Existence and asymptotic behavior of positive solutions for a second-order boundary-value problem

Abstract

We study the boundary-value problem $$\displaylines{ \frac{1}{A(t)}(A(t)u'(t))'=\lambda f(t,u(t))\quad t\in (0,\infty),\cr \lim_{t\to 0^+}A(t)u'(t)=-a\leq 0, \quad \lim_{t\to \infty}u(t)=b>0, }$$ where $\lambda\geq0$ and f is nonnegative continuous and non-decreasing with respect to the second variable. Under some assumptions on the nonlinearity f, we prove the existence of a positive solution for $\lambda$ sufficiently small. Our approach is based on the Schauder fixed point theorem.

Keywords

• Boundary value problem
• positive solution
• fixed point
• Green function