 / 4 v ( 1 − ) = 2 , D 3 / 4 v ( 0 + ) − 3 2 D 3 / 4 v ( 1 − ) = 3 I 3 / 4 u 1 3 + 5 3 u 11 15 + 2 5 u 3 4 .

Here α = 6 / 5 , β = 7 / 4 , a 0 = 1 / 4 , b 0 = − 1 / 4 , a 1 = − 3 / 2 , b 1 = − 3 / 2 , ν = − 2 , μ = 3 , λ 1 = − 4 , λ 2 = 2 , μ 1 = − 6 , μ 2 = − 4 , σ 1 = 5 / 3 , σ 2 = 2 / 5 , η 1 = 1 / 4 , ξ 1 = 1 / 2 , ξ 2 = 2 / 3 , η 2 = 1 / 3 , ζ 1 = 11 / 15 , ζ 2 = 3 / 4 .

Using the given data, it is found that Λ ≃ 70.9531 , ν 1 ≃ 2.9182 , ν 2 ≃ − 6.3910 , ω 1 ≃ − 15.3511 , ω 2 ≃ 15.2944 , δ 1 ≃ 0.7330 , δ 2 ≃ 0.0135 , δ 3 ≃ 0.3100 , δ 1 ∗ ≃ 0.8213 , δ 2 ∗ ≃ 0.0176 , δ 3 ∗ ≃ 0.0869 , N 1 ≃ 12.2414 , N 2 ≃ 3.6569 , N 1 ∗ ≃ 1.1599 , N 2 ∗ ≃ 9.4324 , L 1 = 1 / 160 , L 2 = 1 / 50 , and

L 1 ( N 1 + N 1 ∗ ) + L 2 ( N 2 + N 2 ∗ ) ≃ 0.3455 < 1 .

Thus, all the conditions of Theorem 3.1 are satisfied and hence the problem (33)–(34) has a unique solution on [ 0 , 1 ] .

Example 4.2

Let us consider the problem (33)–(34) with

(35) F ( t , u ( t ) , v ( t ) ) = sin 2 t 2 + t 3 + tan − 1 u ( t ) 2 400 + t + v ( t ) ∣ u ( t ) ∣ 50 ( 1 + ∣ u ( t ) ∣ ) , t ∈ [ 0 , 1 ] , G ( t , u ( t ) , v ( t ) ) = e − t + ∣ u ( t ) ∣ cos u ( t ) 150 + t + v ( t ) 640 1 + sin 2 t , t ∈ [ 0 , 1 ] .

Clearly, ∣ F ( t , u ( t ) , v ( t ) ) ∣ < 1 2 + 1 40 ∣ u ( t ) ∣ + 1 50 ∣ v ( t ) ∣ , G ( t , u ( t ) , v ( t ) ) < 1 + 1 150 ∣ u ( t ) ∣ + 1 640 ∣ v ( t ) ∣ , and so k 0 = 1 / 2 , k 1 = 1 / 40 , k 2 = 1 / 50 , γ 0 = 1 , γ 1 = 1 / 150 , γ 2 = 1 / 640 , k = max { k 1 , k 2 } = 1 / 40 , γ = max { γ 1 , γ 2 } = 1 / 150 . Moreover,

k ( N 1 + N 1 ∗ ) + γ ( N 2 + N 2 ∗ ) ≃ 0.4223 < 1 .

Therefore, by Theorem 3.2, the problem (33)–(34) with F and G given by (35) has at least one solution on [ 0 , 1 ].

5 Conclusion

We have investigated the existence and uniqueness of solutions for a nonlinear system of Riemann-Liouville fractional differential equations, equipped with nonseparated semi-coupled integro-multipoint boundary conditions. We apply Banach contraction mapping principle to establish the existence of a unique solution, while Leray-Schauder alternative is used to obtain the existence result for the problem at hand. We emphasize that the novelty of our results lies on the semi-coupled boundary conditions (2) and enrich the related literature on the topic. Our work also produces some special cases by fixing the parameters involved in the boundary conditions. For example, our results correspond to the ones for nonlocal semi-coupled fractional multipoint boundary conditions by fixing ν = 0 = μ and the results for nonlocal semi-coupled fractional integral boundary conditions follow by taking all μ i = 0 , i = 1 , … , m and σ j = 0 , j = 1 , … , n .

Acknowledgment

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia under grant no. (KEP-PhD-40-130-41). The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also thank the reviewers for their constructive remarks on our work.

  1. Conflict of interest: Authors state no conflict of interest.

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Received: 2021-02-16
Revised: 2021-06-25
Accepted: 2021-06-29
Published Online: 2021-08-05

© 2021 Ahmed Alsaedi et al., published by De Gruyter

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

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
https://doi.org/10.1515/math-2021-0069
Keywords for this article
Riemann-Liouville fractional derivative; integro-differential inclusions; nonlocal multi-point boundary conditions; existence; fixed point theorems
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Sharp conditions for the convergence of greedy expansions with prescribed coefficients
  3. Range-kernel weak orthogonality of some elementary operators
  4. Stability analysis for Selkov-Schnakenberg reaction-diffusion system
  5. On non-normal cyclic subgroups of prime order or order 4 of finite groups
  6. Some results on semigroups of transformations with restricted range
  7. Quasi-ideal Ehresmann transversals: The spined product structure
  8. On the regulator problem for linear systems over rings and algebras
  9. Solvability of the abstract evolution equations in Ls-spaces with critical temporal weights
  10. Resolving resolution dimensions in triangulated categories
  11. Entire functions that share two pairs of small functions
  12. On stochastic inverse problem of construction of stable program motion
  13. Pentagonal quasigroups, their translatability and parastrophes
  14. Counting certain quadratic partitions of zero modulo a prime number
  15. Global attractors for a class of semilinear degenerate parabolic equations
  16. A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation
  17. On sub-class sizes of mutually permutable products
  18. Asymptotic solution of the Cauchy problem for the singularly perturbed partial integro-differential equation with rapidly oscillating coefficients and with rapidly oscillating heterogeneity
  19. Existence and asymptotical behavior of solutions for a quasilinear Choquard equation with singularity
  20. On kernels by rainbow paths in arc-coloured digraphs
  21. Fully degenerate Bell polynomials associated with degenerate Poisson random variables
  22. Multiple solutions and ground state solutions for a class of generalized Kadomtsev-Petviashvili equation
  23. A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces
  24. Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group
  25. Partial sums and inclusion relations for analytic functions involving (p, q)-differential operator
  26. Hodge-Deligne polynomials of character varieties of free abelian groups
  27. Diophantine approximation with one prime, two squares of primes and one kth power of a prime
  28. The equivalent parameter conditions for constructing multiple integral half-discrete Hilbert-type inequalities with a class of nonhomogeneous kernels and their applications
  29. Boundedness of vector-valued sublinear operators on weighted Herz-Morrey spaces with variable exponents
  30. On some new quantum midpoint-type inequalities for twice quantum differentiable convex functions
  31. Quantum Ostrowski-type inequalities for twice quantum differentiable functions in quantum calculus
  32. Asymptotic measure-expansiveness for generic diffeomorphisms
  33. Infinitesimals via Cauchy sequences: Refining the classical equivalence
  34. The (1, 2)-step competition graph of a hypertournament
  35. Properties of multiplication operators on the space of functions of bounded φ-variation
  36. Disproving a conjecture of Thornton on Bohemian matrices
  37. Some estimates for the commutators of multilinear maximal function on Morrey-type space
  38. Inviscid, zero Froude number limit of the viscous shallow water system
  39. Inequalities between height and deviation of polynomials
  40. New criteria-based ℋ-tensors for identifying the positive definiteness of multivariate homogeneous forms
  41. Determinantal inequalities of Hua-Marcus-Zhang type for quaternion matrices
  42. On a new generalization of some Hilbert-type inequalities
  43. On split quaternion equivalents for Quaternaccis, shortly Split Quaternaccis
  44. On split regular BiHom-Poisson color algebras
  45. Asymptotic stability of the time-changed stochastic delay differential equations with Markovian switching
  46. The mixed metric dimension of flower snarks and wheels
  47. Oscillatory bifurcation problems for ODEs with logarithmic nonlinearity
  48. The B-topology on S∗-doubly quasicontinuous posets
  49. Hyers-Ulam stability of isometries on bounded domains
  50. Inhomogeneous conformable abstract Cauchy problem
  51. Path homology theory of edge-colored graphs
  52. Refinements of quantum Hermite-Hadamard-type inequalities
  53. Symmetric graphs of valency seven and their basic normal quotient graphs
  54. Mean oscillation and boundedness of multilinear operator related to multiplier operator
  55. Numerical methods for time-fractional convection-diffusion problems with high-order accuracy
  56. Several explicit formulas for (degenerate) Narumi and Cauchy polynomials and numbers
  57. Finite groups whose intersection power graphs are toroidal and projective-planar
  58. On primitive solutions of the Diophantine equation x2 + y2 = M
  59. A note on polyexponential and unipoly Bernoulli polynomials of the second kind
  60. On the type 2 poly-Bernoulli polynomials associated with umbral calculus
  61. Some estimates for commutators of Littlewood-Paley g-functions
  62. Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials
  63. On the evolutionary bifurcation curves for the one-dimensional prescribed mean curvature equation with logistic type
  64. On intersections of two non-incident subgroups of finite p-groups
  65. Global existence and boundedness in a two-species chemotaxis system with nonlinear diffusion
  66. Finite groups with 4p2q elements of maximal order
  67. Positive solutions of a discrete nonlinear third-order three-point eigenvalue problem with sign-changing Green's function
  68. Power moments of automorphic L-functions related to Maass forms for SL3(ℤ)
  69. Entire solutions for several general quadratic trinomial differential difference equations
  70. Strong consistency of regression function estimator with martingale difference errors
  71. Fractional Hermite-Hadamard-type inequalities for interval-valued co-ordinated convex functions
  72. Montgomery identity and Ostrowski-type inequalities via quantum calculus
  73. Universal inequalities of the poly-drifting Laplacian on smooth metric measure spaces
  74. On reducible non-Weierstrass semigroups
  75. so-metrizable spaces and images of metric spaces
  76. Some new parameterized inequalities for co-ordinated convex functions involving generalized fractional integrals
  77. The concept of cone b-Banach space and fixed point theorems
  78. Complete consistency for the estimator of nonparametric regression model based on m-END errors
  79. A posteriori error estimates based on superconvergence of FEM for fractional evolution equations
  80. Solution of integral equations via coupled fixed point theorems in 𝔉-complete metric spaces
  81. Symmetric pairs and pseudosymmetry of Θ-Yetter-Drinfeld categories for Hom-Hopf algebras
  82. A new characterization of the automorphism groups of Mathieu groups
  83. The role of w-tilting modules in relative Gorenstein (co)homology
  84. Primitive and decomposable elements in homology of ΩΣℂP∞
  85. The G-sequence shadowing property and G-equicontinuity of the inverse limit spaces under group action
  86. Classification of f-biharmonic submanifolds in Lorentz space forms
  87. Some new results on the weaving of K-g-frames in Hilbert spaces
  88. Matrix representation of a cross product and related curl-based differential operators in all space dimensions
  89. Global optimization and applications to a variational inequality problem
  90. Functional equations related to higher derivations in semiprime rings
  91. A partial order on transformation semigroups with restricted range that preserve double direction equivalence
  92. On multi-step methods for singular fractional q-integro-differential equations
  93. Compact perturbations of operators with property (t)
  94. Entire solutions for several complex partial differential-difference equations of Fermat type in ℂ2
  95. Random attractors for stochastic plate equations with memory in unbounded domains
  96. On the convergence of two-step modulus-based matrix splitting iteration method
  97. On the separation method in stochastic reconstruction problem
  98. Robust estimation for partial functional linear regression models based on FPCA and weighted composite quantile regression
  99. Structure of coincidence isometry groups
  100. Sharp function estimates and boundedness for Toeplitz-type operators associated with general fractional integral operators
  101. Oscillatory hyper-Hilbert transform on Wiener amalgam spaces
  102. Euler-type sums involving multiple harmonic sums and binomial coefficients
  103. Poly-falling factorial sequences and poly-rising factorial sequences
  104. Geometric approximations to transition densities of Jump-type Markov processes
  105. Multiple solutions for a quasilinear Choquard equation with critical nonlinearity
  106. Bifurcations and exact traveling wave solutions for the regularized Schamel equation
  107. Almost factorizable weakly type B semigroups
  108. The finite spectrum of Sturm-Liouville problems with n transmission conditions and quadratic eigenparameter-dependent boundary conditions
  109. Ground state sign-changing solutions for a class of quasilinear Schrödinger equations
  110. Epi-quasi normality
  111. Derivative and higher-order Cauchy integral formula of matrix functions
  112. Commutators of multilinear strongly singular integrals on nonhomogeneous metric measure spaces
  113. Solutions to a multi-phase model of sea ice growth
  114. Existence and simulation of positive solutions for m-point fractional differential equations with derivative terms
  115. Bernstein-Walsh type inequalities for derivatives of algebraic polynomials in quasidisks
  116. Review Article
  117. Semiprimeness of semigroup algebras
  118. Special Issue on Problems, Methods and Applications of Nonlinear Analysis (Part II)
  119. Third-order differential equations with three-point boundary conditions
  120. Fractional calculus, zeta functions and Shannon entropy
  121. Uniqueness of positive solutions for boundary value problems associated with indefinite ϕ-Laplacian-type equations
  122. Synchronization of Caputo fractional neural networks with bounded time variable delays
  123. On quasilinear elliptic problems with finite or infinite potential wells
  124. Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials
  125. On a fractional Schrödinger-Poisson system with strong singularity
  126. Parabolic inequalities in Orlicz spaces with data in L1
  127. Special Issue on Evolution Equations, Theory and Applications (Part II)
  128. Impulsive Caputo-Fabrizio fractional differential equations in b-metric spaces
  129. Existence of a solution of Hilfer fractional hybrid problems via new Krasnoselskii-type fixed point theorems
  130. On a nonlinear system of Riemann-Liouville fractional differential equations with semi-coupled integro-multipoint boundary conditions
  131. Blow-up results of the positive solution for a class of degenerate parabolic equations
  132. Long time decay for 3D Navier-Stokes equations in Fourier-Lei-Lin spaces
  133. On the extinction problem for a p-Laplacian equation with a nonlinear gradient source
  134. General decay rate for a viscoelastic wave equation with distributed delay and Balakrishnan-Taylor damping
  135. On hyponormality on a weighted annulus
  136. Exponential stability of Timoshenko system in thermoelasticity of second sound with a memory and distributed delay term
  137. Convergence results on Picard-Krasnoselskii hybrid iterative process in CAT(0) spaces
  138. Special Issue on Boundary Value Problems and their Applications on Biosciences and Engineering (Part I)
  139. Marangoni convection in layers of water-based nanofluids under the effect of rotation
  140. A transient analysis to the M(τ)/M(τ)/k queue with time-dependent parameters
  141. Existence of random attractors and the upper semicontinuity for small random perturbations of 2D Navier-Stokes equations with linear damping
  142. Degenerate binomial and Poisson random variables associated with degenerate Lah-Bell polynomials
  143. Special Issue on Fractional Problems with Variable-Order or Variable Exponents (Part I)
  144. On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems
  145. The Lp dual Minkowski problem about 0 < p < 1 and q > 0
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