An interesting example for a three-point boundary value problem (Q1580695)

The authors study the three-point boundary value problem \[ x''(t)= p(t)x(t)+Aq(t)x'(t)+e(t), \quad x(0)=0, x(1)=\alpha x(\eta), \tag{1} \] with \(\alpha, A \in \mathbb{R}\), \(\eta \in (0,1)\), \(p, q, e \in L^1[0,1]\), \(p(t)\geq 0\), \(q(t)\geq 0\) on \([0,1]\). Provided \(q\) is not the zero function on \([\eta,1]\), the authors prove that there exists an \(A_1\in \mathbb{R}\cup \{-\infty \}\) such that the three-point boundary value problem (1) has a unique solution for \(A_1<A<\infty\). Moreover, \(A_1\leq 0\) if \(\alpha \eta <1\) and \(A_1=-\infty\) if \(\alpha \leq 1\). They also prove the existence theorem for the more general problem \[ x''(t)= f(t,x(t),x'(t))+e(t), \quad x(0)=0, x(1)=\alpha x(\eta),\tag{2} \] with a Carathéodory function \(f\) and \(\alpha \eta \not= 1\). Further, they apply the results to the example \[ x''(t)= t^{-{1\over 4}}x(t)+At^{-{1\over 4}}x'(t)+e(t), \quad x(0)=0, x(1)=\alpha x(\eta), \tag{3} \] where they compute for several given values of \(\alpha\) and \(\eta\) lower bounds for \(A\) which guarantee the unique solvability of (3). The paper extends and completes the authors' earlier results.