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An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems

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Abstract

By using Gerstewitz functions, we establish a new equilibrium version of Ekeland variational principle, which improves the related results by weakening both the lower boundedness and the lower semi-continuity of the objective bimaps. Applying the new version of Ekeland principle, we obtain some existence theorems on solutions for set-valued vector equilibrium problems, where the most used assumption on compactness of domains is weakened. In the setting of complete metric spaces (Z, d), we present an existence result of solutions for set-valued vector equilibrium problems, which only requires that the domain X ⊂ Z is countably compact in any Hausdorff topology weaker than that induced by d. When (Z, d) is a Féchet space (i.e., a complete metrizable locally convex space), our existence result only requires that the domain X ⊂ Z is weakly compact. Furthermore, in the setting of non-compact domains, we deduce several existence theorems on solutions for set-valued vector equilibrium problems, which extend and improve the related known results.

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  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, P. R. China

    Jing Hui Qiu

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Correspondence to Jing Hui Qiu.

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Supported by National Natural Science Foundation of China (Grant Nos. 11471236 and 11561049)

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Qiu, J.H. An equilibrium version of set-valued Ekeland variational principle and its applications to set-valued vector equilibrium problems. Acta. Math. Sin.-English Ser. 33, 210–234 (2017). https://doi.org/10.1007/s10114-016-6184-x

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