Existence of solution for a Langevin equation involving the $\psi$-Hilfer fractional derivative: A variational approach

1.   Introduction

Fractional calculus is a field of mathematical analysis that generalizes the concepts of derivatives and integrals to non-integer orders. It has become increasingly important in numerous scientific and engineering disciplines. As a result, many researchers have investigated its applications across various domains, such as virotherapy (see ^[1]), quantum mechanics (see ^[2]), image processing (see ^[3]), other applications in physics and engineering can be found in the papers ^[4,5]. Given their significance, numerous researchers have focused on addressing problems involving various fractional derivatives. For example, the works ^[6] (for the $\psi$-Caputo derivative), the papers ^[7,8,9,10] (Riemann-Liouville derivative combined with fixed points theorems), the papers ^[11,12,13] (Riemann-Liouville derivative combined with the variational method), and the work ^[14] (Hadamard derivative).

Recently, numerous papers have focused on studying problems involving the $\psi$-Hilfer fractional derivative, which was introduced in the work of Hilfer ^[15]. The $\psi$-Hilfer fractional derivative is based on the generalized Hilfer fractional derivative, which itself combines elements of the Riemann-Liouville and Caputo derivatives. The Hilfer fractional derivative is a more flexible formulation than other fractional derivatives because it incorporates both the memory of a function and its local behavior, thus providing more control over the fractional-order derivative. This fractional derivative type generalizes and extends several classical definitions in fractional calculus and has been applied to model complex processes in physics, biology, and engineering; see, for instance, ^{[16,17,18,19,20]}.

In this work, we continue to address a problem involving the $\psi$-Hilfer fractional derivatives $_{y}D^{\nu, \beta, \psi}_{T}$ and $_{0}D^{\nu, \beta, \psi}_{y}$, which were introduced in Section 2. More precisely, we prove the existence of solutions for the following problem:

where $\nu \in \left(\frac{1}{2}, 1\right)$, $\psi$ is a positive increasing function on $[0, T]$ with $T > 0$ and $\psi'(y) \neq 0$ for all $y \in (0, T)$. The function $g$ is a measurable function on $[0, T] \times \mathbb{R} \times \mathbb{R}$ and satisfies additional conditions that will be stated later. We note that a function $u\in E$ is a solution for problem (1.1), if for every $v\in E$ we have

where $E$ is a functional space introduced in Section 2. We also note that, in the special case where

the problem (1.1) simplifies to a Langevin equation.

It is worth mentioning that the Langevin equation is the fundamental equation in the fields of statistical mechanics and stochastic processes. It describes the time evolution of a system's state, incorporating both deterministic and random forces. Albert Einstein provided a theoretical framework for Brownian motion, which refers to the random motion of particles suspended in a fluid. His work established the foundation for understanding the stochastic behavior of particles. Paul ^[21] further extended the concept of Brownian motion by introducing both a deterministic force and a random force (the thermal or stochastic force). He formulated what is now known as the Langevin equation, which describes the dynamics of a particle in a fluid, accounting for both frictional drag and random thermal fluctuations.

The Langevin equation is given by:

where $\mathfrak{m}$ is the mass of the particle, $\gamma$ is the damping coefficient, and $\eta(y)$ represents the random force following a Gaussian distribution. Recently, significant attention has been given to the study of Langevin equations with various fractional derivatives. Ahmad et al. ^[22,23] applied a combination of the contraction mapping principle and the Krasnoselskii fixed point theorem to establish the existence of solutions for certain Langevin equations involving Caputo fractional derivatives. Almaghamsi ^[24] investigated weak solutions for boundary value problems with fractional Langevin inclusions, employing the Katugampola-Caputo derivative and Pettis integrability, and utilizing the Mönch fixed point theorem along with weak noncompactness. Lim et al. ^[25] examined a Langevin problem involving the Weyl and Riemann-Liouville fractional derivatives, focusing on Gaussian processes and their relationship with fractional Brownian motion, thereby proving several existing results. Most recently, Torres ^[26] studied a Langevin equation incorporating the Riemann-Liouville fractional derivative. He combined a variational method with an iterative technique to demonstrate the existence of solutions for the problem under study.

In this paper, we further explore a Langevin problem involving the $\psi$-Hilfer fractional derivative by using the study of Xie et al. ^[27]. It is important to point out that the presence of the third variable in the function $g$ prevents the application of variational methods to solve this problem. So, we begin by replacing the third variable of a function $g$ with a fixed function in an appropriate function space, and in Section 3, we employ the mountain pass theorem to establish the existence of solutions for this auxiliary problem. Following this, in Section 4 we construct a sequence of solutions for related auxiliary problems and demonstrate that this sequence converges to a solution for the original problem (1.1). It is noted that our study generalizes other works in the literature; for example, if $\psi(x) = x$ and $\beta = 1$, then we obtain the result of Toress ^[26]. To our knowledge, no known work in the literature utilizes the theory of ^[27] to problems involving fractional derivatives with respect to a function.

2.   Preliminaries and variational setting

In this section, we present some foundational background and theoretical concepts related to the $\psi$-Hilfer fractional derivative, which will be applied throughout this paper. We begin by recalling the definition of the fractional integral, as introduced by Kilbas et al. ^[4] and Samko et al. ^[28]. In this section, $\eta$ and $\beta$ are positive real numbers, $\Gamma$ represents the Euler gamma function, and for $-\infty \leq a < b \leq \infty$, $[a, b]$ denotes a finite or infinite interval on the real line. Moreover, $\psi$ is assumed to be an increasing positive function on $[a, b]$, with a continuously differentiable derivative $\psi'(y) \neq 0$ on $(a, b)$.

Definition 2.1. ^[4,28] Let $\chi$ be an integrable function on the interval $[a, b]$, and let $\psi \in \mathfrak{C}^1([a, b], \mathbb{R})$ be a strictly increasing function with $\psi'(y) \neq 0$ for all $y \in [a, b]$. The fractional integrals of $\chi$ with respect to $\psi$, on the left and right sides, are given by:

and

Definition 2.2. ^[19,29] Let $\chi$ be a function that is integrable over the interval $[a, b]$, and let $\psi \in \mathfrak{C}^1([a, b], \mathbb{R})$ be a strictly increasing function such that $\psi'(y) \neq 0$ for all $y \in [a, b]$. Below is the definition of the left and right $\psi$-Hilfer fractional derivatives of type $0 \leq \beta \leq 1$:

and

where $m$ is an integer satisfying $m-1 < \eta \leq m.$

It is important to highlight that the $\psi$-Hilfer fractional derivatives generalize previous concepts, including the $\psi$-Riemann-Liouville and $\psi$-Caputo fractional derivatives. Specifically, the following remark holds:

Remark 2.3. The following statements are true:

(i) From the $\psi$-Hilfer fractional derivatives, as $\beta$ tends to zero, we obtain the $\psi$-Riemann-Liouville fractional derivatives:

and

(ii) As $\beta$ tends to 1, the $\psi$-Hilfer fractional derivatives become equivalent to the $\psi$-Caputo fractional derivatives, defined as:

and

(iii) $\psi$-Hilfer fractional derivatives are directly related to $\psi$-Riemann-Liouville fractional derivatives. Below are the specific results:

and

where $\xi = \eta + \beta(m-\eta)$.

We note that we have an analogue integration by parts formula; for more details, the interested readers can consult [30, Theorem 12].

For $1 \leq s \leq \infty$, $L^s(a, b)$ denotes the set of all measurable functions $\chi$ on $[a, b]$, such that $\int_a^b|\chi(\sigma)|^s\, d\sigma < \infty$. Define the norm as

Remark 2.4. ^[13,19] Let $0 < \eta \leq 1$, $s \geq 1$, and $q = \frac{s}{s-1}$. For each $\chi \in L^s(a, b)$, the following properties hold:

(i) The operator $I^{\eta, \psi}_{a^+} \chi$ is bounded in $L^s(a, b)$, and the following inequality holds:

(ii) If $\frac{1}{s} < \eta < 1$, then $I^{\eta, \psi}_{a^+}\chi$ is Hölder continuous on $[a, b]$ with an exponent of $\eta - \frac{1}{s}$.

(iii) If $\frac{1}{s} < \eta < 1$, then $\lim\limits_{y \to a} I^{\eta, \psi}_{a^+} \chi(y) = 0$, meaning that $I^{\eta, \psi}_{a^+} \chi$ can be continuously extended to zero at $y = a$. Therefore, $I^{\eta, \psi}_{a^+} \chi$ is continuous on $[a, b]$, and the following bound holds:

To establish the variational structure for problem (1.1), we introduce the fractional derivative space \(E_{\eta, \beta, \psi} \), defined as the closure of $C_c^\infty([0, T], \mathbb{R})$ under the norm:

The space \(E_{\eta, \psi} \) can be characterized as follows:

Remark 2.5. ^[13,19] If $0 < \eta \leq 1$ and $0 \leq \beta \leq 1$, then for all $\chi \in E^{\eta, \beta, \psi}_{s}$, the following properties hold:

(i) The space $E^{\eta, \beta, \psi}_{s}$ is a Banach space that is both reflexive and separable.

(ii) If $1 - \eta > \frac{1}{s}$ or $\eta > \frac{1}{s}$, we have the inequality:

(iii) If $\frac{1}{s} < \eta$ and $q = \frac{s}{s-1}$, then the following holds:

Let

which defines a Hilbert space, and from the last remark, $E$ can be equipped with the following equivalent norm:

Furthermore, based on Remark 2.5, there exists a constant $c_1 > 0$, such that for each $\chi \in E$, we have

3.   Auxiliary problem

In this section, we fix a function $\varphi$ in the space $E$ and consider the following auxiliary problem:

We assume the following conditions:

$(H_1)$ We assume that there exist $q > 1$, $0 < \delta < \frac{\Gamma(\nu+1)}{(\psi(T)-\psi(0))^{\nu}},$ and $C_\delta > 0$ such that for all $(y, \xi, \eta) \in [0, T] \times \mathbb{R} \times \mathbb{R}$, the following inequalities hold:

and

$(H_2)$ There exist $\sigma > 2$, and positive constants $c_1$ and $c_2$, such that for all real values $\xi$ and $\eta$, and for each $y \in [0, T]$, we have

where $G$ is the antiderivative of the function $g$ with respect to its second variable and equals zero when evaluated at zero.

$(H_3)$ The function $g$ satisfies the Ambrosetti-Rabinowitz condition, which means that there exists a constant $C > 0$ such that for all $(y, \eta) \in [0, T] \times \mathbb{R}$, we have:

The first main result of this work is stated in the following theorem.

Theorem 3.1. Under the conditions $(H_1)$–$(H_3)$, problem $(P_\varphi)$ has a nontrivial solution.

To prove the existence of solutions for problem $(P_\varphi)$, we will make use of the following theorem.

Theorem 3.2. ^[31] Let $X$ be a real Banach space, and let $J \in \mathfrak{C}^1(E, \mathbb{R})$ be a functional that satisfies the Palais-Smale condition. Assume the following:

(i) $J(0) = 0.$

(ii) There exist constants $\rho > 0$ and $\chi > 0$ such that $J(z) \geq \chi$ for all $z \in E$ with $\|z\| = \rho$.

(iii) There exists $z_1 \in E$ with $\|z_1\| \geq \rho$ such that $J(z_1) < 0.$

Then, $J$ has a critical value $c \geq \sigma$. Furthermore, $c$ can be expressed as

where $\Gamma = \{\gamma \in C([0, 1], X): \gamma(0) = 0, \gamma(1) = z_1\}$.

We recall that a functional $J$ satisfies the Palais-Smale condition if any sequence $\{u_{m}\} \subset X$ such that $J(u_{m})$ is bounded and $J'(u_{m}) \to 0$ in $X'$, possesses a convergent subsequence.

The energy functional associated with problem $(P_\varphi)$ is defined as:

where $G(s, \xi, \eta) = \int_0^\xi g(s, \sigma, \eta) \, d\sigma$.

Since the function $G$ is continuous, then we get $L_\varphi \in C^1(E, \mathbb{R})$. Moreover, for all pairs $(u, v) \in E^2$, we have:

Lemma 3.3. Under assumptions $(H_1)$–$(H_3)$, the functional $L_\varphi$ satisfies the Palais-Smale condition.

Proof. Let $\{u_k\}$ be a sequence in $E$ such that

We first observe that, based on this information, there exists a constant $C_0 > 0$ such that

Next, we show that the sequence $\{u_k\}$ is bounded. To do so, let $r \in \left(\frac{1}{\sigma}, \frac{1}{2}\right)$. Using assumption $(H_3)$, we can derive the necessary bounds for any $k \in \mathbb{N}$.

Since $r < \frac{1}{2}$, it is clear that the sequence $\{u_k\}$ is bounded. Moreover, since $E$ is a reflexive space, this implies that, after possibly passing to a subsequence, there exists $u_* \in E$ such that $u_k$ converges weakly to $u_*$ in $E$. Additionally, due to the compact embedding, $u_k$ converges strongly to $u_*$ in $C([0, T])$. Now, we will show that $\{u_k\}$ converges strongly to $u_*$ in $E$.

From Eq (3.1), we obtain

and

So, we obtain

where

From hypothesis $(H_2)$ and the fact that $u_k$ converges strongly to $u_*$ in $C([0, T]),$ we obtain

Next, since $L'_\varphi(u_k)\to 0$ and $u_k-u_*$ is bounded, then we obtain

Moreover, from the fact that $u_k$ converges strongly to $u_*$ in $C([0, T])$

Hence, one has

In conclusion, by utilizing Eqs (3.4)–(3.6), we can confirm that the sequence $\{u_k\}$ converges strongly to $u_*$ in $E$.

□

Lemma 3.4. Under assumption $(H_1)$–$(H_3)$, if $q > 1$, then, there exist $\rho > 0$ and $\varpi > 0$ such that $L_\varphi(z) \geq \varpi$ for every $z \in E$ with $\|z\| = \rho$.

Proof. Let $z\in E$, and $\delta > 0$. Then using Remark 2.5, hypothesis $(H_1)$, and the decreasing embedding of the Lebegue spaces, we have

Now, if we fix $\delta < \frac{\Gamma(\nu+1)}{(\psi(T)-\psi(0))^{\nu}}$ and

Then, we obtain

So, we can see from (3.7) that if $\|z\| = \rho$, then $L_\varphi(z) > 0$. Thus Lemma 3.4 has been proven. □

Lemma 3.5. Under assumptions $(H_1)$–$(H_3)$, there exists $z_{1} \in E$ with $\|z_{1}\|\geq\rho$ such that $L_\varphi(z_{1}) < 0.$

Proof. Let $\xi > 0$ and $z\in E,$ then from hypothesis $(H_2)$, we obtain

Since we have $\sigma > 2$, then

So, we can choose $\xi$ large enough and $z_1 = \xi u$ so that $\|z_1\|\geq \rho$ and $L_\varphi(z_1) < 0$. □

Proof of Theorem 3.1. The proof of Theorem 3.1 naturally follows from the preceding lemmas in this section. Specifically, Lemma 3.3 confirms that $L_\varphi$ satisfies the Palais-Smale condition, while Lemma 3.4 ensures that condition $(ii)$ of Theorem 3.2 is met. Furthermore, Lemma 3.5 guarantees that condition $(iii)$ of Theorem 3.2 is also fulfilled. Finally, noting that $L_\varphi(0) = 0$, we infer from Theorem 3.2 that $L_\varphi$ possesses a critical point, which corresponds to a weak solution of the problem $(P_\varphi)$. Additionally, Lemma 3.4 confirms that this solution is nontrivial.

For the remainder of this paper, a solution to problem $(P_\varphi)$ will be referred to as $S(P_\varphi)$.

4.   Main result and its proof

In this part, we introduce the second primary finding of this paper, which deals with the existence of solutions for the problem denoted as (1.1). The demonstration relies on the application of the outcome established in Section 3 alongside some iterative techniques. Before revealing the main conclusion of this paper, we assume that the function $g$ adheres to the condition outlined below:

$(H_4)$ There exists $R > 0$ small enough, such that for every pair $(\eta, \eta_1)$ belonging to the real numbers $\mathbb R$, the following holds true:

and

where $\alpha_1$ and $\alpha_2$ are positive constants satisfying the condition:

The second result of this work is the following theorem.

Theorem 4.1. Under the Assumptions $(H_1)$–$(H_4)$, problem (1.1) admits a nontrivial solution.

Proof. To prove Theorem 4.1, we begin by remarking that, since the hypotheses $(H_1)$–$(H_3)$ are satisfied, then, the sequence $\{u_k\}\subset E$ defined by

is well defined, moreover, for each $k\in \mathbb N$, we have

Let $0 < \delta < \frac{\Gamma(\nu+1)}{(\psi(T)-\psi(0))^{\nu}}$, then from Eq (4.2) and hypothesis $(H_1)$, we obtain

So, we obtain

Since $q > 1$, then it follows that

On the other hand, from the characterization of the mountain pass level (see ^[32]), we have

for some $z\in E$ with $\|z\| = 1$.

Now, from the last inequality, and as in Eq (3.8), we obtain

Since $f$ is continuous on $[0, \infty)$, $f(0) = 0$, and $\lim\limits_{s\to \infty}f(s) = -\infty$. Then we can deduce that $f$ is upper bounded, so there exists $M > 0$ such that $|L_{u_{k-1}}(u_k)|\leq M$. Therefore, as in Eq (3.3), we obtain

Now, from the fact that $u_{k+1}$ is a critical point of the functional $L_{u_k}$, and $u_{k}$ is a critical point of the functional $L_{u_{k-1}}$, one has

So, by combining the last information with Eq (4.4), we can deduce the existence of $M_1 > 0$, such that

Now, by combining Eq (4.6) with Eq (4.3), we get

So, if we choose $R = c_1 M_1$, where $R$ is given by hypothesis $(H_4)$, and $c_1$ is given in Eq (2.3), then using Eq (4.7), we obtain

Next, from Eq (4.5), we deduce that

which yields to

So, from Remark 2.5, assumption $(H_4)$, and the Hölder inequality, we obtain

Hence, we deduce that

where $\theta$ is given by Eq (4.1).

Based on the previous inequality, we can infer that the sequence $\{u_{k}\}$ is a Cauchy sequence in the Banach space $E$. Consequently, there is an element $u_*\in E$ to which the sequence $\{u_{n}\}$ converges in $E$. To conclude the proof, it is enough to demonstrate that $u_*$ solves problem (1.1). To achieve this, it is necessary to show that as $k$ approaches infinity, we get

Again, from Remark 2.5, assumption $(H_4)$, and the Hölder inequality, we conclude that

Hence, we deduce that Eq (4.8) holds, and consequently the proof of Theorem 4.1 is completed. □

5.   Example

In this section, we present the following illustrative example:

where $^H D_{1^-}^{\frac{6}{10}}$ and $^H D_{0^+}^{\frac{6}{10}, \frac{1}{2}}$ are the left and right Hilfer derivatives of order $\frac{6}{10}$ and type $\frac{1}{2}$, that is, $\psi(x) = x$, $\nu = \frac{6}{10}, T = 1$, $\beta = \frac{1}{2}$, and

with $c_1 > c_2$ are positive constants that satisfy an appropriate condition fixed later.

A simple calculation shows that, for any $\delta > 0$ small enough, we have

This means that the first inequality in hypothesis $(H_1)$ is satisfied with $q = 2 > 1$. On the other hand, we have

So, hypothesis $(H_1)$ holds.

Now, since $-1\leq sin \mu\leq 1$, then for any $c > 0$, we obtain

which means that hypothesis $(H_1)$ is also satisfied with $\sigma = 3 > 2$.

It is not difficult to see that

so hypothesis $(H_3)$ is satisfied.

Next, we prove that hypothesis $(H_4)$ holds. For this, we recall from [33, Lemme 2.1] the following elementary inequality:

provided that $|a|+|b|\neq 0$.

Now, we applied equation for $p = 2$, $a = \xi$ and $b = \xi_1$, we get

This implies that

on the other hand, we have

To finish, it suffices to prove that Eq (4.1) holds. From the above inequalities, we can see that $\alpha_1 = C(c_1+c_2)$ and $\alpha_2 = c_2 R^2$. Since $\psi(1) = 1$ and $\psi(0) = o$, then we obtain

If we choose $c_1+c_2 < \frac{\left(\Gamma(\frac{8}{5})\right)^2}{C}$ and

then, all hypotheses of Theorem 4.1 hold. Hence, problem $(P)$ admits a nontrivial solution.

6.   Conclusions

In this work, we investigated the existence of solutions for a nonvariational problem involving the $\psi$-Hilfer fractional derivative. Precisely, we fixed a function in the functional space to transform a nonvariational problem into a variational auxiliary problem; after that, we used a mountain pass theorem to prove that the auxiliary problem admits a nontrivial solution. Later, we fix a function $u_0\in E$, and we find a function $u_1$ as a solution for the auxiliary problem associated with $u_0$. by the same way, we determine a function $u_2$ as a solution for the auxiliary problem associated with $u_1$, so a sequence $\{u_k\}$ is constructed. Finally, we proved that this sequence is convergent and its limit is a nontrivial solution for our studied problem. This idea was first introduced by Xie et al. ^[27] and used later by Torres ^[26] for the particular case when $\psi(x) = x$ and $\beta = 1$. We hope to use this idea for a capillarity problem involving the $\psi$-Hilfer derivative and variable exponents, which will also be submitted to Aims Mathematics.

Author contributions

Lamya Almaghamsi: Conceptualization, writing-review & editing, funding acquisition; Aeshah Alghamdi: Conceptualization, writing-review & editing; Abdeljabbar Ghanmi: Conceptualization; Abdeljabbar Ghanmi: Resources. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare that they have no competing interests.