Existence and Nonexistence of Positive Solutions for Singular Fifth-order Three-point BVP

摘要

In this article, we consider the boundary value problem u~((5))(t) + f(t,u(t))= 0, 0 < t < 1, subject to the boundary conditions u(0)= u'(0)= u"(0)= u"'(0)= 0 and u"'(1) - au"'(η)=λ. In the setting, 0 < η < 1 and a e [0, 1/η) are constants and λ ∈ [0, +∞) is parameter. By placing certain restrictions on the nonlinear term f, we prove the existence and nonexistence of at least one positive solution to the boundary value problem with the use of the Krasnosel'skii fixed point theorem. The novelty in our setting lies in the fact that f(t, u) may be singular at t= 0 and t= 1. We conclude with examples illustrating our results obtained in this paper.