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

Abstract

Abstract 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)),00 $\gamma>0$, μj≥0 $\mu_{j}\ge0$, ∀j∈{0,…,m−1} $\forall j \in\{0,\ldots,m-1\} $, νi−γμi≥0 $\nu_{i}-\gamma\mu_{i}\ge0$, ∀i∈{0,…,n−1} $\forall i\in\{0,\ldots,n-1\}$. 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.

Keywords

• Single upper-solution or lower-solution method
• Monotone iterative
• Initial value problem
• Fractional Langevin equation
• Nonnegative solutions