3 - Mathematics and its Applications

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Fixed points theorems in multi-metric spaces 55theorem [19], there exists a complete metric h α on X α such that T (r α ) =T (h α ) and h α (g α (x), g α (y)) ≤ q · h α (x, y) for all x, y ∈ X α . Now we putd α (x, y) = h α (π α (x), π α (y)) for all x, y ∈ X. The proof is complete.Theorem 5.4. Let f : X −→ X be a mapping of a pseudocompact spaceX into itself. The following assertions are equivalent:1. For any positive number q < 1 there exists a family of pseudo-metricsD such that T = T (D) is the topology of the space X and d(f(x), f(y))
56 Mitrofan M. Choban, Laurențiu I. CalmuțchiTheorem 5.6. Let f : X −→ X be a continuous mapping of a compactspace X into itself. The following assertions are equivalent:1. If X 0 = X and X n+1 = f(X n ) for any n ∈ ω, then ∩{X n : n ∈ ω} isa singleton.2. For any positive number q < 1 there exists a family of pseudo-metricsD such that T = T (D) is the topology of the space X and d(f(x), f(y))
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3 - Mathematics and its Applications

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