摘要：The purpose of this paper is to investigate the existence and uniqueness of nonnegative solutions for Langevin equations with two fractional orders: { D t β 0 c ( 0 c D t α − γ ) x ( t ) = f ( t , x ( t ) ) , 0 0 $\gamma>0$ , μ j ≥ 0 $\mu_{j}\ge0$ , ∀ j ∈ { 0 , … , m − 1 } $\forall j \in\{0,\ldots,m-1\} $ , ν i − γ μ i ≥ 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.
关键词：Single upper-solution or lower-solution method ; Monotone iterative ; Initial value problem ; Fractional Langevin equation ; Nonnegative solutions
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