Existence of Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions

Abstract

We study the existence and nonexistence of positive solutions for the following fourth-order two-point boundary value problem subject to nonlinear boundary conditions $u^{\mathrm{\prime \prime \prime \prime }}(t)=\lambda f(t,u(t)),t\in (\mathrm{0,1}),u(\mathrm{0})=\mathrm{0},u^{\prime }(\mathrm{0})=\mu h(u(\mathrm{0})),u^{\mathrm{\prime \prime }}(\mathrm{1})=\mathrm{0},u^{\mathrm{\prime \prime \prime }}(\mathrm{1})=\mu g(u(\mathrm{1}))$, where $\lambda >0, \mu \ge 0$ are parameters, and $f:\left[0, 1\right]\times \left[0,+\infty \right)\to \left(0, +\infty \right), h:\left[0, +\infty \right)\to \left[0, +\infty \right)$, and $g:\left[0, +\infty \right)\to \left(-\infty ,0\right]$ are continuous. By using the fixed-point index theory, we prove that the problem has at least one positive solution for $\lambda ,\mu$ sufficiently small and has no positive solution for $\lambda$ large enough.