Contractibility of Fixed Point Sets of Mean-Type Mappings

and Applied Analysis 3 Lastly, we recall a very well-known fact about the structure of the fixed point sets of quasi-nonexpansive mappings. Definition 13. A normed linear space X is called strictly convex if ‖(x + y)/2‖ < 1 for all x, y ∈ X such that x ̸ = y and ‖x‖ = ‖y‖ = 1; equivalently the boundary of the unit ball does not contain any line segment. Definition 14. Let X be a subset of a normed linear space. A mapping T : X → X is said to be (i) nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ X; (ii) quasi-nonexpansive if ‖Tx−q‖ ≤ ‖x− q‖ for all x ∈ X and q ∈ Fix(T). It is easy to see that nonexpansive mappings are continuous and quasi-nonexpansive while continuous quasinonexpansive mappings may not be nonexpansive. Theorem 15 (see [6]). The fixed point set of a quasinonexpansive mapping defined on a convex subset of a strictly convex space is convex. 3. Main Results In this section, we extend the notions of means and meantype mappings to general vector spaces. Then we prove a convergence theorem as well as the contractibility of fixed point sets for certain continuous quasi-nonexpansive meantype mappings. LetX be a convex subset of a vector space (overR) and p an integer with p ≥ 2. As usual, the diagonal in Xp is simply Δ = {(x1, . . . , xp) ∈ Xp | x1 = ⋅ ⋅ ⋅ = xp}. Definition 16. A function M : Xp → X is said to be a mean if, for each i = 1, . . . , p, there is a function αi : X → [0, 1] such that M(x) = M(x1, . . . , xp) = α1 (x) x1 + ⋅ ⋅ ⋅ + αp (x) xp (18) with α1(x) + ⋅ ⋅ ⋅ + αp(x) = 1 for each x = (x1, . . . , xp) ∈ Xp. For simplicity, we usually write M(x) = ∑n αnxn. We also call M strict if, in addition, αi(x) ∈ (0, 1) for all x ∈ Xp and i = 1, . . . , p. We note that Definition 16 is equivalent to Definition 1 when X is an interval in R. Definition 17. AmappingM : Xp → Xp is said to be ameantype mapping if each coordinate function is a mean; that is M = (M1, . . . ,Mp) , (19) where Mi : Xp → X is a mean for all i = 1, . . . , p. Moreover, M is strict if each Mi is strict. Remark 18. For any mean M : Xp → X, we clearly have M(x, . . . , x) = x, for each x ∈ X. Consequently, for any mean-type mappingM : Xp → Xp, we have Δ ⊆ Fix(M). Example 19. DefineM : (0,∞)2 → (0,∞)2 by M (x, y) = (x + y 2 ,√xy) = (x + y 2 , x√y √x + √y + y√x √x + √y) . (20) Clearly, M is a nonlinear mean-type mapping and it is straightforward to verify thatM is strict. Theorem 20. IfM : Xp → Xp is a strict mean-type mapping, then Fix(M) = Δ. Proof. Let x ∈ Fix(M). We can form the following system of linear equations: x1 = M1 (x) = α11x1 + α12x2 + ⋅ ⋅ ⋅ + α1(p−1)xp−1 + (1 − p−1 ∑ j=1 α1j)xp x2 = M2 (x) = α21x1 + α22x2 + ⋅ ⋅ ⋅ + α2(p−1)xp−1 + (1 − p−1 ∑ j=1 α2j)xp ... xp = Mp (x) = αp1x1 + αp2x2 + ⋅ ⋅ ⋅ + αp(p−1)xp−1 + (1 − p−1 ∑ j=1 αpj)xp, (21) which is equivalent to (((((((( ( 1 − α11 −α12 −α13 ⋅ ⋅ ⋅ −α1(p−1) −α21 1 − α22 −α23 ⋅ ⋅ ⋅ −α2(p−1) −α31 −α32 1 − α33 ⋅ ⋅ ⋅ −α3(p−1) ... ... ... d ... −α(p−1)1 −α(p−1)2 −α(p−1)3 ⋅ ⋅ ⋅ 1 − α(p−1)(p−1) −αp1 −αp2 −αp3 ⋅ ⋅ ⋅ −αp(p−1) )))))))) ) [[[[[[[[[[ x1 − xp x2 − xp x3 − xp ... xp−1 − xp ]]]]]]]]]] = 0. (22) 4 Abstract and Applied Analysis SinceM is strict, we can apply Gauss-Jordan elimination to the coefficient matrix to obtain (((((( ( 1 0 0 ⋅ ⋅ ⋅ 0 0 1 0 ⋅ ⋅ ⋅ 0 0 0 1 ⋅ ⋅ ⋅ 0 ... ... ... d ... 0 0 0 ⋅ ⋅ ⋅ 1 0 0 0 ⋅ ⋅ ⋅ 0 )))))) ) [[[[[[[[[ x1 − xp x2 − xp x3 − xp ... xp−1 − xp ]]]]]]]]] = 0, (23) which implies x1 = ⋅ ⋅ ⋅ = xp. WhenM is nonstrict, Fix(M) may still be the diagonal Δ, or even the whole space Xp. Example 21. Consider Id,M : Xp → Xp defined by Id (x1, . . . , xp) = (x1, x2, . . . , xp) , M (x1, . . . , xp) = (xp, x1, . . . , xp−1) . (24) Clearly, we have Fix(Id) = Xp and Fix(M) = Δ. WhenX is also a metric space, the next theorem surprisingly gives an explicit construction of a continuous meantype mapping, whose fixed point set is any closed subset of Xp containing the diagonal. Recall that the distance between a point x in the metric space (X, d) and 0 ̸ = A ⊆ X is defined to be d (x, A) = inf a∈A d (x, a) . (25) When A is closed, we also have d(x, A) = 0 iff x ∈ A. Theorem 22. Suppose further that (X, d) is a metric space. For any closed subset F of Xp such that Δ ⊆ F, there exists a continuous mean-type mapping M : Xp → Xp such that Fix(M) = F. Proof. Define t : Xp → [0, 1] andM : Xp → Xp by t (x) = d (x, F) 1 + d (x, F) , M (x) = (x1, t (x) x1 + [1 − t (x)] x2, . . . , t (x) x1 + [1 − t (x)] xp) . (26) It is easy to verify that t(x) = 0 iff x ∈ F, and M is continuous with Fix(M) = F. From now on, letX be a convex subset of a normed linear space (E, ‖ ‖), and we will always use the maximum norm 󵄩󵄩󵄩󵄩󵄩(x1, . . . , xp)󵄩󵄩󵄩󵄩󵄩 = max n 󵄩󵄩󵄩󵄩xn󵄩󵄩󵄩󵄩 (27) onXp. Notice thatXp togetherwithmaximumnormmay not be strictly convex. This prevents us from using Theorem 15 to conclude the convexity (and hence contractibility) of fixed point sets of quasi-nonexpansive mean-type mappings. The following theorem shows that, under a simple condition, a mean-type mapping is always quasi-nonexpansive. Theorem 23. Let M : Xp → Xp be a mean-type mapping. If Fix(M) = Δ, thenM is quasi-nonexpansive. In particular, strict mean-type mappings are always quasi-nonexpansive. Proof. Let x = (x1, . . . , xp) ∈ Xp, q = (q, . . . , q) ∈ Δ = Fix(M), and write M = (M1, . . . ,Mp). Consider, for each i = 1, . . . , p, 󵄩󵄩󵄩󵄩Mi (x) − q󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∑n αinxn − ∑n αinq󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 ≤ ∑n αin 󵄩󵄩󵄩󵄩xn − q󵄩󵄩󵄩󵄩 ≤ max n 󵄩󵄩󵄩󵄩xn − q󵄩󵄩󵄩󵄩∑ n αin = max n 󵄩󵄩󵄩󵄩xn − q󵄩󵄩󵄩󵄩 , (28) and hence, 󵄩󵄩󵄩󵄩M (x) − q󵄩󵄩󵄩󵄩 = max i 󵄩󵄩󵄩󵄩Mi (x) − q󵄩󵄩󵄩󵄩 ≤ max n 󵄩󵄩󵄩󵄩xn − q󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩x − q󵄩󵄩󵄩󵄩 . (29) Unfortunately, the converse of the previous theorem is not true (see Example 36 below). Moreover, the last line in the proof of the previous theorem shows that any meantype mapping is continuous on the diagonal. However, the continuity may not hold at other points as in the next example. Example 24. DefineM : R → R by M (x, y) = {{{{{ (2x + y 3 , x + 2y 3 ) ; (x, y) ∈ Q2, (x + 2y 3 , 2x + y 3 ) ; (x, y) ∉ Q2. (30) It is easy to see that M is a strict mean-type mapping. However,M is not continuous at each (x, y) ∉ Δ. Definition 25. For each k ∈ N, λ = (λ0, λ1, . . . , λk) ∈ [0, 1]k+1 with ∑ki=0 λi = 1, and T : X → X, we define the λcombination of T as follows: Tλ = λ0Id + λ1T + ⋅ ⋅ ⋅ + λkTk = k ∑ i=0 λiTi, (31) where T0 denotes the identity mapping (Id). The λcombination of a mean-type mapping M : Xp → Xp is defined similarly. From the above definition, it is easy to verify that the λ-combination of a mean-type mapping M : Xp → Xp is also a mean-type mapping, and it is continuous if M is continuous. Before we establish a convergence theorem, let us recall some basic facts about quasi-nonexpansive mappings and the distance between points in a convex hull. Abstract and Applied Analysis 5 Lemma 26. LetX be a convex subset of a normed linear space, T : X → X a quasi-nonexpansive mapping, p ∈ Fix(T), and x0, y, z ∈ X. For a given λ = (λ0, λ1, . . . , λk) ∈ [0, 1]k+1 with ∑ki=0 λi = 1, define a sequence (xn) ⊆ X by xn = Tλxn−1. (32) (1) Ify and z are limit points of (xn), then ‖y−p‖ = ‖z−p‖. (2) If p is a limit point of (xn), then limn→∞xn = p. Proof. SinceT is quasi-nonexpansive, we have for each n ∈ N, 󵄩󵄩󵄩󵄩xn+1 − p󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 k ∑i=0λiTixn − p 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 k ∑i=0λi (Tixn − p) 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 ≤ k ∑ i=0 λi 󵄩󵄩󵄩󵄩󵄩Tixn − p󵄩󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩xn − p󵄩󵄩󵄩󵄩 . (33)and Applied Analysis 5 Lemma 26. LetX be a convex subset of a normed linear space, T : X → X a quasi-nonexpansive mapping, p ∈ Fix(T), and x0, y, z ∈ X. For a given λ = (λ0, λ1, . . . , λk) ∈ [0, 1]k+1 with ∑ki=0 λi = 1, define a sequence (xn) ⊆ X by xn = Tλxn−1. (32) (1) Ify and z are limit points of (xn), then ‖y−p‖ = ‖z−p‖. (2) If p is a limit point of (xn), then limn→∞xn = p. Proof. SinceT is quasi-nonexpansive, we have for each n ∈ N, 󵄩󵄩󵄩󵄩xn+1 − p󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 k ∑i=0λiTixn − p 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 k ∑i=0λi (Tixn − p) 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 ≤ k ∑ i=0 λi 󵄩󵄩󵄩󵄩󵄩Tixn − p󵄩󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩xn − p󵄩󵄩󵄩󵄩 . (33) (1) If y and z are limit points of (xn), then there exist subsequences (xnk), (xnl) of (xn) such that limk→∞xnk = y and liml→∞xnl = z. From (33), for each l ∈ N, there exist k0, l0 ∈ N such that nl ≤ nk0 ≤ nl0 , and hence 󵄩󵄩󵄩󵄩󵄩󵄩xnl0 − p󵄩󵄩󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩󵄩󵄩xnk0 − p󵄩󵄩󵄩󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩󵄩󵄩xnl − p󵄩󵄩󵄩󵄩󵄩 . (34) This implies, as l → ∞, that ‖z−p‖ ≤ ‖y−p‖ ≤ ‖z−p‖. (2) If p is a limit point of (xn), then there exists a subsequence (xnk) of (xn) such that limk→∞xnk = p. Again, by (33), we must have limn→∞xn = p. For each x = (x1, . . . , xp) ∈ Xp andA ⊆ X, recall that the convex hull of x, the algebraic interior of the convex hull of x, and the diameter ofA are, respectively, defined as follows (see [7] for details): co (x) = {α1x1 + ⋅ ⋅ ⋅ + αpxp : αi ∈ [0, 1] , α1 + ⋅ ⋅ ⋅ + αp = 1} , algint (co (x)) = {α1x1 + ⋅ ⋅ ⋅ + αpxp : αi ∈ (0, 1) , α1 + ⋅ ⋅ ⋅ + αp = 1} , diam (A) = sup {󵄩󵄩󵄩󵄩x − y󵄩󵄩󵄩󵄩 : x, y ∈ A} . (35) Note that diam(A) always exists whenA is nonempty and bounded. Otherwise, let diam(A) = ∞. Lemma 27. Let X be a subset of a normed linear space. co(x) is compact for each x ∈ Xp. Proof. Let H = {(α1, . . . , αp) ∈ [0, 1]p : α1 + ⋅ ⋅ ⋅ + αp = 1}. Define T : H → co(x) by T (α1, . . . , αp) = α1x1 + ⋅ ⋅ ⋅ + αpxp. (36) It is easy to verify that co(x) is the image of the compact set H under the continuous function T. Lemma 28. Let X be a subset of a normed linear space and x ∈ Xp. (1) diam(co(x)) = maxi,j‖xi − xj‖. (2) If x ∉ Δ, y ∈ algint(co(x)) and z ∈ co(x), then ‖y − z‖ < diam(co(x)). Proof. (1) It is clear that maxi,j‖xi − xj‖ ≤ diam(co(x)). Conversely, let y, z ∈ co(x); say y = ∑pn=1 αnxn and z = ∑pn=1 βnxn. Then 󵄩󵄩󵄩󵄩y − z󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∑n αnxn − ∑n βnxn󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 = 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∑n βn∑m αmxm − ∑n βnxn󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩 ≤ ∑ n βn 󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩󵄩∑m αmxm 


Introduction
The theory of mean iteration has been studied long before the 19th century [1].Johann Carl Friedrich Gauss observed the connection between the arithmetic-geometric mean iteration and an elliptic integral.Indeed, if we recursively define the following sequences of positive real numbers we know that both (  ) and (  ) converge to the same limit, say, ( 0 ,  0 ).He found that which is later generalised to However, the convergence above is not coincidental as we will see in the next section.
In 1999, Matkowski introduced the notion of mean-type mappings on a real interval and showed the convergence of its Picard iteration if at most one of its coordinate means is not strict.Later in 2009, he showed the same result for mappings with a weaker condition.The fixed point set of such a mapping is exactly the diagonal; however, the fixed point set of a general (continuous) mean-type mapping only covers the diagonal and may not be contractible.On the other hand, in 2012, Chaoha and Chanthorn introduced the concept of virtually stable (fixed point iteration) schemes to connect topological structures of the convergence set of a scheme to those of the fixed point set via a retraction.Many schemes for nonexpansive-type mappings have been proved to be virtually stable.
With those results in mind, in this work, we first extend the concept of mean-type mappings to vector spaces and then explore their fixed point sets using the notion of virtually stable schemes developed in [2].We are able to establish a convergence theorem for certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces (which immediately covers the result in [3]) and conclude the contractibility of their fixed point sets.This also gives a new result on the structure of fixed point sets of quasinonexpansive mappings outside the strict-convexity setting.

Preliminaries
We begin this section by recalling the notion of means and mean-type mappings from [3].
for all ( 1 , . . .,   ) ∈   − Δ, where Δ denotes the diagonal: Definition 2. A mapping M :   →   is said to be a meantype mapping if each coordinate function is a mean; i.e.,   :   →  is a mean for all  = 1, . . ., , where Moreover, M is strict if all coordinate means are strict.
Definition 3. Let ,  :   →  be means.We say that  and  are comparable if one of the following holds: (i) () ≤ () for all  ∈   .
We are ready to recall the classical and well-known convergence theorem for a 2-dimensional mean iteration.
Again, in 2009, he improved his earlier result to include the larger class of nonstrict mean-type mappings.
Remark 8.If a mean-type mapping M satisfies the condition in Theorem 7, then Next, we recall the concept of virtual stability of fixed point iteration schemes [2].We use this to conclude the contractibility of the fixed point sets of nonexpansive-type mappings.
Let S = (  ) be a sequence of self-maps on a Hausdorff space .Define the fixed point set of S and the convergence set of S as follows: Definition 9 (see [2]).Let (  ) be a sequence of self-maps on  and The sequence Definition 10 (see [2]).Let S = (∏  =1   ) be a scheme.A fixed point  ∈ (S) is called virtually stable if, for each neighbourhood  of , there exist a neighbourhood  of  and a strictly increasing sequence (  ) ⊆ N such that ∏   =   () ⊆  for all  ∈ N and  ≤   .
The scheme S is called virtually stable if all its common fixed points are virtually stable.
Theorem 11 (see [2]).If  is a regular space and S is a virtually stable scheme having a subsequence consisting of continuous mappings, then the function  defined above is continuous and hence (S) is a retract of (S).
Lastly, we recall a very well-known fact about the structure of the fixed point sets of quasi-nonexpansive mappings.Definition 13.A normed linear space  is called strictly convex if ‖( + )/2‖ < 1 for all ,  ∈  such that  ̸ =  and ‖‖ = ‖‖ = 1; equivalently the boundary of the unit ball does not contain any line segment.Definition 14.Let  be a subset of a normed linear space.A mapping  :  →  is said to be (i) nonexpansive if ‖ − ‖ ≤ ‖ − ‖ for all ,  ∈ ; It is easy to see that nonexpansive mappings are continuous and quasi-nonexpansive while continuous quasinonexpansive mappings may not be nonexpansive.
Theorem 15 (see [6]).The fixed point set of a quasinonexpansive mapping defined on a convex subset of a strictly convex space is convex.

Main Results
In this section, we extend the notions of means and meantype mappings to general vector spaces.Then we prove a convergence theorem as well as the contractibility of fixed point sets for certain continuous quasi-nonexpansive meantype mappings.
We note that Definition 16 is equivalent to Definition 1 when  is an interval in R.
Definition 17.A mapping M :   →   is said to be a meantype mapping if each coordinate function is a mean; that is where   :   →  is a mean for all  = 1, . . ., .Moreover, M is strict if each   is strict.
Clearly, M is a nonlinear mean-type mapping and it is straightforward to verify that M is strict.
Proof.Let  ∈ Fix(M).We can form the following system of linear equations: which is equivalent to Since M is strict, we can apply Gauss-Jordan elimination to the coefficient matrix to obtain When M is nonstrict, Fix(M) may still be the diagonal Δ, or even the whole space   .
When  is also a metric space, the next theorem surprisingly gives an explicit construction of a continuous meantype mapping, whose fixed point set is any closed subset of   containing the diagonal.
Recall that the distance between a point  in the metric space (, ) and 0 ̸ =  ⊆  is defined to be When  is closed, we also have (, ) = 0 iff  ∈ .
Theorem 22. Suppose further that (, ) is a metric space.For any closed subset  of   such that Δ ⊆ , there exists a continuous mean-type mapping M :   →   such that Fix(M) = .
From now on, let  be a convex subset of a normed linear space (, ‖ ‖), and we will always use the maximum norm      ( to conclude the convexity (and hence contractibility) of fixed point sets of quasi-nonexpansive mean-type mappings.
The following theorem shows that, under a simple condition, a mean-type mapping is always quasi-nonexpansive.
Theorem 23.Let M :   →   be a mean-type mapping.If Fix(M) = Δ, then M is quasi-nonexpansive.In particular, strict mean-type mappings are always quasi-nonexpansive.
Unfortunately, the converse of the previous theorem is not true (see Example 36 below).Moreover, the last line in the proof of the previous theorem shows that any meantype mapping is continuous on the diagonal.However, the continuity may not hold at other points as in the next example.
Example 24.Define M : R 2 → R 2 by It is easy to see that M is a strict mean-type mapping.However, M is not continuous at each (, ) ∉ Δ.
From the above definition, it is easy to verify that the -combination of a mean-type mapping M :   →   is also a mean-type mapping, and it is continuous if M is continuous.Before we establish a convergence theorem, let us recall some basic facts about quasi-nonexpansive mappings and the distance between points in a convex hull.
For each  = ( 1 , . . .,   ) ∈   and  ⊆ , recall that the convex hull of , the algebraic interior of the convex hull of , and the diameter of  are, respectively, defined as follows (see [7] for details): Note that diam() always exists when  is nonempty and bounded.Otherwise, let diam() = ∞.Lemma 27.Let  be a subset of a normed linear space.co() is compact for each  ∈   .
Lemma 28.Let  be a subset of a normed linear space and  ∈   .