Equilibrium set-valued variational principles and the lower boundedness condition with application to psychology

Abstract

We first give a pre-order principle whose form is very general. Combining the pre-order principle and generalized Gerstewitz functions, we establish a general equilibrium version of set-valued Ekeland variational principle (denoted by EVP), where the objective function is a set-valued bimap defined on the product of quasi-metric spaces and taking values in a quasi-ordered linear space, and the perturbation consists of a subset of the ordering cone multiplied by the quasi-metric. From this, we obtain a number of new results which essentially improve the related results. Particularly, the earlier lower boundedness condition has been weakened. Finally, we apply the new EVPs to Psychology.

Keywords:

• Equilibrium version of Ekeland variational principle
• set-valued perturbation
• quasi-metric space
• pre-order principle
• Gerstewitz function

Mathematics Subject Classifications:

• 49J53
• 90C29

Acknowledgments

The authors are grateful to the reviewers for the favourable comments and helpful suggestions, which have greatly improved our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first and third authors were supported by the National Natural Science Foundation of China (grant numbers 11471236, 12061050).