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On the mixed fractional quantum and Hadamard derivatives for impulsive boundary value problems

  • Somboon Niyoom , Sotiris K. Ntouyas , Chayapat Sudprasert and Jessada Tariboon EMAIL logo
Published/Copyright: December 31, 2021
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Open Mathematics
From the journal Open Mathematics Volume 19 Issue 1

Abstract

In this work, we initiate the study of a new class of impulsive boundary value problems consisting of mixed type fractional quantum and Hadamard derivatives. We will establish existence and uniqueness results by using tools from the functional analysis. We prove the uniqueness result via Banach’s contraction mapping principle, while we will use the Leray-Schauder nonlinear alternative to establish an existence result. We also present examples to illustrate the obtained results.

Keywords: fractional impulsive differential equations; instantaneous impulses; impulses; quantum calculus
MSC 2010: 26A33; 34B37; 39A13

1 Introduction

Fractional calculus is a generalization of classical calculus to an arbitrary real order and has evolved as an interesting and important area of research. Fractional differential equations have gained much attention in the literature because some real-world problems in physics, mechanics, engineering, game theory, stability, optimal control, and other fields can be described better with the help of fractional differential equations. Fractional differential equations constitute a significant branch of nonlinear analysis. The theory and applications of the fractional differential equations have been greatly developed; for more details, one can see the monographs [1,2,3, 4,5,6, 7,8] and references therein.

Impulsive differential equations arise when at certain moments they change their state rapidly and have many applications in physics, engineering, medicine, population dynamics, pharmacology, biotechnology, and economics. There are two types of impulses: the instantaneous impulses in which the duration of these changes is relatively short and the noninstantaneous impulses in which an impulsive jump starts abruptly at any fixed point and continues on a finite interval of time. For the results of instantaneous impulses, see, e.g., the monographs [9,10,11], the papers [12,13,14], and the references cited therein. Noninstantaneous impulsive differential equation was introduced by Hernández and O’Regan in [15], who pointed out that the instantaneous impulses cannot characterize some processes such as evolution processes in pharmacotherapy. In noninstantaneous impulses, impulsive action starts at an arbitrary fixed point and remains active on a finite time interval, which is much different from classical instantaneous impulsive that changes are relatively short compared to the overall duration of the whole process. Let us consider the hemodynamic equilibrium of a person. The introduction of the drugs in the bloodstream and the consequent absorption for the body are a gradual and continuous process. In fact, this situation should be characterized by a new case of impulsive action, which starts at an arbitrary fixed point and stays active on a finite time interval. For more details, see the monograph [16].

For some recent works, we refer the reader to [17, 18,19,20] and references therein.

In [21], the notions of q k -derivative and q k -integral were introduced, and their properties were investigated. New concepts of fractional quantum calculus involving a new q k -shifting operator were introduced in [22]. On the other hand, the analysis of fractional differential equations involving Hadamard fractional derivatives has increased interest in the mathematical analysis; see, for example, the recent monograph [8].

Motivated by the aforementioned papers, in this investigation, we initiate the study of a new class of boundary value problems consisting of impulses and mixed type fractional quantum and Hadamard derivatives. More precisely, we study the following impulsive boundary value problem, in which we combine fractional quantum and Hadamard derivatives of the form

(1) D q i α i s i x ( t ) = f ( t , x ( t ) ) , t ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , D β i t i H x ( t ) = g ( t , x ( t ) ) , t ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m , x ( t i + ) = γ i x ( t i − ) + η i , x ( s i + ) = ξ i x ( s i − ) + θ i , i = 1 , 2 , 3 , … , m , λ 1 x ( 0 ) + λ 2 x ( T ) = λ 3 ,

where D q i α i s i , D β j t j H are the fractional quantum difference and Hadamard fractional derivative in the sense of Caputo type of orders α i , β j ∈ ( 0 , 1 ] with quantum numbers q i ∈ ( 0 , 1 ) , i = 0 , 1 , … , m , j = 1 , 2 , … , m . Let J 1 = ⋃ i = 0 m [ s i , t i + 1 ) , J 2 = ⋃ j = 1 m [ t j , s j ) , and J = J 1 ∪ J 2 ∪ { T } be given intervals. The functions f : J 1 × R → R and g : J 2 × R → R are continuous, and λ 1 , λ 2 , λ 3 , γ i , η i , ξ i , θ i , i = 1 , … , m are constants. In J , the fixed points 0 = s 0 < t 1 < s 1 < t 2 < ⋯ < t m < s m < t m + 1 = T are given.

We establish existence and uniqueness results for the impulsive mixed fractional quantum and Hadamard boundary value problem (1) by using tools from the functional analysis. The main results are presented in Section 3, where Banach’s contraction mapping principle is applied for the uniqueness result, while the Leray-Schauder nonlinear alternative is used to establish an existence result. In Section 2, some basic concepts from quantum calculus and Hadamard derivatives are recalled, and also an auxiliary result concerning a linear variant of the problem (1) is proved. This result is pivotal to transform the problem (1) into a fixed point problem. Illustrative numerical examples are also presented.

2 Preliminaries

In this section, we give some basic concepts of fractional quantum and Hadamard calculus such as derivatives and integrals. For more details, see [2,22, 23,24]. Staring, the q -shifting operator is defined by

(2) Φ q a = q x + ( 1 − q ) a ,

where x ∈ R , a ≥ 0 , and 0 < q < 1 . For any positive integer k , we have

Φ q k a ( x ) = q k x + ( 1 − q k ) a = Φ q k − 1 a ( Φ q a ) and Φ q 0 a ( x ) = x .

The power function of x − y involving q -shifting operator is given by

( x − y ) q ( 0 ) a = 1 , ( x − y ) q ( k ) a = ∏ i = 0 k − 1 ( x − Φ q i a ( y ) ) , k ∈ N ∪ { ∞ } .

In general, if α ∈ R , then we have

( x − y ) q ( α ) a = x ( γ ) ∏ i = 0 ∞ 1 − Φ q i a x ( y / x ) 1 − Φ q α + i a x ( y / x ) .

The q -difference of a function f on interval [ a , b ] can be defined in the term of q -shifting by

( D q a f ) ( t ) = f ( t ) − f ( Φ q a ( t ) ) ( 1 − q ) ( t − a ) , t ≠ a and ( D q a f ) ( a ) = lim t → a ( D q a f ) ( t ) ,

while the q -integral of a function f on interval [ a , b ] is defined as follows:

(3) ( I q a f ) ( t ) = ∫ a t f ( s ) d q a s = ( 1 − q ) ( t − a ) ∑ i = 0 ∞ q i f ( Φ q i a ( t ) ) , t ∈ [ a , b ] .

Now, we present the formulas of Riemann-Liouville fractional q -integral and Caputo fractional q -derivative over the interval [ a , b ] as follows.

Definition 2.1

Let α ≥ 0 and f : [ a , b ] → R be a function. The Riemann-Liouville fractional q -integral is defined by

( I q α a f ) ( t ) = 1 Γ q ( α ) ∫ a t ( t − a Φ q ( s ) ) q ( α − 1 ) a f ( s ) d q a s , α > 0 , t ∈ [ a , b ] ,

and ( I q 0 a f ) ( t ) = f ( t )

Definition 2.2

Let f : [ a , b ] → R be the n -times q -differentiable function. The fractional q -derivative of the Caputo type of order α ≥ 0 on interval [ a , b ] is defined by

( D q α a f ) ( t ) = ( I q n − α a D q n a f ) ( t ) = 1 Γ q ( n − α ) ∫ a t ( t − a Φ q ( s ) ) q ( n − α − 1 ) a D q n a f ( s ) d q a s ,

for α > 0 and ( D q 0 a f ) ( t ) = f ( t ) , where n is the smallest integer greater than or equal to α and D q n a = D q a D q a ⋯ D q a ︸ n times .

Lemma 2.1

Let α > 0 and n be the smallest integer greater than or equal to α . Then, for t ∈ [ a , b ] , we have

I q α a D q α a f ( t ) = f ( t ) − ∑ k = 0 n − 1 ( t − a ) k Γ q ( k + 1 ) D q k a f ( a ) .

Let us introduce some notations and definitions of Hadamard fractional calculus as follows.

Definition 2.3

For at least n -times differentiable function f : [ a , b ] → R , a > 0 , the Caputo type of Hadamard fractional derivative of order α is defined as follows:

(4) D α a H f ( t ) = 1 Γ ( n − α ) ∫ a t log t s n − α − 1 δ n f ( s ) d s s , n − 1 < α < n ,

where the operator δ = t d d t , n = [ α ] + 1 , and [ α ] denotes the integer part of the real number α and log ( ⋅ ) = log e ( ⋅ ) .

Definition 2.4

The Hadamard fractional integral of order α > 0 is defined as follows:

(5) J α a H f ( t ) = 1 Γ ( α ) ∫ a t log t s α − 1 f ( s ) d s s , a > 0 ,

provided the right side integral exists on [ a , b ] .

Lemma 2.2

[23] Let x ∈ C δ n [ a , b ] and α ∈ C . Then, for a > 0 , we have

J α a H ( D α a H ) x ( t ) = x ( t ) − ∑ k = 0 n − 1 δ k x ( a ) k ! log t a k .

Next, let us introduce the future used notation as follows:

⨂ k i ( ξ , γ ) = ∏ r = k i ξ r ( ∏ k < j ≤ i γ j ) .

For examples, ⨂ 3 5 ( ξ , γ ) = ξ 3 ξ 4 ξ 5 γ 4 γ 5 , ⨂ 4 5 ( ξ , γ ) = ξ 4 ξ 5 γ 5 , and ⨂ 5 5 ( ξ , γ ) = ξ 5 , with ∏ 5 < j ≤ 5 γ j = 1 , since the inequality in the subscript is not true. In addition, we assume the constants

Λ 1 ( i ) ≔ ∑ k = 1 i η k ⨂ k i ( ξ , γ ) , Λ 2 ( i ) ≔ ∑ k = 1 i θ k ∏ k < j i ξ j γ j , Ω ≔ λ 1 + λ 2 ∏ k = 1 m ξ k γ k .

The next lemma deals with a linear variant of the boundary value problem (1).

Lemma 2.3

Let f ∗ : J 1 → R and g ∗ : J 2 → R be given functions and let Ω ≠ 0 . In addition, we let α j , q j , j = 0 , 1 , … , m , β i , γ i , η i , ξ i , θ i , i = 1 , 2 , … , m , λ 1 , λ 2 , and λ 3 be given constants, which satisfy problem (1). Then, the linear boundary value problem of mixed type quantum and Hadamard fractional derivatives of the form:

(6) D q i α i s i x ( t ) = f ∗ ( t ) , t ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , D β i t i H x ( t ) = g ∗ ( t ) , t ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m , x ( t i + ) = γ i x ( t i − ) + η i , x ( s i + ) = ξ i x ( s i − ) + θ i , i = 1 , 2 , 3 , … , m , λ 1 x ( 0 ) + λ 2 x ( T ) = λ 3 ,

has a unique solution x on J presented by

x ( t ) = 1 Ω ∏ k = 1 i ξ k γ k λ 3 − λ 2 ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) − λ 2 Λ 1 ( m ) − λ 2 ∑ k = 1 m + 1 ∏ k m ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) − λ 2 Λ 2 ( m ) + ∑ k = 1 i ⨂ k i ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + Λ 1 ( i ) + ∑ k = 1 i ∏ k i ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + Λ 2 ( i ) + ( I q i α i s i f ∗ ) ( t ) , t ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , γ i Ω ∏ k = 1 i − 1 ξ k γ k λ 3 − λ 2 ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) − λ 2 Λ 1 ( m ) − λ 2 ∑ k = 1 m + 1 ∏ k m ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) − λ 2 Λ 2 ( m ) + γ i ∑ k = 1 i − 1 ⨂ k i − 1 ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + Λ 1 ( i − 1 ) + ∑ k = 1 i ∏ k i − 1 ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + Λ 2 ( i − 1 ) + η i + ( J β i t i H g ∗ ) ( t ) , t ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m , ( 7 )

with ∏ a b ( ⋅ ) = 1 , ∑ a b ( ⋅ ) = 0 , if b < a .

Proof

In the first equation in (6) by taking the fractional q 0 -integral of order α 0 from s 0 to t ∈ [ s 0 , t 1 ) , we obtain

I q 0 α 0 s 0 ( D q 0 α 0 s 0 x ) ( t ) = ( I q 0 α 0 s 0 f ∗ ) ( t ) ,

which yields

(8) x ( t ) = c 0 + ( I q 0 α 0 s 0 f ∗ ) ( t ) ,

where c 0 = x ( 0 ) . In the second interval [ t 1 , s 1 ) , we can get x ( t ) by Hadamard fractional integration of order β 1 > 0 as follows:

x ( t ) = x ( t 1 + ) + ( J β 1 t 1 H g ∗ ) ( t ) .

Since x ( t 1 + ) = γ 1 x ( t 1 − ) + η 1 and x ( t 1 − ) = c 0 + ( I q 0 α 0 s 0 f ∗ ) ( t 1 − ) , we have

(9) x ( t ) = γ 1 c 0 + γ 1 ( I q 0 α 0 s 0 f ∗ ) ( t 1 − ) + η 1 + ( J β 1 t 1 H g ∗ ) ( t ) ,

for t ∈ [ t 1 , s 1 ) .

In the third interval [ s 1 , t 2 ) , it is the fractional q 1 -difference of an unknown x ( t ) . Then, we apply the fractional q 1 -integral of order α 1 by

x ( t ) = x ( s 1 + ) + ( I q 1 α 1 s 1 f ∗ ) ( t ) ,

which implies, by impulsive condition, that

x ( t ) = ξ 1 γ 1 c 0 + ξ 1 γ 1 ( I q 0 α 0 s 0 f ∗ ) ( t 1 − ) + ξ 1 η 1 + ξ 1 ( J β 1 t 1 H g ∗ ) ( s 1 − ) + θ 1 + ( I q 1 α 1 s 1 f ∗ ) ( t ) .

For the fourth interval [ t 2 , s 2 ) we can obtain by directing computation that

x ( t ) = γ 2 ξ 1 γ 1 c 0 + γ 2 ξ 1 γ 1 ( I q 0 α 0 s 0 f ∗ ) ( t 1 − ) + γ 2 ξ 1 η 1 + γ 2 ξ 1 ( J β 1 t 1 H g ∗ ) ( s 1 − ) + γ 2 θ 1 + γ 2 ( I q 1 α 1 s 1 f ∗ ) ( t 2 − ) + η 2 + ( J β 2 t 2 H g ∗ ) ( t ) .

Therefore, we can predict the solution x ( t ) of (6) by

(10) x ( t ) = c 0 ∏ k = 1 i ξ k γ k + ∑ k = 1 i ⨂ k i ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 i η k ⨂ k i ( ξ , γ ) + ∑ k = 1 i ∏ k i ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 i θ k ∏ k < j i ξ j γ j + ( I q i α i s i f ∗ ) ( t ) , t ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , γ i c 0 ∏ k = 1 i − 1 ξ k γ k + ∑ k = 1 i − 1 ⨂ k i − 1 ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 i − 1 η k ⨂ k i − 1 ( ξ , γ ) + ∑ k = 1 i ∏ k i − 1 ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 i − 1 θ k ∏ k < j i − 1 ξ j γ j + η i + ( J β i t i H g ∗ ) ( t ) , t ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m .

To claim that our formula (10) is true, we use the mathematical induction by putting i = 0 and i = 1 in the first and second parts of (10), respectively. Then, the initial step holds by (8) and (9). The inductive step will be proved by assuming that the first part of (10) is true for i = n , that is,

x ( t ) = c 0 ∏ k = 1 n ξ k γ k + ∑ k = 1 n ⨂ k n ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 n η k ⨂ k n ( ξ , γ ) + ∑ k = 1 n ∏ k n ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 n θ k ∏ k < j n ξ j γ j + ( I q n α n s n f ∗ ) ( t ) , t ∈ [ s n , t n + 1 ) .

Then, in the consecutive interval [ t n + 1 , s n + 1 ) , we have

x ( t ) = x ( t n + 1 + ) + ( J β n + 1 t n + 1 H g ∗ ) ( t ) = γ n + 1 x ( t n + 1 − ) + η n + 1 + ( J β n + 1 t n + 1 H g ∗ ) ( t ) = γ n + 1 c 0 ∏ k = 1 n ξ k γ k + ∑ k = 1 n ⨂ k n ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 n η k ⨂ k n ( ξ , γ ) + ∑ k = 1 n ∏ k n ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 n θ k ∏ k < j n ξ j γ j + ( I q n α n s n f ∗ ) ( t n + 1 − ) + η n + 1 + ( J β n + 1 t n + 1 H g ∗ ) ( t ) = γ n + 1 c 0 ∏ k = 1 n ξ k γ k + ∑ k = 1 n ⨂ k n ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 n η k ⨂ k n ( ξ , γ ) + ∑ k = 1 n + 1 ∏ k n ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 n θ k ∏ k < j n ξ j γ j + η n + 1 + ( J β n + 1 t n + 1 H g ∗ ) ( t ) ,

which implies that the second part of (10) holds for t ∈ [ t n + 1 , s n + 1 ) . In addition, suppose that the second part of (10) is fulfilled when i = n , that is, t ∈ [ t n , s n ) . Thus, in the consecutive interval [ s n , t n + 1 ) , we get

x ( t ) = x ( s n + ) + ( I q n α n s n f ∗ ) ( t ) = ξ n x ( s n − ) + θ n + ( I q n α n s n f ∗ ) ( t ) = ξ n γ n c 0 ∏ k = 1 n − 1 ξ k γ k + ∑ k = 1 n − 1 ⨂ k n − 1 ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 n − 1 η k ⨂ k n − 1 ( ξ , γ ) + ∑ k = 1 n ∏ k n − 1 ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 n − 1 θ k ∏ k < j n − 1 ξ j γ j + η n + ( J β n t n H g ∗ ) ( s n − ) } + θ n + ( I q n α n s n f ∗ ) ( t ) = ξ n γ n c 0 ∏ k = 1 n − 1 ξ k γ k + ∑ k = 1 n − 1 ⨂ k n − 1 ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 n − 1 η k ⨂ k n − 1 ( ξ , γ ) + ∑ k = 1 n ∏ k n − 1 ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 n − 1 θ k ∏ k < j n − 1 ξ j γ j + ξ n η n + ξ n ( J β n t n H g ∗ ) ( s n − ) + θ n + ( I q n α n s n f ∗ ) ( t ) = c 0 ∏ k = 1 n ξ k γ k + ∑ k = 1 n ⨂ k n ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 n η k ⨂ k n ( ξ , γ ) + ∑ k = 1 n ∏ k n ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 n θ k ∏ k < j n ξ j γ j + ( I q n α n s n f ∗ ) ( t ) .

Therefore, the first part of (10) is true. Hence, the formula (10) holds for all t ∈ [ 0 , T ] .

To take the boundary condition in (6), we put t = T in (10) with i = m . Then, we obtain

(11) x ( T ) = c 0 ∏ k = 1 m ξ k γ k + ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + ∑ k = 1 m η k ⨂ k m ( ξ , γ ) + ∑ k = 1 m ∏ k m ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + ∑ k = 1 m θ k ∏ k < j m ξ j γ j + ( I q m α m s m f ∗ ) ( T ) = c 0 ∏ k = 1 m ξ k γ k + ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) + Λ 1 ( m ) + ∑ k = 1 m + 1 ∏ k m ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) + Λ 2 ( m ) ,

which leads to

(12) c 0 = 1 Ω λ 3 − λ 2 ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H g ∗ ) ( s k − ) − λ 2 Λ 1 ( m ) − λ 2 ∑ k = 1 m + 1 ∏ k m ξ k γ k ( I q k − 1 α k − 1 s k − 1 f ∗ ) ( t k − ) − λ 2 Λ 2 ( m ) .

By substituting the constant c 0 , (12), into (10), we obtain a unique solution of linear mixed fractional quantum and Hadamard impulsive boundary value problem (6).

Conversely, for t ∈ [ s i , t i + 1 ) , we take the operator D q i α i s i on (10), and then we obtain D q i α i s i x ( t ) = f ∗ ( t ) . Applying the Hadamard fractional derivative D β i t i H to the second equation of (10), we have D β i t i H x ( t ) = g ∗ ( t ) . Replacing i by i − 1 and putting t by t i − in the first equation of (10), we get x ( t i + ) after multiplying γ i and adding η i , respectively. Then, x ( s i + ) is obtained by putting t = s i − , and respectively, multiplying ξ i , adding θ i , in the second equation of (10). The value of λ 3 can be computed by putting i = 0 with multiplying λ 1 and i = m , t = T with multiplying λ 2 , in the first equation of (10). Therefore, the proof is completed.□

3 Existence and uniqueness results

In the first step of this section, we define the spaces of functions and the operator, which are related to the investigated problem (1). Let J be an interval defined in (1) and let P C ( J , R ) and P C 1 ( J , R ) be the spaces of piecewise continuous functions defined by P C ( J , R ) = { x : J → R ; x ( t ) is continuous everywhere except for some t k , at which x ( t k + ) and x ( t k − ) exist and x ( t k − ) = x ( t k ) , k = 1 , 2 , … , m } , and P C 1 ( J , R ) = { x ∈ P C ( J , R ) ; x ′ ( t ) is continuous everywhere except for some t k at which x ′ ( t k + ) and x ′ ( t k − ) exist and x ′ ( t k − ) = x ′ ( t k ) , k = 1 , 2 , … , m } . Let E = P C ( J , R ) ∩ P C 1 ( J , R ) be the Banach space with norm ‖ x ‖ = sup { ∣ x ( t ) ∣ , t ∈ J } and now we define the operator on E by

Q x ( t ) = 1 Ω ∏ k = 1 i ξ k γ k λ 3 − λ 2 ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H g x ) ( s k − ) − λ 2 Λ 1 ( m ) − λ 2 ∑ k = 1 m + 1 ∏ k m ξ k γ k ( I q k − 1 α k − 1 s k − 1 f x ) ( t k − ) − λ 2 Λ 2 ( m ) + ∑ k = 1 i ⨂ k i ( ξ , γ ) ( J β k t k H g x ) ( s k − ) + Λ 1 ( i ) + ∑ k = 1 i ∏ k i ξ k γ k ( I q k − 1 α k − 1 s k − 1 f x ) ( t k − ) + Λ 2 ( i ) + ( I q i α i s i f x ) ( t ) , t ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , γ i Ω ∏ k = 1 i − 1 ξ k γ k λ 3 − λ 2 ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H g x ) ( s k − ) − λ 2 Λ 1 ( m ) − λ 2 ∑ k = 1 m + 1 ∏ k m ξ k γ k ( I q k − 1 α k − 1 s k − 1 f x ) ( t k − ) − λ 2 Λ 2 ( m ) + γ i ∑ k = 1 i − 1 ⨂ k i − 1 ( ξ , γ ) ( J β k t k H g x ) ( s k − ) + Λ 1 ( i − 1 ) + ∑ k = 1 i ∏ k i − 1 ξ k γ k ( I q k − 1 α k − 1 s k − 1 f x ) ( t k − ) + Λ 2 ( i − 1 ) + η i + ( J β i t i H g x ) ( t ) , t ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m ,

where abbreviations f x ( t ) , g x ( t ) mean f x ( t ) = f ( t , x ( t ) ) and g x ( t ) = g ( t , x ( t ) ) , respectively. By applying the Banach contraction mapping principle, and Leray-Schauder nonlinear alternative, we are in the position to prove the existence and uniqueness of solutions of the problem (1). The following constants

Φ 1 = ∣ λ 3 ∣ ∣ Ω ∣ ∏ k = 1 m ∣ ξ k γ k ∣ , Φ 2 = ∣ λ 2 ∣ ∣ Ω ∣ ∏ k = 1 m ∣ ξ k γ k ∣ + 1 , Φ 3 = ∑ k = 1 m ⨂ k m ( ξ , γ ) [ log ( s k ) − log ( t k ) ] β k Γ ( β k + 1 ) , Φ 4 = ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( t k − s k − 1 ) α k − 1 Γ q k − 1 ( α k − 1 + 1 ) ,

will be used in our proofs.

Theorem 3.1

Suppose that the two nonlinear functions f : J 1 × R → R and g : J 2 × R → R satisfy the following condition:

  1. There exist constants L 1 , L 2 > 0 such that for all x , y ∈ R ,

    ∣ f ( t , x ) − f ( t , y ) ∣ ≤ L 1 ∣ x − y ∣ , t ∈ J 1 and ∣ g ( t , x ) − g ( t , y ) ∣ ≤ L 2 ∣ x − y ∣ , t ∈ J 2 .

If ( L 1 Φ 4 + L 2 Φ 3 ) Φ 2 < 1 , then the mixed fractional quantum and Hadamard derivatives impulsive boundary value problem (1) has a unique solution on J .

Proof

Let B r be a ball of radius r > 0 , subset of E , defined by B r = { x ∈ E : ‖ x ‖ ≤ r } , where r satisfies

(13) r ≥ Φ 1 + ( ∣ Λ 1 ( m ) ∣ + ∣ Λ 2 ( m ) ∣ + M 1 Φ 4 + M 2 Φ 3 ) Φ 2 1 − ( L 1 Φ 4 + L 2 Φ 3 ) Φ 2 .

We will prove that Q B r ⊂ B r . We set M 1 = sup { ∣ f ( t , 0 ) ∣ , t ∈ J 1 } and M 2 = sup { ∣ g ( t , 0 ) ∣ , t ∈ J 2 } . For t ∈ J 1 and t ∈ J 2 , respectively, we have

∣ Q x ( t ) ∣ ≤ 1 ∣ Ω ∣ ∏ k = 1 i ∣ ξ k γ k ∣ ∣ λ 3 ∣ + ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ∣ g x ∣ ) ( s k − ) + ∣ λ 2 ∣ ∣ Λ 1 ( m ) ∣ + ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x ∣ ) ( t k − ) + ∣ λ 2 ∣ ∣ Λ 2 ( m ) ∣ + ∑ k = 1 i ⨂ k i ( ξ , γ ) ( J β k t k H ∣ g x ∣ ) ( s k − ) + ∣ Λ 1 ( i ) ∣ + ∑ k = 1 i ∏ k i ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x ∣ ) ( t k − ) + ∣ Λ 2 ( i ) ∣ + ( I q i α i s i ∣ f x ∣ ) ( t ) ,

for t ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , and

∣ Q x ( t ) ∣ ≤ ∣ γ i ∣ ∣ Ω ∣ ∏ k = 1 i − 1 ∣ ξ k γ k ∣ ∣ λ 3 ∣ + ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ∣ g x ∣ ) ( s k − ) + ∣ λ 2 ∣ ∣ Λ 1 ( m ) ∣ + ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x ∣ ) ( t k − ) + ∣ λ 2 ∣ ∣ Λ 2 ( m ) ∣ + ∣ γ i ∣ ∑ k = 1 i − 1 ⨂ k i − 1 ( ξ , γ ) ( J β k t k H ∣ g x ∣ ) ( s k − ) + ∣ Λ 1 ( i − 1 ) ∣ + ∑ k = 1 i ∏ k i − 1 ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x ∣ ) ( t k − ) + ∣ Λ 2 ( i − 1 ) ∣ + ∣ η i ∣ + ( J β i t i H ∣ g x ∣ ) ( t )

for t ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m . From triangle inequality and ( H 1 ), we obtain ∣ f x ( t ) ∣ = ∣ f ( t , x ) ∣ ≤ ∣ f ( t , x ) − f ( t , 0 ) ∣ + ∣ f ( t , 0 ) ∣ ≤ L 1 r + M 1 and ∣ g x ( t ) ∣ = ∣ g ( t , x ) ∣ ≤ ∣ g ( t , x ) − g ( t , 0 ) ∣ + ∣ g ( t , 0 ) ∣ ≤ L 2 r + M 2 . Consequently, we have

sup t ∈ J ∣ Q x ( t ) ∣ ≤ 1 ∣ Ω ∣ ∏ k = 1 m ∣ ξ k γ k ∣ ∣ λ 3 ∣ + ∣ λ 2 ∣ ∣ Λ 1 ( m ) ∣ + ∣ λ 2 ∣ ∣ Λ 2 ( m ) ∣ + ( L 2 r + M 2 ) ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ( 1 ) ) ( s k − ) + ( L 1 r + M 1 ) ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ( 1 ) ) ( t k − ) + ( L 2 r + M 2 ) ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ( 1 ) ) ( s k − ) + ∣ Λ 1 ( m ) ∣ + ( L 1 r + M 1 ) ∑ k = 1 m ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ( 1 ) ) ( t k − ) + ∣ Λ 2 ( m ) ∣ + ( L 1 r + M 1 ) ( I q m α m s m ( 1 ) ) ( t m + 1 ) = Φ 1 + ∣ Λ 1 ( m ) ∣ Φ 2 + ∣ Λ 2 ( m ) ∣ Φ 2 + ( L 1 r + M 1 ) Φ 2 Φ 4 + ( L 2 r + M 2 ) Φ 2 Φ 3 = Φ 1 + ∣ Λ 1 ( m ) ∣ Φ 2 + ∣ Λ 2 ( m ) ∣ Φ 2 + r Φ 2 ( L 1 Φ 4 + L 2 Φ 3 ) + Φ 2 ( M 1 Φ 4 + M 2 Φ 3 ) ≤ r .

Thus, ‖ Q x ‖ ≤ r , where r satisfies (13). Therefore, Q B r ⊂ B r holds. In the next step, we will show that Q is a contraction operator. For any x , y ∈ B r , we have

∣ Q x ( t ) − Q y ( t ) ∣ ≤ 1 ∣ Ω ∣ ∏ k = 1 i ∣ ξ k γ k ∣ ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ∣ g x − g y ∣ ) ( s k − ) + ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x − f y ∣ ) ( t k − ) + ∑ k = 1 i ⨂ k i ( ξ , γ ) ( J β k t k H ∣ g x − g y ∣ ) ( s k − ) + ∑ k = 1 i ∏ k i ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x − f y ∣ ) ( t k − ) + ( I q i α i s i ∣ f x − f y ∣ ) ( t ) ,

for t ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , and

∣ Q x ( t ) − Q y ( t ) ∣ ≤ ∣ γ i ∣ ∣ Ω ∣ ∏ k = 1 i − 1 ∣ ξ k γ k ∣ ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ∣ g x − g y ∣ ) ( s k − ) + ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x − f y ∣ ) ( t k − ) + ∣ γ i ∣ ∑ k = 1 i − 1 ⨂ k i − 1 ( ξ , γ ) ( J β k t k H ∣ g x − g y ∣ ) ( s k − ) + ∑ k = 1 i ∏ k i − 1 ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x − f y ∣ ) ( t k − ) + ( J β i t i H ∣ g x − g y ∣ ) ( t ) ,

for t ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m . Then, we have

sup t ∈ J ∣ Q x ( t ) − Q y ( t ) ∣ ≤ 1 ∣ Ω ∣ ∏ k = 1 m ∣ ξ k γ k ∣ ( L 2 ‖ x − y ‖ ) ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ( 1 ) ) ( s k − ) + ( L 1 ‖ x − y ‖ ) ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ( 1 ) ) ( t k − ) + ( L 2 ‖ x − y ‖ ) ∑ k = 1 i ⨂ k i ( ξ , γ ) ( J β k t k H ( 1 ) ) ( s k − ) + ( L 1 ‖ x − y ‖ ) ∑ k = 1 i ∏ k i ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ( 1 ) ) ( t k − ) = ( L 1 Φ 4 + L 2 Φ 3 ) Φ 2 ‖ x − y ‖ ,

which leads to ‖ Q x − Q y ‖ ≤ ( L 1 Φ 4 + L 2 Φ 3 ) Φ 2 ‖ x − y ‖ . Since ( L 1 Φ 4 + L 2 Φ 3 ) Φ 2 < 1 , then, Q is a contraction. Hence, we can conclude, by using Banach’s contraction mapping principle, that the operator Q has a fixed point, which is the solution of the mixed fractional quantum and Hadamard derivatives impulsive boundary value problem (1). This completes the proof.□

The next theorem of Leray-Schauders nonlinear alternative will be used to prove our existence result.

Theorem 3.2

Given E is a Banach space, and B is a closed, convex subset of E . In addition, let G be an open subset of B such that 0 ∈ G . Suppose that Q : G ¯ → B is a continuous, compact (i.e., Q ( G ¯ ) is a relatively compact subset of B ) map. Then either

  1. Q has a fixed point in G ¯ , or

  2. there is a x ∈ ∂ G (the boundary of G in B ) and κ ∈ ( 0 , 1 ) with x = κ Q ( x ) .

Theorem 3.3

Assume that f : J 1 × R → R and g : J 2 × R → R are given functions. Further, we suppose that:

  1. There exist two continuous nondecreasing functions Ψ , Θ : [ 0 , ∞ ) → ( 0 , ∞ ) and continuous functions u , v : J → R + , such that

    ∣ f ( t , x ) ∣ ≤ u ( t ) Ψ ( ∣ x ∣ ) , for each ( t , x ) ∈ J 1 × R ,

    and

    ∣ g ( t , x ) ∣ ≤ v ( t ) Θ ( ∣ x ∣ ) , for each ( t , x ) ∈ J 2 × R .

  2. There exists a positive constant N , such that

    N Φ 1 + ∣ Λ 1 ( m ) ∣ Φ 2 + ∣ Λ 2 ( m ) ∣ Φ 2 + ‖ v ‖ Θ ( N ) Φ 2 Φ 3 + ‖ u ‖ Ψ ( N ) Φ 2 Φ 4 > 1 .

Then, the mixed fractional quantum and Hadamard derivatives impulsive boundary value problem (1) has at least one solution on J .

Proof

Define a ball B ρ = { x ∈ E : ‖ x ‖ ≤ ρ } . It is easy to see that the ball B ρ is a closed, convex subset of E . To use the result in Theorem 3.2, we first will claim that Q is a continuous operator. Define a convergent sequence { x n } such that { x n } → x . Then, we obtain

∣ Q x n ( t ) − Q x ( t ) ∣ ≤ 1 ∣ Ω ∣ ∏ k = 1 i ∣ ξ k γ k ∣ ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ∣ g x n − g x ∣ ) ( s k − ) + ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x n − f x ∣ ) ( t k − ) + ∑ k = 1 i ⨂ k i ( ξ , γ ) ( J β k t k H ∣ g x n − g x ∣ ) ( s k − ) + ∑ k = 1 i ∏ k i ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x n − f x ∣ ) ( t k − ) + ( I q i α i s i ∣ f x n − f x ∣ ) ( t ) → 0 , as n → ∞ ,

for t ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , and

∣ Q x n ( t ) − Q x ( t ) ∣ ≤ ∣ γ i ∣ ∣ Ω ∣ ∏ k = 1 i − 1 ∣ ξ k γ k ∣ ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ∣ g x n − g x ∣ ) ( s k − ) + ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x n − f x ∣ ) ( t k − ) + ∣ γ i ∣ ∑ k = 1 i − 1 ⨂ k i − 1 ( ξ , γ ) ( J β k t k H ∣ g x n − g x ∣ ) ( s k − ) + ∑ k = 1 i ∏ k i − 1 ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ∣ f x n − f x ∣ ) ( t k − ) + ( J β i t i H ∣ g x n − g x ∣ ) ( t ) → 0 , as n → ∞ ,

for t ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m . Then, from the aforementioned two inequalities, the operator Q is continuous.

To prove the compactness of the operator Q , we suppose that x ∈ B ρ and then

(14) ∣ Q x ( t ) ∣ ≤ 1 ∣ Ω ∣ ∏ k = 1 m ∣ ξ k γ k ∣ ∣ λ 3 ∣ + ∣ λ 2 ∣ ∣ Λ 1 ( m ) ∣ + ∣ λ 2 ∣ ∣ Λ 2 ( m ) ∣ + ‖ v ‖ Θ ( ρ ) ∣ λ 2 ∣ ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ( 1 ) ) ( s k − ) + ‖ u ‖ Ψ ( ρ ) ∣ λ 2 ∣ ∑ k = 1 m + 1 ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ( 1 ) ) ( t k − ) + ‖ v ‖ Θ ( ρ ) ∑ k = 1 m ⨂ k m ( ξ , γ ) ( J β k t k H ( 1 ) ) ( s k − ) + ∣ Λ 1 ( m ) ∣ + ‖ u ‖ Ψ ( ρ ) ∑ k = 1 m ∏ k m ∣ ξ k γ k ∣ ( I q k − 1 α k − 1 s k − 1 ( 1 ) ) ( t k − ) + ∣ Λ 2 ( m ) ∣ + ‖ u ‖ Ψ ( ρ ) ( I q m α m s m ( 1 ) ) ( t m + 1 ) = Φ 1 + ∣ Λ 1 ( m ) ∣ Φ 2 + ∣ Λ 2 ( m ) ∣ Φ 2 + ‖ v ‖ Θ ( ρ ) Φ 2 Φ 3 + ‖ u ‖ Ψ ( ρ ) Φ 2 Φ 4 ≔ Φ 5 ,

which implies that ‖ Q x ‖ ≤ Φ 5 . Hence, Q ( B ρ ) is a uniformly bounded set. To prove the equicontinuity of the set Q ( B ρ ) , let t 1 , t 2 ∈ [ 0 , T ] such that t 1 < t 2 . Then, for any x ∈ B ρ , we obtain

∣ Q x ( t 2 ) − Q x ( t 1 ) ∣ = ∣ ( I q i α i s i f ) ( t 2 ) − ( I q i α i s i f ) ( t 1 ) ∣ ≤ ‖ u ‖ Ψ ( ρ ) ∣ ( I q i α i s i ( 1 ) ) ( t 2 ) − ( I q i α i s i ( 1 ) ) ( t 1 ) ∣ = ‖ u ‖ Ψ ( ρ ) Γ q i ( α i + 1 ) [ ( t 2 − s i ) α i − ( t 1 − s i ) α i ] → 0

as t 1 → t 2 when t 1 , t 2 ∈ [ s i , t i + 1 ) , i = 0 , 1 , 2 , … , m , and

∣ Q x ( t 2 ) − Q x ( t 1 ) ∣ = ∣ ( J β i t i H g ) ( t 2 ) − ( J β i t i H g ) ( t 1 ) ∣ ≤ ‖ v ‖ Θ ( ρ ) ∣ ( J β i t i H ( 1 ) ) ( t 2 ) − ( J β i t i H ( 1 ) ) ( t 1 ) ∣ = ‖ v ‖ Θ ( ρ ) Γ ( β i + 1 ) [ [ log ( t 2 ) − log ( t i ) ] β i − [ log ( t 1 ) − log ( t i ) ] β i ] → 0

as t 1 → t 2 , when t 1 , t 2 ∈ [ t i , s i ) , i = 1 , 2 , 3 , … , m . Since the aforementioned two inequalities are convergent to zero independently of x , we can conclude that Q ( B ρ ) is an equicontinuous set. Then, we get that Q B ρ is relatively compact. By applying the Arzelá-Ascoli theorem, the operator Q is completely continuous.

In the final step, we will show that the second condition of Theorem 3.2 does not hold. Assume that x is a solution of the problem (1). Then, for fixed κ ∈ ( 0 , 1 ) , we have x = κ Q x , which implies

‖ x ‖ Φ 1 + ∣ Λ 1 ( m ) ∣ Φ 2 + ∣ Λ 2 ( m ) ∣ Φ 2 + ‖ v ‖ Θ ( ‖ x ‖ ) Φ 2 Φ 3 + ‖ u ‖ Ψ ( ‖ x ‖ ) Φ 2 Φ 4 ≤ 1 ,

by the same method to get (14). From the hypothesis ( H 3 ), there exists a positive constant N , which satisfies ‖ x ‖ ≠ N . Now, we define the open subset of B ρ by G = { x ∈ B ρ : ‖ x ‖ < N } . Note that 0 ∈ G and Q ( G ¯ ) is a relatively compact subset of B ρ . In addition, it is impossible that x ∈ ∂ G such that x = κ Q x for some κ ∈ ( 0 , 1 ) . Therefore, by applying the result in ( i ) of Theorem 3.2, the operator Q has a fixed point x ∈ G ¯ , which is a solution of the problem (1) on J . The proof is completed.□

If condition ( H 2 ) in Theorem 3.3 is replaced by

(15) ∣ f ( t , x ) ∣ ≤ u ( t ) ( a 1 ∣ x ∣ + b 1 ) and ∣ g ( t , x ) ∣ ≤ v ( t ) ( a 2 ∣ x ∣ + b 2 ) ,

where a 1 , a 2 ≥ 0 and b 1 , b 2 > 0 for all ( t , x ) ∈ J × R , then we get the following Corollary.

Corollary 3.1

If (15) holds and if ( ‖ v ‖ a 2 Φ 3 + ‖ u ‖ a 1 Φ 4 ) Φ 2 < 1 , then the problem (1) has at least one solution on J .

In addition, if a 1 = a 2 = 0 , that is, f and g are bounded functions, then the problem (1) has at least one solution on J .

Finally, we consider the existence criteria for the initial vale problem. If we replace λ 1 = 1 and λ 2 = 0 in problem (1), that is,

(16) x ( 0 ) = λ 3 ,

then we obtain an impulsive initial value problem and we also get constants Ω = 1 , Φ 1 = ∣ λ 3 ∣ ∏ k = 1 m ∣ ξ k γ k ∣ ≔ Φ 1 ∗ , and Φ 2 = 1 .

Corollary 3.2

If ( H 1 ) holds and L 1 Φ 4 + L 2 Φ 3 < 1 , then the initial value problem (1)–(16) has a unique solution on J .

Corollary 3.3

If condition ( H 2 ) is satisfied and if there exists a constant N > 0 such that

N Φ 1 ∗ + ∣ Λ 1 ( m ) ∣ + ∣ Λ 2 ( m ) ∣ + ‖ v ‖ Θ ( N ) Φ 3 + ‖ u ‖ Ψ ( N ) Φ 4 > 1 ,

then the initial value problem (1)–(16) has at least one solution on J .

4 Examples

In this section, we give some examples to illustrate the usefulness of our main results.

Example 4.1

Consider the following mixed fractional quantum and Hadamard derivatives impulsive boundary value problem of the form:

(17) D 3 i + 2 3 i + 3 i + 2 i + 3 ( 2 i ) x ( t ) = f ( t , x ( t ) ) , t ∈ [ 2 i , 2 i + 1 ) , i = 0 , 1 , 2 , 3 , D 2 i + 2 2 i + 3 ( 2 i − 1 ) H x ( t ) = g ( t , x ( t ) ) , t ∈ [ 2 i − 1 , 2 i ) , i = 1 , 2 , 3 , x ( ( 2 i − 1 ) + ) = i + 1 i + 2 x ( ( 2 i − 1 ) − ) + 2 i + 1 2 i + 2 , x ( ( 2 i ) + ) = i + 3 i + 4 x ( ( 2 i ) − ) + 2 i + 3 2 i + 4 , i = 1 , 2 , 3 , 1 7 x ( 0 ) + 2 11 x ( 7 ) = 3 13 .

Here, we set α i = ( i + 2 ) / ( i + 3 ) , q i = ( 3 i + 2 ) / ( 3 i + 3 ) , i = 0 , 1 , 2 , 3 , and β i = ( 2 i + 2 ) / ( 2 i + 3 ) , i = 1 , 2 , 3 . Since [ 2 i , 2 i + 1 ) ∪ [ 2 i − 1 , 2 i ) ∪ { 7 } , for i = 0 , 1 , 2 , 3 , then we have J = [ 0 , 7 ] . Further, we let γ i = ( i + 1 ) / ( i + 2 ) , η i = ( 2 i + 1 ) / ( 2 i + 2 ) , ξ i = ( i + 3 ) / ( i + 4 ) , θ i = ( 2 i + 3 ) / ( 2 i + 4 ) , i = 1 , 2 , 3 . Then, previous constants can be used to compute that Ω ≈ 0.1844155844 , Λ 1 ( 3 ) ≈ 1.4833333333 , Λ 2 ( 3 ) ≈ 1.8571428571 , Φ 1 ≈ 0.7436619719 , Φ 2 ≈ 0.4957746479 , Φ 3 ≈ 0.5631729043 , and Φ 4 ≈ 2.1950314210 .

  1. If the two functions f , g are given, respectively, for t ∈ [ 0 , 7 ] , by

    (18) f ( t , x ) = e − t t + 4 x 2 + 2 ∣ x ∣ ( 1 + ∣ x ∣ ) + 3 5 ,

    (19) g ( t , x ) = 1 + 2 cos π t 2 sin ∣ x ∣ + 1 3 ,

    then, from (18)–(19), we obtain ∣ f ( t , x ) − f ( t , y ) ∣ ≤ ( 1 / 2 ) ∣ x − y ∣ and ∣ g ( t , x ) − g ( t , y ) ∣ ≤ ( 3 / 2 ) ∣ x − y ∣ , for x , y ∈ R . By choosing L 1 = 1 / 2 and L 2 = 3 / 2 , we get the relation ( L 1 Φ 4 + L 2 Φ 3 ) Φ 2 ≈ 0.9629307370 < 1 . Theorem 3.1 can be applied to conclude that the mixed fractional quantum and Hadamard derivatives impulsive boundary value problem (17) with f and g given by (18) and (19) has a unique solution on [0,7].

  2. Let the two nonlinear functions f , g are defied on [ 0 , 7 ] as follows:

    (20) f ( t , x ) = 1 + cos π t t + 30 x 26 1 + x 24 + 1 ,

    (21) g ( t , x ) = e − t 2 12 + t ∣ x ∣ 15 1 + ∣ x ∣ 13 + 2 .

    It is obvious to see that

    ∣ f ( t , x ) ∣ ≤ 1 + cos π t t + 30 ( x 2 + 1 )

    and

    ∣ g ( t , x ) ∣ ≤ e − t 2 12 + t ( x 2 + 2 ) .

    Then, we choose for u ( t ) = ( 1 + cos π t ) / ( t + 30 ) , v ( t ) = e − t 2 / ( 12 + t ) , Ψ ( x ) = x 2 + 1 and Θ ( x ) = x 2 + 2 . Thus, we can find that ‖ u ‖ = 1 / 15 and ‖ v ‖ = 1 / 12 and there exists a constant N ∈ ( 4.247616499 , 6.188984869 ) satisfying condition ( H 3 ) . Therefore, applying the result in Theorem 3.3, the impulsive boundary value problem (17) with f and g given by (20) and (21) has at least solution on [ 0 , 7 ] .

  3. If the terms x 26 and ∣ x ∣ 15 in (20) and (21) are replaced by ∣ x ∣ 25 and x 14 , then these are as follows:

    (22) f ( t , x ) = 1 + cos π t t + 30 ∣ x ∣ 25 1 + x 24 + 1 ,

    (23) g ( t , x ) = e − t 2 12 + t x 14 1 + ∣ x ∣ 13 + 2 .

Therefore, we have

∣ f ( t , x ) ∣ ≤ 1 + cos π t t + 30 ( ∣ x ∣ + 1 ) and ∣ g ( t , x ) ∣ ≤ e − t 2 12 + t ( ∣ x ∣ + 2 ) .

Since ‖ v ‖ a 2 Φ 3 + ‖ u ‖ a 1 Φ 4 ≈ 0.1932665035 < 1 , then the problem (17) with f and g given by (22) and (23) has at least one solution on [ 0 , 7 ] , by using Corollary 3.1.

5 Conclusion

In this paper, we initiated the study of a new class of impulsive boundary value problems involving mixed fractional quantum and fractional Hadamard derivatives. Existence and uniqueness results are established by using tools from the fixed point theory. The uniqueness result is proved via Banach’s contraction mapping principle, while the Leray-Schauder nonlinear alternative is used to establish an existence result. Examples illustrating the obtained results are also included. The results of the paper are new and enrich the existing literature on impulsive boundary value problems.

  1. Funding information: This research was funded by Thailand Research and Innovation Fund. Contact no. FRB640064.

  2. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-08-02
Revised: 2021-09-29
Accepted: 2021-10-27
Published Online: 2021-12-31

© 2021 Somboon Niyoom 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-2021-0119
Keywords for this article
fractional impulsive differential equations; instantaneous impulses; impulses; quantum calculus
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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