Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales
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...
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| Format: | Article |
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Wiley
2016-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2016/9636491 |
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| author | Yanning Wang Jianwen Zhou Yongkun Li |
| author_facet | Yanning Wang Jianwen Zhou Yongkun Li |
| author_sort | Yanning Wang |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-516efffd842b494b858b74995aecb365 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
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| series | Advances in Mathematical Physics |
| spelling | doaj-art-516efffd842b494b858b74995aecb3652025-02-03T01:26:43ZengWileyAdvances in Mathematical Physics1687-91201687-91392016-01-01201610.1155/2016/96364919636491Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time ScalesYanning Wang0Jianwen Zhou1Yongkun Li2Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, ChinaDepartment of Mathematics, Yunnan University, Kunming, Yunnan 650091, ChinaDepartment of Mathematics, Yunnan University, Kunming, Yunnan 650091, ChinaUsing 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.http://dx.doi.org/10.1155/2016/9636491 |
| spellingShingle | Yanning Wang Jianwen Zhou Yongkun Li Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales Advances in Mathematical Physics |
| title | Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales |
| title_full | Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales |
| title_fullStr | Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales |
| title_full_unstemmed | Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales |
| title_short | Fractional Sobolev’s Spaces on Time Scales via Conformable Fractional Calculus and Their Application to a Fractional Differential Equation on Time Scales |
| title_sort | fractional sobolev s spaces on time scales via conformable fractional calculus and their application to a fractional differential equation on time scales |
| url | http://dx.doi.org/10.1155/2016/9636491 |
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