Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales - Yanning Wang; Jianwen Zhou; Yongkun Li (Advances in Mathematical Physics)
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 a p-Laplacian conformable fractional differential equation boundary value problem on time scale T:  Tα(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, where Tα(u)(t) denotes the conformable fractional derivative of u of order α at t, σ is the forward jump operator, a,b∈T,  0<a<b,  p>1, and F:[0,T]T×RN→R. 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.