Fractional Sobolev's Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales
Fractional Sobolev's Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales
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基于适形分数阶微积分的分数 Sobolev 时间尺度空间及其在时间尺度分数阶微分方程中的应用
DOI：
10.1155/2016/9636491
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发表时间：
2016
影响因子：
1.2
通讯作者：
Li Yongkun
中科院分区：
文献类型：
--
作者：
Wang Yanning;Zhou Jianwen;Li Yongkun
利用时间尺度上的可适分数阶微积分，首先引入时间尺度上的分数Sobolev空间，对其进行刻画，并定义弱可适分数阶导数。其次，我们证明了在引入的空间中一些范数的等价性，并推导了它们的完备性、自反性、一致凸性和紧性，这些嵌入可以看作是一个新奇项。然后，作为应用，我们利用变分方法和临界点理论，给出了时间尺度et α(t αup- 2t α(u))(t)=∇F(σ(t),u(σ(t)))， Δ-a.e上ap‐laplace可调分数阶微分方程边值问题解的存在性。t∈a,bTκ2,u(a)−u(b) = 0, t α(u)(a)−t α(u)(b) = 0，其中，et α(u)(t)表示阶αatt的适形分数阶导数，σ为前跃算子，，通过建立适当的变分设置，得到了三个存在性结果。最后，通过两个算例说明了存在结果的可行性和有效性。
Using conformable fractional calculus on time scales, we first introduce fractional Sobolev spaces on time scales, characterize them, and define weak conformable fractional derivatives. Second, we prove the equivalence of some norms in the introduced spaces and derive their completeness, reflexivity, uniform convexity, and compactness of some imbeddings, which can be regarded as a novelty item. Then, as an application, we present a recent approach via variational methods and critical point theory to obtain the existence of solutions for ap‐Laplacian conformable fractional differential equation boundary value problem on time scaleTα(Tαup-2Tα(u))(t)=∇F(σ(t),u(σ(t))), Δ-a.e. t∈a,bTκ2,u(a) −u(b) = 0,Tα(u)(a) −Tα(u)(b) = 0, whereTα(u)(t) denotes the conformable fractional derivative ofuof orderαatt,σis the forward jump operator,, and. By establishing a proper variational setting, we obtain three existence results. Finally, we present two examples to illustrate the feasibility and effectiveness of the existence results.