Positive solutions of the nonlinear fourth-order beam equation with three parameters

Abstract

This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation u ( 4 ) ( t ) + η u ″ ( t ) − ζ u ( t ) = λ f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , where f ( t , u ) : [ 0 , 1 ] × [ 0 , + ∞ ) → [ 0 , + ∞ ) is continuous and ζ, η and λ are parameters. We show that there exists a λ * > 0 ˙ such that the above boundary value problem (BVP) has at least two, one and no positive solutions for 0 < λ < λ * , λ = λ * and λ > λ * , respectively. Furthermore, by using the semiorder method on cones of Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution u λ ( t ) of the BVP depends continuously on the parameter λ, i.e., u λ ( t ) is nondecreasing in λ, lim λ → 0 + ‖ u λ ( t ) ‖ = 0 and lim λ → + ∞ ‖ u λ ( t ) ‖ = + ∞ for any t ∈ [ 0 , 1 ] .