Some existence results of efficiency in vector optimization

Abstract

In this paper, we present several existence results for efficient solutions and efficient points in vector optimization problems. Firstly, we apply a corollary of a recently obtained Caristi-Kirk fixed point theorem ([3]) to obtain existence results for efficient solutions of a vector optimization problem, which generalize the existence theorems of efficient solutions in [2] (Theorem 9 and its Corollary). Secondly, we generalize Theorem 10 in [2] to the vector case, obtaining an existence result for efficient points of a vector optimization problem. As a result, an open problem following the Corollary of Theorem 10 in [2] is solved in some way. Finally, the concept of nuclear cones introduced in [5] is extended, somehow answering another open question in [2] (in the Remark following the Corollary of Theorem 9). Applying this concept of generalized nuclear cones, we derive another existence theorem of efficient points.