Solutions to Second-order Three-point Problems on Time Scales

Abstract

In the first part of the paper, we establish the existence of multiple positive solutions to the nonlinear second-order three-point boundary value problem on time scales, u ^▵▿ (t)+f(t,u(t))=0, u(0)=0, 𝛂u(𝛈)=u(T) for t∈[0,T]⊂╥, where ╥ is a time scale, 𝛂>0, η∈(0,p(T)⊂╥, and 𝛂η<T. We employ the Leggett-Williams fixed-point theorem in an appropriate cone to guarantee the existence of at least three positive solutions to this nonlinear problem. In the second part, we establish the existence of at least one positive solution to the related problem u ^▵▿(t)+a(t)f(u(t))=0, u(0)=0, 𝛂u(η)=u(T), again using a fixed-point theorem for operators.