Abstract
In this paper the existence and nonexistence results of positive solutions are obtained for Sturm–Liouville boundary value problem − ( p ( x ) u ′ ) ′ + q ( x ) u = f ( x , u ) , x ∈ ( 0 , 1 ) , a u ( 0 ) − b p ( 0 ) u ′ ( 0 ) = 0 , c u ( 1 ) + d p ( 1 ) u ′ ( 1 ) = 0 , where p ∈ C 1 [ 0 , 1 ] , q ∈ C [ 0 , 1 ] , p ( x ) > 0 , q ( x ) ⩾ 0 for x ∈ [ 0 , 1 ] , f ∈ C ( [ 0 , 1 ] × R + ) , a , b , c , d ⩾ 0 are constants and satisfy ( a + b ) ( c + d ) > 0 . The discussion is based on the positivity estimation for the Green's function of associated linear boundary value problem and the fixed point index theory in cones.
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