The continuity of solution set of a multivalued equation and applications in control problem

Abstract

In this paper, we prove the existence, unbounded continuity of positive set for a multivalued equation containing a parameter of the form $x \in A \circ F(\lambda,x)$ and give applications in the control problem with multi-point boundary conditions and second order derivative operator 
 \begin{equation}
 \left\{ 
 \begin{array}{l}
 u^{\prime \prime }(t) +g(\lambda,t) f(u(t)) =0,\text{ }t\in (0,1) , \\ 
 g(\lambda,t) \in F(\lambda,u(t)) \text{ a.e. on } J
 \\
 u(0) =0, u( 1) =\sum_{i=1}^{m}\alpha_{i} u( \eta _{i})
 \end{array}
 \right. \label{Eq3.1}
 \end{equation}