Positive solutions for a nonlocal fractional differential equation

• Published in: Nonlinear analysis 2011-07, Vol.74 (11), p.3599-3605
• Main Authors: Wang, Yongqing, Liu, Lishan, Wu, Yonghong
• Format: Article
• Language: English
• Subjects: Boundary value problems; Derivatives; Differential equations; Exact sciences and technology; Fixed point index; Fractional differential equation; Global analysis, analysis on manifolds; Mathematical analysis; Mathematics; Nonlinearity; Ordinary differential equations; Positive solution; Sciences and techniques of general use; Singular problem; Singularities; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
• ISSN: 0362-546X; 1873-5215
• DOI: 10.1016/j.na.2011.02.043

In this paper, we study the following singular boundary value problem of a nonlocal fractional differential equation { D 0 + α u ( t ) + q ( t ) f ( t , u ( t ) ) = 0 , 0 < t < 1 , n − 1 < α ≤ n , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 u ( s ) d A ( s ) , where α ≥ 2 , D 0 + α is the standard Riemann–Liouville derivative, ∫ 0 1 u ( s ) d A ( s ) is given by Riemann–Stieltjes integral with a signed measure, q may be singular at t = 0 and/or t = 1 , f ( t , x ) may also have singularity at x = 0 . The existence and multiplicity of positive solutions are obtained by means of the fixed point index theory in cones.