Existence and multiplicity of positive solutions for singular fractional boundary value problems

Abstract In this paper, we discuss the existence and multiplicity of positive solutions for the singular fractional boundary value problem D 0 + α u ( t ) + f ( t , u ( t ) , D 0 + ν u ( t ) , D 0 + μ u ( t ) ) = 0 , u ( 0 ) = u ′ ( 0 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where 3 α ≤ 4 , 0 ν ≤ 1 , 1 μ ≤ 2 , D 0 + α is the standard Riemann–Liouville fractional derivative, f is a Carathedory function and f ( t , x , y , z ) is singular at the value 0 of its arguments x , y , z . By means of a fixed point theorem, the existence and multiplicity of positive solutions are obtained.

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