Solvability of second-order nonlinear three-point boundary value problems

Abstract

We are interested in the existence of nontrivial solutions to the three-point boundary value problem (BVP): (∗) { u ″ ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 u ′ ( 0 ) = 0 , u ( 1 ) = α u ( η ) + β u ′ ( η ) , where 0 < η < 1 , f ( t , u ) ∈ C ( [ 0 , 1 ] × R , R ) and α , β are real constants. Fixed-point theorems and degree theory are frequently used to study such problems. Recently, the authors demonstrated that, in many situations, the shooting method proves to be an effective approach, often leading to better results with shorter proofs. Here we present another such example. Assume that f ( t , 0 ) ≢ 0 and that there exist nonnegative functions k , h ∈ L 1 ( 0 , 1 ) such that | f ( t , w ) | ≤ k ( t ) | w | + h ( t ) for all ( t , w ) ∈ [ 0 , 1 ] × R . Sun and Liu [13] (for the special case β = 0 ), and Sun [14] (for the special case α = 0 ) showed that, if the L 1 norm ‖ k ‖ 1 is sufficiently small, then there exists a nontrivial solution to the BVP (∗). In this paper, their results are improved using the shooting method.