Abstract
We discuss the existence of minimal and maximal positive solutions for fractional differential equations with multipoint boundary value conditions, and new results are given. An example is also given to illustrate the abstract results.
Highlights
1 discussed the existence of positive solutions for the following boundary value problem of fractional order differential equationD0α u t f t, u t 0, 0 < t < 1, u 0 0, D0β u 1 aD0β u ξ, 1.1 where D0α is the standard Riemann-Liouville fractional derivative of order 1 < α ≤ 2, 0 ≤ β ≤ 1, 0 ≤ a ≤ 1, ξ ∈ 0, 1, aξα−β−2 ≤ 1 − β, 0 ≤ α − β − 1 and f : 0, 1 × 0, ∞ → 0, ∞ satisfies Caratheodory-type conditions
E is the Banach space endowed with the norm u sup0≤t≤1|u t | and P is normal cone
By Theorem 2.4, we know that BVP 1.4 has at least one positive solution in Br
Summary
discussed the existence of positive solutions for the following boundary value problem of fractional order differential equation. considered the following nonlinear mpoint boundary value problem of fractional type: Dαx tqtft, x t 0, a.e. on 0, 1 , α ∈ n − 1, n , n ≥ 2, x 0 x 0 x 0 · · · x n−2 0 0, m−2. X k denotes the kth Pseudo-derivative of x, Dα denotes the Pseudo fractional differential operator of order α, q · is a continuous real-valued function on. We consider the existence of minimal and maximal positive solutions for the following multiple-point boundary value problem: D0α u t f t, u t 0, 0 < t < 1, u0. I0α D0α u t u t C1tα−1 C2tα−2 · · · CNtα−N, for some Ci ∈ R, i 1, 2, . . . , N, 1.7 where N is the smallest integer greater than or equal to α
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