Proper approximate solutions and εε-subdifferentials in vector optimization: Basic properties and limit behaviour

The aim of this work is to introduce a proper εε-subdifferential in the sense of Benson for extended mappings between topological linear spaces. It is defined by means of a new concept of a proper εε-efficient solution of a vector optimization problem. The main results are hold by assuming a sort of nearly subconvexlikeness, which is a very general concept of convexity. Some sufficient conditions for the existence of proper εε-subgradients are obtained. Our results generalize other similar results in the literature, and improve them since the limit behaviour when the error εε tends to zero of the involved concepts is more suitable.