$(\mathtt{k},\varphi )$-Hilfer fractional Langevin differential equation having multipoint boundary conditions

Abstract

The primary objective of this manuscript is to investigate the existence and uniqueness of solutions for the Langevin $(\mathtt{k},\varphi )$-Hilfer fractional differential equation of different orders with multipoint nonlocal fractional integral boundary conditions. We consider the generalized version of the Hilfer fractional diferential equation called as $(\mathtt{k},\varphi )$-Hilfer fractional differential equation. We provide some significant outcomes about $(\mathtt{k},\varphi )$-Hilfer fractional Langevin differential equation that requires deriving equivalent fractional integral equation to $(\mathtt{k},\varphi )$-Hilfer Langevin fractional differential equation. The existence result is established using the Krasnoselskii’s fixed-point theorem, while the uniqueness is addressed with the help of Banach contraction principle. Additionally, we investigate the different forms of Ulam stability for the solution of the mentioned problem, under specific conditions. To validate our main outcomes, we present a detailed example at the end of the manuscript.

1 Introduction

Equations containing fractional derivatives are referred to as fractional differential equations. They have become important recently, with rich theoretical implications and extensive physical foundations. Fractional derivatives and integrals find extensive uses in a variety of fields, including chemistry, biology, physics and finance. The updated explanation of nonlocal and nonlinear fractional boundary value problems (BVPs) can be found in [2, 4, 7, 8, 12, 15, 19, 23, 24, 29, 35].

There are different fractional integral operators, which are typically applied to get the solutions of fractional differential equations. Among the mentioned operators, the Riemann–Liouville (RL), Erdelyi–Kober, Caputo, Hadamard, and Hilfer fractional operators [26] are the most important. In [7], Sousa and Oliveira have presented a new definition of fractional derivative with respect to another φ function, called φ-Hilfer fractional derivative. For more related information, we recommend [1, 25, 37, 38]. Daiz and Pariguan [9] introduced the $\mathtt{k}$-gamma function $\Gamma _{\mathtt{k}}(z)={\int _{0}^{\infty}{s}^{z-1}{e}^{ \frac{{-s}^{\mathtt{k}}}{\mathtt{k}}}}\mathtt {d}s, \quad {\mathtt{k}}>0$, which is a generalized Euler’s gamma function with additional parameter $\mathtt{k}$. Mubeen and Habibullah [27] introduced the k-Riemann–Liouville fractional integral operator. Romero [32], introduced a generalized version of $\mathtt{k}$-Riemann–Liouville fractional derivative operators. For more information on the φ-Riemann–Liouville fractional integral operators, see [10]. Kucchi and Mali [20] have presented the most generalized version of a $(\mathtt{k}, \varphi )$-Caputo and $(\mathtt{k}, \varphi )$-Riemann–Liouville fractional derivative. For more related results, we recommend [6, 34].

Since the fractional Langevin equation is regarded as a component of fractional calculus, significant findings have been clarified. An equation of the form $m{\mathtt {d}^{2}}$z/d${t}^{2}={\lambda \mathtt {d}z/ \mathtt {d}{t}+\varrho (t)}$ is known as Langevin equation, investigated by Paul Langevin in 1908. It has been found that the Langevin equation is a useful tool for describing how physical events evolve in a changing environment; for more details, see [3, 11, 22, 31, 33].

The theory relating to the stability of differential equations is known as stability analysis, which has been well investigated for differential equations of arbitrary order and have been extensively explored by researchers. Ulam established the Ulam stability [39] in 1940, and Hyers and Rassias [16, 30] extended it. This concept of stability was named as Ulam–Hyers (UH) and Ulam–Hyers–Rassias (UHR) stability [17, 18, 31, 40].

In [20], the authors analyzed the existence and uniqueness results with the help of Banach’s fixed point theorem for the initial value problem, containing the $(\mathtt{k},\varphi )$-Hilfer fractional derivative:

where is the $({\mathtt{k}},\varphi )$-Hilfer fractional derivative of order α, g is an appropriate real-valued function, and parameter μ, ${0\leq \mu \leq 1,}$ is a continuous function.

In [36], Sitho et al. investigated the system

where ${1<\alpha \leq 2}$, is the φ-Hilfer fractional derivative of order α, with parameter μ, ${0<\mu \leq 1}$, and g is an appropriate function having a continuous derivative.

In [14], the authors have developed criteria to demonstrate the unique solutions of the specific nonlocal BVPs:

where ${\alpha _{i}}, \, 0<{\alpha _{i}}<1$, $^{H}D^{\alpha _{i},\mu _{i};\varphi}$, $i=1,2$, is a φ-Hilfer derivative of order ${\alpha _{i}}$ and parameter ${\mu _{i}}$, $0\leq{\mu _{i}}\leq 1,\, i=1,2, \, 1<{\alpha _{1}+\alpha _{2}}\leq 2, \, {\lambda}\in \mathbb{R}, \, \alpha \geq 0$, $I^{\alpha _{i}}$ is the Riemann–Liouville fractional integral of order ${{\alpha _{i}}>0}$, ${\xi _{i}}\in \mathbbm{{}\mathbb{R}\mathbbm{}}$, ${i}=1,2$, respectively, while g is an appropriate function.

In [28], the authors investigated the following $(\mathtt{k},\varphi )$-Hilfer nonlocal integro-multipoint fractional BVP:

where denotes the $({{\mathtt{k}},\varphi})$-Hilfer fractional derivative of order α, $1<{\alpha}<2$, $\mathtt{k}>0$, is the ${({\mathtt{k}},\varphi )}$-Riemann–Liouville fractional integral of order ${{\phi _{j}}>0}$, ${\xi _{i}},{\zeta _{j}}\in \mathbbm{{}\mathbb{R}\mathbbm{}}$ and $a<{\vartheta _{i}}$, ${\zeta _{j}}< a$, ${i}=1,2,\dots ,{q},\ {j}=1,2,\dots ,{p,}$ respectively.

Motivated by the above mentioned works, we analyze the $(\mathtt{k},\varphi )$-Hilfer–Langevin fractional differential equation having nonlocal integro-multipoint fractional boundary conditions:

(1.1)

where , $i=1,2$, is the $({\mathtt{k},\varphi})$-Hilfer fractional derivative of order ${\alpha _{i}}$, $0<{\alpha _{i}}<1$, and parameter ${\mu _{i}}$, $0\leq{\mu _{i}}\leq 1,\, i=1,2,\, 1<{\alpha _{1}+\alpha _{2}}\leq 2, \,{\lambda}\in \mathbbm{{}\mathbb{R}\mathbbm{}}$, respectively, $g:{(a,b]\times \mathbbm{{}\mathbb{R}\mathbbm{}}}\rightarrow{\mathbbm{{}\mathbb{R}\mathbbm{}}}$ is a continuous function; are the ${(\mathtt{k},\varphi )}$-Riemann–Liouville fractional integrals of order ${{\phi _{j}}>0}$, respectively, ${\xi _{i}},{\zeta _{j}}\in \mathbbm{{}\mathbb{R}\mathbbm{}}$ and $a<{\vartheta _{i}}$, ${\zeta _{j}}< b$, ${i}=1,2,\dots ,{q}, \ {j}=1,2,\dots ,{p}$.

We list the important aspects of this manuscript:

1: In this paper, we investigate nonlocal integro-multipoint fractional BVPs with parameters and differential equations using $(\mathtt{k},\varphi )$-Hilfer–Langevin fractional derivative. The combination is being used in the literature for the first time.

2: In this research, it has been shown that there exists a unique solution to the given problem (1.1). In addition to this, UH stability of the problem is also investigated.

The remainder of the paper is organized as follows: We recall preliminary results in Sect. 2. In Sect. 3, we establish the existence and uniqueness results for the single-valued $(\mathtt{k},\varphi )$-Hilfer nonlocal integro-multipoint BVP. Then we derive a solution for our system equation (1.1). We analyze the existence and uniqueness of solutions and various kinds of Ulam stability in Sect. 4. We offer an illustrative example to confirm the findings of our theoretical results in Sect. 5.

2 Preliminaries

We recall a few definitions and lemmas which will be used for our results.

Definition 2.1

([27])

The $\mathtt{k}$-Riemann–Liouville fractional integral of order α of the function $h\in{ \mathtt {L}^{1}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$, with $\mathtt{k},{\alpha}$ being positive real numbers, is defined as

where ${\Gamma _{\mathtt{k}}}$ is the ${\mathtt{k}}$-Gamma function. For $z\in \mathbb{C}$ with a positive real part and ${\mathtt{k}}\in \mathbbm{{}\mathbb{R}\mathbbm{}}$, the ${\mathtt{k}}$-Gamma function [9] is defined as

$$ \Gamma _{\mathtt{k}}(z)={\int _{0}^{\infty}{s}^{z-1}{e}^{ \frac{{-s}^{\mathtt{k}}}{\mathtt{k}}}}\mathtt {d}s, \quad {\mathtt{k}}>0. $$

Furthermore, the following relations hold:

$$ \Gamma (\phi )=\lim _{{\mathtt{k}}\rightarrow 1}{\Gamma _{\mathtt{k}}}( \phi ),\quad \Gamma _{\mathtt{k}}(\phi )={\mathtt{k}}^{ \frac{\phi}{{\mathtt{k}}}-1}\Gamma{\Big(\frac{\phi}{{\mathtt{k}}}} \Big),\quad \text{and} \quad \Gamma _{\mathtt{k}}(\phi +{\mathtt{k}})={ \phi}{\Gamma _{\mathtt{k}}(\phi )}. $$

Definition 2.2

([10])

The ${\mathtt{k}}$-Riemann–Liouville fractional derivative of order α for the function $h\in{ \mathtt {L}^{1}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$ is given as

$$ ^{{\mathtt{k}},{RL}}D_{a{+}}^{\alpha}h({t})=\Big({{\mathtt{k}}} \frac{{\mathtt {d}}}{{\mathtt {d}}{t}}\Big)^{{n}} { ^{{\mathtt{k}}}I_{a_{+}}^{ {n}\mathtt{k}-\alpha}} , \quad {n}=\Big[ \frac{\alpha}{{\mathtt{k}}}\Big]. $$

The expression $[\frac{\alpha}{\mathtt{k}}]$ is known as the ceiling function of $\frac{\alpha}{\mathtt{k}}$.

Definition 2.3

([19])

The φ-Riemann–Liouville fractional integral of order α for the function $h\in{ \mathtt {L}^{1}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$ corresponding to an increasing function $\varphi :[a,b]\rightarrow \mathbbm{{}\mathbb{R}\mathbbm{}}$, with ${\varphi '{(t)}}\neq 0$ for all $t\in [a,b]$, is defined as

$$ I_{a_{+}}^{\alpha ;\varphi}h(t)=\frac{1}{\Gamma (\alpha )}\int _{ {n}}^{t} {\varphi '({u})}\big({\varphi (t)-\varphi ({u})} \big)^{\alpha -1}h({u})\mathtt {d}{u}. $$

Definition 2.4

([19])

The φ-Riemann–Liouville fractional derivative of order α for the function $h\in \mathtt {C}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$, with ${n}-1<\alpha \leq {n}$, $\varphi \in {\mathtt {C}^{{n}}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$, $t\in [a,b]$, ${\varphi '{(t)}}\neq 0$, is defined as

$$ ^{RL}D_{a_{+}}^{\alpha ;\varphi}h({t})= \left [ \frac{1}{\varphi '({t})}\frac{\mathtt {d}}{\mathtt {d}t}\right ]^{ {n}} {I^{{n}-\alpha ;\varphi}_{a^{+}}}h({t}). $$

Definition 2.5

([5])

The φ-Caputo fractional derivative of order α for the function $h\in{ \mathtt {C}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$, with $\varphi \in {\mathtt {C}^{{n}}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$, ${n}-1<\alpha \leq {n}$, $t\in [a,b], {\varphi '{(t)}}\neq 0$, is presented as

$$ ^{C}D_{a_{+}}^{\alpha ;\varphi}h({t})={I^{{n}-\alpha ;\varphi}_{a^{+}}} \left [\frac{1}{\varphi '({t})}\frac{\mathtt {d}}{\mathtt {d}t}\right ]^{ {n}}h({t}). $$

Definition 2.6

([7])

The φ-Hilfer fractional derivative for the function $h\in{ \mathtt {C}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$ of order $\alpha \in ({n}-1,{n}]$ with $\mu \in [0,1]$, ${n}-1<\alpha \leq {n}$, $\varphi \in {\mathtt {C}^{{n}}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$, ${\varphi '(t)}\neq 0$, $t\in [a,b]$, is defined by

$$ ^{H}D_{a_{+}}^{\alpha ,\mu ;\varphi}h({t})={I^{\mu ({n}-\alpha ); \varphi}_{a^{+}}} \left [\frac{1}{\acute{\varphi}({t})} \frac{\mathtt {d}}{\mathtt {d}t}\right ]^{{n}}{I^{(1-\mu )({n}- \alpha );\varphi}_{a^{+}}}h({t}). $$

Definition 2.7

([21])

The $({\mathtt{k}},\varphi )$-Riemann–Liouville fractional integral of order α for the function $h\in{ \mathtt {L}^{1}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$ is defined by

Definition 2.8

([20])

The $({\mathtt{k}},\varphi )$-Hilfer fractional derivative (HFD) of order α and μ for the function $h\in{ \mathtt {C}^{{n}}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$, with $\alpha ,{\mathtt{k}}=(0,\infty )$, $\mu \in [0,1]$, $\varphi \in {\mathtt {C}^{{n}}}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$, ${{\varphi '{(t)}}}\neq 0$, $t\in [a,b]$, is given by

(2.1)

Remark 2.1

Note that:

1. When ${\mu =0}$, the $({\mathtt{k}},\varphi )$-Hilfer fractional derivative (2.1) reduces to ${({\mathtt{k}},\varphi )}$-Riemann–Liouville fractional derivative operator.

On the other hand, when ${\mu =1}$, the $({\mathtt{k}},\varphi )$-Hilfer fractional derivative (2.1) reduces to ${({\mathtt{k}},\varphi )}$-Caputo fractional derivative operator [20].

2. If ${\varphi ({t})}={t}^{\rho}$, then the $({\mathtt{k}},\varphi )$-Hilfer fractional derivative (2.1) reduces to ${\mathtt{k}}$-Hilfer–Katugampola fractional operator.

3. When we set ${{\varphi ({{t}})}}=\log{t}$, then $({\mathtt{k}},\varphi )$-HFD (2.1) becomes ${\mathtt{k}}$-Hilfer–Hadamard fractional derivative operator.

Remark 2.2

The ${({\mathtt{k}},\varphi )}$-Hilfer fractional derivative can be defined in the form of a ${({\mathtt{k}},\varphi )}$-Riemann–Liouville fractional derivative as

for $(1-\mu )({n}{{\mathtt{k}}}-\alpha )=n{\mathtt{k}}-\theta _{ \mathtt{k}}$. Note that for $\mu \in [0,1]$ and ${n}-1<{\frac{\alpha}{{\mathtt{k}}}}\leq \, {n}$, we have ${n}-1<{\frac{\theta _{\mathtt{k}}}{{\mathtt{k}}}}\leq {n}$.

Lemma 2.1

([20])

Let $\alpha _{i},{\mathtt{k}}\in (0,\infty )$ and ${n}=[{\frac{\alpha}{\mathtt{k}}}]$. Consider $h\in{ \mathtt {C}^{{n}}}\big([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}}\big)$ and . Then

Lemma 2.2

([20])

Let $\alpha ,{\mathtt{k}}\in (0,\infty )$ be such that $\alpha <{\mathtt{k}}$, $\mu \in [0,1]$, and $\theta _{\mathtt{k}}=\alpha +\mu ({\mathtt{k}}-\alpha )$. Then

Lemma 2.3

([20])

Suppose $\zeta ,{\mathtt{k}}\in (0,\infty )$ and $\vartheta \in \mathbbm{{}\mathbb{R}\mathbbm{}}$ so that ${\frac{\vartheta}{\mathtt{k}}}>-1$. Then

Lemma 2.4

([20])

Suppose $\alpha _{i},{\mathtt{k}}\in (0,\infty )\ (i=1,2)$. Then we have

Theorem 2.1

(Krasnoselskii’s fixed point theorem, [13])

Let p be a bounded, closed, convex and nonempty subset of a Banach space and let $\mathbf{A}, \,\mathbf{B}$ be operators such that

1. $\mathbf{A}z+\mathbf{B}y\in {p}$ whenever $z,y\in {p}$.

2. A is continuous and compact.

3. B is a contraction.

Then we can find $v\in {p}$ such that $v=\mathbf{A}v+\mathbf{B}v$.

3 Uniqueness and existence results

Lemma 3.1

Let $a< b$, $0<\alpha _{i}\leq 1$, $\mu _{i}\in [0,1]$, $\theta _{\mathtt{k}}=\alpha _{i}+\mu _{i}({\mathtt{k}}-\alpha _{i})$, ${i={1,2}}$, where ${\mathtt{k}}>0$, ${g\in \mathtt {C}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})}$, and

$$\begin{aligned} \Lambda ={}&\bigg( \frac{(\varphi (b)-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}-1}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2})} \bigg)-{\mathtt{k}}\sum _{i=1}^{{n}} \xi{i}\bigg( \frac{(\varphi (\vartheta )-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2}+{\mathtt{k}})} \bigg)\\ &{}-\bigg( \frac{(\varphi (z_{j})-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\phi _{j}+\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\phi _{j}+\alpha _{2})} \bigg). \end{aligned}$$

The solution of the Hilfer nonlocal integro-multipoint BVP,

(3.1)

is

Proof

Suppose $z({t})$ represents the solution of (1.1). Applying the fractional integral operator on both sides of equation (1.1), we get

(3.2)

where

By using Riemann–Liouville fractional integral of order $\alpha _{2}$ in equation (3.2), we have

where . Also

The above implies

(3.3)

where $c_{0}$ and $c_{1}$ are unknown arbitrary constants.

By using boundary conditions, $z(a)=0$ in equation (3.3), since $(\frac{\theta _{\mathtt{k}}}{\mathtt{k}})-1<0$ and, from Remark 2.2, we get $c_{1}=0$. Now, using the second boundary condition,

we get,

(3.4)

Putting ${t}=b$ in (3.4), we obtain

Similarly,

Now,

where

$$\begin{aligned} \Lambda ={}&\bigg( \frac{(\varphi (b)-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}-1}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2})} \bigg)-{\mathtt{k}}\sum _{i=1}^{{n}} \xi{i}\bigg( \frac{(\varphi (\vartheta )-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2}+{\mathtt{k}})} \bigg)\\ &{}-\bigg( \frac{(\varphi (z_{j})-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\phi _{j}+\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\phi _{j}+\alpha _{2})} \bigg) \end{aligned}$$

and

Putting the values of $c_{0}, c_{1}$ into (3.4), we obtain

 □

By Lemma 3.1, we define a function $\digamma :\sigma \rightarrow \sigma$ by

(3.5)

For further analysis, the following assumptions hold:

$\mathbf{(N_{3})}$:

$g:[a,b]\times \mathbbm{{}\mathbb{R}\mathbbm{}}\rightarrow \mathbbm{{}\mathbb{R}\mathbbm{}}$ is a continuous function such that $(g({t},z))\leq \sigma ({t})$, $\forall ({{t}},z)\in [a,b]\times \mathbbm{{}\mathbb{R}\mathbbm{}}$, with $\sigma \in \mathtt {C}([a,b];\mathbbm{{}\mathbb{R}\mathbbm{}})$.

$\mathbf{(N_{4})}$:

There exists a positive real constant L such that, for each ${{t}}\in [a,b]$ and $z,y\in \mathbbm{{}\mathbb{R}\mathbbm{}}$,

$$\begin{aligned} &|g({t},z)-g({t},y)|\leq \mathtt {L}|z-y|. \end{aligned}$$

$\mathbf{(N_{5})}$:

$r\in \mathtt {C}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$ is an increasing function such that there exists $\Omega _{r}>0$ satisfying

$$\begin{aligned} I^{\alpha _{1}+\alpha _{2};\varphi}r(t)\leq \Omega _{r} r(t). \end{aligned}$$

We define the Banach space of all continuous functions by $X=\mathtt {C}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$ and consider the norm $||z||=\sup \{|z({{t}})|;{{t}}\in [a,b]\}$. To simplify the computations, we use the following notations:

$$\begin{aligned} Q_{1}={}& \frac{(\varphi (b)-\varphi (a))^{\frac{\alpha _{1}+\alpha _{2}}{k}}}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+k)} \\ &{}+ \frac{(\varphi (b)-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}-1}}{{\Lambda}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2})}} \bigg[\sum _{i=1}^{{n}}\xi _{i}{ \frac{\mathtt{k}}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+2{\mathtt{k}})}} {(\varphi (\vartheta _{i})-\varphi (a))^{ \frac{\alpha _{1}+\alpha _{2}}{\mathtt{k}}+1}} \\ &{} +\sum _{j=1}^{{p}}\zeta _{j}{ \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+\phi _{j}+{\mathtt{k}})}} {(\varphi (z_{j})-\varphi (a))^{ \frac{\alpha _{1}+\alpha _{2}+\phi _{j}}{\mathtt{k}}}} \\ &{}- \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+{\mathtt{k}})}( \varphi (b)-\varphi (a))^{\frac{\alpha _{1}+\alpha _{2}}{\mathtt{k}}} \bigg], \end{aligned}$$

(3.6)

$$\begin{aligned} Q_{2}={}&|\lambda |\bigg[\bigg( \frac{(\varphi (b)-\varphi (a))^{\frac{\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\alpha _{2}+{\mathtt{k}})} \bigg) +\bigg( \frac{{(\varphi (b)}-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}-1}}{{\Lambda}\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2})} \bigg) \\ &{}\times \bigg[\sum _{i=1}^{{n}}\xi _{i}{ \frac{\mathtt{k}}{\Gamma _{\mathtt{k}}(\alpha _{2}+2{\mathtt{k}})}( \varphi (\vartheta _{i})-\varphi (a))^{\frac{\alpha _{2}}{\mathtt{k}}+1}} \\ &{}-\sum _{j=1}^{{p}}\zeta _{j}{ \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{2}+{\mathtt{k}})}(\varphi (z_{j})- \varphi (a))^{\frac{\alpha _{2}}{\mathtt{k}}}}+{ \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{2}+{\mathtt{k}})}(\varphi (b)- \varphi (a))^{\frac{\alpha _{2}}{\mathtt{k}}}}\bigg]\bigg]. \end{aligned}$$

(3.7)

First, we use the well-known fixed-point theorem of Krasnoselskii to prove that there is at least one solution.

Theorem 3.1

Let $Q_{2}<1$ and suppose $\mathbf{(N_{3})}$ holds true. Then there exist at least one solution for the BVP (1.1) on $[a,b]$.

Proof

We will show that the operator Ϝ satisfies the assumptions of Krasnoselskii’s fixed point theorem. We split the operators Ϝ into the sum of two operators $\digamma _{1}$ and $\digamma _{2}$ on the closed ball $B_{\rho}=\{{z\in X;||z||\leq \rho}\}$, with $\rho \geq ||X||{\frac{Q_{1}}{1-Q_{2}}}$, $\sup _{{{t}}\in [a,b]}X({n})=||X||$, where

(3.8)

(3.9)

For $z,y\in {B}_{\rho}$, we have

Therefore, $||\digamma _{1}z{-}\digamma _{2}y||\leq \rho$ implies that $\digamma _{1}z{-}\digamma _{2}y\in {B}_{\rho}$.

Next, by using $\mathbf{(N_{3})}$, we prove that $\digamma _{2}$ is a contraction mapping. Letting $z,y\in X$, for ${{t}}\in [a,b]$,

This shows that $||\digamma _{2}z{-}\digamma _{2}y||\leq Q_{2}||z-y||$. Then, by using $\mathbf{(N_{3})}$, $\digamma _{2}$ is a contraction mapping. The operator $\digamma _{1}$ is continuous, given that g is also continuous. It is uniformly bounded on ${B}_{\rho}$ since $||\digamma _{1}z||\leq Q_{1}||X||$. Now, we prove that operator $\digamma _{1}$ is compact. Setting $\ \sup _{({t},z)\in [a,b]\times {B}_{\rho}}|g({t},z)|=\bar{g}< \infty$ and considering ${t}_{1},{t}_{2}\in [a,b], \ {t}_{1}<{t}_{2}$, we have

The right-hand side will tend to zero as ${{t}}_{2}-{{t}}_{1}\rightarrow 0$, independently of $z\in B_{\rho}$. By Arzela–Ascoli theorem, $\digamma _{1}$ is compact on $B_{\rho}$. Thus, by Krasnoselskii’s fixed point theorem, the problem (1.1) has at least one solution on the interval $[a,b]$. □

Theorem 3.2

Let $\mathbf{(N_{4})}$ hold and suppose ${L}Q_{1}+Q_{2}<1$, where $Q_{1},Q_{2}$ are given by (3.6) and (3.7), respectively. Then, the problem (1.1) has a unique solution on $[a,b]$.

Proof

The operator Ϝ is defined in (3.5). The problem (1.1) is subsequently transformed to a fixed point problem $z=\digamma z$. By using Banach contraction principle, we will show that Ϝ has a unique fixed point. We set ${\ \sup _{{{t}}\in [a,b]}=|g({{t}},0)|={p}<\infty}$ and choose $r>0$ such that

$$\begin{aligned} r\geq{\frac{{p}Q_{1}}{1-\mathtt {L}Q_{2}-Q_{2}}}. \end{aligned}$$

Now, we prove that ${\digamma {B}_{r}\subset {B}_{r}}$ where ${B}_{r}=\{z\in X$ $([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}});||z||\leq r\}$. For $z\in {B}_{r}$, we get

which implies that $\digamma {B}_{r}\subset {B}_{r}$. Next, consider $z,y\in X([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$. For ${{t}}\in [a,b]$, we obtain

which implies that $||(\digamma z)({{t}})-(\digamma y)({{t}})||\leq (\mathtt {L}Q_{1}+Q_{2})||z-y||$.

The operator Ϝ is a contraction, due to the inequality $\mathtt {L}Q_{1} + Q_{2} < 1$. According to Banach contraction principle, the operator Ϝ has a fixed point, which is the only solution to problem (1.1). □

4 Ulam–Hyers stability result

We give some definitions related to Ulam–Heyrs stability in this section.

Definition 4.1

The problem (1.1) is Ulam–Hyers stable (UHS) if there is a constant $h>0$ such that for every solution $\epsilon >0$ and for any solution $z \in X$ of the inequality

(4.1)

there exist a solution $\hat{z} \in X$ of (1.1) such that

$$\begin{aligned} |z({{t}})-\hat{z}({{t}})\leq h\epsilon ,\quad {{t}}\in (a,b]. \end{aligned}$$

Definition 4.2

The problem (1.1) is Ulam–Hyers–Rassias stable (UHRS) with respect to $\nu \in \mathtt {C}([a, b], \mathbbm{{}\mathbb{R}\mathbbm{}})$, if there is a real constant $h_{\nu }> 0$ such that for each $\epsilon > 0$ and for any solution $z \in X$ of the inequality

(4.2)

there exists a solution ${\hat{z}\in X}$ of (1.1) such that

$$\begin{aligned} |z({{t}})-\hat{z}({{t}})|\leq h_{\nu }\nu ({{t}}),\quad {{t}}\in (a,b]. \end{aligned}$$

Remark 4.1

Let us consider that $z \in X$ is a solution of the inequality (4.1), if there is a function $\hat{g}({t}) \in \mathtt {C}([a, b], \mathbbm{{}\mathbb{R}\mathbbm{}})$ (dependent on z) such that

• 1. $|\hat{g}({t})|\leq \epsilon \, \text{for all} \ {{t}}\in (a,b]$.

• 2. .

Lemma 4.1

If $z \in (\mathtt {C}[a,b], \mathbbm{{}\mathbb{R}\mathbbm{}})$ is a solution of the inequality (4.1), and $0 < \alpha _{i}, \mu _{i} \leq 1$ for $i=1,2$, then z fulfills the following integral inequality:

$$\begin{aligned} |z({{t}})-\varepsilon ({{t}})|\leq Q_{1}\epsilon , \end{aligned}$$

where

Proof

Consider that z is a solution of the inequality (4.1). Thus, in light of Remark 4.1.2, we get

(4.3)

For ${{t}}\in (a,b]$, the solution of (4.3) has the form

(4.4)

For computational ease, we use $\varepsilon ({t})$ for the sum of terms which are independent of ĝ:

So from the above, we obtain

From Remark 4.1.1 and (4.1), we get

$$\begin{aligned} |z({t})-\varepsilon ({t})|\leq Q_{1}\epsilon \end{aligned}$$

where

$$\begin{aligned} Q_{1}={}& \frac{(\varphi (b)-\varphi (a))^{\frac{\alpha _{1}+\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+{\mathtt{k}})}+ \frac{(\varphi ({t})-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}-1}}{{\Lambda}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2})}}\\ &{}\times\bigg[\sum _{i=1}^{{n}}\xi _{i}{ \frac{\mathtt{k}}{\alpha (\alpha _{1}+\alpha _{2}+2{\mathtt{k}})}} {( \varphi (\vartheta _{i})-\varphi (a))^{ \frac{\alpha _{1}+\alpha _{2}}{\mathtt{k}}+1}} \\ &{} +\sum _{j=1}^{{p}}{ \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+\phi _{j}+{\mathtt{k}})}} {(\varphi (z_{j})-\varphi (a))^{ \frac{\alpha _{1}+\alpha _{2}+\phi _{j}}{\mathtt{k}}}}\\ &{}- \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+{\mathtt{k}})}( \varphi (b)-\varphi (a))^{\frac{\alpha _{1}+\alpha _{2}}{\mathtt{k}}} \bigg]. \end{aligned}$$

 □

Theorem 4.1

Under the hypotheses $\mathbf{(N_{3})}$ and $\mathbf{(N_{4})}$ with

$$\begin{aligned} {L}Q_{1}+Q_{2}\leq 1, \end{aligned}$$

the system (1.1) is UHS.

Proof

Let $z\in \mathtt {C}((a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$ be a solution of (4.1) and let the unique solution of the problem, ẑ, be given as

(4.5)

Then, for ${{t}}\in (a,b]$, the solution of (4.5) is

Consider

$$\begin{aligned} |z({{t}})-\hat{z}({{t}})|\leq |z({{t}})-\varepsilon (t)|+| \varepsilon ({{t}})-\hat{z}({{t}})|. \end{aligned}$$

(4.6)

By utilizing Lemma 4.1 in (4.6), we obtain

(4.7)

where $w_{1},\hbar _{1}\in \mathtt {C}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})$,

$$\begin{aligned} w_{1}({{t}})=g({{t}},z({{t}})),\quad \hbar _{1}({{t}})=g({{t}}, \hat{z}({{t}})). \end{aligned}$$

Using $\mathbf{(N_{4})}$, we get

$$\begin{aligned} |w_{1}({{t}})-\hbar _{1}({{t}})|&=|g({{t}},z({{t}}))-g({{t}},\hat{z}({{t}}))|, \end{aligned}$$

then

$$\begin{aligned} |w_{1}({{t}})-\hbar _{1}({{t}})|\leq {L}|z({{t}})-\hat{z}({{t}})|. \end{aligned}$$

(4.8)

By using (4.8) in (4.7), we have

$$\begin{aligned} ||z-\hat{z}||_{X}\leq Q_{1}\epsilon +[Q_{1}L+Q_{2}]||z-\hat{z}||_{X}. \end{aligned}$$

The latter implies

$$\begin{aligned} ||z-\hat{z}||_{X}\leq {\frac{Q_{1}}{1-({L}Q_{1}+Q_{2})}} \epsilon =h_{\epsilon}, \end{aligned}$$

(4.9)

where

$$\begin{aligned} h={\frac{Q_{1}}{1-({L}Q_{1}+Q_{2})}} \end{aligned}$$

is such that

$$\begin{aligned} {L}Q_{1}+Q_{2}< 1. \end{aligned}$$

Hence, the considered problem (1.1) is UHS. □

Lemma 4.2

Assume that the hypothesis $\mathbf{(N_{5})}$ holds and let $z \in \mathtt {C}([a,b], \mathbbm{{}\mathbb{R}\mathbbm{}})$ be the solution of the inequality (4.2). Then z is the solution of the following inequality:

$$\begin{aligned} |z({{t}})-\varepsilon ({{t}})|\leq Q_{1}\Omega _{r} r({{t}})\epsilon . \end{aligned}$$

(4.10)

Proof

From Lemma 4.1, we obtain

By (4.1), Remark 4.1.2 and $\mathbf{(N_{5})}$, we get

$$\begin{aligned} |z({{t}})-\hat{z}({{t}})|\leq Q_{1}\Omega _{r} r({{t}})\epsilon , \end{aligned}$$

where $Q_{1}$ is given by (3.5). □

Theorem 4.2

Under the hypotheses $\mathbf{(N_{3})}$–$\mathbf{(N_{5})}$ and

$$\begin{aligned} {L}Q_{1}+Q_{2}< 1, \end{aligned}$$

(4.11)

the considered system (1.1) is UHRS.

Proof

Consider $z \in \mathtt {C}([a,b], \mathbbm{{}\mathbb{R}\mathbbm{}})$ as an approximate solution of (4.2) and let ẑ be the unique solution

(4.12)

Then, for ${{t}}\in (a,b]$, the solution of (4.12) is

Consider

$$\begin{aligned} |z({{t}})-\hat{z}({{t}})|\leq |z({{t}})-\varepsilon ({{t}})|+| \varepsilon ({{t}})-\hat{z}({{t}})|. \end{aligned}$$

(4.13)

Using $\mathbf{(N_{4})}$ in similar way as in Theorem 4.1, we have

$$\begin{aligned} |\varepsilon ({{t}})-\hat{z}({{t}})|\leq ({L}Q_{1}+Q_{2})|z({{t}})- \hat{z}({{t}})|. \end{aligned}$$

(4.14)

Now by Lemma 4.2 and (4.14), (4.13) becomes

$$\begin{aligned} ||z-\hat{z}||_{\omega}\leq Q_{1}\Omega _{r} r({{t}})\epsilon +( {L}Q_{1}+Q_{2})||z({{t}})-\hat{z}({{t}})||_{\omega}, \end{aligned}$$

which shows

$$\begin{aligned} ||z-\hat{z}||_{\omega}\leq{\frac{1}{1-({L}Q_{1}+Q_{2})}}Q_{1} \Omega _{r} r({{t}})\epsilon . \end{aligned}$$

 □

5 Example

An example is provided to verify our results.

Example 5.1

Consider

(5.1)

where $\mathtt{k}=\frac{5}{6}$, $\alpha _{1}=\frac{1}{2}$, $\alpha _{2}=\frac{2}{3}$, $\mu _{1}=\mu _{2}=\frac{1}{3}$, $\lambda =\frac{1}{20}$, $\varphi ({t})={{t}}e^{\frac{-{t}}{3}}$, $q=2$, $\xi _{1}=\frac{3}{32}$, $\xi _{2}=\frac{1}{14}$, $\vartheta _{1}=\frac{3}{4}$, $\vartheta _{2}=\frac{4}{5}$, $p=2$, $\zeta _{1}=\frac{9}{92}$, $\zeta _{2}=\frac{5}{72}$, $\theta _{1}=\frac{7}{52}$, $\theta _{2}=\frac{6}{7}$, $z_{1}=\frac{4}{5}$, $z_{2}=\frac{3}{12}$, ${a}=0$, ${b}=1$, $\phi _{\mathtt{k}}=\alpha _{1}+\mu _{1}({\mathtt{k}}-\alpha _{1})= \frac{17}{18}$, and ${L}=\frac{1}{3}$, while

$$\begin{aligned} \Lambda ={}&\bigg( \frac{(\varphi (b)-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}-1}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2})} \bigg)-{\mathtt{k}}\sum _{i=1}^{{n}} \xi _{i}\bigg( \frac{(\varphi (\vartheta )-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2}+{\mathtt{k}})} \bigg)\\ &{}-\bigg( \frac{(\varphi (z_{j})-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\phi _{j}+\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\phi _{j}+\alpha _{2})} \bigg), \\ &|\Lambda |\simeq 0.458799, \end{aligned}$$

$$\begin{aligned} Q_{1}={}& \frac{(\varphi (b)-\varphi (a))^{\frac{\alpha _{1}+\alpha _{2}}{k}}}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+k)} \\ &{}+ \frac{(\varphi ({t})-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}-1}}{{\Lambda}{\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2})}} \bigg[\sum _{i=1}^{{n}}\xi _{i}{ \frac{\mathtt{k}}{\Gamma _{k}(\alpha _{1}+\alpha _{2}+2{\mathtt{k}})}} {(\varphi (\vartheta _{i})-\varphi (a))^{ \frac{\alpha _{1}+\alpha _{2}}{\mathtt{k}}+1}} \\ &{} +\sum _{j=1}^{{p}}\zeta _{j}{ \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+\phi _{j}+{\mathtt{k}})}} {(\varphi (z_{j})-\varphi (a))^{ \frac{\alpha _{1}+\alpha _{2}+\phi _{j}}{\mathtt{k}}}} \\ &{}- \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{1}+\alpha _{2}+{\mathtt{k}})}( \varphi (b)-\varphi (a))^{\frac{\alpha _{1}+\alpha _{2}}{\mathtt{k}}} \bigg] \\ \simeq{}& -0.559602< 1, \end{aligned}$$

(5.2)

$$\begin{aligned} Q_{2}={}&|\lambda |\bigg[\bigg( \frac{(\varphi (b)-\varphi (a))^{\frac{\alpha _{2}}{\mathtt{k}}}}{\Gamma _{\mathtt{k}}(\alpha _{2}+{\mathtt{k}})} \bigg)+\bigg( \frac{(\varphi ({t})-\varphi (a))^{\frac{\theta _{\mathtt{k}}+\alpha _{2}}{\mathtt{k}}-1}}{{\Lambda}\Gamma _{\mathtt{k}}(\theta _{\mathtt{k}}+\alpha _{2})} \bigg) \\ &{} \times \bigg[\sum _{i=1}^{{n}}\xi _{i}{ \frac{\mathtt{k}}{\Gamma _{\mathtt{k}}(\alpha _{2}+2{\mathtt{k}})}( \varphi (\vartheta _{i})-\varphi (a))^{\frac{\alpha _{2}}{\mathtt{k}}+1}}- \sum _{j=1}^{{p}}\zeta _{j}{ \frac{1}{\Gamma _{\mathtt{k}}(\alpha _{2}+{\mathtt{k}})}(\varphi (z_{j})- \varphi (a))^{\frac{\alpha _{2}}{\mathtt{k}}}} \\ &{} +{\frac{1}{\Gamma _{\mathtt{k}}(\alpha _{2}+{\mathtt{k}})}( \varphi (b)-\varphi (a))^{\frac{\alpha _{2}}{\mathtt{k}}}}\bigg] \bigg]\simeq 0.134205< 1, \end{aligned}$$

(5.3)

and

$$\begin{aligned} |g({t},z)-g({t},y)|\leq \frac{1}{3}|z-y|, \quad z,y \in \mathbbm{{}\mathbb{R}\mathbbm{}}. \end{aligned}$$

Since ${L}Q_{1}+Q_{2}=-0.0523294<1$, the problem (1.1) has a unique solution on $[0,1]$ and satisfies the conditions of Theorem 3.2. Furthermore, the condition of Theorem 4.1 holds, namely

$$\begin{aligned} {L}Q_{1}+Q_{2}\approx -0.0523294< 1. \end{aligned}$$

Hence, by Theorem 4.1, the given problem (5.1) is UHS.

6 Conclusion

In this manuscript, we studied the $(\mathtt{k}, \varphi )$-Hilfer Langevin fractional system having multipoint boundary conditions and involving a fractional integral. We also investigated the existence and uniqueness of solution of proposed problem (1.1) by Krasnoselskii’s fixed point theorem and Banach contraction principle, respectively. We also determined various kinds of Ulam–Hyers stability for problem (1.1), by using the techniques of nonlinear functional analysis. At the end, an example was provided to validate our main theoretical results. In our future work, we will study $(\mathtt{k}, \varphi )$-Hilfer fractional Langevin differential equation with impulses and different types of delay, particularly state-dependent delay. Our results are significant and contribute to the existing material on $(\mathtt{k}, \varphi )$-Hilfer fractional Langevin differential equation, supplemented with nonlocal multipoint boundary conditions. Further generalization of the $(\mathtt{k}, \varphi )$-Hilfer fractional derivatives will be interesting investigations.