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Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces

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Abstract

In this paper, we introduce the weakly relaxed α-pseudomonotone mapping which generalizes the class of pseudomonotone mappings. Some examples are given which show that the weakly relaxed α-pseudomonotonicity is a proper generalization of pseudomonotonicity for both scalar and vector valued bi-functions. Using the KKM technique, we establish the existence of solutions of equilibrium problems and vector equilibrium problems with weakly relaxed α-pseudomonotonicity in the reflexive Banach spaces. The present work extends some corresponding results of the variational-like inequalities (Fang and Huang in J. Optim. Theory Appl. 118:327–338, 2003; Lee and Lee in J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 11: 231–242, 2004) to equilibrium problems.

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Acknowledgements

The authors would like to express their deep gratitude to the journal editor and referees for their careful reviews and valuable comments which helped to improve the paper. The work of the authors was partially supported by CSIR, New Delhi, grant 25(0163)/08/EMR-II.

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  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    N. K. Mahato & C. Nahak

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  1. N. K. Mahato
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  2. C. Nahak
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Correspondence to C. Nahak.

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Mahato, N.K., Nahak, C. Weakly relaxed α-pseudomonotonicity and equilibrium problem in Banach spaces. J. Appl. Math. Comput. 40, 499–509 (2012). https://doi.org/10.1007/s12190-012-0584-6

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