Analysis of positive solutions for a fourth-order beam equation with sign-changing Green's function (Q6633363)

The paper deals with the existence of positive solutions of the fourth order three point boundary value problem \(u^{(4)}(x)-\lambda u''(x)=\mu h(x)f(u(x)),\) \(x\in[0,1]\), \(u(0)=u''(0)=u(1)=0,\) \(\delta u''(1)-u'''(\zeta)+\lambda u'(\zeta)=0,\) where \(\lambda>-\pi^2,\) \(\mu>0\), \(\delta\in[0,1])\) and \(\zeta\in(0,1).\) Also, the function \(h\) is Lebesgue integrable on the interval \([0,1]\) and \(f:[0,+\infty)\to\mathbb{R}\) is continuous. The major step in the problem is the calculation of the Green's function, which is not of constant sign. Then the existence results are obtained under some sufficient conditions by applying the Leray-Schauder fixed point theorem. In section 4 the authors give a Ulam-Hyers stability result of the problem and then they close the work by giving some examples to illustrate the main results.