We propose a definition of a generalized type of Knaster–Kuratowski–Mazurkiewicz (KKM) mappings, called a weak T-KKM mapping, and a corresponding weak KKM property. A new extension of the Fan–Glicksberg fixed-point …
We propose a definition of a generalized type of Knaster–Kuratowski–Mazurkiewicz (KKM) mappings, called a weak T-KKM mapping, and a corresponding weak KKM property. A new extension of the Fan–Glicksberg fixed-point theorem is established. Sufficient conditions for the existence of a continuous selection, a fixed point of a composition, and a coincidence point are also provided. Then, we use the obtained results to study the existence of solutions to various optimization-related problems. Discussions and detailed examples are included as well to compare our results with existing ones and to explain their advantages in many situations.
We establish theorems of the Knaster-Kuratowski-Mazurkiewicz (KKM) type on topological spaces with topologically-based generalized convex structures and apply them to the existence study for variational relation problems, variational inequalities, minimax …
We establish theorems of the Knaster-Kuratowski-Mazurkiewicz (KKM) type on topological spaces with topologically-based generalized convex structures and apply them to the existence study for variational relation problems, variational inequalities, minimax inequalities and saddle points, and to alternative theorems. Our first main KKM-type theorem is a necessary and sufficient condition, not only a sufficient one as usual. The results improve and extend corresponding contributions in the literature in many aspects.
The purpose of this paper is to present a refinement of earlier directional variational principles and solution existence in optimization. By using the so-called directional minimal time function, we first …
The purpose of this paper is to present a refinement of earlier directional variational principles and solution existence in optimization. By using the so-called directional minimal time function, we first provide a directional invariant point theorem. As direct consequences, we obtain several directional variational principles (with quite different formulations). Then, applying these results, we establish sufficient conditions for the existence of solutions for two general models of directional variational relation and inclusion problems. We also include corresponding consequences for particular cases.
This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. …
This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. We introduce, study, and utilize a novel notion of quasi-near convexity of sets that is an infinite-dimensional extension of the widely acknowledged notion of near convexity. Quasi-near convexity is associated with the quasi-relative interior of sets, which is investigated in the paper together with other generalized relative interior notions for sets, not necessarily convex. In this way, we obtain new results on generalized relative interiors for graphs of set-valued mappings in convexity and generalized convexity settings.
This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. …
This paper focuses on investigating generalized relative interior notions for sets in locally convex topological vector spaces with particular attentions to graphs of set-valued mappings and epigraphs of extended-real-valued functions. We introduce, study, and utilize a novel notion of quasi-near convexity of sets that is an infinite-dimensional extension of the widely acknowledged notion of near convexity. Quasi-near convexity is associated with the quasi-relative interior of sets, which is investigated in the paper together with other generalized relative interior notions for sets, not necessarily convex. In this way, we obtain new results on generalized relative interiors for graphs of set-valued mappings in convexity and generalized convexity settings.
The purpose of this paper is to present a refinement of earlier directional variational principles and solution existence in optimization. By using the so-called directional minimal time function, we first …
The purpose of this paper is to present a refinement of earlier directional variational principles and solution existence in optimization. By using the so-called directional minimal time function, we first provide a directional invariant point theorem. As direct consequences, we obtain several directional variational principles (with quite different formulations). Then, applying these results, we establish sufficient conditions for the existence of solutions for two general models of directional variational relation and inclusion problems. We also include corresponding consequences for particular cases.
We establish theorems of the Knaster-Kuratowski-Mazurkiewicz (KKM) type on topological spaces with topologically-based generalized convex structures and apply them to the existence study for variational relation problems, variational inequalities, minimax …
We establish theorems of the Knaster-Kuratowski-Mazurkiewicz (KKM) type on topological spaces with topologically-based generalized convex structures and apply them to the existence study for variational relation problems, variational inequalities, minimax inequalities and saddle points, and to alternative theorems. Our first main KKM-type theorem is a necessary and sufficient condition, not only a sufficient one as usual. The results improve and extend corresponding contributions in the literature in many aspects.
We propose a definition of a generalized type of Knaster–Kuratowski–Mazurkiewicz (KKM) mappings, called a weak T-KKM mapping, and a corresponding weak KKM property. A new extension of the Fan–Glicksberg fixed-point …
We propose a definition of a generalized type of Knaster–Kuratowski–Mazurkiewicz (KKM) mappings, called a weak T-KKM mapping, and a corresponding weak KKM property. A new extension of the Fan–Glicksberg fixed-point theorem is established. Sufficient conditions for the existence of a continuous selection, a fixed point of a composition, and a coincidence point are also provided. Then, we use the obtained results to study the existence of solutions to various optimization-related problems. Discussions and detailed examples are included as well to compare our results with existing ones and to explain their advantages in many situations.