Single upper-solution or lower-solution method for Langevin equations with two fractional orders

The purpose of this paper is to investigate the existence and uniqueness of nonnegative solutions for Langevin equations with two fractional orders: {Dtβ0c(0cDtα−γ)x(t)=f(t,x(t)),0<t<1,x(k)(0)=μk,0≤k<l,x(α+k)(0)=νk,0≤k<n,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textstyle\begin{cases} {}^{c}_{0}D^{\beta}_{t}({}^{c}_{0}D^{\alpha}_{t}-\gamma)x(t)=f(t,x(t)),& 0< t< 1, \\ x^{(k)}(0)=\mu_{k},& 0\leq k< l, \\ x^{(\alpha+k)}(0)=\nu_{k},& 0\leq k< n, \end{cases} $$\end{document} where Dtα0c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}^{c}_{0}D^{\alpha}_{t}$\end{document} and Dtβ0c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${}^{c}_{0}D^{\beta}_{t}$\end{document} denote the Caputo fractional derivatives, f:[0,1]×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f:[0,1]\times \mathbf{R}\rightarrow\mathbf{R}$\end{document} is a continuous function, and m−1<α≤m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m-1<\alpha\leq m$\end{document}, n−1<β≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n-1<\beta\leq n$\end{document}, l=max{m,n}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l=\max\{ m,n\}$\end{document}, n,m∈N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${n, m}\in \mathbf{N}$\end{document}, γ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\gamma>0$\end{document}, μj≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mu_{j}\ge0$\end{document}, ∀j∈{0,…,m−1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\forall j \in\{0,\ldots,m-1\} $\end{document}, νi−γμi≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nu_{i}-\gamma\mu_{i}\ge0$\end{document}, ∀i∈{0,…,n−1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\forall i\in\{0,\ldots,n-1\}$\end{document}. By using a single upper-solution or lower-solution method and monotone iterative approach, several existence and uniqueness results of nonnegative solutions are obtained. Moreover, an example is given to illustrate the main results.


Introduction
In 1908, Paul Langevin gave an elaborate description of Brownian motion, and thus Langevin equations were proposed, see [9,12]. Langevin equations can also describe many stochastic problems in fluctuating environments. In 1966, Kube gave a generalized Langevin equation for modeling anomalous diffusive processes in a complex and viscoelastic environment [10,11]. An important extension of the topic is fractional Langevin equation, which was introduced by Mainardi and collaborators [16,17] in the early 1990s. A lot of fractional Langevin equations have been established, e.g., fractional Langevin equations for modeling of single-file diffusion [6] and for a free particle driven by power law type of noise [18]. So fractional Langevin equations have been studied widely, see [1-9, 13, 15, 16, 19-26] for example. Recently, there have been many papers considering fractional Langevin equations involving two fractional orders, see [1-6, 8, 13, 21, 24-26]. Most of these articles studied the existence and uniqueness of solutions for Langevin equations, and some good results have been given by using the Banach contraction principle, Krasnoselskii's fixed point theorem, Schauder's fixed point theorem, Leray-Schauder nonlinear alternative, Leray-Schauder degree, and so on.
In [26], by using the Leray-Schauder nonlinear alternative, the authors studied the following initial value problem of Langevin equations with two fractional orders: The existence of solutions was given. Further, the uniqueness of solutions was also obtained by means of the Banach contraction principle. Recently, the author [4] studied this problem by introducing a new norm for a measurable function f : [0, 1] → R and got the existence, uniqueness of solutions for this problem via the Banach contraction principle.
We can find that there are few papers devoted to the study of nonnegative solutions for Langevin equations involving two fractional orders. In this paper we use a single uppersolution or lower-solution method and a monotone iterative approach to consider the following initial value problem of Langevin equations involving two fractional orders: is a continuous function, and some initial conditions are given: In [14], by using e-positive operators and Altman fixed point theory, we gave some existence and uniqueness results of solutions for (1.1). Different from the above-mentioned results, in this paper we establish the existence and uniqueness of nonnegative solutions for problem (1.1), which are new results on initial value problems for Langevin equations. It should be pointed out that we only use single lower-solution or single upper-solution to get the existence and uniqueness of nonnegative solutions for problem (1.1). This method is novel and our results are new.
In this paper, we always assume that the function f satisfies the following two conditions:

Preliminaries
In order to obtain our results, we first list necessary definitions, lemmas, and basic results.

Definition 2.1 ([4, 26]) For a function x(t), the Riemann-Liouville fractional integral of order
where [α] denotes the integer part of the real number α.

Lemma 2.1 ([26]) x(t) is a solution of problem (1.1) if and only if x(t) is a solution of the integral equation
Define an operator T :

Main results
In this section, we apply a single upper-solution or lower-solution method and a monotone iterative approach to study problem (1.1), and we obtain some new results on the existence results of unique nonnegative solutions. 1] |x(t)|, and θ denotes the zero element in E. Given the usual normal cone where We know that L is a positive linear bounded operator, and its norm Therefore, Because T is increasing in D 1 , we get and thus, Since τ ∈ (0, 1), {x n } is a Cauchy sequence in D 1 . Because D 1 is a close set in E, so it is complete. Hence, there exists x * ∈ D 1 such that x n → x * as n → ∞. By (3.3), x n ≤ x * and then Tx n ≤ Tx * , that is, x n+1 ≤ Tx * , which implies Also, Because x n → x * as n → ∞, we get Tx *x * = 0, and thus Tx * = x * . That is, x * is a fixed point of T. Therefore, x * is a nonnegative solution of problem (1.1).
In the following, we show that the solution x * of problem (1.1) is a unique solution in D 1 .
Suppose that x ∈ D 1 is the other solution of problem (1.1). Then x is a fixed point of T in D 1 . Since x ≥ x 0 , we get Tx ≥ Tx 0 , that is, x ≥ x 1 . In general, x ≥ x n (n = 0, 1, 2, . . .). Let n → ∞, we have x ≥ x * , and by (3.2), So xx * = 0, and thus x = x * .

From (3.4), (3.5), we have
Tv so from the fact that T is increasing on D 2 , we can easily get Further, by (3.2), Note that τ ∈ (0, 1), {v n } is a Cauchy sequence in D 2 . Because D 2 is a close set in E, so it is complete. Hence, there exists v * ∈ D 2 such that v n → v * as n → ∞. From (3.6), v n ≥ v * and then Tv n ≥ Tv * , that is, v n+1 ≥ Tv * . So we have θ ≤ Tv n -Tv * ≤ L x *x n , then we can obtain Hence, from v n → v * as n → ∞, we have Tv *v * = 0, and thus Tv * = v * . That is, v * is a fixed point of T. Therefore, v * is a nonnegative solution of problem (1.1).
In the following, we show that v * is a unique solution of problem (1.1) in D 2 .
Suppose that v is the other solution of problem (1. and by (3.2), So vv * = 0, and thus v = v * .
where φ, R are given as in (2.4) and (3.4), respectively. From Theorems 3.1 and 3.2, we can obtain the following.

An example
We present an example to better illustrate our main results.