Solutions of Sign-Changing Fractional Differential Equation with the Fractional Derivatives

• Published in: Abstract and Applied Analysis Vol. 2012; pp. 485 - 500-632
• Main Authors: Wu, Tunhua, Zhang, Xinguang, Lu, Yinan
• Format: Journal Article
• Language: English
• Published: New York Hindawi Limiteds 01.01.2012; Hindawi Publishing Corporation; Hindawi Limited; Wiley
• Subjects: Banach spaces; Boundary value problems; Derivatives; Differential equations; Mathematical analysis; Mathematical models; Nonlinearity; Science; Studies; Topology
• ISSN: 1085-3375; 1687-0409
• DOI: 10.1155/2012/797398

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.