Acta mathematica scientia, Series B >
VECTORIAL EKELAND´S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS
Received date: 2011-06-02
Online published: 2012-11-20
Supported by
Supported by the National Natural Science Foundation of China (10871141).
By using the properties of w-distances and Gerstewitz´s functions, we first give a vectorial Takahashi´s nonconvex minimization theorem with a w-distance. From this, we deduce a general vectorial Ekeland´s variational principle, where the objective function is from a complete metric space into a pre-ordered topological vector space and the perturbation contains a w-distance and a non-decreasing function of the objective function value. From the general vectorial variational principle, we deduce a vectorial Caristi´s fixed point theorem with a w-distance. Finally we show that the above three theorems are equivalent to each other. The related known results are generalized and improved. In particular, some conditions in the theorems of [Y. Araya, Ekeland´s variational principle and its equivalent theorems in vector optimization, J. Math. Anal. Appl. 346(2008), 9–16] are weakened or even completely relieved.
QIU Jing-Hui , LI Bo , HE Fei . VECTORIAL EKELAND´S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS[J]. Acta mathematica scientia, Series B, 2012 , 32(6) : 2221 -2236 . DOI: 10.1016/S0252-9602(12)60172-6
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