Remarks on positive solutions of some three point boundary value problems
The author investigates the existence of positive solutions of the second-order differential equation \[ u''(t)+g(t)f(u(t))=0 \tag{1} \] subjected to one of the following three-point boundary conditions \[ u'(0)=0, \quad \alpha u(\eta)=u(1), \tag{2} \] or \[ u(0)=0, \quad \alpha u(\eta)=u(1), \tag{3} \] where \(\eta \in (0,1)\),\ \(f:[0,\infty) \to [0,\infty)\) is continuous,\ \(g\in L[0,1]\) is nonnegative and \(\int^b_ag(s)\,ds>0\) on some subinterval \([a,b]\). A solution is found as a fixed-point of a corresponding integral operator. The classical theory of fixed-point index for compact mappings in cones is used. Conditions which guarantee the existence of one or two positive solutions of (1),(2) or (1),(3) are given in terms of the quantities \[ f^\alpha = \limsup_{u\to \alpha}{f(u)\over u}, \quad f_\alpha = \liminf_{u\to \alpha}{f(u)\over u}, \quad \alpha\in \{0,\infty\}, \] \[ f_{c\rho,\rho} = \min \biggl\{ {f(u)\over \rho} :u\in [c\rho,\rho]\biggr\}, \quad f^{0,\rho} = \max \biggl\{ {f(u)\over \rho} :u\in [0,\rho]\biggr\},\quad c\in (0,1],\;\rho>0, \] \[ m(a,b)=\biggl(\max_{a\leq t \leq b} \int_a^bk(t,s)g(s)\,ds\biggr)^{-1},\quad M(a,b)=\biggl(\min_{a\leq t \leq b} \int_a^bk(t,s)g(s)\,ds\biggr)^{-1},\;[a,b]\subset [0,1]. \] Here, \(k(t,s)\) is the Green function to the boundary value problem \(u''=0\), (2) or \(u''=0\), (3). Optimal relations among the above quantities are discussed from the point of view of the solvability of problem (1), (2) or problem (1), (3).
scientific article; zbMATH DE number 2046267