Positive Solutions for a System of Fractional Integral Boundary Value Problems of Riemann–Liouville Type Involving Semipositone Nonlinearities

Abstract

In this work by the index of fixed point and matrix theory, we discuss the positive solutions for the system of Riemann−Liouville type fractional boundary value problems D 0 + α u ( t ) + f 1 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ∈ ( 0 , 1 ) , D 0 + α v ( t ) + f 2 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ∈ ( 0 , 1 ) , D 0 + α w ( t ) + f 3 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , D 0 + p u ( t ) | t = 1 = ∫ 0 1 h ( t ) D 0 + q u ( t ) d t , v ( 0 ) = v ′ ( 0 ) = ⋯ = v ( n − 2 ) ( 0 ) = 0 , D 0 + p v ( t ) | t = 1 = ∫ 0 1 h ( t ) D 0 + q v ( t ) d t , w ( 0 ) = w ′ ( 0 ) = ⋯ = w ( n − 2 ) ( 0 ) = 0 , D 0 + p w ( t ) | t = 1 = ∫ 0 1 h ( t ) D 0 + q w ( t ) d t , where α ∈ ( n − 1 , n ] with n ∈ N , n ≥ 3 , p , q ∈ R with p ∈ [ 1 , n − 2 ] , q ∈ [ 0 , p ] , D 0 + α is the α order Riemann−Liouville type fractional derivative, and f i ( i = 1 , 2 , 3 ) ∈ C ( [ 0 , 1 ] × R + × R + × R + , R ) are semipositone nonlinearities.

Keywords

• riemann–liouville type fractional problem
• positive solutions
• the index of fixed point
• matrix theory