Generalized Gerstewitz’s Functions and Vector Variational Principle for ϵ-Efficient Solutions in the Sense of Németh

Abstract

In this paper, we establish a partial order principle, which is useful to deriving vector Ekeland variational principle (denoted by EVP). By using the partial order principle and extending Gerstewitz's functions, we obtain a vector EVP for ǫ-efficient solutions in the sense of Németh, which essentially improves the earlier results by removing a usual assumption for boundedness of range of the objective function. From this, we also deduce several special vector EVPs, which improve and generalize the related known results.

Minimal element theorems revisited

“…(b) Theorems 2.1 in [25], [26] and [27] follow applying Theorem 3.4 for W := S(x 1 ) with As O. Cârjȃ mentions in [2, p. 120], the use of strict monotonicity of the sequence (S(x n )) in the condition corresponding to (Ab) of the usual version of the BB-principle is due to Corneliu Ursescu.…”

“…A prominent result of this type is the Brezis-Browder principle [1] (BB-principle) where the existence of a real-valued function is assumed which is bounded from below and increasing wrt the order relation. Further examples for such results can be found in [25,27] and also in [31]. Of course, BB-type theorems can also be used to obtain the results on metric spaces discussed in the previous paragraph with a suitable monotone function; the proof of the BB-principle also involves a countable induction argument (see the first proof of Theorem 3.2 below).…”

A variational principle, coupled fixed points and market equilibrium

“…In the same article [15], various applications of the variational principle in different fields of mathematics are presented. Ekeland's variational principle has many applications and generalizations [3,6,10,26,27]. It is well known that fixed point theorems and variational principles are closely related [7,15].…”

“…Thus we cannot apply Ekeland's variational principle as it is done in [15]. A similar approach was used in [27], where variational principles in partially ordered metric spaces were obtained and used to investigated problems, otherwise impossible to solve with the known variational principles.…”