Abstract
Sufficient conditions are given for the existence of solutions of the following nonlinear boundary value problem with nonhomogeneous multi-point boundary condition u ″ + f ( t , u , u ′ ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) − ∑ i = 1 m a i u ( t i ) = λ 1 , u ( 1 ) − ∑ i = 1 m b i u ( t i ) = λ 2 . We prove that the whole plane R 2 is divided by a “continuous decreasing curve” Γ into two disjoint connected regions Λ E and Λ N such that the above problem has at least one solution for ( λ 1 , λ 2 ) ∈ Γ , has at least two solutions for ( λ 1 , λ 2 ) ∈ Λ E ∖ Γ , and has no solution for ( λ 1 , λ 2 ) ∈ Λ N . We also find explicit subregions of Λ E where the above problem has at least two solutions and two positive solutions, respectively.
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