A multiplicity result in finite-dimensional vector spaces

Abstract

Let f 1 , … , f n be n ( n ≥ 2 ) continuous real-valued functions on R such that lim | t | → + ∞ ∫ 0 t f k ( s ) d s t 2 = − ∞ for all k = 1 , … , n . This sole condition is far from ensuring the existence of multiple solutions for the classical problem { − ( x k + 1 − 2 x k + x k − 1 ) = f k ( x k ) k = 1 , … , n , x 0 = x n + 1 = 0. However, as a by-product of a much more general result, we get the following: for each ρ ∈ R and for each i = 1 , … , n , there exists ( λ , μ ) ∈ R 2 such that the problem { − ρ ( x k + 1 − 2 x k + x k − 1 ) = f k ( x k ) k = 1 , … , n , k ≠ i − ρ ( x i + 1 − 2 x i + x i − 1 ) = f i ( x i ) + λ x i + μ x 0 = x n + 1 = 0 has at least three solutions.