Coincidence points in the cases of metric spaces and metric maps

Coincidence points in the cases of metric spaces and metric maps tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án...

JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.1 (1-21) Topology and its Applications ••• (••••) •••–••• Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Coincidence points in the cases of metric spaces and metric maps Thi Hong Van Nguyen a , B.A Pasynkov b,∗ a b Vietnam National University, Hanoi, Viet Nam Moscow State University, Moscow, Russian Federation a r t i c l e i n f o Article history: Received 31 December 2014 Accepted 21 May 2015 Available online xxxx MSC: 54H25 54C10 54E35 a b s t r a c t In the first half of the paper, we are concerned with the problems of existence (and searching) of coincidence points and the common preimage of a closed subset (in particular, a common root) in the case of a finite system of mappings of one metric space to another one The second half of the paper is devoted to fiberwise variants of Arutyunov’s theorem on coincidence points Obtaining the main results of the paper is based on the use of the class of almost exactly (α, β)-search functionals that is wider than Fomenko’s class of (α, β)-search functionals © 2015 Published by Elsevier B.V Keywords: Metric space Metric mapping (Almost exactly) (α, β)-search functional Fixed point Coincidence point Continuous section Perfect section Introduction We will use 1) “space” instead of “metric space” in Sections and and instead of “topological space” in Section 3; 2) “map” instead of “continuous mapping” For a metric space (X, ρ) and x ∈ X, by Or x ≡ O(x, r) (more detailed, OrX x ≡ OX (x, r)) denote the r-neighborhood of x in X If we use B instead of O in this notation, we will get the notation for the closed ball with the center x of radius r For metric spaces (X, ρX ) and (Y, ρY ), we consider only the metric ρX + ρY on X × Y Note also that for a mapping f : X → Y and f X ⊂ Y ⊂ Y , the corestriction cor Y f of f to Y is the mapping of X to Y such that (cor Y f )(x) = f x for any x ∈ X If Y = f X then cor f is used instead of cor f X f * Corresponding author E-mail addresses: nguyenhongvan@mail.ru (T.H.V Nguyen), bpasynkov@gmail.com (B.A Pasynkov) http://dx.doi.org/10.1016/j.topol.2015.12.026 0166-8641/© 2015 Published by Elsevier B.V JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.2 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• In Sections and 2, we consider the problems of the existence and searching of coincidence points and the common preimage of a closed subset (in particular, the common root) in the case of a finite system of mappings of one metric space to another one In [7], T.N Fomenko used (α, β)-search functionals (i.e mappings to [0, +∞)) on metric spaces to solve the problems mentioned above An (α, β)-search functional ϕ on a metric space X allows to obtain, for any n→∞ x ∈ X, a fundamental sequence of points x0 = x, x1 , , xn , in X such that ϕ(xn ) −−−−→ Under some additional conditions (for example, if X is complete and ϕ is continuous), there exists ξ = lim xn ∈ X n→∞ such that ϕ(ξ) = and ρ(x, ξ) ϕ(x) α−β (In particular: for a mapping f : X → X and ϕ(x) = ρ(x, f (x)), x ∈ X, the equality ϕ(ξ) = means that ξ is a fixed point for f ; for mappings f, g : X → Y and ϕ(x) = ρ(f (x), g(x)), x ∈ X, the equality ϕ(ξ) = means that ξ is a coincidence point for f and g.) In Section 1, the class of almost exactly (α, β)-search functionals on metric spaces is defined This class is wider than the class of (α, β)-search functionals It is used in Section to obtain (in more general situations) results that are similar to ones in [7] In Section 3, a fiberwise variants of Arutyunov’s theorem on coincidence points ([2], Theorem 1) are obtained The proofs of our theorems are based on the use of almost exactly (α, β)-search functionals We consider only one-valued mappings Search functionals on metric spaces Fix a space (X, ρ) Let F (X) be the set of all (not necessarily continuous) mappings of X to itself and CF (X) the set of all continuous mappings of X to itself For A, B ∈ F (X), set dF (A, B) = sup{ρ(Ax, Bx) : x ∈ X} For A ∈ F (X) and C ∈ CF (X), set F (X, A) = {B ∈ F (X) : dF (A, B) < +∞} and CF (X, A) = {D ∈ CF (X) : dF (C, D) < +∞} It is evident that: for A, B ∈ F (X), either F (X, A) ∩ F (X, B) = ∅ or F (X, A) = F (X, B) and this equality is equivalent to the inequality dF (A, B) < +∞; for C, D ∈ CF (X), either CF (X, C) ∩ CF (X, D) = ∅ or CF (X, C) = CF (X, D) and this equality is equivalent to the inequality dF (C, D) < +∞; for every A ∈ F (X) (respectively, A ∈ CF (X)), the function dF is a metric on F (X, A) (respectively, on CF (X, A)) The sets of type F (X, A) (respectively, CF (X, A)) will be called metric parts of F (X) (respectively, of CF (X, A)) It is easy to see that every metric part of F (X) (respectively, of CF (X)) is a complete space (with the metric dF ) if the space X is complete Recall the definition of (α, β)-search functionals given by T.N Fomenko in [7] Further, let R+ = {x ∈ R : x 0} and N+ = {0} ∪ N JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.3 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• Definition 1.1 A functional ϕ : X → R+ is called (α, β)-search, α, β ∈ R, < β < α, if (*) for any x ∈ X, there exists a point x = x (x) ∈ X such that ϕ(x) and ϕ(x ) α ρ(x, x ) β ϕ(x) α (1.1) The following assertion is evident Proposition 1.1 A functional ϕ : X → R+ is (α, β)-search, < β < α, if and only if (∗A ) there exists a mapping A : X → X (i.e., A ∈ F (X)) such that for any x ∈ X, β ϕ(x) α ϕ(x) and ϕ(Ax) α ρ(x, Ax) (1.1A ) We extend the class of (α, β)-search functionals in the following way Definition 1.2 A functional ϕ : X → R+ is called almost exactly (α, β)-search for α, β ∈ R, < β < α, if (**) for any x ∈ X and every δ > there is a point x = x (x, δ) ∈ X such that ρ(x, x ) ϕ(x) + δ and ϕ(x ) α β ϕ(x) + δ α (1.1δ ) The next assertion is also evident Proposition 1.2 For a functional ϕ : X → R+ and α, β ∈ R, < β < α, the following conditions are equivalent: ϕ is almost exactly (α, β)-search; for any δ > 0, there exists a mapping Aδ ∈ F (X) such that for any x ∈ X, ρ(x, Aδ x) ϕ(x) + δ and ϕ(Aδ x) α β ϕ(x) + δ; α (1.1Aδ ) for any k ∈ N, there exist Ak ∈ F (X) and δk > such that for any x ∈ X, ρ(x, Ak x) ϕ(x) + δk , ϕ(Ak x) α β ϕ(x) + δk and δk → as k → +∞ α (1.1Ak ) Remark 1.1 An almost exactly (α, β)-search functional ϕ is (α, β)-search if and only if in Item (respectively, in Item 3) of Proposition 1.2, for all δ > (respectively, for all k), one can take the same A ∈ F (X) instead of all Aδ (respectively, Ak ) For any functional ϕ(x) on X, let ϕ(x) = min(ϕ(x), 1), x ∈ X Definition 1.3 For an almost exactly (α, β)-search functional ϕ : X → R+ , a number ε > 0, a convergent ∞ ε(α − β) , a sequence of points series Σ = σk with nonnegative terms and the sum σ > and γ = σ k=1 x0 = x, x1 , , xk , of X is called (ϕ, Σ, x, ε)-generated if (for ϕ0 = ϕ(x)) (ak ) ρ(xk−1 , xk ) ϕ(xk−1 ) + α α β α k−1 · γσk ϕ(xk−1 ), JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.4 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• (bk ) β ϕ(xk−1 ) + α ϕ(xk ) β α k · γσk ϕ(xk−1 ), k ∈ N Evidently, (a1 ) ρ(x0 , x1 ) (b1 ) ϕ(x1 ) 1 (ϕ0 + γσ1 ϕ(x0 )) (ϕ0 + γσ1 ), α α β β (ϕ0 + γσ1 ϕ(x0 )) (ϕ0 + γσ1 ) α α By induction, it is easy to show that (ak ) ρ(xk−1 , xk ) (bk ) ϕ(xk ) β α α β α k−1 α · (ϕ0 + γ(σ1 + + σk )) k β α · (ϕ0 + γ(σ1 + + σk )) β α k−1 · (ϕ0 + γσ), k · (ϕ0 + γσ), k ∈ N Remark 1.2 It follows from (ak ) and (bk ) that if ϕ(xk−1 ) = 0, then xl = xk−1 and ϕ(xl ) = for any l k In particular, it is true for k − = 0, i.e., if ϕ(x) = ϕ(x0 ) = 0, then xk = x and ϕ(xk ) = for all k ∈ N+ Theorem 1.1 If a functional ϕ : X → R+ is almost exactly (α, β)-search, then for every convergent series ∞ Σ= σk with nonnegative terms and the sum σ > 0, any ε > and any x ∈ X, k=1 (0) (1) (2) (3) there exists at least one (ϕ, Σ, x, ε)-generated sequence; any (ϕ, Σ, x, ε)-generated sequence x0 = x, x1 , , xk , is fundamental; for this sequence, ϕ(xk ) → as k → ∞; if there is the limit r(x) of this sequence, then ρ(xk , r(x)) k β α ϕ(x) + ε , k ∈ N+ , and ρ(x, r(x)) α−β ϕ(x) + ε; α−β (1.2) (4) if x ∈ ϕ−1 (0), then the limit r(x) from (3) exists and r(x) = x Proof (0) Points of the required sequence x0 = x, x1 , , xk , must be chosen in the following way: if ϕ(xk−1 ) = 0, then xk is taken so that (1.1δ ) is true for x = xk−1 , x = xk−1 and for δ that is equal to the minimum of the second summands in the right parts of (ak ) and (bk ); if ϕ(xk−1 ) = 0, then xk = xk−1 (1) Let p, q ∈ N+ , p < q Then (see (ak )) q−1 ρ(xp , xq ) ρ(xp , xp+1 ) + + ρ(xq−1 , xq ) i=p α β α ∞ i (ϕ0 + γσ) (ϕ0 + γσ) α i=p p β α = = (ϕ0 + γσ) β α 1− α β α p ϕ0 + γσ = α−β We have proved that our sequence is fundamental β α p ϕ0 +ε α−β p→∞ −−−−→ β α i JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.5 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• (2) follows from (bk ) (3) If there is a limit r(x) of the sequence, then ρ(xk , r(x)) (ϕ0 + γσ) α ∞ i β α i=k = k β α · ϕ0 + γσ = α−β β α k ϕ0 +ε α−β ϕ(x) +ε α−β In particular, ρ(x, r(x)) = ρ(x0 , r(x)) (4) follows from Remark 1.2 ✷ Definition 1.4 For an almost exactly (α, β)-search functional ϕ : X → R+ , a number ε > 0, a convergent ∞ ε(α − β) ), a sequence of series Σ = σk with nonnegative terms and with the sum σ > (and γ = σ k=1 A(k) ∈ F (X), k ∈ N+ , where A(0) = id X , will be called (ϕ, Σ, ε)-generated if for any x ∈ X, the sequence xk = A(k) x, k ∈ N+ , is (ϕ, Σ, x, ε)-generated Corollary 1.1 If ϕ is an almost exactly (α, β)-search functional on X, then ∞ (0 ) for any ε > and any convergent series Σ = σk with nonnegative terms and the sum σ > 0, k=1 there exists a (ϕ, Σ, ε)-generated sequence of A(k) ∈ F (X), k ∈ N+ ; (5) if a sequence s = {A(k) ∈ F (X) : k ∈ N+ } is (ϕ, Σ, ε)-generated and the functional ϕ is bounded, then all A(k) are contained in the metric part F (X, id X ) of F (X) and s is fundamental with respect to dF Suppose now that a (ϕ, Σ, ε)-generated sequence A(k) ∈ F (X), k ∈ N+ , pointwise converges to a mapping r ∈ F (X) Then (6) r(X) ⊃ ϕ−1 (0) and r|ϕ−1 (0) = id X |ϕ−1 (0) ; (7) if for O ⊂ X, the function ϕO = ϕ|O is bounded, then the sequence (A(k) )|O , k ∈ N+ , converges uniformly on O to r|O ; if, additionally, all A(k) are continuous, then the mapping r|O is continuous; (8) if ϕ is locally bounded (for example, continuous), and all A(k) are continuous, then r is continuous; if, additionally, r(X) = ϕ−1 (0) (for example, if the functional ϕ is continuous), then r is a retraction of X onto ϕ−1 (0); (9) if ϕ is bounded, then the sequence A(k) , k ∈ N+ , converges uniformly on X to r Proof (0 ) For each x ∈ X, take (see point (0) of Theorem 1.1) a (ϕ, Σ, x, ε)-generated sequence xk , k ∈ N+ , and let A(k) (x) = xk , x ∈ X, k ∈ N+ The sequence {A(k) : k ∈ N+ } is required (5) Suppose that ϕ(x) C > for any x ∈ X Then (as in the proof of point (1) of Theorem 1.1) for any p, q ∈ N+ , p < q, and any x ∈ X, ρ(A(p) (x), A(q) (x)) Hence (for p = 0) dF (id X , Aq ) p β α p ϕ(x) +ε α−β β α p C +ε α−β p→∞ −−−−→ C + ε and so Aq ∈ F (X, id X ), q ∈ N+ , and dF (A(p) , A(q) ) α−β C β p→∞ + ε −−−−→ Thus the sequence s is fundamental in F (X, id X ) Point (5) is proved α α−β Point (6) follows from point (4) of Theorem 1.1 (7) Suppose that ϕ(x) C > for every x ∈ O Then for all such points (see point (3) of Theorem 1.1), the inequalities ρ(A(p) (x), r(x)) β α p ϕ(x) +ε α−β β α p C +ε α−β JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.6 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• hold We have obtained the first assertion of point (7) The continuity of all A(k) and the uniform convergence of A(k) , k ∈ N, to r on O imply the continuity of r on O (8) It follows from point (7) that r is locally continuous and so r is continuous If ϕ is continuous, then (see point of Theorem 1.1) ϕ(r(x)) = lim ϕ(A(k) (x))) = for every x ∈ X Hence r(X) ⊂ ϕ−1 (0) It k→∞ follows from point (6) that r(X) = ϕ−1 (0), hence, r is a retraction of X onto ϕ−1 (0) Point (9) follows from point (7) (when O = X) ✷ Definition 1.5 An almost exactly (α, β)-search functional ϕ : X → R+ will be called: effective if for any x ∈ X, ε > and convergent series Σ with nonnegative terms and the positive sum, ( ) there exists a convergent (ϕ, Σ, x, ε)-generated sequence s such that for its limit r(s), ϕ(r(s))) = 0; completely effective if for any x ∈ X, ε > and convergent series Σ with nonnegative terms and the positive sum, ( ) any (ϕ, Σ, x, ε)-generated sequence s is convergent and for its limit r(s), ϕ(r(s)) = Remark 1.3 For an almost exactly (α, β)-search functional ϕ : X → R+ the following assertions are equivalent: ϕ is effective; for any ε > and any convergent series Σ with nonnegative terms and the positive sum, there is a (ϕ, Σ, ε)-generated sequence of mappings Ak ∈ F (X), k ∈ N+ , that pointwise converges to a mapping r ∈ F (X) such that ϕ(r(x)) = for every x ∈ X (i.e r(X) ⊂ ϕ−1 (0)) (Implication ⇒ is proved as point (0 ) of Corollary 1.1 Implication ⇒ is obvious.) The following definitions were given by T.N Fomenko [7] The graph G of a functional ϕ : X → R+ is called: (a) 0-closed if (ξ, 0) ∈ cl(G) implies (ξ, 0) ∈ G; (b) 0-complete if for any fundamental sequence of points xk ∈ X, k ∈ N+ , such that lim ϕ(xk ) = 0, the k→∞ sequence (xk , ϕ(xk )), k ∈ N, converges to a point (ξ, η) ∈ G in the product X × R (i.e., there is a limit ξ of the sequence xk in X, k ∈ N, and ϕ(ξ) = η = 0) Theorem 1.2 Let G be the graph of an almost exactly (α, β)-search functional ϕ : X → R If (a) G is 0-closed and the space X is complete, or (b) G is 0-complete, then ϕ is completely effective Proof Take a (ϕ, Σ, x, ε)-generated sequence xk ∈ X, k ∈ N+ In the case (a), by the completeness of X, the limit ξ ∈ X of this sequence exists Then (ξ, 0) is the limit of the sequence (xk , ϕ(xk )), k ∈ N+ , in X × R Hence, (ξ, 0) ∈ cl G So (ξ, 0) ∈ G, i.e., ϕ(ξ) = In the case (b) the assertion is evident ✷ Corollary 1.2 If the space X is complete and an almost exactly (α, β)-search functional ϕ : X → R is continuous, then ϕ is completely effective If X is a compactum, then for every x ∈ X there is a point ϕ(x) η = η(x) such that ϕ(η) = and ρ(x, η) α−β Proof The assertion of the first sentence of the corollary is evident JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.7 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• Let X be a compactum Take x ∈ X For each n ∈ N one can find a point ξn such that ϕ(ξn ) = and ϕ(x) + Select from the sequence ξn , n ∈ N, a convergent subsequence ξn(k) , k ∈ N If η is ρ(x, ξn ) α−β n the limit of the subsequence, then by the continuity of ϕ and ρ, ϕ(η) = and ρ(x, η) ϕ(x) α−β ✷ Corollary 1.3 If an almost exactly (α, β)-search functional ϕ : X → R+ is effective, then (see Remark 1.3) for any ε > and convergent series Σ with nonnegative terms and the positive sum, there is a (ϕ, Σ, ε)-generated sequence of mappings A(k) ∈ F (X), k ∈ N+ , that pointwise converges to a mapping r ∈ F (X), k ∈ N+ , ϕ(x) + ε, x ∈ X, and (see (6) from Corollary 1.1 and moreover, (see (3) from Theorem 1.1) ρ(x, r(x)) α−β Remark 1.3) r(X) = ϕ−1 (0), r|ϕ−1 (0) = id X |ϕ−1 (0) If all mappings A(k) are continuous and ϕ is locally bounded (for example, continuous), then (see (8) from Corollary 1.1) r (is continuous and) is a retraction of X onto ϕ−1 (0) Definition 1.6 For an (α, β)-search functional ϕ : X → R+ , a mapping A ∈ F (X) is called ϕ-generated if for every x ∈ X, ρ(x, A(x)) ϕ(x) and ϕ(A(x)) α β ϕ(x) α (1.3) Obviously, for any (α, β)-search functional ϕ, there always exists a ϕ-generated mapping Theorem 1.3 Let ϕ : X → R+ be an (α, β)-search functional Then for any ϕ-generated mapping A and x∈X the sequence Ak x, k ∈ N+ , (note that, A0 = id X ) is fundamental; ϕ(Ak x) → as k → ∞; if there exists the limit r(x) of the sequence Ak x, k ∈ N+ , then ρ(Ak (x), r(x)) β α k ϕ(x) , in particular, ρ(x, r(x)) α−β ϕ(x) ; α−β (1.4) if x ∈ ϕ−1 (0), then the limit r(x) (from point 3) exists and r(x) = x; if ϕ is bounded, then Ak ∈ F (X, id X ), k ∈ N+ , and the sequence Ak , k ∈ N+ , is fundamental in the metric space (F (X, id X ), dF ) Now, suppose that for a ϕ-generated mapping A, the sequence Ak , k ∈ N+ , pointwise converges to a mapping r ∈ F (X) Then: r(X) ⊃ ϕ−1 (0) and r|ϕ−1 (0) = id X |ϕ−1 (0) ; if for O ⊂ X, the function ϕO = ϕ|O is bounded, then the sequence AkO = Ak |O , k ∈ N+ , uniformly converges to r|O on O; if, in addition, A is continuous, then r|O is also continuous; if ϕ is locally bounded (for example, continuous), and A is continuous, then r is continuous and it is a retraction of X onto ϕ−1 (0) in the case of the continuity of ϕ; if ϕ is bounded, then the sequence Ak , k ∈ N+ , converges uniformly on X to r ϕ(Ak (x)) β and ϕ(Ak+1 (x)) ϕ(Ak (x)), k ∈ N+ Points 1–3 are α α proved as Theorem 1.3 from [7] or as Theorem 1.1 for γ = Point follows from the first inequality (1.3) Proof Obviously, ρ(Ak (x), Ak+1 (x)) JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.8 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• Prove point Suppose that ϕ(x) C > for every x ∈ X Then (as in Proof of Theorem 1.1 for γ = 0) one can show that for every p, q ∈ N+ , p < q, and every x ∈ X, the next relations are true ρ(Ap (x), Aq (x)) β α p · ϕ(x) α−β β α p · C α−β (1.5) p C β C p→∞ and dF (Ap , Aq ) · −−−−→ Point is proved α−β α α−β Point follows from point Prove point If ϕ(x) C > for all x ∈ O, then for all x ∈ O, the first inequality (1.4) is true, and so k β C , k ∈ N+ is true too It gives us the first assertion of point the inequality ρ(Ak (x), r(x)) · α α−β The continuity of A (and all Ak ) and the uniform convergence of Ak |O , k ∈ N, to r|O on O imply the continuity of r|O Point is proved Points and are proved as points (8) and (9) of Corollary 1.1 ✷ Hence, dF (id X , Aq ) Definition 1.7 An (α, β)-search functional ϕ : X → R+ will be called effective (respectively, completely effective) if there exists a ϕ-generated mapping A (respectively, if any ϕ-generated mapping A) has the following property for any x ∈ X, there exists the limit r(x) of the sequence Ak (x), k ∈ N+ , and ϕ(r(x)) = The next assertion was obtained (in other terms) by T.N Fomenko in [7] Theorem 1.2 If the graph G of an (α, β)-search functional ϕ : X → R+ is (a) 0-closed (for example, ϕ is continuous), and X is complete, or (b) 0-complete, then ϕ is completely effective Corollary 1.4 If a functional ϕ : X → R+ is (α, β)-search and effective (completely effective), then for a (for every) ϕ-generated mapping A, the sequence Ak , k ∈ N+ , pointwise converges to a mapping r ∈ F (X) and points 6.–9 of Theorem 1.3 are true Moreover, in point 8., the requirement of the continuity of ϕ can be removed without losing the property of r to be a retraction of X onto ϕ−1 (0) (since, by the effectiveness of ϕ, the equality ϕ(r(x)) = is not lost for every x ∈ X) Definition 1.8 A functional ϕ : X → R+ is called continuously (α, β)-search, < β < α, if there is a continuous mapping A : X → X such that ρ(x, Ax) ϕ(x) and ϕ(Ax) α β ϕ(x) α Remark 1.4 Similarly, one can define an almost exactly continuously (α, β)-search functional Corollary 1.5 If X is complete, and the functional ϕ : X → R+ is continuous and continuously (α, β)-search, then there is a continuous mapping A : X → X such that for every x ∈ X the sequence Ak x, k ∈ N+ , has ϕ(x) and r is a retraction of X onto ϕ−1 (0) the limit r(x), ρ(x, r(x)) α−β Remark 1.5 Obviously, Banach’s fixed point theorem follows from Corollary 1.5 (if α = and ϕ(x) = ρ(x, Ax)) Some applications of search functionals in the case of metric spaces Let fi be mappings of a space X to a space Y , i = 1, , n, n ∈ N, and H = ∅ be a closed subset of Y JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.9 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• Theorem 2.1 (Theorem on the common preimage of a closed set for a finite system of mappings.) If a functional ϕ(x) = max{ρ(f1 x, f2 x), , ρ(f1 x, fn x), ρ(f1 x, H)}, x ∈ X, is almost exactly (α, β)-search and effective (for example, if (a) the graph of ϕ is 0-closed and X is complete or (b) the graph of ϕ is 0-complete), then for every ε > and every convergent series terms and the sum σ > 0, there is a mapping r ∈ F (X) such that (ε) (r) ∞ σk with nonnegative k=1 ϕ(x) + ε, x ∈ X; α−β r(X) = ϕ−1 (0) = {x ∈ X : f1 (x) = = fn (x) ∈ H} and r|ϕ−1 (0) = id X |ϕ−1 (0) ρ(x, r(x)) Moreover, r(x) is the limit of a (ϕ, Σ, x, ε)-generated sequence, x ∈ X If the functional ϕ is (α, β)-search and effective, then there is a ϕ-generated mapping A such that the sequence Ak , k ∈ N+ , pointwise converges to a mapping r ∈ F (X), the equalities (r) are true and instead of ϕ(x) , x ∈ X, hold If A is continuous the inequalities (ε), the following more exact inequalities ρ(x, r(x)) α−β and ϕ is locally bounded, then r is a retraction of X onto ϕ−1 (0) Proof The first part of the theorem follows from Corollary 1.3 and Definition 1.4 The second part follows from Theorem 1.3 ✷ Theorem 2.1 must be compared with Theorem 1.20 from [7] Consider partial cases of Theorem 2.1 If in Theorem 2.1: the set H consists of one point c, then we obtain a sufficient condition for the existence of roots of the system of equations fi (x) = c, x ∈ X, i = 1, , n (This result must be compared with Corollary 1.21 in [7]) n = 1, then we obtain a sufficient condition for searching the preimage of a closed subset H of the space Y under the mapping f1 (This result must be compared with Theorem 1.4 from [7].) H = Y , then ϕ(x) = max{ρ(f1 x, f2 x), , ρ(f1 x, fn x)}, x ∈ X) we obtain a sufficient conditions for searching coincidence points of mappings f1 , , fn (This result must be compared with Theorems 1.7, 1.8, 1.11 from [7].) Y = X, n > 1, f1 = id X and α = 1, then ϕ(x) = max{ρ(x, f2 x), , ρ(x, fn x)}) and we obtain a sufficient condition for searching common fixed points of mappings f2 , , fn (This result must be compared with Theorems 1.12–1.14 from [7].) Add the following assertion to Theorem from [6] Proposition 2.1 Let A be a continuous mapping of a complete space X to itself Suppose that there is a number β, < β < 1, such that ρ(Ax, A2 x) βρ(x, Ax) for any x ∈ X Then the functional ϕ(x) = ρ(x, Ax) on X is continuously (α, β)-search for α = It is completely effective and the sequence Ak , k ∈ N+ , pointwise converges to a retraction r : X → ϕ−1 (0) = Fix(A) on X (where Fix(A) is the set of all fixed points of A) JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.10 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• 10 ϕ(x) β and ϕ(Ax) = ρ(Ax, A2 x) βρ(x, Ax) = ϕ(x), ϕ is (1, β)-search, Proof Since ρ(x, Ax) = ϕ(x) α α and A is a ϕ-generated mapping Since A is continuous, ϕ is continuous too and its graph is closed Since X is complete, ϕ is completely effective The rest follows from point of Theorem 1.3 ✷ Pass to coincidence points Definition 2.1 A mapping Ψ of a space X to a space Y is called α-covering (respectively, open-α-covering) for a positive number α if for any x ∈ X and r > 0, B(Ψx, αr) ⊂ Ψ(B(x, r)) (respectively, O(Ψx, αr) ⊂ Ψ(O(x, r)) The notion of α-covering mapping is well-known, the notion of open-α-covering mapping was introduced by A.V Arutyunov in [4], where he noted that every α-covering mapping is open-α-covering but there exist open-α-covering mappings that are not α-covering Recall that a mapping Φ of a space X to a space Y is called β-Lipschitz, β > 0, if ρ(Φx1 , Φx2 ) βρ(x1 , x2 ) for any x1 , x2 ∈ X In [2], Theorem (see also [3]), A.V Arutyunov obtained the following assertion Arutyunov theorem Let a map Ψ : X → Y be α-covering, a mapping Φ : X → Y β-Lipschitz, X complete and < β < α Then for any x ∈ X and ε > 0, there exists ξ = ξ(x, ε) ∈ X such that Ψ(ξ) = Φ(ξ) and ρ(x, ξ) ϕ(x) α−β In [7], T.N Fomenko proved that if a mapping Ψ : X → Y is α-covering, a mapping Φ : X → Y is β-Lipschitz and < β < α, then the functional ϕ(x) = ρ(Ψ(x), Φ(x)), x ∈ X, is (α, β)-search Lemma 2.1 Let a mapping Ψ : X → Y be open-α-covering, a mapping Φ : X → Y be β-Lipschitz and < β < α Then the functional ϕ(x) = ρ(Ψ(x), Φ(x)) on X is almost exactly (α, β)-search δ ) Consider the case ϕ(x) > only Since Ψ is open-α-covering, β ϕ(x) ϕ(x) + δ )) Hence there exists x ∈ O(x, + δ ) such B(Ψ(x), ϕ(x)) ⊂ O(Ψ(x), ϕ(x) + αδ ) ⊂ Ψ(O(x, α α that Ψ(x ) = Φ(x) It follows from this that Proof Fix x ∈ X and δ > Let δ = min(δ, ρ(x, x ) < ϕ(x) ϕ(x) +δ ≤ +δ α α and (since Φ is β-Lipschitz) ϕ(x ) = ρ(Ψ(x ), Φ(x )) = ρ(Φ(x), Φ(x )) βρ(x, x ) β ϕ(x) + βδ α δ β ϕ(x) + β α β β ϕ(x) + δ α ✷ JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.11 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• 11 Definition 2.2 A mapping Ψ of a space X to a space Y is called α-filling for a positive number α if for any x ∈ X and r > 0, B(Ψx, αr) ⊂ cl Ψ(B(x, r)) Evidently, for any α-filling mapping Ψ : X → Y , cl ΨX = Y Note that the identical embedding of Q in R is a 1-filling but this mapping is not α-covering for any α Lemma 2.2 Let a mapping Ψ : X → Y be α-filling, a mapping Φ : X → Y be β-Lipschitz and < β < α Then the functional ϕ(x) = ρ(Ψ(x), Φ(x)), x ∈ X, is almost exactly (α, β)-search Proof Fix x ∈ X and δ > Consider the case ϕ(x) > only Since Ψ is α-filling, B(Ψ(x), ϕ(x)) ⊂ ϕ(x) ϕ(x) Hence there exists x ∈ B x, such that ρ(Φ(x), Ψ(x )) < δ Then cl Ψ B x, α α ρ(x, x ) Since Φ is β-Lipschitz, ρ(Φ(x), Φ(x )) ϕ(x ) = ρ(Ψ(x ), Φ(x )) ϕ(x) and so ρ(x, x ) α βρ(x, x ) ϕ(x) α ϕ(x) + δ α β ϕ(x) Hence, α ρ(Ψ(x ), Φ(x)) + ρ(Φ(x), Φ(x )) δ+ β ϕ(x) α ✷ Corollary 2.1 Let a mapping Ψ : X → Y be open-α-covering or α-filling, a mapping Φ : X → Y be β-Lipschitz and < β < α Then the functional ϕ(x) = ρ(Ψ(x), Φ(x)), x ∈ X, on X is almost exactly (α, β)-search and if ϕ is effective (for example, if (a) the graph G of ϕ is 0-closed and X is complete or (b) the graph G is 0-complete (see Theorem 1.2)), then for any x ∈ X and ε > 0, there exists ξ = ξ(x, ε) ∈ X such that Ψ(ξ) = Φ(ξ) and ρ(x, ξ) ϕ(x) + ε α−β Corollary 2.2 If a space X is complete, a mapping Ψ of X to a space Y is continuous and either open-α-covering or α-filling and a mapping Φ : X → Y is β-Lipschitz, < β < α, then the functional ϕ(x) = ρ(Ψx, Φx) on X is almost exactly (α, β)-search and effective and for every x ∈ X and ε > 0, there exists ξ = ξ(x, ε) ∈ X such that Ψ(ξ) = Φ(ξ) and ρ(x, ξ) ϕ(x) + ε α−β Indeed, since every β-Lipschitz mapping is continuous, ϕ is also continuous and so effective (by the completeness of X) Evidently, Corollaries 2.1 and 2.2 are analogs of Arutyunov theorem T.N Fomenko noted that a functional ϕ : X → R+ on a space X is almost exactly (α, β)-search if and only if it is (α , β)-search for all α ∈ (β, α) Some applications of search functionals in the case of metric mappings As it was noted in the beginning of the paper, “space” means “topological space” in this section Recall (see [5]) that a metric on a mapping f of a set X to a space (Z, θ) is a pseudometric ρ on X such that it is a metric on every fiber f −1 z of f , z ∈ Z The topology τ (f, ρ) on f generated by the metric ρ on JID:TOPOL 12 AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.12 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• f is the topology on X with the base τρ ∧ f −1 θ = {U ∩ f −1 O : U ∈ τρ , O ∈ θ}, where τρ is the topology on X generated by the pseudometric ρ A pair (f, ρ) consisting of a mapping f of a set to a space and of a metric ρ on f is called a metric mapping Evidently, for every metric mapping (f, ρ), the mapping f : (X, τ (f, ρ)) → Z is continuous (For any metric mapping (f, ρ) : X → Z, we will consider X with the topology τ (f, ρ) and so all metric mappings are continuous.) A metric mapping (f, ρ) is called fiberwise complete if ρ is a complete metric on every fiber of f A map between spaces f : (X, τ ) → Z is called metrizable if there exists a metric ρ on f such that τ (f, ρ) = τ (we say in this situation that ρ metrizes f ) It is not difficult to prove that for any metric mapping (f, ρ) : X → Z and Y ⊂ X, (τ (f, ρ))|Y = τ (f |Y , ρ|Y ) (i.e the metric ρ|Y on f |Y generates the topology (τ (f, ρ))|Y on f |Y ) From this moment, let (f, ρ) : X → Z be an onto metric mapping Recall that a map s : Z → X is called a continuous section (section, for short) of f if f ◦ s = id Z Note that, firstly, for every section s : Z → X of f , the restriction f |sZ is a homeomorphism onto Z, and, secondly, if for a subset S of X, the restriction f |S is a homeomorphism onto Z, then the mapping s = (f |S )−1 is a section of f such that sZ = S So, further, we consider every section of f as a subset s of X such that f |s is a homeomorphism onto Z (but for z ∈ Z, the unique point of the set s ∩ f −1 z will be denoted by the symbol s(z)) Let S(f ) be the set of all sections of f Evidently, for any s ∈ S(f ), the set S(f, s) of all s ∈ S(f ) such that d(s, s ) = sup{ρ(x, x ) : x ∈ s, x ∈ s , f x = f x } < +∞, (3.1) is a metric space with the metric df (d, for short) such that d(s , s ) = sup{ρ(x , x ) : x ∈ s , x ∈ s , f x = f x } Note that for any s, s ∈ S(f ), the sets S(f, s) and S(f, s ) either coincide or not intersect Further, we will identify S(f, s) and S(f, s ) if d(s, s ) < +∞ Any set S(f, s) with the metric d on it will be called a metric parts of S(f ) Evidently, if f is fiberwise complete, then any metric space S(f, s) is complete Definition 3.1 For x ∈ X, z = f x and ε > 0, the set Oεf x ≡ Of (x, ε) = {x ∈ f −1 z : ρ(x, x ) < ε} and Bεf x ≡ B f (x, ε) = {x ∈ f −1 z : ρ(x, x ) ε} are called f -fiberwise ε-neighborhood of x and f -fiberwise ε-ball with the center x (f -fiberwise (ε, x)-ball, for short) respectively For a section s ∈ S(f ) the sets (Oε s ≡ OεX s ≡ O(s, ε) ≡ OX (s, ε)) = {Of (s(z)) : z ∈ Z} and (Bε s ≡ BεX s ≡ B(s, ε) ≡ B X (s, ε)) = {B f (s(z), ε) : z ∈ Z} will be called a f -fiberwise ε-neighborhood of s and f -fiberwise ε-ball around s respectively Proposition 3.1 For any section s ∈ S(f ) and ε > 0, the set B f (s, ε) is closed, and the set Of (s, ε) is open in X Moreover, for every δ ∈ (0, ε) and z ∈ Z there exists a neighborhood Oδ z of z such that Uzδ = f −1 Oδ z ∩ O(s(z), ε − δ) ⊂ Of (s, ε) (3.2) Proof Take z ∈ Z and δ ∈ (0, ε) Since s is continuous, there exists a neighborhood Oδ z of z such that s(z ) ∈ O(s(z), δ) ∩ f −1 Oδ z for every z ∈ Oδ z Suppose that x ∈ Uzδ Then z = f x ∈ Oδ z, x ∈ O(s(z), ε − δ) and ρ(s(z ), x ) ρ(s(z ), s(z)) + f f ρ(s(z), x ) < δ + ε − δ = ε Hence, x ∈ O (s(z ), ε) ⊂ O (s, ε) We have proved (3.2) Hence Of (s(z), ε) ⊂ {Uzδ : δ ∈ (0, ε)} ⊂ Of (s, ε) and O(s, ε) = {Uzδ : δ ∈ (0, ε), z ∈ Z} Thus Of (s, ε) is open To prove that B f (s, ε) is closed it is sufficient to prove that C = X\B f (s, ε) is open JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.13 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• 13 (ρ(x, s(z)) − ε) Take a neighborhood V δ z of z such that s(z ) ∈ f −1 V δ z ∩ O(s(z), δ) for every z ∈ V δ z If x ∈ Ox = f −1 V δ z ∩ O(x, δ), then z = f x ∈ V δ z, x ∈ O(x, δ) and ρ(x , s(z )) ρ(x, s(z)) − ρ(x, x ) − ρ(s(z ), s(z)) > 2δ + ε − δ − δ = ε Hence, x ∈ C and Ox ⊂ C ✷ Let x ∈ C, z = f x and δ = Definition 3.2 A set K ⊂ X is called f -fiberwise closed if for every z ∈ Z the intersection K ∩ f −1 z is closed in f −1 z For K ⊂ X, the union of all closures of K ∩ f −1 z in f −1 z, z ∈ Z, is called the f -fiberwise closure of K and is denoted by cl f K Note that a space P with dim P will be called 0-dimensional (≡ zero-dimensional) Lemma 3.1 Let Z be a 0-dimensional paracompactum, O an open set in X, K ⊂ O, f K = Z and the mapping f |K open Then for any ε > 0, there exist an open set K in K and an open disjoint cover λ of Z such that f K = Z, diam(cl f (K ∩ f −1 V )) < ε for every V ∈ λ and cl f K ⊂ O ε and neighborhoods Oz K X −1 of z in Z and O (x(z), 3δ) (in K) such that O (x(z), 3δ) ∩ f Oz ⊂ O Take Ox(z) = OK (x(z), δ) Then diam(Ox(z)) ≤ 2δ < ε, diam(cl f (Ox(z) ∩ f −1 Oz)) ≤ 2δ < ε and cl f (Ox(z) ∩ f −1 Oz) ⊂ OX (x(z), 3δ) ∩ f −1 Oz) ⊂ O Let U x(z) = Ox(z) ∩ f −1 Oz and U z = f (U x(z)) ⊂ Oz The open cover {U z : z ∈ Z} of Z has an open disjoint refinement λ For each V ∈ λ, fix z(V ) ∈ Z so that V ⊂ U z(V ) Let UV = U x(z(V )) ∩ f −1 V and K = {UV : V ∈ λ} Since f is surjective, f (UV ) = f (U x(z(V )) ∩ f −1 V ) = f (U x(z(V ))) ∩ V = U z(V ) ∩ V = V Since the cover λ is disjoint, K ∩ f −1 V = UV It follows from this that f K = ∪λ = X Besides, cl f (K ∩ f −1 V ) = cl f (UV ) = cl f (U x(z(V )) ∩f −1 V ⊂ cl f (Ox(z(V )) ∩f −1 Oz(V ) ∩f −1 U z(V ) ⊂ cl f (Ox(z(V )) ∩f −1 Oz(V ) ∩f −1 Oz(V )) = cl f (Ox(z(V )) ∩ f −1 Oz(V ) Hence Proof Take ε For every z ∈ Z, take x(z) ∈ f −1 z ∩ K, a positive number δ < diam cl f (K ∩ f −1 V ) ≤ diam cl f (Ox(z(V )) ∩ f −1 Oz(V ) < ε, V ∈ λ, and cl f K = cl f = {UV : V ∈ λ} = cl f {K ∩ f −1 V ) : V ∈ λ} {cl f (K ∩ f −1 V ) : V ∈ λ} ⊂ {cl f (Ox(z(V )) ∩ f −1 Oz(V )) : V ∈ λ} ⊂ O ✷ Proposition 3.2 Let f be fiberwise complete, Z a 0-dimensional paracompactum, G be a Gδ -set in X, K ⊂ G, f K = Z and the mapping f |K open Then f |G has a section If, in addition, K is fiberwise closed in X, then f |K also has a section Proof Let G be the intersection of open sets On in X, n ∈ N By Lemma 3.1, there exists an open set K1 in K and an open disjoint cover λ1 of Z such that f K1 = Z, diam(cl f (K1 ∩ f −1 V )) < 2−1 for every V ∈ λ1 and cl f K1 ⊂ O1 Using Lemma 3.1, by induction, one can find open sets Kn in Kn−1 and open disjoint covers λn of Z such that f Kn = Z, diam(cl f (Kn ∩ f −1 V )) < 2−n for every V ∈ λn and cl f Kn ⊂ On , n = 2, 3, Hence for every z ∈ Z and every n ∈ N, (a) diam cl f (Kn ∩ f −1 z) < 2−n , (b) cl f (Kn ∩ f −1 z) ⊃ cl f (Kn+1 ∩ f −1 z) and (c) cl f (Kn ∩ f −1 z) ⊂ On JID:TOPOL 14 AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.14 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• By the fiberwise completeness of f , the intersection {cl f (Kn ∩ f −1 z) : n ∈ N} consists of a unique point s(z) ∈ f −1 z From (c) it follows that s(z) ∈ G Hence we have an injective mapping s : Z → G Besides, f ◦ s = id Z Prove the continuity of s Let U z be a neighborhood of z in Z, ε > and Os(z) = O(s(z), ε) ∩ f −1 U z Suppose that 2−n < ε and V ∈ λn contains z Then s(U z ∩ V ) ⊂ cl f (Kn ∩ f −1 V ) and diam s(U z ∩ V ) < diam(cl f (Kn ∩ f −1 V )) < 2−n < ε Hence s(U z ∩ V ) ⊂ Os(z) and s is continuous We have proved that sZ ⊂ G is a section of the map f |G If K is fiberwise closed in X, then all sets cl f Kn are contained in K and so sZ ⊂ K ✷ Corollary 3.1 Let f be fiberwise complete, Z a 0-dimensional paracompactum, G ⊂ X be a Gδ -set in X, f G = Z and f |G open Then f |G has a section Remark Proposition 3.2 and Corollary 3.1 may be compared with Corollary 1.3 in [8] ( ) Additionally, we will consider an onto metric mapping (g, σ) : Y → Z For maps f and g, a map Ψ : X → Y will be called a map-morphism Ψ : f → g (a map-morphism of f to g) if f = g ◦ Ψ Evidently, if Ψ : f → g is a map-morphism then Ψ(f −1 z) ⊂ g −1 z for any z ∈ Z Hence for any z ∈ Z, the equalities Ψz (x) = Ψ(x), x ∈ f −1 z, define the map Ψz : f −1 z → g −1 z Evidently, for any map-morphism Ψ : f → g and any section s of f , the image Ψ(s) (for the sake of convenience, it will be denoted by sΨ ) is a section of g Hence, we have defined the mapping Ψ∗ : S(f ) → S(g), such that Ψ∗ (s) = sΨ ( ) Fix a map-morphism Ψ : f → g For a metric part S of S(f ) and a metric part T of S(g), let SΨT = S ∩ (Ψ∗ )−1 T and Ψ∗ST be the corestriction to T of the restriction of Ψ to SΨT Definition 3.3 For α > 0, the map-morphism Ψ is called: fiberwise open-α-covering (respectively, fiberwise α-covering) if for any z ∈ Z, the map Ψz is open-α-covering (respectively, α-covering); open-α-covering if for any x ∈ X, any ε > and any neighborhood U z of z = f x, there exists a neighborhood V z ⊂ U z such that Ψ(O(x, ε) ∩ f −1 U z) ⊃ O(Ψx, αε) ∩ g −1 V z Note that (as in the case of mappings between metric spaces) every fiberwise α-covering map-morphism Ψ is a fiberwise open-α-covering map-morphism Lemma 3.2 Any open-α-covering map-morphism Ψ : f → g is an open mapping of the space X to the space Y Proof Let O be open in X and x ∈ O There are ε > and a neighborhood U of z = f x such that Ox = O(x, ε) ∩ f −1 U ⊂ O Since Ψ is an open-α-covering map-morphism, there is a neighborhood V ⊂ U of z such that O(Ψx, αε) ∩ g −1 V ⊂ Ψ(Ox) ⊂ Ψ(O) Hence, Ψ(O) is open in Y ✷ Lemma 3.3 If the mapping f is open and the map-morphism Ψ is fiberwise open-α-covering, then Ψ is α open- -covering Proof Take a point x ∈ X, a neighborhood U z of z = f x and ε > Let y = Ψx By the continuity of Ψ, ε there exist δ > and a neighborhood W z ⊂ U z of z such that δ < and Ψ(O(x, δ) ∩ f −1 W z) ⊂ O(y, α ε) ∩ g −1 U z JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.15 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• 15 If V z = f (O(x, δ)) ∩ W z then V z = f (O(x, δ) ∩ f −1 V z) = g(Ψ(O(x, δ) ∩ f −1 V z)) ⊂ g(O(y, α ε)) Since f is open, V z is open too α Take y ∈ O(y, ε) ∩ g −1 V z and z = g(y ) ∈ V z Then there exists x ∈ O(x, δ) ∩ f −1 W z ⊂ O(x, δ) α α such that f x = z and y = Ψx ∈ O(y, ε) ∩ g −1 U z ⊂ O(y, ε) 4 Since ε σ(y , y) + σ(y, y ) < α , 2) Ψ is fiberwise open-α-covering, ε 3) δ + < ε, 1) σ(y , y ) ε ε O(x , ) ⊂ O(x, ε) and there exists x ∈ O(x , ) ∩ f −1 z (⊂ O(x, ε) ∩ f −1 z ), such that Ψx = y Hence, 2 α Ψ(O(x, ε) ∩ f −1 U z) ⊃ Ψ(O(x, ε) ∩ f −1 V z) ⊃ O(y, ε) ∩ g −1 V z α Thus, Ψ is open- -covering ✷ Lemmas 3.2 and 3.3 imply the following assertion Corollary 3.2 If the mapping f is open and the map-morphism Ψ is fiberwise open-α-covering, then Ψ is an open mapping of X to Y Lemma 3.4 Let the mapping f be open and fiberwise complete, the map-morphism Ψ be fiberwise open-α-covering, Z be a 0-dimensional paracompactum, s ∈ S(f ), t ∈ S(g) and d = d(sΨ , t) < +∞ Then for every δ > 0, there exists a section s ∈ S(f, s) such that Ψ(s ) = t and d(s , s) < d+δ α Proof Fix δ > d+δ It follows from Proposition 3.1 that O is open in X α Since the map-morphism Ψ is fiberwise open-α-covering, Let O = Of s, Ψ(O) ⊃ Og (sΨ , d + δ) ⊃ t It follows from Corollary 3.2 that the mapping Ψ is open If X = Ψ−1 t, then the set O = X ∩ O is open in X , Ψ(O ) = t and the mappings Ψ = cor(Ψ|X ) and ΨO = Ψ |O are also open Since Ψ is a map-morphism, any fiber of Ψ is contained in one of fibers of f and closed in it The restriction ρ = ρ|X of the pseudometric ρ to X is a metric on Ψ that metrizes the map Ψ (Indeed, if θ is the topology on Z, θ = (g|t )−1 θ and f is the restriction of f to X , then (f )−1 θ = (Ψ )−1 θ ) From the fiberwise completeness of f it follows that the metric mapping (Ψ , ρ ) is fiberwise complete Since the section t is homeomorphic to Z, it is a 0-dimensional paracompactum Since O is open in X , it follows from Corollary 3.1 that there exists a section s ⊂ O ⊂ O of Ψ Since cor(Ψ|s ) is a homeomorphism onto t and g|t is a homeomorphism onto Z, f |s = g|t ◦ Ψ|s also is a homeomorphism d+δ and Ψ(s ) = t ✷ onto Z So s ⊂ O ⊂ O is a section of f Hence d(s, s ) < α ( ) Further, we will use one more map-morphism Φ : f → g JID:TOPOL 16 AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.16 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• Definition 3.4 A map-morphism Φ : f → g is called β-Lipschitz for a number β > if for every z ∈ Z and x, x ∈ f −1 z, ρ(Φx, Φx ) βρ(x, x ) Remark 3.1 Evidently, if the map-morphism Φ : f → g is β-Lipschitz, then for any metric part S of S(f ) there is a metric part T of S(g) such that Φ∗ (S) ⊂ T Besides, if s, s ∈ S and d(s, s ) < ∞, then d(sΦ , sΦ ) = sup{σ(Φx, Φx ) : x, x ∈ f −1 z, z ∈ Z} ≤ sup{{βρ(x, x ) : x, x ∈ f −1 z, z ∈ Z} = βd(s, s ) Lemma 3.5 Let f be open and fiberwise complete, Z a 0-dimensional paracompactum, the map-morphism Ψ : f → g be open-α-covering, the map-morphism Φ : f → g be β-Lipschitz, < β < α, s ∈ S(f ) and ϕ(s) = d(sΨ , sΦ ) < +∞ Then for any δ > 0, there is a section s ∈ S(f ) such that Ψ(s ) = Φ(s), ϕ(s) β + δ and ϕ(s ) = d(Ψ(s ), Φ(s )) < ϕ(s) + δ d(s, s ) < α α δ } < δ It follows from Lemma 3.4 that there is a section α ϕ(s) ϕ(s) + δ < + δ Then ϕ(s ) = d(Ψ(s ), Φ(s )) = s ∈ S such that Ψ(s ) = Φ(s) and d(s, s ) < α α β β d(Φ(s), Φ(s )) β · d(s, s ) < ϕ(s) + βδ < ϕ(s) + δ ✷ α α Proof Take δ > and δ > so that max{βδ , The following theorem is a fiberwise analog of Arutyunov theorem and a partial fiberwise generalization of Corollary 2.2 (see the case |Z| = 1) Theorem 3.1 Let f : X → Z and g : Y → Z be metric mappings onto a 0-dimensional paracompactum Z and f be open and fiberwise complete If a map-morphism Ψ : f → g is fiberwise open-α-covering, a map-morphism Φ : f → g is β-Lipschitz, < β < α, and the metric part S of S(f ) and the metric part T of S(g) are such that Φ∗ (S) ⊂ T (see Remark 3.1) and SΨT = ∅, then the functional ϕ(s) = d(Ψ(s), Φ(s)) on SΨT is almost exactly (α, β)-search, completely effective and for any ε > and s ∈ S, there is a section ξ = ξ(s, ε) ∈ S such that Ψ|ξ = Φ|ξ and d(s, ξ) ϕ(s) +ε α−β (3.3) (thus the section ξ consists of coincidence points of the maps Ψ and Φ) Proof Let s ∈ SΨT It follows from Lemma 3.5 that, for any δ > and δ = min(δ, δ ), there is s ∈ S(f ) β ϕ(s) ϕ(s) +δ + δ Since the map-morphism Φ is β-Lipschitz, α α β β ϕ(s) + βδ + δ Hence, the functional ϕ is ϕ(s ) = d(Ψ(s ), Φ(s )) = d(Φ(s), Φ(s )) βd(s, s ) α α almost exactly (α, β)-search such that Ψ(s ) = Φ(s) and d(s, s ) Prove that ϕ is completely effective Take arbitrarily ε > 0, s ∈ SΨT , a convergent series Σ = ∞ σk k=1 with nonnegative terms and the sum σ > and a (ϕ, Σ, s, ε)-generated sequence s0 = s, s1 , , sk , (see Theorem 1.1, (0)) Then (see Theorem 1.1, (1), (2)) the sequence sk , k ∈ N+ , is fundamental and the sequence ϕ(sk ), k ∈ N+ , converges to It follows from the fiberwise completeness of f that the metric part S is complete Hence there exists the limit ξ ∈ S of the sequence sk , k ∈ N+ Then ξΦ ∈ T Fix z ∈ Z It follows from the continuity of Φ that Φ(ξ(z)) = Φ( lim sk (z)) = lim Φ(sk (z)) = lim (sk )Φ (z) Since the k→∞ k→∞ k→∞ sequence d((sk )Ψ , (sk )Φ ) = ϕ(sk ), k ∈ N+ , converges to and Ψ is continuous, Φ(ξ(z)) = lim ((sk )Φ (z)) = k→∞ JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.17 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• 17 lim ((sk )Ψ (z)) = lim (Ψ(sk (z))) = Ψ( lim sk (z)) = Ψ(ξ(z)) Hence Ψ(ξ) = Φ(ξ) = ξΦ ∈ T and ξ ∈ SΨT , k→∞ k→∞ k→∞ ϕ(ξ) = The inequality (3.3) follows from Theorem 1.1, (3) ✷ Corollary 3.3 Let f : X → Z and g : Y → Z be metric mappings onto a 0-dimensional paracompactum Z and f open and fiberwise complete If a map-morphism Ψ : f → g is fiberwise open-α-covering, a map-morphism Φ : f → g is β-Lipschitz, < β < α, and d(Ψ(s), Φ(s)) < +∞ for s ∈ S(f ), then for any ε > 0, there exists ξ = ξ(s, ε) ∈ S(f ) such that Ψ|ξ = Φ|ξ and d(s, ξ) d(Ψ(s), Φ(s)) + ε α−β Corollary 3.4 Suppose that in Theorem 3.1, the condition (a) the map-morphism Ψ is fiberwise open-α-covering and the map-morphism Φ is β-Lipschitz, < α < β, is changed in the following way: (b) for every z ∈ Z, there is a neighborhood Oz of z and numbers α(z), β(z), < β(z) < α(z), such that map-morphisms (ΨOz = cor(Ψ|f −1 Oz )) : f −1 Oz → g −1 Oz and (ΦOz = cor(Φ|f −1 Oz )) : f −1 Oz → g −1 Oz) of fz = cor(f |f −1 Oz ) to gz = cor(g|g−1 Oz ) are fiberwise open-α(z)-covering and β(z)-Lipschitz respectively, then for any section s ∈ S(f ) and ε > 0, there exists a section ξ = ξ(s) ∈ S(f ) such that Ψ|ξ = Φ|ξ and there exists an open disjoint cover λ = {Oj : j ∈ J} of Z and for every j ∈ J, there exists a point z(j) ∈ Z such that Oj ⊂ Oz(j) and for sj = s ∩ f −1 (Oj ) and ξj = ξ ∩ f −1 (Oj ), d(sj , ξj ) d(Ψ(sj ), Φ(sj )) + ε α(zj ) − β(zj ) Proof Since dim Z and the mappings Ψ and Φ are continuous, the cover {Oz : z ∈ Z} has an open disjoint refinement λ = {Oj : j ∈ J} such that the function ρ(sΨ (z), sΦ (z)) is bounded on every Oj For any j take z(j) ∈ Z so that Oj ⊂ Oz(j) The existence of required sections ξj follows from the previous corollary (note that every Oj is a paracompactum) Finally, take ξ = {ξj : j ∈ J} ✷ Definition 3.5 For the mappings f , g and for positive functions α(z) and β(z), z ∈ Z, a map-morphism Ψ : f → g is called functionally open-α-covering, if for any point z ∈ Z, the corestriction to g −1 z of the restriction of Ψ to f −1 z is open-α(z)-covering; a map-morphism Φ : f → g is called functionally β-Lipschitz, if for any z ∈ Z and x, x ∈ f −1 z, ρ(Φx, Φx ) β(z)ρ(x, x ) Note that for a continuous function γ : Z → R, α(z) < γ(z) < β(z), z ∈ Z, and for any z ∈ Z, there exists a neighborhood Oz of z such that < β(z )) < γ(z) < α(z ) for all z ∈ Oz This remark allows to obtain the next assertion (see Corollary 3.4) Corollary 3.5 Let f : X → Z and g : Y → Z be metric mappings onto a 0-dimensional paracompactum Z and f open and fiberwise complete If a map-morphism Ψ : f → g is functionally open-α-covering for a continuous function α on Z, a map-morphism Φ : f → g is functionally β-Lipschitz for a continuous function β on Z, < β(z) < α(z) for all z ∈ Z, then for any section s ∈ S(f ) and ε > 0, there exists a section ξ = ξ(s) ∈ S(f ) such that Ψ|ξ = Φ|ξ JID:TOPOL 18 AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.18 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• and there exists an open disjoint cover λ = {Oj : j ∈ J} of Z and for every j ∈ J, there exists a point z(j) ∈ Z such that Oj ⊂ Oz(j) and for sj = s ∩ f −1 (Oj ), ξj = ξ ∩ f −1 (Oj ), d(sj , ξj ) d(Ψ(sj ), Φ(sj )) + ε α(zj ) − β(zj ) Theorem 3.1 allows to obtain the second fiberwise analog of Arutyunov theorem (see Theorem 3.2) Recall (see [1], ch 1, §2) that for maps fi : Xi → X0 , i = 1, 2, the fan product of spaces X1 and X2 with respect to f1 and f2 is a subset X = {(x1 , x2 ) ∈ X1 × X2 : f1 x1 = f2 x2 } of the topological product Π = X1 × X2 The restriction pi = pri |X of the projection pri of Π onto the factor Xi is called the short projection of the fan product X to the factor Xi , i = 1, Evidently, f1 ◦ p1 = f2 ◦ p2 (3.4) The mapping p = fi ◦ pi , i = 1, 2, is called the long projection of the fan product X In the first and the second lemmas ”on parallel straight lines” ([1]), it is proved that: if x1 ∈ X1 and x0 = f1 x1 , then the corestriction to the fiber f2−1 x0 of the restriction of the map p2 to −1 the fiber p−1 x1 = {x1 } × f2 x0 is a homeomorphism; if the mapping f2 is open (perfect), then the projection p1 is open (perfect) Lemma 3.6 Let f : X → Z and h0 : Z0 → Z be continuous mappings onto Z and let X0 be the fan product of Z0 and X with respect to h0 and f with short projections f0 : X0 → Z0 and hX : X0 → X If there is a map h00 : Z0 → X, such that h0 = f ◦ h00 , then there is a section s0 of f0 such that h00 ◦ (f0 |s0 ) = hX |s0 and h0 ◦ (f0 |s0 ) = f ◦ (hX |s0 ) (3.5) Proof Let prZ0 and prX be projections of the product Z0 × X onto the factors Z0 and X respectively If s0 ⊂ Z0 × X is the graph of h00 , then prZ0 |s0 is a homeomorphism of s0 onto Z0 and prX |s0 = h00 ◦ (prZ0 |s0 ) If for z0 ∈ Z0 , x0 = (z0 , h00 z0 ) ∈ s0 , then h0 (z0 ) = f (h00 (z0 )) Hence, x0 ∈ X0 Then hX (x0 ) = prX (x0 ) = h00 (prZ0 (x0 )) = h00 (f0 (x0 )) Thus, the first equality (3.5) follows from definition of the fan products The second equality (3.5) follows from (3.4) ✷ Recall that we fixed the metric ρ on the map (f, ρ) : X → Z Lemma 3.7 Let h0 be a map of a space Z0 onto Z; X0 the fan product of X and Z0 with respect to h0 and f ; f0 : X0 → Z0 and hX : X0 → X the short projections of this fan product Then one can define a metric ρ0 on the map f0 (i.e., τ (f0 , ρ0 ) coincides with the topology of X0 ) such that ρ(hX x0 , hX x0 ) = ρ0 (x0 , x0 ), x0 , x0 ∈ X0 , (3.6) and for every z0 ∈ Z0 , the corestriction to the fiber f −1 (h0 (z0 )) of the restriction of hX to the fiber f0−1 z0 is an isometry of f0−1 z0 onto f −1 (h0 (z0 )) Proof Consider on (X0 )2 a function ρ0 defined by equalities (3.6) Obviously, ρ0 is a pseudometric on the set X0 As noted before Lemma 3.6, for every z0 ∈ Z0 , hXz0 is a one-to-one mapping (even a homeomorphism) of f0−1 z0 onto f −1 (h0 (z0 )) Hence ρ0 is a metric on f0 and, evidently, for every z0 ∈ Z0 , hXz0 is an isometry of the fiber f0−1 z0 onto the fiber f −1 (h0 (z0 )) From the continuity of f and h0 it follows that the short projections f0 and hX are continuous If O is a neighborhood of a point (z0 , x) ∈ X0 ⊂ Z0 × X in X0 , then (by the definition of the topology of topological products) there exist neighborhoods Oz0 of z0 in Z0 and Ox of x in X such that (z0 , x) ∈ JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.19 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• 19 f0−1 Oz0 ∩ h−1 X Ox ⊂ O By the definition of the topology τ (f, ρ), there exist a neighborhood Oz of z = f x and ε > such that f −1 Oz ∩ O(x, ε) ⊂ Ox By the continuity of h0 , one can suppose that Oz0 ⊂ h−1 Oz It −1 −1 −1 follows from the definition of ρ0 that h−1 O(x, ε) = O((z , x), ε) Hence (see (3.5)) h Ox ⊃ h (f Oz ∩ X X X −1 −1 −1 −1 −1 −1 O(x, ε)) = hX f Oz ∩ hX O(x, ε) = f0 h0 Oz ∩ O((z0 , x), ε) ⊃ f0 Oz0 ∩ O((z0 , x), ε) and (z0 , x) ∈ f0−1 Oz0 ∩ O((z0 , x), ε) ⊂ f0−1 Oz0 ∩ h−1 X Ox ⊂ O Thus, τ (f0 , ρ0 ) coincides with the topology of X0 ✷ A subset P s of a space A will be called a perfect section of a map κ : A → C if κ(P s) = C and the map κ|P s is perfect Let P S(κ) be the set of all perfect sections of a metric mapping κ : A → C For P s and P s from P S(κ), let H(P s, P s ) = sup{Hc (P s ∩ κ−1 c, P s ∩ κ−1 c) : c ∈ C}, where Hc is the Hausdorff metric on the set of all compact subsets of the fiber κ−1 c For a perfect section P s of κ, let P S(κ, P s) be the set of all P s ∈ P S(f ) such that H(P s, P s ) < +∞ Evidently, P S(κ, P s) = P S(κ, P s ) if and only if H(P s, P s ) < +∞ It is also clear that for every P s ∈ P S(κ), the function H(P s , P s ), P s , P s ∈ P S(κ, P s), is a metric on P S(κ, P s) Any subset P of P S(κ) will be called a metric part of P S(κ), if P = P S(κ, P s) for some P s ∈ P S(κ) Evidently, for every section s of κ, the metric part S(κ, s) of S(κ) is a subset of the metric part P S(κ, s) of P S(κ) Note that for any perfect section P s of the map f and for any map-morphism Γ : f → g, the image Γ(P s) of P s is a perfect section of g Theorem 3.2 Let (f, ρ) : X → Z and (g, σ) : Y → Z be metric mappings onto a paracompactum Z and f be open and fiberwise complete If a map-morphism Ψ : f → g is fiberwise α-covering, a map-morphism Φ : f → g is β-Lipschitz, < β < α, and for a perfect section P s of f , the inequality H(Ψ(P s), Φ(P s)) < +∞ holds, then for every ε > 0, there exists a perfect section P ξ = P ξ(P s, ε) of f such that Ψ|P ξ = Φ|P ξ and H(P s, P ξ) H(Ψ(P s), Φ(P s)) + ε α−β Proof Take P s ∈ P S(f ) and ε > Since Z is a paracompactum and f |P s is perfect, P s is also a paracompactum So there is a perfect mapping h00 of a 0-dimensional paracompactum Z0 onto P s Then h0 = f ◦ h00 = (f |P s ) ◦ h00 is a perfect mapping Z0 onto Z It follows from the previous lemma and the properties of the fan products formulated before Lemma 3.6 that 1) for the fan product X0 of the spaces Z0 and X with respect to the maps h0 and f , the short projection f0 : X0 → Z0 of this product is (continuous and) open and the short projection hX : X0 → X is (continuous and) perfect and 2) for the fan product Y0 of the spaces Z0 and Y with respect to the mappings h0 and g, the short projection hY : Y0 → Y is (continuous and) perfect (and the short projection g0 : Y0 → Z0 is continuous) Moreover, f ◦ hX = h0 ◦ f0 , g ◦ hY = h0 ◦ g0 and (see Lemma 3.7) there exist metrics ρ0 and σ0 on the continuous mappings f0 and g0 respectively such that ρ(hX x0 , hX x0 ) = ρ0 (x0 , x0 ), x0 , x0 ∈ X0 , and σ(hY y0 , hY y0 ) = σ0 (y0 , y0 ), y0 , y0 ∈ Y0 ; (iso) for every z0 ∈ Z0 , the mappings cor(hX |f0−1 z0 ) : f0−1 z0 → f −1 (h0 (z0 )) and cor(hY |g0−1 z0 ) : g0−1 z0 → g −1 (h0 (z0 )) are isometries JID:TOPOL 20 AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.20 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• Hence, the mapping f0 is fiberwise complete Since P s ⊂ X, one can consider h00 as a mapping to X Then it follows from Lemma 3.6 that there exists a section s0 of f0 such that h00 ◦ (f0 |s0 ) = hX |s0 and h0 ◦ (f0 |s0 ) = f ◦ (hX |s0 ) It follows from this that hX (s0 ) = P s Define map-morphisms Ψ0 , Φ0 : f0 → g0 Let x0 = (z0 , x) ∈ X0 ⊂ Z0 × X Set Ψ0 (x0 ) = (z0 , Ψx) ∈ Z0 × Y Since x0 ∈ X0 , h0 (z0 ) = f x, and since Ψ is a map-morphism, g(Ψx) = f x = h0 (z0 ) Hence Ψ0 (x0 ) ∈ Y0 Thus, the mapping Ψ0 : X0 → Y0 is defined The continuity of Ψ implies the continuity of Ψ0 Since g0 (Ψ0 (x0 )) = g0 (z0 , Ψ(x)) = z0 = f0 x0 , g0 ◦ Ψ0 = f0 Hence Ψ0 is a map-morphism of f0 to g0 Similarly, define the mapping Φ0 : X0 → Z0 × Y so that for any x0 = (z0 , x) ∈ X0 , Φ0 (x0 ) = (z0 , Φx) As above, we can prove that Φ0 is a map-morphism of f0 to g0 We need the following two equalities Ψ ◦ hX = hY ◦ Ψ0 and Φ ◦ hX = hY ◦ Φ0 (3.7) Indeed, if x0 = (z0 , x) ∈ X0 , then hY (Ψ0 (x0 )) = hY (z0 , Ψ(x)) = Ψ(x) = Ψ(hX (x0 )) The first equality (3.7) has been proved The second one may be proved in the same way Show that the map-morphism Ψ0 is fiberwise open-α-covering Fix z0 ∈ Z0 , x0 = (z0 , x) ∈ X0 and ε > Since (see (iso)) cor(hX |f −1 z0 ) : f0−1 z0 → f −1 (z = h0 (z0 )) and cor(hY |g−1 z0 ) : g0−1 z0 → g −1 (z)) are 0 isometries, and the map-morphism Ψ is fiberwise open-α-covering, Ψ0 (Of0 (x0 , ε)) = {z0 } × Ψ(Of (x, ε)) ⊃ {z0 } × Og (Ψx, αε) = Og0 (Ψ0 x0 , αε) We have proved that Ψ0 is a fiberwise open-α-covering map-morphism Prove that Φ0 is a β-Lipschitz map-morphism Take x0 = (z0 , x) and x0 = (z0 , x ) in X0 Then βρ0 (x0 , x0 ) = βρ(hX x0 , hX x0 ) σ(Φ(hX x0 ), Φ(hX x0 )) = (see (3.7)) σ(hY (Φ0 (x0 )), hY (Φ0 (x0 ))) = σ0 (Φ0 (x0 ), Φ0 (x0 )) We have proved that Φ0 is a β-Lipschitz map-morphism Suppose that H(Ψ(P s), Φ(P s)) = d < +∞ Then for x0 = (z0 , x) ∈ s0 , hX x0 ∈ P s and σ(Ψ(hX (x0 )), Φ(hX (x0 ))) d By (iso), σ0 (Ψ0 (x0 ), Φ0 (x0 )) d Hence d(Ψ0 (s0 ), Φ0 (s0 )) d By Theorem 3.1, there is a continuous section ξ of f0 such that d(s0 , ξ) d + ε and Ψ0 |ξ = Φ0 |ξ α−β (3.8) Since f0 |ξ is a homeomorphism and h0 is perfect, the maps f ◦ (hX |ξ ) = h0 ◦ (f0 |ξ ), hX |ξ and f |hX (ξ) are perfect Hence, P ξ = hX (ξ) is a perfect section of f Let x ∈ P ξ Take x0 ∈ ξ so that hX (x0 ) = x Then (see (3.8) and (3.7)) Ψ(x) = Ψ(hX (x0 )) = hY (Ψ0 (x0 )) = hY (Φ0 (x0 )) = Φ(hX (x0 )) = Φ(x) Hence Ψ|P ξ = Φ|P ξ Besides, for {x0 } = (f0−1 f0 x0 ) ∩ s0 d + ε and ρ(x, x ) = ρ(hX (x0 ), hX (x0 )) = and x = hX (x0 ) ∈ P s, we have inequalities ρ0 (x0 , x0 ) α−β d + ε Since f0 (x0 ) = f0 (x0 ), we obtain f x = f x Similarly, for every x ∈ P s there is a ρ0 (x0 , x0 ) α−β d +ε Hence the inequality Hz (P s ∩f −1 z, P ξ ∩f −1 z) point x ∈ P ξ such that f x = f x and ρ(x, x ) α−β d d + ε holds for every z ∈ Z and so H(P s, P ξ) + ε ✷ α−β α−β JID:TOPOL AID:5663 /FLA [m3L; v1.169; Prn:29/12/2015; 14:36] P.21 (1-21) T.H.V Nguyen, B.A Pasynkov / Topology and its Applications ••• (••••) •••–••• 21 Corollary 3.6 Let (f, ρ) : X → Z and (g, σ) : Y → Z be metric mappings onto a paracompactum Z and the map f be open and fiberwise complete If map-morphisms Ψ, Φ : f → g and a perfect section P s of f are such that for every z ∈ Z, there is a neighborhood Oz of z with the following properties: the corestriction to g −1 Oz of the restriction Ψ (respectively, Φ) to f −1 Oz is a fiberwise open-α(z)-covering (respectively, β(z)-Lipschitz) map-morphism, < β(z) < α(z); H(Ψ(P s ∩ f −1 Oz), Φ(P s ∩ f −1 Oz)) < +∞, then there exists a perfect section P ξ = P ξ(P s) of f such that Ψ|P ξ = Φ|P ξ Proof Take a locally finite closed refinement λ = {Fj : j ∈ J} of the open cover {Oz : z ∈ Z} of the paracompactum Z For each j ∈ J, the set P sj = P s ∩ f −1 Fj is a perfect section of the map fj that is the corestriction to Fj of the restriction of f to f −1 Fj and the inequality H(Ψ(P sj ), Φ(P sj )) < +∞ holds By Theorem 3.2 there exists a perfect section P ξj of fj such that Ψ|P ξj = Φ|P ξj Set P ξ = {P ξj : j ∈ J} Then Ψ|P ξ = Φ|P ξ Obviously, f (P ξ) = Z f (A ∩ f −1 Fj ) = fj (A ∩ f −1 Fj ) Prove that the map f |P ξ is perfect If A is closed in P ξ, then f A = j∈J j∈J is closed in Z because fj is closed for any j ∈ J and the system λ is locally finite Hence the map f |P ξ is closed Since every z ∈ Z is contained in a finite number of elements of λ, f −1 z ∩ P ξ is the union of a finite number of compacta fj−1 z, j ∈ J Hence the set f −1 z ∩ P ξ is compact Thus, the mapping f |P ξ is perfect ✷ Corollary 3.7 Let (f, ρ) : X → Z and (g, σ) : Y → Z be metric mappings onto a paracompactum Z and the mapping f is open and fiberwise complete If map-morphisms Ψ, Φ : f → g and continuous real-valued functions α and β on Z are such that < β(z) < α(z), z ∈ Z, and for every z ∈ Z, the mapping cor(Ψ|f −1 z ) : f −1 z → g −1 z is fiberwise open-α(z)-covering, and the mapping cor(Φ|f −1 z ) : f −1 z → g −1 z is β(z)-Lipschitz, then for every perfect section P s of f , there is a perfect section P ξ = P ξ(P s) of f such that Ψ|P ξ = Φ|P ξ Acknowledgements The authors are grateful to the referee and A.V Arutyunov for their very useful remarks References [1] P.S Alexandroff, B.A Pasynkov, Introduction to Dimension Theory, Nauka, Moskva, 1973 (in Russian) [2] A.V Arutyunov, Covering mappings in metric spaces and fixed points, Dokl Akad Nauk 416 (2) (2007) 151–155 (in Russian), English translation: Dokl Math 76 (2) (2007) 665–668 [3] A.V Arutyunov, Stability of coincidence points and properties of covering mappings, Math Notes 86 (1–2) (2009) 153–158 [4] A.V Arutyunov, Lectures on Convex and Multi-Valued Analysis, Fizmatlit, M., 2014 (in Russian) [5] B.A Pasynkov, On metric mappings, Vestn Mosk Univ., Ser 1, Math., Mech (1999) 29–32 (in Russian), English translation: Mosc Univ Math Bull 54 (3) (1999) 29–32 [6] T.N Fomenko, Approximation of coincidence points and common fixed points of a collection of mappings of metric spaces, Mat Zametki 86 (1) (2009) 110–125 (in Russian), English translation: Math Notes 86 (1) (2009) 107–120 [7] T.N Fomenko, Cascade search principle and its applications to the coincidence problems of n one-valued or multi-valued mappings, Topol Appl 157 (4) (2010) 760–773 [8] E Michael, A generalization of a theorem on continuous selections, Proc Am Math Soc 105 (1) (1989) 236–243  ... Topology and its Applications ••• (••••) •••–••• In Sections and 2, we consider the problems of the existence and searching of coincidence points and the common preimage of a closed subset (in particular,... pointwise converges to a mapping r ∈ F (X) and points 6.–9 of Theorem 1.3 are true Moreover, in point 8., the requirement of the continuity of ϕ can be removed without losing the property of. .. bounded (for example, continuous), and A is continuous, then r is continuous and it is a retraction of X onto ϕ−1 (0) in the case of the continuity of ϕ; if ϕ is bounded, then the sequence Ak , k ∈