A function is locally -invertible at if for any point in there exists a unique element in such that If is locally -invertible at each then we say that is locally -invertible.
For any x in H, there exists a unique element in C, which is denoted by P C x, such that ||x - P C x|| ≤ ||x - y|| for all y in C. We call P C the metric projection of H onto C. It is well-known that P C is a nonexpansive mapping from H onto C, and ⟨ x - P C x, P C x - y ⟩ ≥ 0 for all x ∈ H, y ∈ C ; (1).
A lower spring ordered transversal space is a nonempty partially ordered set X (with ordering ) together with a given lower spring ordered transverse A on X, where every increasing sequence ({u_{n}}_{ninBbb{N}}) of elements in ((a, b]) has a unique element u in ((a, b]) as limit (in notation (u_{n}to u) ((ntoinfty))).
Then, for any (x in E), there exists a unique element (z in K) such that (|x - z| le|x - y|), (forall y in K).
Let C be a nonempty, closed, and convex subset of a strictly convex and reflexive Banach space E. Then we know that, for any (x in E), there exists a unique element (z in C) such that (|x-z| leq|x-y|) for all (y in C).
(1) If E is a reflexive and strictly convex Banach space and C is a nonempty closed and convex subset of E, then for each (x in E) there exists a unique element (v in C) such that (Vert x - v Vert = inf { Vert x - y Vert :y in C}).
If E is a reflexive and strictly convex Banach space and C is a nonempty closed and convex subset of E, then for each (x in E) there exists a unique element (v in C) such that (Vert x - v Vert = inf { Vert x - y Vert :y in C}).
Let E be a real reflexive, strictly convex, and smooth Banach space and C be a nonempty closed and convex subset of E, then for (forall x in E), there exists a unique element (x_{0} in C) satisfying (varphi (x_{0},x) = inf { varphi (y, x) :y in C}).
Such an element v is denoted by (P_{C}x) and (P_{C}) is called the metric projection of E onto C. (2) Let E be a real reflexive, strictly convex, and smooth Banach space and C be a nonempty closed and convex subset of E, then for (forall x in E), there exists a unique element (x_{0} in C) satisfying (varphi (x_{0},x) = inf { varphi (y, x) :y in C}).
An upper spring ordered transversal space is a nonempty partially ordered set X (with ordering ) together with a given upper spring ordered transverse A on X, where every decreasing sequence ({u_{n}}_{ninBbb{N}}) of elements in ([a, b)) has a unique element (uin[a, b)) as limit (in notation (u_{n}to u) ((ntoinfty))). The element (ain[a, b) subset P) is called spring of space X (cf. [5]).
