Analytic Solutions for Hilfer Type Fractional Langevin Equations with Variable Coefficients in a Weighted Space

: In this work, analytic solutions of initial value problems for fractional Langevin equations involving Hilfer fractional derivatives and variable coefficients are studied. Firstly, the equivalence of an initial value problem to an integral equation is proved. Secondly, the existence and uniqueness of solutions for the above initial value problem are obtained without a contractive assumption. Finally, a formula of explicit solutions for the proposed problem is derived. By using similar arguments, corresponding conclusions for the case involving Riemann–Liouville fractional derivatives and variable coefficients are obtained. Moreover, the nonlinear case is discussed. Two examples are provided to illustrate theoretical results


Introduction
In 1908, Langevin introduced an integer-order equation mx ′′ (t) = −λx ′ (t) + F(t), where m is the mass of a Brownian particle, x(t) is the particle's position, −λx ′ (t) is the viscous force with coefficient λ and F(t) is the fluctuating force.It is regarded as an effective tool to describe the evolution of physical phenomena in fluctuating environments [1].
Fractional differential equations are important tools for investigating many practical problems in physics, chemistry, biology, etc.Many scholars have conducted extensive research on equations involving Riemann-Liouville or Caputo fractional derivatives-for details, see [2][3][4][5].
For more complex physical phenomena, some researchers have generalized Langevin equations from integer order to fractional order.In 1996, the above classical Langevin equation was generalized by Mainardi and Pironi [6] to the fractional Langevin equation (FLE) where a is the particle's radius, ν is the fluid's viscosity, 1 σ is the friction coefficient for unit mass, F(t, x(t)) = − 1 σ x ′ (t) + 1 m R(t) and R(t) is a random force.FLEs have attracted many scholars to study properties of solutions for FLEs-for instance, the existence and uniqueness of solutions for FLEs with Caputo or Riemann-Liouville fractional derivatives [7,8], boundary value problems for FLEs [9][10][11][12], etc. Baghani and Nieto [13] studied the following Langevin differential equation with two different fractional orders: c D ξ ( c D ν + λ)x(t) = h(t, x(t)).
Moreover, we refer to some recent works [17,18] that deal with a qualitative analysis of FLEs with Hilfer fractional derivatives.
In [17], the authors studied a nonlinear fractional Langevin dynamical system with impulse as follows: In [18], the authors investigated the existence and uniqueness of solutions to the following system of Hilfer FLEs: Due to the complexity of variable-coefficient functions, it is very hard to obtain representations of solutions of FLEs with variable coefficients.Recently, some methods have been presented to deal with linear fractional differential equations with continuous coefficients, such as power series methods [19,20] and the Banach fixed point theorem [21].However, to the best of our knowledge, there are very few studies on FLEs with Hilfer derivatives and negative power function coefficients.
Motivated by previous research, in this article, we study the initial value problem (IVP) for FLEs with Hilfer derivatives and variable coefficients: where 0 This paper is split into six sections.Some definitions and properties of fractional derivatives are provided in Section 2. We study the existence and uniqueness of solutions for the linear case and the nonlinear case in Sections 3 and 4, respectively.In Section 5, we present two examples to illustrate our results and provide an approximate result.Finally, in Section 6, we present the conclusions.

Preliminaries
Firstly, we recall the basic definitions and properties of fractional derivatives and weighted spaces.Definition 1 ([5]).Let α ∈ (0, 1).The Riemann-Liouville fractional integral I α a + f and derivative RL D α a + f are defined by where Γ(•) is the Gamma function, provided that the right-hand side of (3) exists.
A modified Hilfer derivative was presented in [20].
It is not difficult to see that the conditions to guarantee the existence of the Riemann-Liouville derivative in (5) are weaker than those needed for the Hilfer fractional derivative in (4) [20].
is the weighted space of functions x, which are continuously differentiable on [a, b] up to order n − 1 and have the derivative of order n on (a, b] such that x (iv) We denote the weighted space Then, for φ ∈ C σ , the following assertions are valid.
From Lemma 6, it follows that thus,

Equivalence with an Integral Equation
We consider the following IVP for FLEs with Hilfer derivatives and variable coefficients: We define the following constants ν, µ > 0 such that satisfies ( 9) and ( 10) if and only if x satisfies the following equation satisfy ( 9) and (10); then, H D By Theorem 1, Applying I α 2 a + to (9) and in view of ( 13), one obtains By Theorem 2, we obtain x 0 . Applying x 0 , which means that x(t) satisfies (12).If x(t) satisfies (12), then Clearly, (I 1−γ 1 a + x)(a + ) = x 0 and Hence, H D α 1 ,β 1 a + x(t) exists and belongs to C 1−γ 1 .Then, Taking into account the fact that (9).The results are proved completely.Theorem 4. Let δ(t) ∈ C ν and f (t) ∈ C µ .Then, there exists a unique solution x(t) ∈ C α 1 ,β 1 1−γ 1 to Problems ( 9) and (10) given by Proof.We define an operator F : It is easy to see that F is a well-defined operator whose fixed point determines the solution of Equation (12).
Next, we consider the following IVP for FLEs with Riemann-Liouville derivatives.
In this case, the constants µ, ν satisfy satisfies ( 17) and ( 18) if and only if x satisfies the following equation: satisfy ( 17) and ( 18); then, RL D By Theorem 1, we have Applying I α 2 a + to (17) and taking (20) into account, one obtains Applying I α 1 a + to (21) and in view of Lemma 6, one obtains x 0 , which means that x(t) satisfies (19).If x(t) satisfies ( 19), then Obviously, (I 1−α 1 a + x)(a + ) = x 0 and which means that RL D α 1 a + x(t) exists and belongs to C 1−α 1 .Hence, Similar to the arguments of Theorem 4, we have the following conclusion.
and this solution has the form where

Nonlinear Case
We consider the following IVP for nonlinear FLEs with Hilfer derivatives and variable coefficients: where µ < η < 1.
We define the constants ν, µ > 0 as (11) and let δ(t) ∈ C ν .Similar to the arguments of Theorem 3, we have the following result.22) and (23) if and only if x satisfies the following equation: Set the operator G : clearly, G is a well-defined operator, whose fixed point is the solution of Equation ( 24).given by (24).
In particular, we take the following IVP for Hilfer-type fractional Langevin integrodifferential equations with variable coefficients into account: where µ < η < 1.
Theorem 9. Let l(t) ∈ C µ and f (t) ∈ C µ .Then, there exists a unique solution x(t) ∈ C Proof.It follows from Theorem 8 that (25) and ( 26) have a unique solution Similar to the arguments of Theorem 4, one has Similar to the arguments of Theorem 4, we can deduce the corresponding conclusions of the IVP for Riemann-Liouville-type FLEs.We set α 2 < µ < 1 − α 1 + α 2 and 0 < ν < µ + α 1 − 1.
Theorem 10.Let f (t, y(t)) ∈ C µ for any y ∈ C µ .If there exists a non-negative function l is the unique solution to the following IVP In particular, if δ(t) = 0 and f (t, (I .

Conclusions
In this paper, the existence and uniqueness results of solutions of IVPs for FLEs with Hilfer derivatives and variable coefficients are obtained in weighted spaces.The variablecoefficient function δ(t)(∈ C ν ) is not necessarily continuous on a closed interval.Our technique is also useful to solve more general equations.Moreover, the boundary value problem for corresponding equations can be studied.

Theorem 8 .
Let f (t, y(t)) ∈ C µ for any y ∈ C µ .If there exists a non-negative function l(t) ∈ C µ