Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives
We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term -𝒟αx(t)=p(t)f(t,x(t),𝒟μ1x(t),𝒟μ2x(t),…,𝒟μn-1x(t)),0<t<1,𝒟μix(0)=0,1≤i≤n-1,𝒟μn-1+1x(0)=0, 𝒟μn-1x(1)=∑j=1p-2aj𝒟μn-1x(ξj), where n-1<α≤n, n∈ℕ and n≥3 with 0<μ1<μ2<⋯<μn-...
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| Format: | Article |
| Language: | English |
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Wiley
2012-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2012/797398 |
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| author | Tunhua Wu Xinguang Zhang Yinan Lu |
| author_facet | Tunhua Wu Xinguang Zhang Yinan Lu |
| author_sort | Tunhua Wu |
| collection | DOAJ |
| description | We study the singular fractional-order boundary-value problem with a sign-changing nonlinear term -𝒟αx(t)=p(t)f(t,x(t),𝒟μ1x(t),𝒟μ2x(t),…,𝒟μn-1x(t)),0<t<1,𝒟μix(0)=0,1≤i≤n-1,𝒟μn-1+1x(0)=0,
𝒟μn-1x(1)=∑j=1p-2aj𝒟μn-1x(ξj), where n-1<α≤n, n∈ℕ and n≥3 with 0<μ1<μ2<⋯<μn-2<μn-1 and n-3<μn-1<α-2, aj∈ℝ,0<ξ1<ξ2<⋯<ξp-2<1 satisfying 0<∑j=1p-2ajξjα-μn-1-1<1, 𝒟α is the standard Riemann-Liouville derivative, f:[0,1]×ℝn→ℝ is a sign-changing continuous function and may be unbounded from below with respect to xi, and p:(0,1)→[0,∞) is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field. |
| format | Article |
| id | doaj-art-dae36db04fbd4e2fa118cbd0f79dd843 |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2012-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-dae36db04fbd4e2fa118cbd0f79dd8432025-08-20T03:24:25ZengWileyAbstract and Applied Analysis1085-33751687-04092012-01-01201210.1155/2012/797398797398Solutions of Sign-Changing Fractional Differential Equation with the Fractional DerivativesTunhua Wu0Xinguang Zhang1Yinan Lu2School of Information and Engineering, Wenzhou Medical College, Zhejiang, Wenzhou 325035, ChinaSchool of Mathematical and Informational Sciences, Yantai University, Shandong, Yantai 264005, ChinaInformation Engineering Department, Anhui Xinhua University, Anhui, Hefei 230031, ChinaWe study the singular fractional-order boundary-value problem with a sign-changing nonlinear term -𝒟αx(t)=p(t)f(t,x(t),𝒟μ1x(t),𝒟μ2x(t),…,𝒟μn-1x(t)),0<t<1,𝒟μix(0)=0,1≤i≤n-1,𝒟μn-1+1x(0)=0, 𝒟μn-1x(1)=∑j=1p-2aj𝒟μn-1x(ξj), where n-1<α≤n, n∈ℕ and n≥3 with 0<μ1<μ2<⋯<μn-2<μn-1 and n-3<μn-1<α-2, aj∈ℝ,0<ξ1<ξ2<⋯<ξp-2<1 satisfying 0<∑j=1p-2ajξjα-μn-1-1<1, 𝒟α is the standard Riemann-Liouville derivative, f:[0,1]×ℝn→ℝ is a sign-changing continuous function and may be unbounded from below with respect to xi, and p:(0,1)→[0,∞) is continuous. Some new results on the existence of nontrivial solutions for the above problem are obtained by computing the topological degree of a completely continuous field.http://dx.doi.org/10.1155/2012/797398 |
| spellingShingle | Tunhua Wu Xinguang Zhang Yinan Lu Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives Abstract and Applied Analysis |
| title | Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives |
| title_full | Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives |
| title_fullStr | Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives |
| title_full_unstemmed | Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives |
| title_short | Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives |
| title_sort | solutions of sign changing fractional differential equation with the fractional derivatives |
| url | http://dx.doi.org/10.1155/2012/797398 |
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