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The existence of zeros of set-valued mappings in reflexive Banach spaces

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Abstract

In this paper, we establish the equivalence between the existence of zeros for set-valued mappings and the solvability of variational inequality, under some conditions involving the generalized projection operator. Basing on this result, we obtain some existence theorems of zeros for quasimonotone set-valued mappings. As application, we derive several fixed point theorems for generalized inward mappings in Hilbert spaces.

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Acknowledgments

The author thanks the anonymous referees for their valuable remarks and suggestions that helped to improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Guangxi Normal University, Guilin, Guangxi , 541004, People’s Republic of China

    Ren-you Zhong, Xiang Liu & Jiang-Hua Fan

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  1. Ren-you Zhong
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  2. Xiang Liu
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  3. Jiang-Hua Fan
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Corresponding author

Correspondence to Jiang-Hua Fan.

Additional information

This work was supported by the National Natural Science Foundation of China (11061006), the National Natural Science Foundation of China (11226224), the Program for Excellent Talents in Guangxi Higher Education Institutions, the Guangxi Natural Science Foundation (2012GXNSFBA053008) and the Initial Scientific Research Foundation for PHD of Guangxi Normal University.

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Zhong, Ry., Liu, X. & Fan, JH. The existence of zeros of set-valued mappings in reflexive Banach spaces. Optim Lett 8, 1741–1751 (2014). https://doi.org/10.1007/s11590-013-0696-4

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