Bifurcation from Interval and Positive Solutions of a Nonlinear Second-Order Dynamic Boundary Value Problem on Time Scales

Abstract

Let $𝕋$ be a time scale with $0,T\in 𝕋$. We give a global description of the branches of positive solutions to the nonlinear boundary value problem of second-order dynamic equation on a time scale $𝕋$, $u^{\mathrm{\Delta }\mathrm{\Delta }}(t)+f(t,u^{\sigma }(t))=0,t\in [0,T{]}_{𝕋},u(0)=u({\sigma }^2(T))=0$, which is not necessarily linearizable. Our approaches are based on topological degree theory and global bifurcation techniques.