In the present manuscript we analyze non-linear multi-order fractional differential equation $$Lleft( D ight)uleft( t ight) = fleft( {t,uleft( t ight)} ight), t in left[ {0,T} ight], T > 0,$$ where $$Lleft( D ight) = lambda _n ^c D^{lpha _n } + lambda _{n - 1} ^c D^{lpha _{n - 1} } + cdots + lambda _1 ^c D^{lpha _1 } + lambda _0 ^c D^{lpha _0 } ,lambda _i in mathbb{R}left( {i = 0,1, cdots ,n} ight),lambda _n e 0, 0 leqslant lpha _0 < lpha _1 < cdots < lpha _{n - 1} < lpha _n < 1,$$ and c D α denotes the Caputo fractional derivative of order α. We find the Greens functions for this equation corresponding to periodic/anti-periodic boundary conditions in terms of the two-parametric functions of Mittag-Leffler type. Further we prove existence and uniqueness theorems for these fractional boundary value problems.
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机译：在本论文中，我们分析非线性多阶分数阶微分方程$$ L left（D right）u left（t right）= f left（{t，u left（t right）} right），t in left [{0，T} right]，T> 0，$$其中$$ L left（D right）= lambda _n ^ c D ^ { alpha _n} + lambda _ {n-1} ^ c D ^ { alpha _ {n-1}} + cdots + lambda _1 ^ c D ^ { alpha _1} + lambda _0 ^ c D ^ { alpha _0 }， lambda _i in mathbb {R} left（{i = 0,1， cdots，n} right）， lambda _n ne 0，0 leqslant alpha _0 < alpha _1 < cdots < alpha _ {n-1} < alpha _n <1，$$和c Dα表示阶数的Caputo分数导数。我们根据Mittag-Leffler类型的两参数函数，找到了对应于周期/反周期边界条件的该方程的Greens函数。进一步，我们证明了这些分数边值问题的存在性和唯一性定理。
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