Existence and properties of solutions for boundary value problems based on the nonlinear reactor dynamics

We deal with the existence of positive solutions for the following class of nonlinear equation u ''(t)+Au '(t)+g(t,u(t),v(t))=0 a.e. in (0, 1), with boundary conditions u '(0)=0, u '(1)+Au(1)=0, where v is a functional parameter. The form of the problem is associated with the classical model described by Markus and Amundson. We show the existence of at least one positive solution of this problem and discuss its properties. Moreover we describe conditions that guarantee the continuous dependence of solution on parameter v also in the case of the lack of the uniqueness of a solution. The results are based on the clasical fixed point methods. Our approach allows us to consider both sub and superlinear nonlinearities which may be singular with respect to the first variable.