## Existence of Solutions for Fractional Langevin Equations with Boundary Conditions on an Infinite Interval

**Authors:** Zaid Laadjal

In this paper, we investigate the existence and uniqueness of solutions for the following fractional Langevin equations with boundary conditions $$\left\{\begin{array}{l}D^{\alpha}( D^{\beta}+\lambda)u(t)=f(t,u(t)),\text{ \ \ \ }t\in(0,+\infty),\\ \\u(0)=D^{\beta}u(0)=0,\\ \\ \underset{t\rightarrow+\infty}{\lim}D^{\alpha-1}u(t)=\underset{t\rightarrow+\infty}{\lim}D^{\alpha +\beta-1}u(t)=au(\xi),\end{array}\right.$$ where $1<\alpha \leq2$ and$\ 0<\beta \leq1,$ such that $1<\alpha +\beta \leq2,$ with $\ a,b\in\mathbb{R},$ $\xi \in\mathbb{R}^{+},$\ and $D^{\alpha}$, $D^{\beta }$ are the Riemman-Liouville fractional derivative. Some new results are obtained by applying standard fixed point theorems.

**Comments:** 5 Pages.

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### Submission history

[v1] 2018-10-10 15:21:57

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