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Existence of Solutions to Fractional Differential Equations with Fractional-order Derivative Terms
Ai Sun,You-Hui Su,Qingchun Yuan,Tongxiang Li
Keywords:Fractional differential equation, Green""s function, the fixed point theorems, numerical simulation.
      The study in this paper is made on the nonlinear fractional differential equation whose nonlinearity involves the explicit fractional order $D_{0^+}^{\beta}u(t)$. The corresponding Green""s function is derived first, and then the completely continuous operator is proved. Besides, based on the Schauder""s fixed point theorem and the Krasnosel""skii""s fixed point theorem, the sufficient conditions for at least one or two existence of positive solutions are established. Furthermore, several other sufficient conditions for at least three, $n$ or $2n-1$ positive solutions are also obtained by applying the generalized Avery-Henderson fixe point theorem and the Avery-Peterson fixed point theorem. Finally, several simulation examples are provided to illustrate the main results of the paper. In particularly, a novel efficient iterative method is employed for simulating the examples mentioned above, that is, the interesting point of this paper is that the approximation graphics for the solutions are given by using the iterative method.