Global structure of positive solutions for superlinear second-order periodic boundary value problems

Abstract

We consider periodic boundary value problems of nonlinear second order ordinary differential equations of the form. u ″ - ρ 2 u + λ a ( t ) f ( u ) = 0 , 0 < t < 2 π , u ( 0 ) = u ( 2 π ) , u ′ ( 0 ) = u ′ ( 2 π ) , where ρ > 0 is a constant, a ∈ C([0, 1], [0, ∞)) with a( t 0) > 0 for some t 0 ∈ [0, 2 π], f ∈ C([0, ∞), [0, ∞)) and f( s) > 0 for s > 0, and f 0 = ∞, where f 0 = lim s → 0 + f ( s ) / s . We investigate the global structure of positive solutions by using the Rabinowitz’s global bifurcation theorem.