Coexistence of positive solutions of nonlinear three-point boundary value and its conjugate problem (Q874947)

This paper deals with a three-point boundary value problem of the form \[ -u''= f(t,u(t)), \quad u'(0)=0, \;u(1)=\alpha u(\eta), \] together with its conjugate BVP \[ -v''= f(s,v(s)), \; v'(0)=0, \; v'_+(\eta)-v'_-(\eta)=\alpha v'(1), \; v(1)=0. \] It is interesting to note that the linear problems associated with the two different boundary conditions have the same first eigenvalue \(\lambda\). Assuming \[ \limsup_{x\to 0^+} \max_{0\leq t \leq 1} f(t,x)/x < \lambda < \liminf_{x\to +\infty} \min_{0\leq t \leq 1} f(t,x)/x \] or \[ \limsup_{x\to +\infty} \max_{0\leq t \leq 1} f(t,x)/x < \lambda < \liminf_{x\to 0} \min_{0\leq t \leq 1} f(t,x)/x, \] the existence of at least one positive solution is proved. The other main result provides (technical) sufficient conditions for the existence of at least two positive solutions. The proof is developed in the framework of fixed point index theory in cones.