3 - Mathematics and its Applications

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Fixed points theorems in multi-metric spaces 59Corollary 6.1 permits to study the construction of the general dilation ofthe multi-metric space. Let (X, P) be a non-empty multi-metric space andk = (k ρ : ρ ∈ P) > 0 be a positive P-number. Suppose that f : X −→ Xis a mapping and ρ(f(x), f(y)) = k ρ f(x, y) for all x, y ∈ X and ρ ∈ P. Weput P ′ = {ρ ∈ P : k ρ < 1}, P ′′ = {ρ ∈ P : k ρ > 1} and P 0 = {ρ ∈ P :k ρ = 1}. If P 0 ≠ 0, then we say that f is a general dilation. For any ρ ∈ Pconsider the mapping f ρ : X/ρ −→ X/ρ, where π ρ (f(x)) = f ρ (π ρ (x)) and¯ρ(f ρ (π ρ (x))), f ρ (π ρ (y)) = k ρ · ρ(x, y) for all x, y ∈ X. If k ρ = 1, then f ρ is anisometry of the metric space (X/ρ, ¯ρ). If k ρ ≠ 1, then f ρ is a dilation withthe coefficient k ρ .Fix ρ ∈ P ′ . We put N (ρ,f) (u) = f ρ (u),u ∈ X/ρ.Fix ρ ∈ P 0 . We put N (ρ,f (u) = Q (ρ,f (u) = u.u ∈ X/ρ.S (ρ,f)(u) = q (ρ,f) (u) = u for anyS (ρ,f) = f ρ (u) for anyNow we consider the embedding e P : X −→ Π{X/ρ : ρ ∈ P} and themappings N f (x) = e −1P (N (ρ,f)(π ρ (x)) : ρ ∈ P), S f (x) = e −1P (S (ρ,f)(π ρ (x)) :ρ ∈ P), Q f (x) = e −1P (Q (ρ,f)(π ρ (x)) : ρ ∈ P).The mappings N f , S f , Q f are homeomorphisms of the space (X, P) associatedwith the mapping f. By construction, we have the following fourproperties.Property 1. f = S f · N f · Q f = N f · S f · Q f = Q f · N f · S f = S f · Q f · N f .Property 2. ρ(S f (x), S f (y)) = ρ(x, y) for any x, y ∈ X and ρ ∈ P.Property 3. ρ(N f (x), N f (y)) = k ρ ρ(x, y) for any ρ ∈ P ′ and ρ(N f (x),N f (y)) = ρ(x, y) for any ρ ∈ P \ P ′ and all x, y ∈ X.Property 4. ρ(Q f (x), Q f (y)) = k ρ (x, y) for any ρ ∈ P ′′ and ρ(N f (x),N f (y)) = ρ(x, y) for any ρ ∈ P \ P ′′ and all x, y ∈ X.We say that S f is the isometrically component of the mapping f, N f isthe negative component of the mapping f and Q f is the positive componentof the mapping f.Assume that P 0 = ∅, i.e. f is a dilation. In this case D f = N f ◦ Q −1fis a contraction mapping. By construction, F ix(f) = F ix(D f ). Thus fromCorollary 6.1 it follows the next assertion.Corollary 6.2. Let (X, P) be a sequentially complete multi-metric spaceand f : X −→ X be a dilation. Then the mapping f admits one and onlyone fixed point.
60 Mitrofan M. Choban, Laurențiu I. Calmuțchi7 Dilations of E-metric spacesFix an m-scale E.Let (X, d, E) be an E-metric space.A mapping f : X −→ X is a dilation mapping if there exist an elementq ∈ E (+,1) and an element k ≥ 0 such that q · k = 1 and d(f(x), f(y)) =k·d(x, y) for all x, y ∈ X. Every dilation mapping f is a uniformly continuoushomeomorphism and f −1 is a contraction with the Lipschitz constant q ∈E (+,1) .From Corollary 3.2 it followsCorollary 7.1. Let (X, d, E) be a sequentially complete E-metric spaceand f : X −→ X be a dilation mapping. Then the mapping f admits oneand only one fixed point b ∈ X. Moreover, b = lim n→∞ n(x, f −1 ) for anypoint x ∈ X.8 On M-complete multi-metric spacesWe say that the pseudo-metric d on a space X is M-complete if it is continuous,the metric space (X/d, ¯d) is complete and the sequence {x n : n ∈ N}has an accumulation point provided lim n→∞ d(x, x n ) = 0 for some x ∈ X.Remark 8.1. The pseudo-metric d on a space X is M-complete ifand only if the metric space (X/d, ¯d) is complete and the projection π d :X −→ X/d is a continuous closed mapping with the countably compactfibers π −1d(u). Thus a space with a complete pseudo-metric is an M-space[20]. The spaces related to M-spaces were studied in [4, 5, 9, 10].A multi-metric space (X, P) is said to be an M-complete multi-metricspace if there exists some M-complete pseudo-metric d ∈ P. Denote by s(P)the set of all M-complete pseudo-metrics d ∈ P.Theorem 8.2. Any M-complete multi-metric space (X, P) is sequentiallycomplete.Proof. Fix d ∈ s(P). Let L = {x n ∈ X : n ∈ N} be a fundamentalsequence. Since the metric space (X/d, ¯d) is complete, there exists a pointb ∈ X such that lim n→∞ d(b, x n ) = 0. In this case F = {x ∈ X : d(b, x) = 0}and H = F ∪ L are countably compact closed subsets of the space X. Thusthe sequence {x n : n ∈ N} is convergent in the subspace H. The proof iscomplete.
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