Exceptional Family Of Elements Research Articles

Sufficient conditions on nonemptiness and boundedness of the solution set of the P0 function nonlinear complementarity problem

Recently, the P 0 function nonlinear complementarity problem (NCP) has attracted a lot of attention among researchers. Various assumed conditions, which ensure that the NCP has a solution have been proposed. In this paper, by using the notion of an exceptional family of elements we develop a sufficient condition which ensures that the solution set of the P 0 function NCP is nonempty and bounded. In particular, we prove that many existing assumed conditions imply this sufficient condition. Thus, these conditions imply that the solution set of the P 0 function NCP is nonempty and bounded. In addition, we also prove directly that a few existence conditions imply that the solution set of the P 0 function NCP is bounded.

Solvability of Implicit Complementarity Problems

In this paper, a new notion of exceptional family of elements (EFE) for a pair of functions involved in the implicit complementarity problem (ICP) is introduced. Based upon this notion and the Leray–Schauder Alternative, a general alternative is obtained which gives more general existence theorems for the implicit complementarity problem. Finally, via the techniques of continuous selections, these existence theorems are extended to the multi-valued implicit complementarity problems (MIPS).

Leray-Schauder type alternatives and the solvability of complementarity problems

We present in this paper several existence theorems for nonlinear complementarity problems in Hilbert spaces. Our results are based on the concept of ``exceptional family of elements'' and on Leray-Schauder type altrenatives.

Exceptional Family of Elements, Leray–Schauder Alternative, Pseudomonotone Operators and Complementarity

In this paper, the property of a function to be without exceptional family of elements (EFE) is investigated. We show that, for a pseudomonotone operator, the solvability of a complementarity problem is equivalent to the property of the function to be without EFE. Finally, we study the strict feasibility of a complementarity problem making use of the Leray-Schauder alternative and the notion of EFE.

Exceptional Families and Existence Results for Nonlinear Complementarity Problem

In this paper we establish several sufficient conditions for the existence of a solution to the linear and some classes of nonlinear complementarity problems. These conditions involve a notion of the ``exceptional family of elements'' introduced by Smith [19] and Isac, Bulavski and Kalashnikov [4], where the authors have shown that the nonexistence of the ``exceptional family of elements'' implies solvability of the complementarity problem. In particular, we establish several sufficient conditions for the nonexistence as well as for the existence of the exceptional family of elements.

Exceptional Family of Elements and the Solvability of Variational Inequalities for Unbounded Sets in Infinite Dimensional Hilbert Spaces

Exceptional Family of Elements and the Solvability of Variational Inequalities for Unbounded Sets in Infinite Dimensional Hilbert Spaces

Quasi-P*-Maps, P(τ, α, β)-Maps, Exceptional Family of Elements, and Complementarity Problems

Quasi-P*-maps and P(τ, α, β)-maps defined in this paper are two large classes of nonlinear mappings which are broad enough to include P*-maps as special cases. It is of interest that the class of quasi-P*-maps also encompasses quasimonotone maps (in particular, pseudomonotone maps) as special cases. Under a strict feasibility condition, it is shown that the nonlinear complementarity problem has a solution if the function is a nonlinear quasi-P*-map or P(τ, α, β)-map. This result generalizes a classical Karamardian existence theorem and a recent result concerning quasimonotone maps established by Hadjisawas and Schaible, but restricted to complementarity problems. A new existence result under an exceptional regularity condition is also established. Our method is based on the concept of exceptional family of elements for a continuous function, which is a powerful tool for investigating the solvability of complementarity problems.

Exceptional Families of Elements, Feasibility and Complementarity

Feasibility is an important property for a complementarity problem. A complementarity problem is solvable if it is feasible and some supplementary assumptions are satisfied. In this paper, we introduce the notion of (α, β)-exceptional family of elements for a continuous function and we apply this notion to the study of feasibility of nonlinear complementarity problems.

An alternative theorem for generalized variational inequalities and solvability of nonlinear quasi- [formula omitted]-complementarity problems

In this paper, an alternative theorem, and hence a sufficient solution condition, is established for generalized variational inequality problems. The concept of exceptional family for generalized variational inequality is introduced. This concept is general enough to include as special cases the notions of exceptional family of elements and the D-orientation sequence for continuous functions. Particularly, we apply the alternative theorem for investigating the solvability of the nonlinear complementarity problems with so-called quasi- P M ∗ -maps, which are broad enough to encompass the quasi-monotone maps and P ∗ -maps as the special cases. An existence theorem for this class of complementarity problems is established, which significantly generalizes several previous existence results in the literature.

D-orientation sequences for continuous functions and nonlinear complementarity problems

We introduce the new concept of d-orientation sequence for continuous functions. It is shown that if there does not exist a d-orientation sequence for a continuous function, then the corresponding complementarity problem (CP) has a solution. We believe that such a result characterizes an intrinsic property of CPs. As the concept of “exceptional family of elements”, the notion of “d-orientation sequence of a function” is also a powerful tool for investigating the existence theorems of CPs. We use this new tool to establish, among other things, a new existence result for a class of P ∗ -mapping CPs.

Exceptional Family of Elements for a Variational Inequality Problem and its Applications

This paper introduces a new concept of exceptional family of elements (abbreviated, exceptional family) for a finite-dimensional nonlinear variational inequality problem. By using this new concept, we establish a general sufficient condition for the existence of a solution to the problem. Such a condition is used to develop several new existence theorems. Among other things, a sufficient and necessary condition for the solvability of pseudo-monotone variational inequality problem is proved. The notion of coercivity of a function and related classical existence theorems for variational inequality are also generalized. Finally, a solution condition for a class of nonlinear complementarity problems with so-called P_* -mappings is also obtained.

Exceptional Families of Elements for Continuous Functions: Some Applications to Complementarity Theory

Using the topological degree and the concept of exceptional family of elements for a continuous function, we prove a very general existence theorem for the nonlinear complementarity problem. This result is an alternative theorem. A generalization of Karamardian‘s condition and the asymptotic monotonicity are also introduced. Several applications of the main results are presented.

Functions Without Exceptional Family of Elements and Complementarity Problems

In Ref. 1, Isac, Bulavski, and Kalashnikov introduced the concept of exceptional family of elements for a continuous function f: Rn→Rn. It is known that, if there does not exist an exceptional family of elements for f, then the corresponding complementarity problem has a solution. In this paper, we show that several classes of nonlinear functions, known in complementarity theory or other domains, are functions without exceptional family of elements and consequently the corresponding complementarity problem is solvable. It is evident that the notion of exceptional family of elements provides an alternative way of determining whether or not the complementarity problem has a solution.

Exceptional Families, Topological Degree and Complementarity Problems

By using the topological degree we introduce the concept of ’’exceptional family of elements‘‘ specifically for continuous functions. This has important consequences pertaining to the solvability of the explicit, the implicit and the general order complementarity problems. In this way a new direction for research in the complementarity theory is now opened.