Multiple unbounded positive solutions for three-point BVPs with sign-changing nonlinearities on the positive half-line (Q966480)

The authors consider the following second-order nonlinear three-point boundary value problem on the positive half-line \[ -x''+cx'+\lambda x=f(t,x(t), x'(t)), \quad t\in (0,\infty), \] \[ x(0)-\alpha x(\eta)=a_0, \quad \lim_{t\to \infty}\frac{x'(t)}{re^{rt}}=b_0, \] where \(a_0, b_0\) are nonnegative real numbers, \(\alpha\geq 0,\) \(\eta>0,\) \(c,\lambda\) are real positive constants, \(r\in (0,c),\) and \(f: (0,\infty)\times [0,\infty)\times {\mathbb R}\to {\mathbb R}\) is a Carathéodory function which may change sign. Under general polynomial growth conditions on \(f\) the existence of nontrivial single and multiple unbounded positive solutions is proved via fixed point theorems in a cone in a special weighted Banach space.