Global structure of positive solutions for nonlocal boundary value problems involving integral conditions

Abstract

In this paper, we consider the nonlinear eigenvalue problems u ″ + λ h ( t ) f ( u ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 u ( s ) d A ( s ) , where ∫ 0 1 u ( s ) d A ( s ) is a Stieltjes integral with A nondecreasing and A ( t ) is not a constant on ( 0 , 1 ) ; h ∈ C ( ( 0 , 1 ) , [ 0 , ∞ ) ) and h ( t ) ≢ 0 on any subinterval of ( 0 , 1 ) ; f ∈ C ( [ 0 , ∞ ) , [ 0 , ∞ ) ) and f ( s ) > 0 for s > 0 , and f 0 = f ∞ = 0 , f 0 = lim s → 0 + f ( s ) / s , f ∞ = lim s → + ∞ f ( s ) / s . We investigate the global structure of positive solutions by using global bifurcation techniques.