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An order theoretic fixed point theorem with application to multivalued variational inequalities with nonsmooth bifunctions

  • Christoph TietzEmail author
Article
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Abstract

We present an order-theoretic fixed point theorem for increasing multivalued operators and its application to the following multivalued variational inequality: find \(u\in K\) such that
$$\begin{aligned} \langle Au,\varphi -u\rangle + \langle \eta ,\varphi -u\rangle \ge 0\quad \text {for all }\varphi \in K \text { and some } \eta \subset f(\cdot ,u,u), \end{aligned}$$
where \(A:W_0^{1,p} \rightarrow (W_0^{1,p})'\) is a Leray–Lions operator of divergence form \(u\mapsto -{\text {div}} a(\cdot ,\nabla u)\) and \(K\subset W_0^{1,p}\) is a closed convex set. The multivalued bifunction \(f:\Omega \times {\mathbb {R}}\times {\mathbb {R}} \rightarrow {\mathcal {P}}({\mathbb {R}})\) is assumed to be upper semicontinuous in the second and increasing in the third argument. The main existence result states that there are smallest and greatest solutions between an appropriately defined pair of sub-supersolutions.

Keywords

Fixed point ordered space discontinuous multifunction variational inequality sub-supersolution extremal solution 

Mathematics Subject Classification

Primary 47H10 47J20 Secondary 47H04 47H07 35J87 35R70 

Notes

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Institut für MathematikMartin-Luther-Universität Halle-WittenbergHalle (Saale)Germany

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