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Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 3
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Articles

The degree theory for set-valued compact perturbation of monotone-type mappings with an application

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Pages 616-635 | Received 24 Aug 2011, Accepted 09 Oct 2011, Published online: 14 Nov 2011
 

Abstract

Degree theory has been developed as a tool for checking the solution existence of nonlinear equations. Hu and Parageorgiou [S.C. Hu, N.S. Parageorgiou, Generalisation of Browders degree theory, Trans. Amer. Math. Soc. 347 (1995), pp. 233–259] generalized the results of Browder [F.E. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. 9 (1983), pp. 1–39] on the degree theory to mappings of the form f + T + G, where f is a bounded and demicotinuous mapping of class (S)+ from a bounded open set in a reflexive Banach space X into its dual X*, T is a maximal monotone mapping with 0 ∈ T(0) from X into X*, and G is an u.s.c. compact set-valued mapping from X into X*. In this article we continue to generalize and extend the results of Browder on the degree theory to mappings of the form f + T + G. By enlarging the class of maximal monotone mappings and pseudo-monotone homotopies we obtain some new results of the degree theory for such mappings. As an application, an existence result of solutions for generalized mixed variational inequalities is given under some suitable conditions.

AMS Subject Classifications::

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11171237) and the Key Program of NSFC (Grant No. 70831005).

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