On the Relationship Between Topological Boundedness and the Ordered Boundedness for Solid Cones via Scalarizing

Abstract

In this paper, first we prove some lemmas, then by using the nonlinear scalarization function, we prove that there is no difference between the topological boundedness and the ordered boundedness. As an application, a fixed point theorem which is a new version of the main result obtained by Zangenehmehr et al (Positivity, 19, 333-340, 2015), by relaxing or weakening some assumptions, is established.

Existence of solutions for a new version of generalized operator equilibrium problems

“…For each i ∈ I, let C i : Π j∈I K j → 2 Y i be a set-valued mapping such that, for each f ∈ Π j∈I K j , C i ( f ) is closed, pointed convex cone such that e i ∈ intC i ( f ) (we recall that a subset C i ( f ) of Y i is convex cone and pointed whenever λC i ( f ) + (1 − λ)C i ( f ) ⊆ C i ( f ), for all 0 < λ < 1, 2C i ( f ) ⊆ C i ( f ) and C i ( f ) ∩ −C i ( f ) = {0 Y i }, resp. ), for more details see [7]. Also for each i ∈ I, let F i : Π j∈I K j × K i → 2 Y i be a set-valued mapping.…”