Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses

*Correspondence: wxmsuda03@163.com 2School of Mathematics & Computer Science, Shangrao Normal University, Shangrao, China Full list of author information is available at the end of the article Abstract In this paper, we consider a nonlocal boundary value problem of nonlinear implicit fractional Langevin equation with noninstantaneous impulses. We study the existence, uniqueness and generalized Ulam–Hyers–Rassias stability of the proposed model with the help of fixed point approach, over generalized complete metric space. We give an example which supports our main result.


Introduction
At Wisconsin university, Ulam raised a question about the stability of functional equations in 1940. The question of Ulam was: Under what conditions does there exist an additive mapping near an approximately additive mapping?; see [30]. In 1941, Hyers was the first mathematician who gave a partial answer to Ulam's question [12] in a Banach space. Since then, stability of such form is known as Ulam-Hyers stability. In 1978, Rassias [23] provided a remarkable generalization of the Ulam-Hyers stability of mappings by considering variables. For more information about the topic, we refer the reader to [3, 14-16, 24, 28, 31, 40, 42].
An equation of the form m d 2 X dt 2 = λ dX dt + η(t) is called Langevin equation, introduced by Paul Langevin in 1908. Langevin equations have been widely used to describe stochastic problems in physics, chemistry and electrical engineering. For example, Brownian motion is well described by the Langevin equation when the random fluctuation force is assumed to be white noise. For the removal of noise, mathematicians used fractional order differential equations, which also perform well in reducing the staircase effects compared to integer order differential equations. Thus it is very important to study Langevin equations with fractional derivatives; see, for instance, [2,10,20,21].
Fractional order differential equations are generalizations of the classical integer order differential equations. Fractional calculus has become a fast developing area, and its applications can be found in diverse fields ranging from physical sciences, porous media, electrochemistry, economics, electromagnetics, medicine and engineering to biological sciences. Progressively, fractional differential equations play a very important role in thermodynamics, statistical physics, viscoelasticity, nonlinear oscillation of earthquakes, defence, optics, control, electrical circuits, signal processing, astronomy, etc. There are some outstanding articles which provide the main theoretical tools for the qualitative analysis of this research field, and at the same time, show the interconnection as well as the distinction between integral models, classical and fractional differential equations; see [1,5,13,17,19,22,[25][26][27]29].
Impulsive fractional differential equations are used to describe both physical and social sciences. Also they describe many practical dynamical systems such as evolutionary processes, characterized by abrupt changes of the state at certain instants. In the last few decades, the theory of impulsive fractional differential equations were well utilized in medicine, mechanical engineering, ecology, biology and astronomy, etc. There are some remarkable monographs [8, 11, 18, 32, 33, 35-37, 39, 41], which consider fractional differential equations with impulses.
Recently, many mathematicians devoted considerable attention to the existence, uniqueness and different types of Hyers-Ulam stability of the solutions of nonlinear implicit fractional differential equations with Caputo fractional derivative, see [4,6,7].

Solution framework of linear impulsive fractional Langevin equation
Let J = [0, τ ] and C(J, R) be the space of all continuous functions from J to R, and the piecewise continuous function space In the current section, we create a uniform framework to originate an appropriate formula for the solution of impulsive fractional differential equation of the form: We recall some definitions of fractional calculus from [17] as follows.

Definition 2.1
The fractional integral of order α from 0 to t for the function f is defined by where Γ (·) is the Gamma function.

Definition 2.2
The Riemann-Liouville fractional derivative of fractional order α from 0 to t for a function f can be written as where Γ (·) is the Gamma function.

Definition 2.3
The Caputo derivative of fractional order α from 0 to t for a function f can be defined as

Definition 2.4
The general form of classical Caputo derivative of order α of a function f can be given as (ii) In Definition 2.4, the integrable function f can be discontinuous. This fact can lead us to consider impulsive fractional problems in the sequel. Then Proof Let x be a solution of problem (2.1).
After using fractional integrals I α and I β for the solution of the above fractional Langevin equation, we get Using boundary conditions, we obtain For t ∈ (s 0 , Since x(t 1 ) = g 1 (t 1 ), Eq. (2.2) is of the following type: Using boundary conditions, we get Generally speaking, for t ∈ (s k-1 , t k ], x(t k ) = g k (t).
Case 3. For t ∈ (t k , s k ], we consider , with x(t k ) = g k (t k ) and x(T) = θ I p x(η).
Conversely, one can verify the fact by proceeding the standard steps to complete the proof.

Generalized Ulam-Hyers-Rassias stability
Using the ideas of stability in [24,31], we can generate a generalized Ulam-Hyers-Rassias stability concept for Eq. (1.1). Let , ψ ≥ 0 and for a nondecreasing ϕ ∈ PC(J, R + ) consider , then x is a solution of the following integral inequality: In fact, by Remark 3.1, we get Clearly, the solution of Eq. (3.3) is given by For t ∈ (t k , s k ], k = 0, 1, . . . , m, we get Proceeding as above, we derive

Main results via fixed point methods
In order to apply a fixed point theorem of the alternative for contractions on a generalized complete metric space to achieve our main result, we want to collect the following facts.   (Tu, u). We can introduce some assumptions as follows: (H 2 ) There exists a positive constant L f such that (H 3 ) g k ∈ C((s k-1 , t k ] × R, R) and there are positive constant L gk , k = 1, 2, . . . , m such that for all t ∈ J if 0 < α < β < 1, with Proof Consider the space of piecewise continuous functions V = g : J → R such that g ∈ PC(J, R) , endowed with the generalized metric on V , defined by where It is easy to verify that (V , d) is a complete generalized metric space [19]. Define an operator Λ : for all x belongs to V and t ∈ J. Obviously, according to (H 1 ), Λ is a well-defined operator.