VECTORIAL EKELAND´S VARIATIONAL PRINCIPLE WITH A W-DISTANCE AND ITS EQUIVALENT THEOREMS

doi:10.1016/S0252-9602(12)60172-6

Abstract

Abstract:

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.

Key words: Takahashi´s minimization theorem, Ekeland´s variational principle, Caristi´s fixed point theorem, Gerstewitz´s function, w-distance

CLC Number:

58E30

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.

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