 Research
 Open access
 Published:
\((\mathtt{k},\varphi )\)Hilfer fractional Langevin differential equation having multipoint boundary conditions
Boundary Value Problems volume 2024, Article number: 113 (2024)
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 fixedpoint 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 φHilfer fractional derivative. For more related information, we recommend [1, 25, 37, 38]. Daiz and Pariguan [9] introduced the \(\mathtt{k}\)gamma function \(\Gamma _{\mathtt{k}}(z)={\int _{0}^{\infty}{s}^{z1}{e}^{ \frac{{s}^{\mathtt{k}}}{\mathtt{k}}}}\mathtt {d}s, \quad {\mathtt{k}}>0\), which is a generalized Euler’s gamma function with additional parameter \(\mathtt{k}\). Mubeen and Habibullah [27] introduced the kRiemann–Liouville fractional integral operator. Romero [32], introduced a generalized version of \(\mathtt{k}\)Riemann–Liouville fractional derivative operators. For more information on the φRiemann–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 realvalued 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 φHilfer 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 φHilfer 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 integromultipoint 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 integromultipoint fractional boundary conditions:
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 integromultipoint 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 singlevalued \((\mathtt{k},\varphi )\)Hilfer nonlocal integromultipoint 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
Furthermore, the following relations hold:
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
The expression \([\frac{\alpha}{\mathtt{k}}]\) is known as the ceiling function of \(\frac{\alpha}{\mathtt{k}}\).
Definition 2.3
([19])
The φRiemann–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
Definition 2.4
([19])
The φRiemann–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
Definition 2.5
([5])
The φCaputo 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
Definition 2.6
([7])
The φHilfer 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
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
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
The solution of the Hilfer nonlocal integromultipoint BVP,
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
where
By using Riemann–Liouville fractional integral of order \(\alpha _{2}\) in equation (3.2), we have
where . Also
The above implies
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,
Putting \({t}=b\) in (3.4), we obtain
Similarly,
Now,
where
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
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 \(z,y\in \mathbbm{{}\mathbb{R}\mathbbm{}}\),
$$\begin{aligned} &g({t},z)g({t},y)\leq \mathtt {L}zy. \end{aligned}$$  \(\mathbf{(N_{5})}\):

\(r\in \mathtt {C}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})\) 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:
First, we use the wellknown fixedpoint 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 \(\rho \geq X{\frac{Q_{1}}{1Q_{2}}}\), \(\sup _{{{t}}\in [a,b]}X({n})=X\), where
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 \(\digamma _{2}z{}\digamma _{2}y\leq Q_{2}zy\). 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 righthand 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
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 \((\digamma z)({{t}})(\digamma y)({{t}})\leq (\mathtt {L}Q_{1}+Q_{2})zy\).
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
there exist a solution \(\hat{z} \in X\) of (1.1) such that
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
there exists a solution \({\hat{z}\in X}\) of (1.1) such that
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:
where
Proof
Consider that z is a solution of the inequality (4.1). Thus, in light of Remark 4.1.2, we get
For \({{t}}\in (a,b]\), the solution of (4.3) has the form
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
where
□
Theorem 4.1
Under the hypotheses \(\mathbf{(N_{3})}\) and \(\mathbf{(N_{4})}\) with
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
Then, for \({{t}}\in (a,b]\), the solution of (4.5) is
Consider
By utilizing Lemma 4.1 in (4.6), we obtain
where \(w_{1},\hbar _{1}\in \mathtt {C}([a,b],\mathbbm{{}\mathbb{R}\mathbbm{}})\),
Using \(\mathbf{(N_{4})}\), we get
then
By using (4.8) in (4.7), we have
The latter implies
where
is such that
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:
Proof
From Lemma 4.1, we obtain
By (4.1), Remark 4.1.2 and \(\mathbf{(N_{5})}\), we get
where \(Q_{1}\) is given by (3.5). □
Theorem 4.2
Under the hypotheses \(\mathbf{(N_{3})}\)–\(\mathbf{(N_{5})}\) and
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
Then, for \({{t}}\in (a,b]\), the solution of (4.12) is
Consider
Using \(\mathbf{(N_{4})}\) in similar way as in Theorem 4.1, we have
Now by Lemma 4.2 and (4.14), (4.13) becomes
which shows
□
5 Example
An example is provided to verify our results.
Example 5.1
Consider
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
and
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
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 statedependent 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.
Data Availability
No datasets were generated or analysed during the current study.
References
Abdeljawad, T., Thabet, S.T.M., Kedim, I., VivasCortez, M.: On a new structure of multiterm Hilfer fractional impulsive neutral Levin–Nohel integrodifferential system with variable time delay. AIMS Math. 9(3), 7372–7395 (2024)
Agrawal, O.P.: Formulation of Euler–Lagrange equations for fractional variational problem. J. Math. Anal. Appl. 272, 368–379 (2002)
Ahmad, B., Nieto, J.J., Alsaedi, A., ElShahed, M.: A study of nonlinear Langevin equation involving two fractional orders in different intervals. Nonlinear Anal., Real World Appl. 13(2), 599–602 (2012)
Ahmed, A., Mokhtar, K., Ahmad, F.Z., Ahmad, B.: On nonexistence of solutions to some time space fractional evolution equations with transformed space argument. Bull. Math. Sci. 13(2), Article ID 2250009 (2023)
Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)
Boutiara, A., Etemad, S., Thabet, S.T.M., Ntouyas, S.K., Rezapour, S., Tariboon, J.: A mathematical theoretical study of a coupled fully hybrid \((k, \phi )\)fractional order system of BVPs in generalized Banach spaces. Symmetry 15(2), 1041 (2023)
da Vanterler, J., Sousa, C., Capelas de Oliveira, E.: On the φHilfer fractional derivative. Commun. Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)
Daniele, C., Lele, D.: Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs. Adv. Nonlinear Anal. 12(4), Article ID 20230103 (2023)
Diaz, R., Pariguan, E.: On hypergeometric functions and Pochhammer Ksymbol. Divulg. Mat. 2, 179–192 (2007)
Dorrego, G.A.: An alternative definition for the KRiemann–Liouville fractional derivative. Appl. Math. Sci. 9(10), 481–491 (2015)
Fa, K.S.: Generalized Langevin equation with fractional derivative and longtime correlation function. Phys. Rev. E 73(6), 061104 (2006)
Fritz, M., Khristenko, U., Wohlmuth, B.: Equivalence between a timefractional and an integerorder gradient flow: the memory effect reflected in the energy. Adv. Nonlinear Anal. 12(1), Article ID 20220262 (2023)
Granas, A., Dugundji, J.: Fixed Point Theory. Springer, New York (2005)
Hilal, K., Kajouni, A., Lmou, H.: Boundary value problems for the Langevin equation inclusion with the Hilfer fractional derivative. Int. J. Differ. Equ. (2022)
Hilfer, R.: Applications of Fractional Calculus in Physics, vol. 35. World Scientific, Singapore (2000)
Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27(4), 222–224 (1941)
Khan, H.N.A., Zada, A., Khan, I.: Analysis of a coupled system of φCaputo fractional derivatives with multipoint–multistrip integral type boundary conditions. Qual. Theory Dyn. Syst. 23(3), 1–41 (2024)
Khan, I., Zada, A.: Analysis of abstract partial impulsive integrodifferential system with delay via integrated resolvent operator. Qual. Theory Dyn. Syst. 23(3), 1–15 (2024)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. NorthHolland Mathematics Studies., vol. 204. Elsevier Science B.V., Amsterdam (2006)
Kucche, K.D., Mali, A.D.: On the nonlinear \((k,\varphi )\)Hilfer fractional differential equations. Chaos Solitons Fractals 152, 111–335 (2021)
Kwun, Y.C., Farid, G., Nazeer, W., Ullah, S., Kang, S.M.: Generalized Riemann–Liouville Kfractional integrals associated with Ostrowski type inequalities and error bounds of Hadamard inequalities. IEEE Access 6, 64946–64953 (2018)
Lim, S.C., Li, M., Teo, L.P.: Langevin equation with two fractional orders. Phys. Lett. A 372(42), 6309–6320 (2008)
Marek, M.: On the existence of optimal solutions to the Lagrange problem governed by a nonlinear Goursat–Darboux problem of fractional order. Opusc. Math. 43(4), 547–558 (2023)
Mohammed, A.A., Omar, B., Satish, P.K., Askar, S.S., Georgia, O.I.: Analytical study of two nonlinear coupled hybrid system involving generalized Hilfer fractional operators. Fractal Fract. 5(4), 178 (2021)
Mohammed, A.A., Satish, P.K.: Some properties of implicit impulsive coupled system via φHilfer fractional operator. Bound. Value Probl. 2021, Article ID 57 (2021)
Mohammed, A.A., Satish, P.K., Khaled, A., Mansor, L.: On the explicit solution of φHilfer integrodifferential nonlocal Cauchy problem. Prog. Fract. Differ. Appl. 9(1), 65–77 (2023)
Mubeen, S., Habibullah, G.M.: Kfractional integrals and applications. Int. J. Contemp. Math. Sci. 7, 89–94 (2012)
Ntouyas, S.K., Ahmad, B., Teriboon, J.: \((k,\varphi )\)Hilfer nonlocal integromultipoint boundary value problems for fractional differential equations and inclusion. Mathematics 10, 2615 (2022)
Podlubny, I.: Fractional Differential Equation. Academic Press, San Diego (1999)
Rassias, T.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Rizwan, R.: Existence theory and stability analysis of fractional Langevin equation. Int. J. Nonlinear Sci. Numer. Simul. 20(7–8), 833–848 (2019)
Romero, L.G., Luque, L.L., Dorrego, G.A., Cerutti, R.A.: On the kRiemann–Liouville fractional derivative. Int. J. Contemp. Math. Sci. 8(1), 41–51 (2013)
Salim, A., Alghamdi, B.: Multistrip and multipoint boundary conditions for fractional Langevin equation. Fractal Fract. 4(2), 18 (2020)
Salim, A., Thabet, S.T.M., Kedim, I., VivasCortez, M.: On the nonlocal hybrid \((k,\phi )\)Hilfer inverse problem with delay and anticipation. AIMS Math. 9(8), 22859–22882 (2024)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon Breach, Yverdon (1993)
Sitho, S., Ntouyas, S.K., Samadi, A., Teriboon, J.: Boundary value problems for φHilfer type sequential fractional differential equations and inclusions with integral multipoint boundary conditions. Mathematics 9, 1001 (2021)
Thabet, S.T.M., Kedim, I., Rafeeq, A.S., Rezapour, S.: Analysis study on multiorder ρHilfer fractional pantograph implicit differential equation on unbounded domains. AIMS Math. 8(8), 18455–18473 (2023)
Thabet, S.T.M., VivasCortez, M., Kedim, I., Samei, M.E., Ayari, M.I.: Solvability of a ρHilfer fractional snap dynamic system on unbounded domains. Fractal Fract. 7(8), 607 (2023)
Ulam, S.M.: A Collection of Mathematical Problems. Interscience publishers, New York (1968)
Waheed, H., Zada, A., Popa, I.L., Etemad, S., Rezapour, S.: On a system of sequential Caputotype pLaplacian fractional BVPs with stability analysis. Qual. Theory Dyn. Syst. 23(3), 1–28 (2024)
Acknowledgements
The authors extend their appreciation to the Deanship of Research and Graduate Studies at King Khalid University for funding this work through Large Research Project under grant number RGP2/215/45.
Funding
Not applicable.
Author information
Authors and Affiliations
Contributions
All the authors contributed equally to this article.
Corresponding author
Ethics declarations
Consent for publication
Not applicable.
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License, which permits any noncommercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/byncnd/4.0/.
About this article
Cite this article
Cheng, H., Naila, Zada, A. et al. \((\mathtt{k},\varphi )\)Hilfer fractional Langevin differential equation having multipoint boundary conditions. Bound Value Probl 2024, 113 (2024). https://doi.org/10.1186/s13661024019183
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661024019183