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

## Keywords:

Multi-valued, Fixed point index, Feedback control, Multi-point boundary## 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 \(\lambda\).

### Additional Files

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## How to Cite

*Letters on Applied and Pure Mathematics*,

*1*(1), 23–31. Retrieved from https://lapmjournal.com/index.php/lapm/article/view/lapm.2023v1n1-3

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Copyright (c) 2023 Letters on Applied and Pure Mathematics

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