Some Existence Results of Positive Solution to Second-Order Boundary Value Problems

Abstract

We study the existence of positive and monotone solution to the boundary value problem $u^{\prime \prime }(t)+f(t,u(t))=0$, $0⩽\mathrm{t}⩽1$, $u(0)=\xi u(1)$, $u\mathrm{\text{'}}(1)=\eta u\mathrm{\text{'}}(0)$, where $\xi ,\eta \in (\mathrm{0,1})\cup (1,\mathrm{\infty })$. The main tool is the fixed point theorem of cone expansion and compression of functional type by Avery, Henderson, and O’Regan. Finally, four examples are provided to demonstrate the availability of our main results.