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}
\begin{cases}
u”(t) + g(\lambda, t)f(u(t)) = 0, & t \in (0,1),\\[6pt]
g(\lambda, t) \in F(\lambda, u(t)) \text{ a.e. on } J,\\[6pt]
u(0) = 0, \quad u(1) = \displaystyle\sum_{i=1}^{m} \alpha_i u(\eta_i),
\end{cases}
\tag{1}
\end{equation}
where the functions and parameters satisfy suitable conditions.