Solution of a class of equilibrium problems and variational inequalities in FC spaces

Generalized monotone mixed equilibrium problems in FC-spaces: Existence and approximation

“…The beauty of the problem considered in this paper is that it encompasses several important type of problems as special cases. The approach developed in this paper generalizes and improve the recent results obtained in [13,18].…”

“…Theorem 4.1 improves Theorem 2 in[13]. Moreover, by the approach developed in this paper, we give a new proof of Theorems 2 and 5 in[13].…”

“…where N : E × E → E * is a nonlinear operator with E * as the topological dual of E, b : E × E → R is a bifunction and η : K × K → E is a vector-valued bifunction. Problem (5.15) was introduced and studied recently by Pany-Mohapatra-Pani [13] in the setting of FC-spaces. Numerous mixed variational inequalities studied earlier can be obtained as special cases from inequality problem (5.15), see, e.g., [11,16] and the references therein.…”

“…Let E be a finitely continuous topological space and let K be a finitely continuous subset of E. We consider the following extended generalized mixed equilibrium problem: Findx ∈ K such that φ (y,x, z) + b(x, z) − b(x,x) ≥ 0, for all y, z ∈ K, (1.1) where φ : K × K → R is a trifunction and b : E × E → R is a bifunction. Recently, the extended generalized mixed equilibrium problem has been studied by Pany-Mohapatra-Pani [13] when K is an FC-subspace of a reflexive Banach space E. It involves particularly the problem studied in FC-spaces by Wang [18]. If b ≡ 0 and φ (y, x, z) = ψ(x, η(z, x)), where η : K × K → E is a mapping, then problem (1.1) is reduced to the following problem studied by Liu-Zeng in [12]:…”

“…The following nonlinear mixed variational-like inequality problem findx ∈ K such that N(x, y), η(z,x) + b(x, z) − b(x,x) ≥ 0, ∀y, z ∈ K studied in [13,14], can be obtained as a special case of (1.1) if φ (y, x, z) = N(x, y), η(z, x) , where N : E × E → E * and η : K × K → E. Here, E denotes a topological vector space with its dual space E * and K is an FC-subspace of E.…”