Denote by (Kinmathbb{R}^{n}) a given nonempty convex subset.
Let ( X, ∥ ⋅ ∥ ) be a Banach space and K be a given nonempty closed subset of X.
The set of all relative interior points of a given nonempty convex set (mathcal{C}) is denoted by (operatorname{ri}(mathcal{C})).
The metric projection P C onto a given nonempty, closed, and convex set C (⊂ H), satisfies the nonexpansivity with Fix(P C ) = C [[22], Theorem 3.1.4(i ], [[29], p. 371], [[30], Theorem 2.4-3].
For a given (nonempty) Hausdorff space X and a sequence (mathcal{S}=(s_{n})) of mappings on X, let (operatorname{Fix}(mathcal{S}) = bigcap_{n=1}^{infty} operatorname{Fix}(s_{n})) and (mathrm{C}(mathcal{S}) = {x : lim_{n toinfty} s_{n}(x) mbox{ exists}}) denote the fixed point set and the convergence set of (mathcal{S}), respectively.
For a given nonempty subset (mathcal {P}) of X and a set-valued function (F Xrightarrow2^{X}), we assume that (H0) there exist (u_{0}, v_{0}inmathcal{P}) with (u_{0}preceq v_{0}) such that (F[X]=bigcup_{xin X}F x)subset[u_{0}, v_{0}]); (H1) if (pinmathcal{P}), then (min F(p)) and (max F(p)) exist and belong to (mathcal{P}).
The effective domain of a closed proper function (f: mathcal{X} rightarrow -infty,+infty]) is defined as (operightarrow -infty:= {xinmathcal{X}|f(x) < +infty}), and the symbol (operatorname{ri}(mathcal{C})) denotes the set of all relative interior points of a given nonempty convex set (mathcal{C}).
Let be a nonempty subset of, and be two given nonempty set-valued maps.
Given nonempty subsets A and B of a metric space, we recall the following notations and notions, which will be used in the sequel.
