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1.   Introduction

In the recent years, it has been realized that fractional calculus has an important role in various scientific fields. Fractional differential equations (FDE), which is a consequence of the development of fractional calculus, have attracted the attention of many researchers working in different disciplines (^[28]). Scientific literature has witnessed the appearance of several kinds of fractional derivatives, such as the Riemann-Liouville fractional derivative, Caputo fractional derivative, Hadamard fractional derivative, Grünwald-Letnikov fractional derivative and Caputo-Fabrizio etc (for more details, see ^{[11,13,16,21,22,44,48,51,54]}). It is worthy mentioning here that almost all researches have been conducted within Riemann-Liouville or Caputo fractional derivatives, which are the most popular fractional differential operators.

J. Hadamard suggested a construction of fractional integro-differentiation which is a fractional power of the type $\left(t\frac{\mathrm{d}}{\mathrm{d}t}\right)^{\alpha}$. This construction is well suited to the case of the half-axis and is invariant relative to dilation ([53,p. 330]). The dilation is interpreted in various forms in relation to the field of application. Furthermore, Riemann-Liouville fractional integro-differentiation is formally a fractional power $\left(\frac{\mathrm{d}}{\mathrm{d}t}\right)^{\alpha }$ of the differentiation operator $\left(\frac{\mathrm{d}}{ \mathrm{d}t}\right)$ and is invariant relative to translation if considered on the whole axis. On the other hand, the investigations in terms of Hadamard or Grünwald-Letnikov fractional derivatives are comparably considered seldom.

The boundary value problems defined by FDE have been extensively studied over the last years. Particularly, the study of solutions of fractional differential and integral equations is the key topic of applied mathematics research. Many interesting results have been reported regarding the existence, uniqueness, multiplicity and stability of solutions or positive solutions by means of some fixed point theorems, such as the Krasnosel'skii fixed point theorem, the Schaefer fixed point theorem and the Leggett-Williams fixed point theorem. However, most of the considered problems have been treated in the frame of fractional derivatives of Riemann-Liouville or Caputo types (^{[12,14,15,41,45]}). The qualitative investigations with respect to Hadamard derivative have gained less attention compared to the analysis in terms of Riemann-Liouville and Caputo settings. Recent results on Hadamard FDE can be consulted in (^{[1,4,5,7,10,17,42,43,52]}).

The physical phenomena in fluctuating environments are adequately described using the so called Langevin differential equation (LDE) which was proposed by Langevin himself in [31,1908] to give an elaborated interpretation of Brownian motion. Indeed, LDE is a powerful tool for the study of dynamical properties of many interesting systems in physics, chemistry and engineering (^[9,32,57]). The generalized LDE was introduced later by Kubo in [29,1966], where a fractional memory kernel was incorporated into the equation to describe the fractal and memory properties. Since then the investigation of the generalized LDE has become a hot research topic. As a result, various generalizations of LDE have been offered to describe dynamical processes in a fractal medium. One such generalization is the generalized LDE which incorporates the fractal and memory properties with a disruptive memory kernel. This gives rise to study fractional Langevin equation (^[36]). As the intensive development of fractional derivative, a natural generalization of the LDE is to replace the ordinary derivative by a fractional derivative to yield fractional Langevin equation (FLE). The FLE was introduced by Mainardi and Pironi in earlier 1990s (^[40]). Afterwards, different types of FLE were introduced and studied in ^{[2,3,8,19,30,34,37,38,39,47,50,60,61,62]}. In ^[3], the authors studied a nonlinear LDE involving two fractional orders in different intervals with three-point boundary conditions. The study of FLE in frame of Hadamard derivative has comparably been seldom; see the papers ^[27,56] in which the authors discussed Sturm-Liouville and Langevin equations via Caputo-Hadamard fractional derivatives and systems of FLE of Riemann-Liouville and Hadamard types, respectively.

In the paper by Kiataramkul et al. ^[27]: Generalized Sturm–Liouville and Langevin equations via Hadamard fractional derivatives with anti-periodic boundary conditions. In particular, the authors initiate the study of the existence and uniqueness of solutions for the generalized Sturm-Liouville and Langevin fractional differential equations of Caputo-Hadamard type (^[21]), with two-point nonlocal anti-periodic boundary conditions, by applying the Banach contraction mapping principle. Moreover, two existence results are established via Leray-Schauder nonlinear alternative and Krasnosleskii's fixed point theorem. In addition, the article by W. Sudsutad et al. ^[56]: Systems of fractional Langevin equations of Riemann-Liouville and Hadamard types subject to the nonlocal Hadamard and standard Riemann-Liouville with multi-point and multi-term fractional integral boundary conditions, respectively. In particular, the authors also studied the existence and uniqueness results of solutions for coupled and uncoupled systems are obtained by Banach's contraction mapping principle, Leray-Schauder's alternative.

In the present work, we study the existence, uniqueness and stability of solutions for the following FLE with Hadamard fractional derivatives involving local boundary conditions

where $0 < \alpha < 1$, $\lambda, \beta > 0$, such that

$\mathcal{D}_{1}^{\alpha}$ denotes the Hadamard fractional derivative of order $\alpha$, $\mathrm{D}$ is the ordinary derivative and

is a continuous function.

Our approach is new and is totally different from the ones obtained in ^[27,56] in the sense that different fractional derivatives, ordinary and Hadamard fractional order, are accommodated. Different boundary conditions are associated to problem (1.1) such as three point local boundary conditions and associating different fixed point theorems. It is worthwhile to mention that the nonlinear term $f$ in papers ^[27,56] is independent of fractional derivative of unknown function $x(t)$. But the opposite case is more difficult and complicated. The dependence of the solution on the parameters is discussed, which has not been investigated in ^[27,56]. It is worth mentioning here that Ulam and generalized Ulam-Hyers-Rassias stability results have not been considered in ^[27,56]. Furthermore, the presented work illustrates a numerical simulation obtained through a discretization methods for the evaluation of the Hadamard derivative.

Our method differs from that used by ^[27,56] in our emphasis on the Schaefer fixed point theorem is utilized to investigate existence results for problem (1.1). We also employ the generalization Gronwall inequality techniques to prove the Ulam stability for problem (1.1), and we use important classical and fractional techniques such as: integration by parts in the settings of Hadamard fractional operators, right Hadamard fractional integral, method of variation of parameters, mean value theorem, Dirichlet formula, differentiating an integral, incomplete Gamma function and discretization methods. To the best of the authors' knowledge, there is no work in literature which treats local boundary value problems on mixed type ordinary differential equations involving the Hadamard fractional derivative using the above mentioned techniques.

The rest of the paper is organized as follows: In Section 2, we introduce some notations, definitions and lemmas that are essential in our further analysis. In Section 3, we systemically analyze problem (1.1). An equivalent integral equation is constructed for problem (1.1) and some infra structure are furnished for the use of fixed point theorems. The main results of existence and stability are discussed in Sections 4 and 5, respectively. We prove the main results via the implementation of some fixed point theorems and Ulam's approach. We study the solution's dependence on parameters in Section 6. Indeed, we give an affirmative response to the question on how the solution varies when we change the order of differential operator, the initial values or the nonlinear term $f$. In Section 7, some illustrative examples along with graphical representations are presented to prove consistency with our theoretical findings.

2.   Fundamental definitions, lemmas and remarks

In this section we introduce notations, lemmas, definitions and preliminary facts which are used throughout this paper. In terms of the familiar Gamma function $\Gamma \left(t\right)$, the incomplete Gamma function $\gamma \left(\alpha, t\right)$ and its complement $\Gamma \left(\alpha, t\right)$ are defined by (see, for details, ^[16,20])

and

for all complex $t$. For fixed $\alpha$, $\gamma \left(\alpha, t\right)$ is an increasing function of $t$ with $\lim_{t\rightarrow \infty }\gamma \left(\alpha, t\right) = \Gamma \left(\alpha \right)$. The classical Riemann-Liouville fractional integral of order $\alpha$ for suitable function $x$ is defined as

for $0 < a < t$ and $\Re e(\alpha) > 0$. The corresponding left-sided Riemann-Liouville fractional derivative of order $\alpha$ is defined by

for $\alpha \in \left[n-1, n\right)$. However, the left and right Hadamard fractional integrals of order $\Re e(\alpha) > 0$, for suitable function $x$, introduced essentially by $\mathcal{J}$. Hadamard fractional integral in [18,1892], are defined by

and

respectively. Definition (2.3) is based on the generalisation of the $n$th integral

where $n = [\Re e(\alpha)]+1$ and $[\Re e(\alpha)]$ means the integer part of $\Re e(\alpha)$. Hadamard also proposed ^[18,53] a definition of the fractional integral as

It should be emphasized that expression (2.5) contains $x\left(ts\right)$ in place of $x\left(s\right)$. Therefore we can consider the term $s > 0$ as a variable that describes dilation. As a consequence, using the change of variables $\tau = ts$, would results in the definition of the classical Riemann-Liouville fractional integral. It should be noted that in order to describe the change of dilation we can use the operator $\Upsilon_{s}$ (see [53,p. 330]) such that $\left(\Upsilon _{s}x\right) \left(t\right) = x\left(\exp \left(ts\right) \right)$ where $s > 0$. It is known that the dilation of Euclidean geometric figures changes in size while the shape is unchanged. The connection

allows us to extend various properties of operators $J_{a}^{\alpha }$ to the case of operators $\mathcal{J}_{a}^{\alpha }$. It is directly checked that such connections for the operators (2.5) and (2.1) are given by the relations (2.6). The corresponding left-sided Hadamard fractional derivative of order $\alpha$ is defined by

where $\alpha \in \left[n-1, n\right)$ and $\delta^{n} = \left(t\mathrm{D} \right)^{n}$ is the so-called $\delta$-derivative and $\mathrm{D}\mathcal{\equiv } \frac{\mathrm{d}}{\mathrm{d}t}$.

Firstly, from the above definitions, we see the difference between Hadamard derivative and the Riemann–Liouville one. As a clarification, the aforementioned derivatives differ in the sense that the kernel of the integral in the definition of the Hadamard derivative contains a logarithmic function, while the Riemann-Liouville integral contains a power function. On the other hand, the Hadamard derivative is viewed as a generalization of the operator $\left(t\mathrm{D} \right)^{n}$, while the Riemann–Liouville derivative is considered as an extension of the classical Euler differential operator $\left(\mathrm{D} \right)^{n}$. Secondly, we observe that formally the relationship between Hadamard-type derivatives and Riemann-Liouville derivatives is given by the change of variable $t\to \ln \left(t\right)$, leading to the logarithmic kernel.

Supposedly one can reduce the theorems and results to the corresponding ones of Hadamard-type derivatives by a simple change of variables and functions. It is possible to reduce a formula by such a change of operations but not the precise hypotheses under which a formula is valid. As an illustration, the function $x(t) = \sin t$ is obviously uniformly continuous, but not $\ln$-uniformly continuous on $\mathbb{R}_{+}$, while the function $x(t) = \sin (\ln t)$ is $\ln$-uniformly continuous but not uniformly continuous on $\mathbb{R}_{+}$. However, the two notions are equivalent on every bounded interval $[a, b]$ with $a > 0$. Besides, the Hadamard derivative (also integral) starts at the initial time a which is bigger than zero, but the Riemann–Liouville derivative (also integral) often begins at the origin (or any other real number). Under certain precise conditions, an equivalence could be obtained between a problem involving Hadamard derivative to another defined using a Riemann Liouville derivative.

Lemma 2.1. ^[28] Let $\Re e(\alpha) > 0$, $n = [\Re e(\alpha)]+1$ and $x\in C[a, +\infty)\cap L^{1}[a, +\infty)$, then the Hadamard fractional differential equation $\mathcal{D}_a^{ \alpha }\left[x\right] (t) = 0$, has a solution

Further, the following formulas hold

where $c_{k}\in \mathbb{R}, \left(k = 1, 2, \ldots, n\right)$ are arbitrary constants.

Lemma 2.2. (^[21]) If $0 < \alpha < 1$, then

Theorem 2.3. (^[6]) Consider the continuous function $x: [a, b] \to \mathbb{R}$ belongs to $C^2[a, +\infty)$ and let $\Delta T = \frac{1}{n} \ln \frac{b}{a}$ for $n \geq 1$. Denote the time and space grid by

and $x_N = x(t_N)$ for $N\in \{0, 1, 2, \cdots, n\}$. Then for all $N \in \{1, 2, \cdots, n\}$,

where

and $\lim_{\Delta T \to 0} O (\Delta T) = 0$, here $\left(\tau_{k}^\alpha \right) = k^{1-\alpha} - (k-1)^{1- \alpha}$ and

Lemma 2.4. (^[26]) If $\alpha, \beta > 0$, then the following equality holds

where $a > 0$ is the starting point in the interval. In particular, for $a = 0$,

The following discussion is essential for our further investigation.

Remark 2.5. If $\alpha, \beta > 0$, for $t\in [1, e]$. Then

i) It is easy to verify that

ii) The function $\mathcal{J}_{1}^{\alpha } \left[\sin \lambda \left(t-1\right) \right]$ is continuous as a result of the continuity of $\sin$ function. Furthermore and according to (2.3), we have

Note that

iii) From Lemma 2.2, we have

Remark 2.6. If $\alpha > 0$, for $t\in \left[1, e\right]$. Then, using the elementary inequality $\left(\ln s\right) ^{\alpha }\leq s^{\alpha }$, we obtain the inequality

Utilizing the particular case of the Fubini's theorem, one can deduce that

Indeed, interchanging the order of integration with the help of (2.3) and (2.4) and it follows that

If we take $v = \frac{s}{\tau },$ then

Following ^[48], we bring a formula generalizing the well-known rule of differentiating an integral with respect to its upper limit which serves also as a parameter of the integrand

From (2.7), we have for $\alpha \in \left(0, 1\right)$ and $t\in \left(a, b\right)$ that

Interchanging the order of integration and applying Dirichlet formula, we obtain

In particular, we get

To simplify the presentation, we let

In virtue of equation (2.14), we deduce that

Applying a suitable shift in the fractional operators with lower terminal $\tau$, we deduce the next property ^[23,24].

Property 2.1. Let $0 < \alpha < 1, \ \mathcal{D}_{1}^{\alpha }\left[g\right] \in L^{1}\left(1, e\right)$ and $\mathcal{J}_{1}^{\alpha }\left[f_{x} \right] \in C\left(1, e\right)$. Then we have

In the literature, we can read the following Schaefer fixed point theorem.

Lemma 2.7. ^[21,55] Let $E$ be a Banach space and assume that $\Psi : E\to E$ is a completely continuous operator. If the set

is bounded, then $\Psi$ has a fixed point in $E$.

The next result is a generalization of Gronwall inequality due to Pachpatte (^[46]).

Lemma 2.8. Let $u\in C(I, \mathbb{R}^{+})$, $\tilde{a}(t, s),$ $\tilde{b }(t, s)\in C(D, \mathbb{R}^{+})$ and $\tilde{a}(t, s),$ $\tilde{b}(t, s)$ are nondecreasing in $t$ for each $s\in I$, where $I = [\tilde{\alpha}, \tilde{\beta }]$, $\mathbb{R}^{+} = [0, \infty)$,

and suppose that

for $t\in I$, where $k\geq 0$ is a constant. If

for $t\in I$, then

The following hypotheses will be used in the sequel:

H1: There exist a constant $N_{i} > 0\ \left(i = 1, 2\right)$ such that

for each $t\in \left[1, e\right]$ and all $x_{i}, \tilde{x}_{i}\in \mathbb{R}$.

H2: There exists a constant $L > 0$ such that $\left \vert f(t, x, \tilde{ x})\right \vert \leq L$, for each $t\in \left[1, e\right]$ and all $x, \tilde{x}\in \mathbb{R}$.

3.   The nonlinear boundary value problem (1.1)

In order to study the nonlinear problem (1.1), we first consider the associated linear problem and obtain its solution:

for $0 < \alpha \leq 1$, where $h$ is a continuous function on $\left[1, e\right]$.

Lemma 3.1. The general solution of the linear differential equation

for $t\in \left[1, e\right]$, is given by

where $c_{1}, c_{2}$ are unknown arbitrary constants.

Proof. Assume that $x\left(t\right)$ satisfies (3.1), then the method of variation of parameters implies the desired results.

Lemma 3.2. Let $0 < \alpha < 1, \ h\in C([1, e], \mathbb{R})$. Then the unique solution of the linear problem

for $t\in (1, e)$, is equivalent to the integral equation

Proof. Applying Lemma 2.1, we may reduce (3.2)-a to an equivalent integral equation

where $c_{0}\in\mathbb{R}$. In view of the boundary condition $\tilde{x}(1) = 0$, we have $c_{0} = 0$, thus (3.3) holds.

Lemma 3.3. Let $h\in C([1, e], \mathbb{R}), \alpha \in (0, 1]$ and$\ 1 < \xi < e$. Then the fractional problem

has a unique solution given by

where

Proof. Assuming

and then applying Lemma 3.1 when $0 < \alpha < 1,$ we get

By the boundary condition $x\left(1\right) = 0$ and privous equation, we conclude that

On the other hand, $x(e) = \beta x(\xi)$, combining with

and

yield

where $\Delta$ is given by (3.6). If $\lambda = \frac{(2k+1)\pi }{2}$, $k = 0, 1, \ldots$, then $c_{2} = 0,$ and by (3.7), we get

otherwise, we find

The above two expressions of $c_{1}$ are equivalent for the particular choice of $\lambda$. Substituting these values of $c_{1}$ and $c_{2}$ in (3.7) and applying Lemma 3.2, we finally obtain (3.5). So, the unique solution of problem (3.4) is given by (3.5). Conversely, let $x(t)$ be given by formula (3.5), operating $\mathrm{D}^{2}$ on both sides and using (2.13), we get

Hence

Operating $\mathcal{D}_{1}^{\alpha }$ on the above relation and using (2.8), we obtain the first equation of (3.4). Further, it is easy to get that all conditions in (3.4) are satisfied. The proof is completed.

By virtue of Lemma 3.3, we get the following result.

Lemma 3.4. Let $0 < \alpha < 1$, $\lambda > 0$. Then the problem (1.1) is equivalent to the integral equation

For convenience, we define the following functions

and

Then, the integral equation (3.8) can be written as

From the expressions of (3.5) and (3.8), we can see that if all conditions in Lemmas 3.3 and 3.4 are satisfied, then the solution is a continuous solution of the boundary value problem (1.1). Let $\mathcal{C} = C([1, e], \mathbb{R})$ be a Banach space of all continuous functions defined on $[1, e]$ endowed with the usual supremum norm. Consider the space defined by

equipped with the norm $\left\Vert x \right\Vert_{E} = \left\Vert x \right\Vert +\left\Vert \mathcal{D}_{1}^{ \alpha }\left[x\right] \right\Vert$, then $\left(E, \left\Vert.\right\Vert_{E} \right)$ is a Banach space. On this space, by virtue of Lemma 3.4, we may define the operator $\Psi : E\to E$ by

where $g\left(t-1\right)$, $\phi_{x} \left(t\right)$ and $H_{x}\left(\xi, \beta \right)$ defined by (2.15), (3.9) and (3.10) respectively. Then

By virtue of Property 2.1 and Eq (2.10) in Remark 2.5, we get the following

The continuity of the functional $f$ would imply the continuity of $\Psi x$ and $\mathcal{D}_{1}^{\alpha }\left[\Psi x\right]$. Hence the operator $\Psi$ maps the Banach space $E$ into itself. This operator will be used to prove our main results. Next section, we employ fixed point theorems to prove the main results of this paper. In view of Lemma 3.4, we transform problem (1.1) as

Observe that problem (1.1) or (3.8) has solutions if the operator $\Psi$ in (3.14) has fixed points. For computational convenience, we set the notations:

and

where

and

4.   Results of existence and uniqueness

In this section, we establish the existence and uniqueness results via fixed point theorems.

Theorem 4.1. Assume that $f:[1, e]\times\mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a continuous function that satisfies (H1). If we suppose

where $Q$ is defined in (3.18), then problem (3.14) has a unique solution in $E$.

Proof. To prove this theorem, we need to prove that the operator $\Psi$ has a fixed point in $E$. So, we shall prove that $\Psi$ is a contraction mapping on $E$. For any $x, \tilde{x}\in E$ and for each $t\in [1, e]$, we have

where $x\left(t\right)$ and $\tilde{x} \left(t\right)$ are defined in Lemma 3.4. From assumption (H1) and Eqs (3.9) and (4.1), we obtain

where $\rho_{\alpha }\left(t\right)$ is given by (2.11) and

Similarly, we can obtain $\left\vert \phi_{\tilde{x}}\left(\xi \right) -\phi_{x}\left(\xi \right) \right\vert$ and $\left\vert \phi_{\tilde{x} }\left(e\right) -\phi_{x}\left(e\right) \right\vert$. Then

Linking (4.2), (4.3) and (4.4), for every $x, \ \tilde{x} \in E$, we get

where $\rho \left(t\right)$ is given by (3.15). Consequently, it yields that

with

On the other hand, we observe that

By (3.13), we have

Taking into account that

we have,

where $\sigma_{\alpha }\left(t\right)$ is given by (3.16). Therefore, from (4.7), (4.7) and (4.10), we have

where

This gives

with

By (4.5) and (4.13), we can write

with $Q\geq Q_{1}+Q_{2}$. Combining (4.1) with (4.15), we conclude that $\Psi$ is contractive on $E$. As a consequence of Banach fixed point theorem, we deduce that $\Psi$ has a unique fixed point which is a solution of our problem in $E$.

Corollary 4.2. Let the assumptions of the Theorem 4.1 be fulfilled. If we suppose that (4.1) holds, with $Q$ is defined as

then, problem (3.14) has a unique solution in $E.$

Let $B_{r}\subset E$ be bounded, i.e., there exists a positive constant $r > 0$ such that $\left \Vert x\right \Vert_{E} < r$ for all $x \in B_{r}$. then $B_r$ is a closed ball in the Banach space $E$, hence it is also a Banach space. The restriction of $\Psi$ on $B_r$ is still a contraction by Theorem 4.1. Then, problem (3.14) has a unique solution in $B_r$ if $\Psi \left(B_{r}\right) \subset$ $B_{r}$.

Theorem 4.3. Assume that $f:[1, e]\times \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a continuous function that satisfies (H1). If we suppose that (4.1) holds, with $Q$ is defined in (3.18), then problem (3.14) has a unique solution in $B_r$.

Proof. Now we show that $\Psi \left(B_{r}\right) \subset B_{r},$ that is $\left\Vert \Psi x \right \Vert_{E}\leq r$ whenever $\left\Vert x\right\Vert_{E}\leq r$. Denoting

where $L_{0} = \max \left\{ \left\vert f(t, 0, 0\right\vert :t\in \left[1, e\right] \right\}$. Observe that

So, we have

and

Then $\left\vert \Psi x(t)\right\vert \leq \frac{\rho \left(t\right) }{\Gamma \left(1+\alpha \right) }L_{b}$. Therefore,

where $Q_{1}$ is given by (4.6). On the other hand, we have

Thanks to (H1), it yields that

This gives

where $\sigma_{\alpha }\left(t\right)$ is given by (3.16). Consequently, by (4.17), (4.19) and (4.20), we have

where $Q_{2}$ is given by (4.14). Using (4.18) and (4.21), we obtain

and we find that $\left\Vert \Psi x \right\Vert_{E} \leq Q\left(Nr+L_{0}\right) \leq r$, where we choose $r\geq L_{0}\left(Q^{-1}-N\right)^{-1}$. Hence, the operator $\Psi$ maps bounded sets into bounded sets in $B_{r}$, therefore $\Psi$ is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.

Corollary 4.4. Assume that $f:[1, e]\times \mathcal{C} \times\mathcal{C} \to \mathcal{C}$ is a continuous function that satisfies (H1). If we suppose $N = \max \left\{N_{1}, N_{2} \right\}$ and

where $Q$ is defined in (4.16). Then, problem (3.14) has a unique solution in $B_{r}$.

Our second result will use the Scheafer fixed point theorem.

Theorem 4.5. The problem (3.14) has at least one solution defined on $E$, whenever assumption (H2) be hold.

Proof. The proof will be given in several steps.

Step 1: We show that $\Psi$ is continuous. Let us consider a sequence $\left\{ x_{n}\right\} \in E$ converging to $x$. For each $t\in \lbrack 1, e],$ we have

where

Similarly, we can obtain

Thus, from (4.19), (4.22) and (4.23), we have

If $\left(t, x\right) \in \left[1, e\right] \times E$, $x_{n}\to x$ as $n\to \infty$ and $f$ is continuous, then (4.24) gives

as $n\to \infty$. On the other hand, from (4.11) we observe that

where $R_{12}$ is given by (4.12). Thus

as $n\to \infty$. Since the convergence of a sequence implies its boundedness, therefore, there exists $r > 0$ such that $\left\Vert x_{n}\right\Vert \leq r, \left\Vert x\right\Vert \leq r$ and hence $f$ is uniformly continuous on the compact set

By (4.25) and (4.26), we can write $\left\Vert \left[\Psi x_{n}\right] - \left[\Psi x\right] \right\Vert_{E}\to 0$ as $n\to \infty$. This shows that $\Psi$ is continuous.

Step 2: Now we show that the operator $\Psi : E \to E$ maps bounded sets into bounded sets in $E$. Let $B_{r}\subset E$ be bounded, i.e., there exists a positive constant $r > 0$ such that $\left\Vert x\right\Vert_{E}\leq r$ for all $x \in B_{r}$. Let

then, for $x\in B_{r}$, we have

and

Then from (4.27) and (4.28), we get $\left\vert \Psi x(t)\right\vert \leq \frac{\rho \left(t\right) }{\Gamma \left(1+\alpha \right) }L$. Therefore,

According to Property 2.1, we should have

Consequently, by (3.12), (4.28) and (4.30), we have

Using (4.29) and (4.31), we obtain $\left\Vert \Psi x\right\Vert_{E}\leq QL$. Hence, the operator $\Psi$ maps bounded sets into bounded sets in $E$. Next we show that $\Psi$ maps bounded sets into equicontinuous sets of $B_{r}$.

Step 3: In this step, we show that $\Psi \left(B_{r}\right)$ is equicontinuity. Let $t_{1}, t_2\in [1, e]$ such that $t_{1} < t_{2}$. Then we obtain

We can show that

Hence

It is easy to find that

Therefore by (4.28), (4.32), (4.33) and (4.34) we have

We have also,

Thus, we obtain

We find that

Note that

is independent of $x$. Therefore

In accordance with (4.35), (4.36), (4.37) and (4.38), we deduce that

as $\left\vert t_{2}-t_{1}\right\vert \to 0$. Hence the sets of functions $\left\{ \Psi x(t) : x \in B_{r}\right\}$ and

are bounded in $B_{r}$ and equicontinuous on $[1, e]$. Thus, by the Arzelá–Ascoli Theorem, the mapping $\Psi$ is completely continuous on $E$.

Step 4: In the last step, it remains to show that the set defined by

is bounded. Let $x$ be a solution. Then, for $t\in [1, e]$ and using the computations in proving that $\Psi$ is bounded, we have $\left\vert x\left(t\right) \right\vert = \left\vert \mu \left(\Psi x\right) \left(t\right) \right\vert$. Let $x\in \Lambda$, $x = \mu \Psi x$ for some $0 < \mu < 1$. Thus, by (4.29), for each $t\in [1, e]$, we have

for $\mu \in \left(0, 1\right)$. On the other hand, by (4.31), we have

It follows from (4.39) and (4.40) that $\left\Vert x\right\Vert_{E} = \left\Vert x\right\Vert +\left\Vert \mathcal{D}_{1}^{\alpha }\left[x\right] \right\Vert \leq QL < \infty$, where $Q$ defined by (3.18). This implies that the set $\Lambda$ is bounded independently of $\mu \in \left(0, 1\right)$. Therefore, $\Lambda$ is bounded. As a conclusion of Schaefer fixed point theorem, we deduce that $\Psi$ has at least one fixed point, which is a solution of (3.14). The proof is completed.

Remark 4.6. Let $\varepsilon > 0$ and choose a number $\eta > 0$ such that

where

Assume $t_{1}, t_{2}\in \left[1, e\right]$; $t_{1} < t_{2}$ such that $t_{2}-t_{1}\leq \eta$. It obvious that $t_{2} > \eta$ and there are two possibilities for $t_{1}$ and $\eta$.

Case 1: For $\eta \leq t_{1} < t_{2}\leq e$, by means of mean value theorem of differentiation implies that there exists $t\in \left(t_{1}, t_{2}\right)$, such that

whence, we obtain

Case 2: For $1\leq t_{1} < \eta < t_{2} \leq e$ and so $t_{2} < 2\eta$. These imply that

Combining Case 1 and 2, we obtain

Now, it is obvious by (4.35) and (4.41) that $\left\vert \Psi x(t_{2})-\Psi x(t_{1})\right\vert \leq \frac{\varepsilon }{2}$. A similar argument can be applied to obtain

Corollary 4.7. Suppose that the conditions of Theorem 4.5 hold, with $Q$ is defined as (4.16). Then, problem (3.14) has at least one solution defined on $\left[1, e\right]$.

5.   Results of stability

For the study of Hyers-Ulam-Rassias and generalized Ulam-Hyers-Rassias stabilities of problem (1.1) on a compact interval $[1, e]$, we adopt the following definitions ^{[22,35,49,58,59]}.

Definition 5.1. Problem (1.1) is Ulam-Hyers-Rassias stable with respect to $\varphi \in C\left(\left[1, e\right], \mathbb{R}^{+}\right)$ if for each $\epsilon > 0$ and each solution $\tilde{x}$ of the inequality

for $t\in \left[1, e\right]$, there exists a real number $c_{\varphi } > 0$ and a solution $x$ of problem (1.1) such that

for $t\in \left[1, e\right]$. Particularly, in the case that $\varphi$ is identity function on $\left[1, e \right]$, problem (1.1) is called Ulam-Hyers stable. Moreover, if there exists $\psi \in C\left(\mathbb{R}^{+}, \mathbb{R}^{+}\right), \psi (0) = 0$, such that $\left\vert \tilde{x} (t) - x(t) \right\vert \leq \psi (\epsilon)$, for $t\in \left[1, e \right]$, then problem (1.1) is called generalized Ulam-Hyers stable.

Definition 5.2. Problem (1.1) is generalized Ulam-Hyers-Rassias stable with respect to a function $\varphi \in C\left(\left[1, e\right], \mathbb{R}^{+}\right)$ if for each solution $\tilde{x}$ of the inequality

for $t\in \left[1, e\right]$, there exist a real number $c_{\varphi } > 0$ and a solution $x$ of problem (1.1) such that $\left\vert \tilde{x}(t)-x(t)\right\vert \leq c_{\varphi }\varphi \left(t\right)$, for $t\in \left[1, e\right]$.

Remark 5.3. A function $\tilde{x}\in \mathcal{C}$ is a solution of the inequality (5.1) if and only if there exists a function $h\in\mathcal{C}$ (which depends on $\tilde{x}$) such that for all $t\in \left[1, e\right]$,

(i) $\left\vert h\left(t\right) \right\vert \leq \epsilon \varphi \left(t\right)$.

(ii) $\mathcal{D}_{1}^{\alpha }\left(\mathrm{D}^{2}+\lambda^{2}\right) \tilde{x}(t) = f(t, \tilde{x}(t), \mathcal{D}_{1}^{\alpha }\left[\tilde{x}\right] (t))+h\left(t\right)$.

Lemma 5.4. Let $0 < \alpha < 1$, if $\tilde{x}\in \mathcal{C}$ is a solution of the inequality (5.1)(or (5.2)) then $\tilde{x}$ is a solution of the following integral inequality

for $t\in \left[1, e\right]$, where

for $\xi \in (1, e]$ and

Proof. By using Remark 5.3-ii, we have

In accordance with Lemma 3.4, we deduce $\tilde{x}(t) = \frac{1}{\lambda }\phi_{\tilde{x}, h}\left(t\right) +H_{\tilde{x}, h}\left(\xi, \beta \right) g\left(t-1\right)$, where

and $H_{\tilde{x}, h}\left(\xi, \beta \right) = \frac{1}{\Delta }\left(\beta \phi_{\tilde{x}, h}\left(\xi \right) -\phi_{\tilde{x}, h}\left(e\right) \right)$. Hence

Accordingly, we easily deduce equation (5.3).

By virtue of Remark 5.3-i, it can be easily seen that

Similar arguments can be applied as in (4.3) to deduce that

Now, it is obvious that

and

Theorem 5.5. Assume that the conditions of Theorem 4.1 and (5.1), (5.4) hold. Then, problem (1.1) is Ulam-Hyers-Rassias stable with respect to positive constant functions. Particularly, problem (1.1) is Ulam-Hyers stable and generalized Ulam-Hyers stable.

Proof. Using Theorem 4.1, there exists a unique solution $x\in \mathcal{C}$ of problem (1.1) that is given by integral equation (3.11). Let $\tilde{x}\in\mathcal{C}$ be any solution of the inequality (5.1), then by (5.5) and (5.6), we have

where $x\left(t\right)$ and $\rho \left(t\right)$ are as in (3.8) and (3.15). Thus

since by (3.18), we get $0 < \frac{NM_{\rho }}{ \Gamma \left(1+\alpha \right) } < 1$. Then, for each $t\in \left[1, e\right]$,

Accordingly, to satisfy the inequality $\left\vert \tilde{x}(t)-x\left(t\right) \right\vert \leq \epsilon c_{\varphi } \varphi \left(t\right)$, we have to pose that $\varphi$ is a constant function on $[1, e]$. Hence, if $\varphi (t) = c > 0,$ $t\in [1, e]$, then any finite positive constant

will satisfy the problem. Thus, the fractional boundary value problem (1.1) is Ulam-Hyers-Rassias with respect to a constant function. The Ulam-Hyers stability can be obtained by putting $\varphi = 1,$ and hence generalized Ulam-Hyers stable with $\psi$ as identity function.

In the next result, we prove the (generalized) Ulam-Hyers-Rassias stability in terms of a function.

Theorem 5.6. Assume that the conditions of Theorem 4.1 and (5.1) hold. Then, problem (1.1) is Ulam-Hyers-Rassias stable with respect to $\varphi$ provided that

and $\sup_{t\in \lbrack 1, e]}p(t) < 1,$ where

Proof. Let us denote by $x\in \mathcal{C}$ the unique solution of the problem (1.1). Let $\tilde{x}\in \mathcal{C}$ be a solution of the inequality (5.1), with

By modifying the estimate (5.5), we have

The above inequality implies

Taking into account (2.12), (5.9) and (5.10), lead to

Hence

where

and

In virtue of Lemma 2.8, we deduce that

for $t\in \lbrack 1, e]$. Problem (1.1) is Ulam-Hyers-Rassias stable with respect to $\varphi \geq \frac{1}{\epsilon }\left\vert h\right\vert$, $\varphi$ must satisfy the inequality (5.7). In this case, we get $\left\vert \tilde{x}(t)-x\left(t\right) \right\vert \leq c_{\varphi }\epsilon \varphi (t)$, where

This completes the proof.

Theorem 5.7. Assume the conditions of Theorem 4.1 and (5.2) hold. Then, problem (1.1) is generalized Ulam-Hyers-Rassias stable with respect to $\varphi$ provided for any $t \in \left[1, e\right]$ that

Proof. Let us denote by $x\in C\left(\left[1, e\right], \mathbb{R} \right)$ the unique solution of the problem (1.1). Let $\bar{x}\in \mathcal{C}$ be a solution of the inequality (5.2), with (5.8). It follows

On the other hand, we have, for each $t\in \left[1, e\right]$,

Hence by Lemma 5.4, for each $t\in \left[1, e\right]$, we get

where

Then (see (3.17))

Accordingly, we get

We are going now to get an estimate for $\left\Vert \mathcal{D}_{1}^{\alpha } \left[\bar{x}\right] -\mathcal{D}_{1}^{\alpha }\left[x\right] \right\Vert.$ It is obvious that

where

Also, we get

Hence, we deduce that

Then

If $\left\vert h\right\vert \leq \varphi$, we get

Hence by the given condition (5.11), the equation (1.1) is generalized Ulam-Hyers-Rassias stable with respect to $\varphi$.

6.   Dependence of solution on the parameters

For $f$ Lipschitz in the second and the third variables, the solution's dependence on the order of the differential operator, the boundary values and the nonlinear term $f$ are discussed in this section. We show that the solutions of two equations with neighbouring orders will (under suitable conditions on their right hand sides $f$) lie close to one another.

Theorem 6.1. Suppose that the conditions of Theorem 4.1 hold. Let $x\left(t\right)$, $x_{\epsilon }\left(t\right)$ be the solutions, respectively, of problems (1.1) and

for $t\in \left(0, 1\right)$ and $\epsilon > 0$, with the boundary conditions (1.1)-b, where $0 < \alpha -\epsilon < \alpha < 1$. Then there exists a constant $k_{\epsilon } > 0$ such that

where $\left\| f\right\|_* = \sup_{ \epsilon } \left\Vert f_{x_{\epsilon }}\right\Vert$ and $f_{x_{\epsilon }} (t): = f \left(t, x_{\epsilon }(t), \mathcal{D}_{1}^{ \alpha }\left[x_{ \epsilon }\right] (t) \right)$.

Proof. By Lemma 3.4 and equation (3.11), we can obtain

is the solution of (6.1) with the boundary conditions in (1.1), where

and $H_{x_{\epsilon }}\left(\xi, \beta \right) = \frac{1}{ \Delta }\left(\beta \phi_{x_{\epsilon }} \left(\xi \right) +\phi_{x_{\epsilon }} (e) \right)$. Then

This leads to

with

In a similar manner, we can get

Then

with $\varrho \left(t\right) = \frac{1}{\lambda }\varrho_{\epsilon }\left(t\right) +\frac{1}{\left\vert \Delta \right\vert }\left[\beta \varrho_{\epsilon }\left(\xi \right) + \varrho_{\epsilon }\left(e\right) \right]$. On the other hand,

By (4.8), we have

Then

with

Then the expression above becomes

Then

with

Moreover, from (6.3), (6.5), we deduce that

Finally, we get the inequality

which is exactly the required inequality (6.2), where

Theorem 6.2. Suppose that the conditions of Theorem 4.1 hold. Let $x\left(t\right), x_{\epsilon }\left(t\right)$ be the solutions, respectively, of the problems (1.1) and

for $t\in \left(1, e\right)$ and $h_{\epsilon }\in \mathcal{C}$, with boundary conditions (1.1)-b, where $\epsilon < 0.$ Then $\left\Vert x-x_{\epsilon }\right\Vert_{E} = O\left(\epsilon \right)$.

Proof. In accordance with Lemma 3.4, we have

and

and

From (6.8) and (6.9), we derive

On the other hand,

and

where $R_{11}$ is given by (4.9). Hence, we obtain

and

Consequently

where $Q$ is given by (3.18) and $\left\| h\right\|_* = \sup_{0 < \epsilon }\left\Vert h_{\epsilon }\right\Vert$. It is obvious that $\left\Vert x-x_{\epsilon }\right\Vert_{E} = O\left(\epsilon \right)$.

Let us introduce small perturbation in the boundary conditions of (1.1) such that

for $\xi \in (1, e]$.

Theorem 6.3. Assume the conditions of Theorem 4.1 hold. Let $x\left(t\right)$, $x_{\epsilon }\left(t\right)$ be respective solutions, of the problems (1.1) and the boundary conditions (1.1)-a with (6.10). Then

Proof. Similar arguments as in the proof of Lemma 3.4, may lead to the solution of equations (1.1)-a and (6.10) that has the following form

Therefore

and $\Delta \cos \lambda \neq 0$, where

and $H_{x_{\epsilon }}\left(\xi, \beta \right) = \frac{1}{\Delta }\left(\beta \phi_{x_{\epsilon }}\left(\xi \right) +\phi_{x_{\epsilon }} \left(e\right) \right)$. As before, we find that

and

Hence

Consequently

It is obvious that $\left\Vert x-x_{\epsilon }\right\Vert_{E} = O\left(\epsilon \right)$.

7.   Illustrative examples and applications

In this section, we present some examples to illustrate the validity and applicability of the main results.

Example 7.1. Consider problem (1.1) with

Then $f$ fulfills the Lipschitz condition (H1) such that $\lambda = 2$, $\beta = 2$, $\xi = \frac{3}{2}$,

In Table 1, we show values of $\alpha$, $t$ and $Q$. Thus $NQ < 1$. Hence, by Theorem 4.1, the problem (1.1) with (7.1) has a unique solution on $\left[1, e\right]$.

Example 7.2. Consider problem (1.1) with

or

Then $f$ fulfills the Lipschitz condition (H1), where $\lambda = 4$, $\beta = 2$, $\xi = \frac{3}{2}$, $N = \max \left\{ N_{1}, N_{2} \right\} = \frac{5}{4}$ or $\frac{7}{6}$. In Table 2, we show values of $\alpha$, $t$, $Q$, $\frac{5}{4}Q$ and $\frac{7}{6}Q$. Thus, the condition (4.1) holds. Again, taking $N = \max \left\{N_{1}, N_{2} \right\} = \frac{5}{4}$ or $\frac{7}{6}$, we have $NQ < 1$. Note that all the assumptions of the Theorem 4.3 holds. Therefore problem (1.1) has a unique solution on $E$.

Example 7.3. Consider problem (1.1) with

for $d_{i} \in C\left[1, e\right]$ with $i = 1, 2, 3, 4$, that fulfils (H1) with

for example $d_1 (t) = d_3 (t) = \frac{1}{4}$, $d_2(t) = d_4 (t) = 2$. Thus, we can put $\beta = 2$, $\xi = \frac{3}{2}$, $N = 1$, $L_0 = \frac{1}{4}$. In Table 3, one can find some values of $\alpha$, $\lambda$, $t$, $Q$ and $r$, where $r$ and $Q$ are as defined in Theorem 4.3.

Hence, by Theorem 4.3, problem (1.1) with (7.4) has a unique solution on $B_{r}$.

Example 7.4. (Illustrative example on stability) Consider the FLE problem (1.1) with Hadamard fractional derivatives involving nonlocal boundary conditions:

with

where $\alpha = \frac{3}{8}$, $\lambda = \pi$, $\beta = 2$ and $\xi = \frac{11}{5}$. It is obvious that

So

See the Figure 1. Also, we have

for each $t\in \left[1, \frac{15}{4} \right]$ and all $x_{i}, \tilde{x}_{i} \in \mathbb{R}$, here $N_{1} = \frac{16}{135}$ and $N_{2} = \frac{1}{49}$. Put $\varphi (t) = \frac{t^2}{t^2+1}$ and

By using Eq (2.11), we obtain

and using MatLab program,

Also, by applying Eq (3.17), we obtain

Then we get

Then the assumptions of Theorem 5.5 are satisfied. Then, problem (1.1) is Ulam-Hyers stable and generalized Ulam-Hyers stable.

Example 7.5. (Illustrative example on solution dependence) Consider the FLE problem (1.1) with Hadamard fractional derivatives involving nonlocal boundary conditions:

and

for $t\in \left(0, 1\right)$ and $\epsilon > 0$ with

where $\alpha = \frac{7}{9}$, $\lambda = \frac{4\pi}{3}$, $\beta = \frac{7}{3}$, $\xi = \frac{9}{4}$ and $0 < \frac{7}{9} -\epsilon < \alpha < 1$. It is obvious that

So

See Figure 2. Also, we have

for each $t\in \left[1, \frac{12}{5} \right]$ and all $x_{i}, \tilde{x}_{i} \in \mathbb{R}$, here $N_{1} = \frac{4}{121}$ and $N_{2} = \frac{2}{75}$. Put $\varphi (t) = \frac{t^2}{t^2 + 1}$,

By using Eq (2.11), we obtain

and

Also, by applying Eq (3.17), we obtain

Then for $\epsilon = 0.1$, by using MatLab program, we get

and

Then the assumptions of Theorem 6.1 are satisfied. In addition to, by applying Eqs (6.6) and (6.7), we can calculate $C_{22}(t)$, $C_{33}(t)$ and $k_\epsilon$.

Now, we describ discretization method and use Theorem 2.3 for this example. Fix $n \geq 1$ for $N \in \{1, \cdots, n \}$, define

with

and $[a, b] = [1, \frac{12}{5}]$.

Also,

and

Thus, for $f\left(t, x(t), \mathcal{D}_1^{ \frac{7}{9}} \left[x\right] (t) \right)$, we get

where

8.   Conclusions

The Langevin equation has been proposed to describe dynamical processes in a fractal medium in which the fractal and memory properties with a dissipative memory kernel are incorporated. However, it has been realized that the classical Langevin equation failed to describe the complex systems. Thus, the consideration of LDE in frame of fractional derivatives becomes compulsory. As a result of this interest, several results have been revealed and different versions of LDE have been under study. In this paper, we have presented some results dealing with the existence and uniqueness of solutions for boundary value problem of nonlinear Langevin equation involving Hadamard fractional order. As a first step, the boundary value problem is transformed to a fixed point problem by applying the tools of Hadamard fractional calculus. Based on this, the existence results are established by means of the Schaefer's fixed point theorem and Banach contraction principle.

We claim that the results of this paper is new and generalize some earlier results. For instance, by taking $\alpha = 1$ in the results of this paper which can be considered a special case of a simple Jerk Chaotic circuit equation see ^[33]. The paper presented a discuss on the Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stabilities of the solution of the FLD using the generalization for the Gronwall inequality. We present an example to demonstrate the consistency to the theoretical findings. We also analyze the continuous dependence of solutions all on its right side function, initial value condition and the fractional order for FDE. Using these results, the properties of the solution process can be discussed through numerical simulation. We hope to consider this problem in a future work.

Availability of data and material

Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.

Authors' contributions

The authors declare that the study was realized in collaboration with equal responsibility. All authors read and approved the final manuscript.

Acknowledgments

The second author was supported by Bu-Ali Sina University. J. Alzabut would like to acknowledge the support of Prince Sultan University for paying the Article Processing Charges (APC) of this publication.

Conflict of interest

The authors declare that they have no competing interests.

Supplementary