The continuity of solution set of a multi-point boundary problem with a control system

Abstract

In this paper, we prove the unbounded continuity of the positive set of the inclusion \(x\in A\circ T(\lambda, x)\) and apply it to the problem that finds \((\lambda, u)\) satisfying
\begin{equation}
\left\{
\begin{array}{l}
D^{2 }u(t) +q(\lambda,t) f(t,u(t)) =0,\text{ }t\in (0,1) , \\
q(\lambda,t) \in F(\lambda,u(t)) \text{ a.e. } t\in [0,1]\\
u(0)=0\,\, ({\rm resp., }\, Du(0) =0), u( 1) =\sum_{j=1}^{m}\gamma_{j} u( \eta _{j}).
\end{array}
\right. \label{Eq0.1}
\end{equation}
Here, \(D^n\) is the derivative of n order \((D\equiv D^1)\). To obtain results, we use topological degree theories and the monotone lower evaluation for multivalued mapping in the infinite neighborhood of \(\lambda\).