Monotone iterative method for fractional differential equations

Bai, Zhanbing
Zhang, Shuo
Sun, Sujing
Yin, Chun
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Texas State University, Department of Mathematics
In this article, by using the lower and upper solution method, we prove the existence of iterative solutions for a class of fractional initial value problem with non-monotone term Dα0+ u(t) = ƒ(t, u(t)), t ∈ (0, h), t1-α u(t)|t=0 = u0 ≠ 0, where 0 < h < +∞, ƒ ∈ C([0, h] x ℝ, ℝ), Dα0+ u(t) is the standard Riemann-Liouville fractional derivative, 0 < α < 1. A new condition on the nonlinear term is given to guarantee the equivalence between the solution of the IVP and the fixed-point of the corresponding operator. Moreover, we show the existence of maximal and minimal solutions.
Fractional initial value problem, Lower and upper solution method, Existence of solutions
Bai, Z., Zhang, S., Sun, S., & Yin, C. (2016). Monotone iterative method for fractional differential equations. <i>Electronic Journal of Differential Equations, 2016</i>(06), pp. 1-8.