Existence of Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions - Ruikuan Liu; Ruyun Ma (Journal of Applied Mathematics)
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′′′′(t)=λf(t,u(t)),  t∈(0,1),u(0)=0,  u′(0)=μh(u(0)),  u′′(1)=0,  u′′′(1)=μg(u(1)), where λ>0, μ≥0 are parameters, and f:0, 1×0,+∞→0, +∞, h:0, +∞→0, +∞, and g:0, +∞→-∞,0 are continuous. By using the fixed-point index theory, we prove that the problem has at least one positive solution for λ,  μ sufficiently small and has no positive solution for λ large enough.