Search

In this paper, we consider the existence of at least three positive solutions for the nonlinear n th-order m -point boundary value problem { u ( n ) ( t ) + f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 , u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 k i u ( ξ i ) , where n ≥ 2 , k i > 0 ( i = 1 , 2 , … , m − 2 ) , 0 ξ 1 ξ 2 ⋯ ξ m − 2 1 . The associated Green’s function for the n th-order m -point boundary value problem is first given, and growth conditions are imposed on the nonlinearity f which yield the existence of multiple positive solutions by using the Leggett–Williams fixed point theorem.

In this paper a new fixed point theorem in a cone is applied to obtain the existence of at least one positive solution for second-order quasilinear m -point boundary value problem ( ϕ p ( x ′ ( t ) ) ) ′ + a ( t ) f ( t , x , x ′ ) = 0 , 0 ⩽ t ⩽ 1 , x ′ ( 0 ) = 0 , x ( 1 ) = ∑ i = 1 m - 2 α i x ( ξ i ) , where f is a nonnegative continuous function and ϕ p ( x ) = ∣ x ∣ p - 2 x , p > 1 . A simple example is presented to illustrate the applications of the obtained results.

By using a new fixed-point theorem introduced by Avery and Peterson (2001), we obtain sufficient conditions for the existence of at least three positive solutions for the equationΔ2x(k−1)+q(k)f(k,x(k),Δx(k))=0, fork∈{1,2,…,n−1}, subject to the following two boundary conditions:x(0)=x(n)=0orx(0)=Δx(n−1)=0, wheren≥3.