Solvability of implicit complementarity problems

In this paper, we introduce some new notions of (α,γ)(α,γ)-exceptional family of elements (in short, (α,γ)(α,γ)-(EFE)) and (α,β,γ)(α,β,γ)-exceptional family of elements (in short, (α,β,γ)(α,β,γ)-(EFE)) for a pair of continuous functions involved in the implicit complementarity problem (in short, ICP). Based upon these notions and the topological degree theory, we studied the feasibility and strictly feasibility of (ICP) in RnRn and an infinite-dimensional Hilbert space HH, respectively. As special cases, we obtain the feasibility and strictly feasibility of complementarity problems and partly answered the second open problem (P2) proposed by Isac [G. Isac, Exceptional families of elements, feasibility and complementarity, J. Optim. Theory Appl. 104 (2000) 577–588].