Sentence examples for for any given nonempty from inspiring English sources
For any given nonempty finite subset of.
Next, we claim that the family has the finite intersection property, and then the whole intersection is nonempty and any element in the intersection is a solution of (SEP) I, for any given nonempty finite subset of.
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}).
More precisely, for two given nonempty closed subsets A and B of a complete metric space ((X,d)), a non-self contraction (T : Ato B) does not necessarily have a fixed point.
By using viscosity approximation methods for a finite family of nonexpansive mappings, it is shown that for any given contractive mapping, where is a nonempty closed convex subset of and for any given the iterative scheme is strongly convergent to a solution of (CFP), if and only if and satisfy certain conditions, where and is a sunny nonexpansive retraction of onto.
Let be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space, let be an equilibrium bifunction satisfying conditions, and let for any given.
For any given, let (4.3).
