ON THE EXISTENCE AND UNIQUENESS OF EXTENSIONS OF SEMI-H

. Let ( X, d ) be a quasi-metric space, y 0 ∈ X a ﬁxed element and Y a subset of X such that y 0 ∈ Y . Denote by (Λ α, 0 ( Y, d ) , (cid:107) · | αY,d ) the asymmetric normed cone of real-valued d -semi-H¨ o lder functions deﬁned on Y of exponent α ∈ (0 , 1], vanishing in y 0 , and by (Λ α, 0 ( Y, ¯ d ) , (cid:107) · | αY, ¯ d ) the similar cone if d is replaced by conjugate ¯ d of d . One considers the following claims:

Then the function d is called a quasi-metric on X and the pair (X, d) is called quasi-metric space ( [13]).
Because, in general, d(x, y) = d(y, x), for x, y ∈ X one defines the conjugate quasi-metric d of d, by the equality d(x, y) = d(y, x), for all x, y ∈ X.
Let Y be a nonvoid subset of (X, d) and α ∈ (0, 1] a fixed number. Definition 1. a) A function f : Y → R is called d-semi-Hölder (of exponent α) if there exists a constant K Y (f ) ≥ 0 such that for all x, y ∈ Y.
The smallest constant K Y (f ) in ( 1) is denoted by f | α Y,d and one shows that Definition 2. ( [14]).Let (X, d) be a quasi-metric space and Y ⊆ X a nonempty set.The function f : The set of all ≤ d -increasing functions on Y is denoted by R Y ≤ d and it is a cone in the linear space R Y of all real-valued functions on Y.
The set < ∞} is also a cone, called the cone of d-semi-Hölder functions on Y.
is called the asymmetric normed cone of dsemi-Hölder real-valued function on Y (compare with [14]).
Analogously, one defines the asymmetric normed cone ). of all d-semi-Hölder real-valued functions on Y , vanishing at the fixed point y 0 ∈ Y.
By the above definitions it follows that It follows that Λ α,0 (Y ) is a linear subspace.The following, theorem holds.
By Theorem 2 and Remark 3 in [11] it follows that the functions defined by the formulae: are elements of Λ α,0 (X, d) and, respectively, the functions given by For f ∈ Λ α,0 (Y ) let us consider the following (nonempty) sets of extensions: The sets E d (f ) and E d (f ) are convex and ( 14) Let (X, d) be a quasi-metric space, y 0 ∈ X fixed and Y ⊆ X such that y 0 ∈ Y.
In the sequel we prove a result of Phelps type ([1], [10], [12]) concerning the existence and uniqueness of the extensions preserving the smallest semi-Hölder constants and a problem of best approximation by elements of Y ⊥ d and Y ⊥ d , respectively.
Let (X, |) be an asymmetric norm (see [13], [14]) and let M be a nonempty set of X.The set M is called proximinal for x ∈ X iff there exists at least one element m 0 ∈ M such that If M is proximinal for x, then the set P M (x) = {m 0 ∈ M : x − m 0 | = ρ(x, M )} is called the set of elements of best approximations for x in M. If card P M (x) = 1 then the set M is called Chebyshevian for x.
The set M is called proximinal if M is proximinal for every x ∈ X, and Chebyshevian if M is Chebyshevian for every x ∈ X.Now, consider the following two problems of best approximation: The following theorem holds. and Taking into account the first part of the proof it follows Analogously, one obtains By the equalities ( 22) and (23) it follows.
Remark 6. Observe that the linear space Λ α,0 (X) = Λ α,0 (X, d)∩Λ α,0 (X, d) is a Banach space with respect to the norm (28) In fact this space in the space of all real-valued Lipschitz functions defined on the quasi-metric space (X, d α ), vanishing at a fixed point y 0 ∈ X. Obviously, (29) The set of all extensions of f ∈ Λ α,0 (Y ) preserving the norm f α Y (of the form (29), is denoted by E(f ), i.e., (30) the annihilator of the set Y in Banach space Λ α,0 (X), and one considers the following problem of best approximation: Corollary 8.The subspace Y ⊥ is proximinal in Λ α,0 (X) and the set of elements of best approximation for F ∈ Λ α,0 (X) is The distance of F to Y ⊥ is given by The subspace Y ⊥ is Chebyshevian for For f in the linear space Λ α,0 (Y ), the equalities F (f )(x) = F (f )(x), x ∈ X and G(f and, consequently, By Theorem 3 in [14], it follows that Λ a,0 (Y ) is a Banach space and (Y, d α ) is a metric space.