Nonlocal boundary value problems of fractional order at resonance with integral conditions
Abstract Based upon the well-known coincidence degree theory of Mawhin, we obtain some new existence results for a class of nonlocal fractional boundary value problems at resonance given by { D 0 + α u ( t ) = f ( t , u ( t ) , D 0 + α − 1 u ( t ) , D 0 + α − 2 u ( t ) ) , t ∈ ( 0 , 1 ) , I 0 + 3 −...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-10-01
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| Series: | Advances in Difference Equations |
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| Online Access: | http://link.springer.com/article/10.1186/s13662-017-1379-8 |
| _version_ | 1819208197623250944 |
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| author | Hai-E Zhang |
| author_facet | Hai-E Zhang |
| author_sort | Hai-E Zhang |
| collection | DOAJ |
| description | Abstract Based upon the well-known coincidence degree theory of Mawhin, we obtain some new existence results for a class of nonlocal fractional boundary value problems at resonance given by { D 0 + α u ( t ) = f ( t , u ( t ) , D 0 + α − 1 u ( t ) , D 0 + α − 2 u ( t ) ) , t ∈ ( 0 , 1 ) , I 0 + 3 − α u ( 0 ) = u ′ ( 0 ) = 0 , D 0 + β u ( 1 ) = ∫ 0 1 D 0 + β u ( t ) d A ( t ) , $$ \textstyle\begin{cases} D_{0+}^{\alpha}u(t)=f(t,u(t),D_{0+}^{\alpha-1}u(t),D_{0+}^{\alpha-2}u(t)),\quad t\in(0,1), \\ I_{0^{+}}^{3-\alpha}u ( 0 ) =u' ( 0 ) =0,\quad\quad D_{0+} ^{\beta}u(1)=\int_{0}^{1}D_{0+}^{\beta}u(t)\,dA(t), \end{cases} $$ where α, β are real numbers with 2 < α ≤ 3 $2<\alpha\leq3$ , 0 < β ≤ 1 $0<\beta\leq1$ , D 0 + α $D_{0+}^{\alpha}$ and I 0 + α $I_{0+}^{\alpha}$ respectively denote Riemann-Liouville derivative and integral of order α, f : [ 0 , 1 ] × R 3 → R $f:[0,1]\times\mathbb{R}^{3}\rightarrow\mathbb{R}$ satisfies the Carathéodory conditions, ∫ 0 1 D 0 + β u ( t ) d A ( t ) $\int_{0}^{1}D_{0+}^{\beta}u(t)\,dA(t)$ is a Riemann-Stieltjes integral with ∫ 0 1 t α − β − 1 d A ( t ) = 1 $\int_{0}^{1}t^{\alpha-\beta-1}\,dA(t)=1$ . We also present an example to demonstrate the application of the main results. |
| first_indexed | 2024-12-23T05:35:34Z |
| format | Article |
| id | doaj.art-bc584c24d75e4d1d9046cdf3b0b4da76 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-23T05:35:34Z |
| publishDate | 2017-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-bc584c24d75e4d1d9046cdf3b0b4da762022-12-21T17:58:21ZengSpringerOpenAdvances in Difference Equations1687-18472017-10-012017111210.1186/s13662-017-1379-8Nonlocal boundary value problems of fractional order at resonance with integral conditionsHai-E Zhang0Department of Basic Teaching, Tangshan UniversityAbstract Based upon the well-known coincidence degree theory of Mawhin, we obtain some new existence results for a class of nonlocal fractional boundary value problems at resonance given by { D 0 + α u ( t ) = f ( t , u ( t ) , D 0 + α − 1 u ( t ) , D 0 + α − 2 u ( t ) ) , t ∈ ( 0 , 1 ) , I 0 + 3 − α u ( 0 ) = u ′ ( 0 ) = 0 , D 0 + β u ( 1 ) = ∫ 0 1 D 0 + β u ( t ) d A ( t ) , $$ \textstyle\begin{cases} D_{0+}^{\alpha}u(t)=f(t,u(t),D_{0+}^{\alpha-1}u(t),D_{0+}^{\alpha-2}u(t)),\quad t\in(0,1), \\ I_{0^{+}}^{3-\alpha}u ( 0 ) =u' ( 0 ) =0,\quad\quad D_{0+} ^{\beta}u(1)=\int_{0}^{1}D_{0+}^{\beta}u(t)\,dA(t), \end{cases} $$ where α, β are real numbers with 2 < α ≤ 3 $2<\alpha\leq3$ , 0 < β ≤ 1 $0<\beta\leq1$ , D 0 + α $D_{0+}^{\alpha}$ and I 0 + α $I_{0+}^{\alpha}$ respectively denote Riemann-Liouville derivative and integral of order α, f : [ 0 , 1 ] × R 3 → R $f:[0,1]\times\mathbb{R}^{3}\rightarrow\mathbb{R}$ satisfies the Carathéodory conditions, ∫ 0 1 D 0 + β u ( t ) d A ( t ) $\int_{0}^{1}D_{0+}^{\beta}u(t)\,dA(t)$ is a Riemann-Stieltjes integral with ∫ 0 1 t α − β − 1 d A ( t ) = 1 $\int_{0}^{1}t^{\alpha-\beta-1}\,dA(t)=1$ . We also present an example to demonstrate the application of the main results.http://link.springer.com/article/10.1186/s13662-017-1379-8fractional differential equationresonanceRiemann-Stieltjes integralcoincidence degree theory |
| spellingShingle | Hai-E Zhang Nonlocal boundary value problems of fractional order at resonance with integral conditions Advances in Difference Equations fractional differential equation resonance Riemann-Stieltjes integral coincidence degree theory |
| title | Nonlocal boundary value problems of fractional order at resonance with integral conditions |
| title_full | Nonlocal boundary value problems of fractional order at resonance with integral conditions |
| title_fullStr | Nonlocal boundary value problems of fractional order at resonance with integral conditions |
| title_full_unstemmed | Nonlocal boundary value problems of fractional order at resonance with integral conditions |
| title_short | Nonlocal boundary value problems of fractional order at resonance with integral conditions |
| title_sort | nonlocal boundary value problems of fractional order at resonance with integral conditions |
| topic | fractional differential equation resonance Riemann-Stieltjes integral coincidence degree theory |
| url | http://link.springer.com/article/10.1186/s13662-017-1379-8 |
| work_keys_str_mv | AT haiezhang nonlocalboundaryvalueproblemsoffractionalorderatresonancewithintegralconditions |