The Existence of Countably Many Positive Solutions for Nonlinear nth-Order Three-Point Boundary Value Problems. (4th November 2009) - Ji, Yude; Guo, Yanping
Abstract : We consider the existence of countably many positive solutions for nonlinearn th-order three-point boundary value problemu ( n ) ( t ) + a ( t ) f ( u ( t ) ) = 0, t ∈ ( 0, 1 ), u ( 0 ) = α u ( η ), u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0, u ( 1 ) = β u ( η ), wheren ≥ 2, α ≥ 0, β ≥ 0, 0 < η < 1, α + ( β − α ) η n − 1 < 1, a ( t ) ∈ L p [ 0, 1 ] for somep ≥ 1 and has countably many singularities in[ 0, 1 / 2 ) . The associated Green's function for then th-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearityf which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.