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ATTRACTIVE AND MEAN CONVERGENCE THEOREMS FOR TWO COMMUTATIVE NONLINEAR MAPPINGS IN BANACH SPACES.

1. INTRODUCTION

Let H be a real Hilbert space and let C be a nonempty subset of H. Let T be a mapping of C into H. Then we denote by F(T) the set of fixed points of T and by A(T) the set of attractive points [27] of T, i.e.,

(i) F(T) = {z [member of] C: Tz = z};

(ii) A(T) = {z [member of] H : [parallel]Tx - z[parallel] [less than or equal to] [parallel]x - z[parallel], [for all]x [member of] C}.

We know from [27] that A(T) is closed and convex. This property is important for proving mean convergence theorems. Such a concept of attractive points was defined in a Banach space; see [20]. A mapping T : C [right arrow] H is said to be nonexpansive [4] if [parallel]Tx - Ty[parallel] [less than or equal to] [parallel]x - y[parallel] for all x,y [member of] C. Baillon [2] proved the first mean convergence [parallel]Tx - Ty[parallel] [less than or equal to] [parallel]x - y[parallel] for all x,y [member of] C. Baillon [2] proved the first mean convergence theorem in a Hilbert space.

Theorem 1.1 ([2]). Let C be a bounded, closed and convex subset of H and let T:C [right arrow] C be nonexpansive. Then for any x [member of] C,

[S.sub.n]x = 1/n [n-1.summation over (k=0)] [T.sup.k] x

converges weakly to an element z [member of] F(T).

This theorem for nonexpansive mappings has been extended to Banach spaces by many authors; see, for example, [3, 5]. On the other hand, in 2010, Kocourek, Takahashi and Yao [13] defined a broad class of nonlinear mappings in a Hilbert space: Let H be a Hilbert space and let C be a nonempty subset of H. A mapping T : C [right arrow] H is called generalized hybrid [13] if there exist [alpha], [beta] [member of] R such that

(1.1) [mathematical expression not reproducible]

for all x,y [member of] C. Such a mapping T is called ([alpha], [beta])- generalized hybrid. Notice that the class of generalized hybrid mappings covers several well-known mappings. For example, a (1,0)-generalized hybrid mapping is nonexpansive, i.e.,

[parallel]Tx - Ty[parallel] [less than or equal to] [parallel]x - y[parallel], [for all] x,y [member of] C.

It is nonspreading [17, 18] for [alpha] = 2 and [beta] = 1, i.e.,

[mathematical expression not reproducible].

It is also hybrid [25] for [alpha] = 3/2 and [beta] = 1/2, i.e.,

[mathematical expression not reproducible].

In general, nonspreading and hybrid mappings are not continuous; see [10]. The mean convergence theorem by Baillon [2] for nonexpansive mappings has been extended to generalized hybrid mappings in a Hilbert space by Kocourek, Takahashi and Yao [13]. Furthermore, Takahashi and Takeuchi [27] proved the following mean convergence theorem without convexity in a Hilbert space.

Theorem 1.2 ([27]). Let H be a Hilbert space and let C be a nonempty subset of H. Let T be a generalized hybrid mapping from C into itself. Assume that {[T.sup.n]z} for some z [member of] C is bounded and define

[S.sub.n]x = 1/n [n-1.summation over (k=0)] [T.sup.k]x

for all x [member of] C and n [member of] N. Then {[S.sub.n]x} converges weakly to [u.sub.0] G A(T), where [u.sub.0] = [lim.sub.n[right arrow][infinity]] [P.sub.A(T)][T.sup.n]x and [P.sub.A(T)] is the metric projection of H onto A(T).

Maruyama, Takahashi and Yao [21] also defined a more broad class of nonlinear mappings called 2-generalized hybrid which covers generalized hybrid mappings in a Hilbert space. Let C be a nonempty subset of H and let T be a mapping of C into H. A mapping T:C [right arrow] H is 2-generalized hybrid [21] if there exist [[alpha].sub.1], [[alpha].sub.2], [[beta].sub.1], [[beta].sub.2] [member of] R such that

(1.2) [mathematical expression not reproducible]

for all x, y [member of] C.

Recently, Hojo, Takahashi and Takahashi [6] proved attractive and mean convergence theorems without convexity for commutative 2-generalized hybrid mappings in a Hilbert space. These results generalize Takahashi and Takeuchi's theorem (Theorem 1.2) and Kohsaka's theorem [15] which is a mean convergence theorem for commutative [lambda]-hybrid mappings in a Hilbert space.

In this paper, using the class of 2-generalized nonspreading mappings which was defined by [29] in a Banach space and covers 2-generalized hybrid mappings in a Hilbert space, we prove an attractive point theorem in a Banach space. This theorem generalizes Hojo, Takahashi and Takahashi's attractive point theorem [6] in a Hilbert space. Then we prove a mean convergence theorem of Baillon's type [2] without convexity for commutative 2-generalized nonspreading mappings in a Banach space. This result is a general mean convergence theorem which extends Baillon's theorem (Theorem 1.1) to a Banach space.

2. PRELIMINARIES

Let E be a real Banach space with norm [parallel]*[parallel] and let [E.sub.*] be the topological dual space of E. We denote the value of [y.sup.*] [member of] [E.sup.*] at x [member of] E by <x, [y.sup.*]>. When {[x.sub.n]} is a sequence in E, we denote the strong convergence of {[x.sub.n]} to x G [member of] by [x.sub.n] [right arrow] x and the weak convergence by [x.sub.n] [??] x. The modulus [delta] of convexity of E is defined by

[mathematical expression not reproducible]

for every [epsilon] with 0 [less than or equal to] [epsilon] [less than or equal to] 2. A Banach space E is said to be uniformly convex if [delta]([epsilon]) > 0 for every [epsilon] > 0. A uniformly convex Banach space is strictly convex and reflexive. Let C be a nonempty subset of a Banach space E. A mapping T : C [right arrow] E is nonexpansive if [parallel]Tx - Ty[parallel] [less than or equal to] [parallel]x - y[parallel] for all x, y [member of] C. A mapping T : C [right arrow] E is quasi-nonexpansive if F(T) = 0 and [parallel]Tx - y[parallel] [less than or equal to] [parallel]x - y[parallel] for all x [member of] C and y [member of] F(T), where F(T) is the set of fixed points of T. If C is a nonempty, closed and convex subset of a strictly convex Banach space E and T : C [right arrow] E is quasi- nonexpansive, then F(T) is closed and convex; see Itoh and Takahashi [11].

Let E be a Banach space. The duality mapping J from E into [mathematical expression not reproducible] is defined by

Jx = {[x.sup.*] [member of] [E.sup.*]: <x,[x.sup.*]> = [[parallel]x[parallel].sup.2] = [[parallel][x.sup.*][parallel].sup.2]}

for every x [member of] E. Let U = {x [member of] E: [parallel]x[parallel] = 1}. The norm of E is said to be Gateaux differentiable if for each x, y [member of] U, the limit

(2.1) [mathematical expression not reproducible]

exists. In this case, E is called smooth. We know that E is smooth if and only if J is a single-valued mapping of E into [E.sup.*]. We also know that E is reflexive if and only if J is surjective, and E is strictly convex if and only if J is one-to-one. Therefore, if E is a smooth, strictly convex and reflexive Banach space, then J is a single- valued bijection. The norm of E is said to be uniformly Gateaux differentiable if for each y [member of] U, the limit (2.1) is attained uniformly for x E U. It is also said to be Frechet differentiable if for each x [member of] U, the limit (2.1) is attained uniformly for y [member of] U. A Banach space E is called uniformly smooth if the limit (2.1) is attained uniformly for x,y [member of] U. It is known that if the norm of E is uniformly Gateaux differentiable, then J is uniformly norm to weak * continuous on each bounded subset of E, and if the norm of E is Frechet differentiable, then J is norm to norm continuous. If E is uniformly smooth, J is uniformly norm to norm continuous on each bounded subset of E. For more details, see [23, 24].

Lemma 2.1 ([23, 24]). Let E be a smooth Banach space and let J be the duality mapping on E. Then <x - y, Jx - Jy> [greater than or equal to] 0 for all x,y [member of] E. Furthermore, if E is strictly convex and <x - y, Jx - Jy> = 0, then x = y.

Let E be a smooth Banach space. The function [phi]: E x E [right arrow](- [infinity], [infinity]) is defined by

(2.2) [phi](x,y) = [[parallel]x[parallel].sup.2] - 2<x, Jy> + [[parallel]y[parallel].sup.2]

for x, y [member of] E, where J is the duality mapping of E; see [1] and [12]. We have from the definition of f that

(2.3) [phi](x y) = [phi](x z) + [phi](z, y) + 2<x - z, Jz - Jy>

for all x, y, z [member of] E. From [([parallel]x[parallel] - [parallel]y[parallel]).sup.2] [less than or equal to] [phi](x,y) for all x,y [member of] E, we can see that [phi] (x, y) [greater than or equal to] 0. Furthermore, we can obtain the following equality:

(2.4) 2<x - y, Jz - Jw> = [phi](x, w) + [phi](y, z) - [phi](x, z) - [phi](y, w)

for x, y, z, w [member of] E. If E is additionally assumed to be strictly convex, then from

Lemma 2.1 we have

(2.5) [phi] (x, y) = 0 [??] x = y.

The following lemma is in Xu [33].

Lemma 2.2 ([33]). Let E be a uniformly convex Banach space and let r > 0. Then there exists a strictly increasing, continuous and convex function g : [0, [infinity]) [right arrow] [0, [infinity]) such that g(0) = 0 and

[mathematical expression not reproducible]

for all x, y [member of] [B.sub.r] and [lambda] with 0 [less than or equal to] [lambda] [less than or equal to] 1, where [B.sub.r] = {z [member of] E : [parallel]z[parallel] [less than or equal to] r}.

Using Lemma 2.2, we have the following lemma by Kamimura and Takahashi [12].

Lemma 2.3 ([12]). Let E be a smooth and uniformly convex Banach space and let r > 0. Then there exists a strictly increasing, continuous and convex function g : [0, 2r] [right arrow] R such that g(0) = 0 and

g([parallel]x - y[parallel]) [less than or equal to] [phi](x,y)

for all x, y [member of] Br, where [B.sub.r] = {z [member of] E : [parallel]z[parallel] [less than or equal to] r}.

Let E be a smooth Banach space. Let C be a nonempty subset of E and let T be a mapping of C into E. We denote by A(T) the set of attractive points of T, i.e., A(T) = {z [member of] E: [phi](z, Tx) [less than or equal to] [phi](z, x), [for all]x [member of] C}; see [20].

Lemma 2.4 ([20]). Let E be a smooth Banach space and let C be a nonempty subset of E. Let T be a mapping from C into E. Then A(T) is a closed and convex subset of E.

Let E be a smooth Banach space and let C be a nonempty subset of E. Then a mapping T : C [right arrow] E is called generalized nonexpansive [8] if F(T) [not equal to] 0 and

[phi](Tx,y) [less than or equal to] [phi](x, y)

for all x [member of] C and y [member of] F(T); see also [32]. Let D be a nonempty subset of a Banach space E. A mapping R : E [right arrow] D is said to be sunny if

R(Rx + t(x - Rx)) = Rx

for all x [member of] E and t [greater than or equal to] 0. A mapping R : E [right arrow] D is said to be a retraction or a projection if Rx = x for all x [member of] D. A nonempty subset D of a smooth Banach space E is said to be a generalized nonexpansive retract (resp. sunny generalized nonexpansive retract) of E if there exists a generalized nonexpansive retraction (resp. sunny generalized nonexpansive retraction) R from E onto D; see [8] for more details. The following results are in Ibaraki and Takahashi [8].

Lemma 2.5 ([8]). Let C be a nonempty closed sunny generalized nonexpansive retract of a smooth and strictly convex Banach space E. Then the sunny generalized nonexpansive retraction from E onto C is uniquely determined.

Lemma 2.6 ([8]). Let C be a nonempty closed subset of a smooth and strictly convex Banach space E such that there exists a sunny generalized nonexpansive retraction R from E onto C and let (x, z) [member of] E x C. Then the following hold:

(i) z = Rx if and only if <x - z, Jy - Jz> [less than or equal to] 0 for all y [member of] C;

(ii) [phi](Rx, z) + [phi](x, Rx) [less than or equal to] [phi](x, z).

In 2007, Kohsaka and Takahashi [16] proved the following results:

Lemma 2.7 ([16]). Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty closed subset of E. Then the following are equivalent:

(a) C is a sunny generalized nonexpansive retract of E;

(b) C is a generalized nonexpansive retract of E;

(c) JC is closed and convex.

Lemma 2.8 ([16]). Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty closed sunny generalized nonexpansive retract of E. Let R be the sunny generalized nonexpansive retraction from E onto C and let (x, z) [member of] E x C. Then the following are equivalent:

(i) z = Rx;

(ii) [phi](x,z) = [min.sub.y [member of] C] f(x, y).

Ibaraki and Takahashi [9] also obtained the following result concerning the set of fixed points of a generalized nonexpansive mapping.

Lemma 2.9 ([9]). Let E be a reflexive, strictly convex and smooth Banach space and let T be a generalized nonexpansive mapping from E into itself. Then F(T) is closed and JF(T) is closed and convex.

The following theorem is proved by using Lemmas 2.7 and 2.9.

Lemma 2.10 ([9]). Let E be a reflexive, strictly convex and smooth Banach space and let T be a generalized nonexpansive mapping from E into itself. Then F(T) is a sunny generalized nonexpansive retract of E.

Using Lemma 2.7, we also have the following result.

Lemma 2.11 ([26]). Let E be a smooth, strictly convex and reflexive Banach space and let {[C.sub.i]: i [member of] I} be a family of sunny generalized nonexpansive retracts of E such that [[intersection].sub.i [member of] I][C.sub.i] is nonempty. Then [[intersection].sub.i [member of] I] [C.sub.i] is a sunny generalized nonexpansive retract of E.

Let [l.sup.[infinity]] be the Banach space of bounded sequences with supremum norm. Let [mu] be an element of [([l.sup.[infinity]]).sup.*] (the dual space of [l.sup.[infinity]]). Then, we denote by [mu](f) the value of [mu] at f = ([x.sub.1], [x.sub.2], [x.sub.3], ...) G [l.sup.[infinity]]. Sometimes, we denote by [[mu].sub.n]([x.sub.n]) the value [mu](f). A linear functional [mu] on [l.sup.[infinity]] is called a mean if [mu](e) = [parallel]a[parallel] = 1, where e = (1,1,1, ...). A mean [mu] is called a Banach limit on [l.sup.[infinity]] if [[mu].sub.n]([x.sub.n+1]) = [[mu].sub.n]([x.sub.n]). We know that there exists a Banach limit on [l.sup.[infinity]]. If [mu] is a Banach limit on [l.sup.[infinity]], then for f = ([x.sub.1], [x.sub.2], [x.sub.3], ...) [member of] [l.sup.[infinity]],

[mathematical expression not reproducible].

In particular, if f = ([x.sub.1], [x.sub.2], [x.sub.3], ...) [member of] [l.sup.[infinity]] and [x.sub.n] [right arrow] a [member of] R, then we have [mu](f) = [[mu].sub.n]([x.sub.n]) = a. For the proof of existence of a Banach limit and its other elementary properties, see [23].

3. FIXED POINT THEOREMS

Let E be a smooth Banach space and let C be a nonempty subset of E. Then a mapping T : C [right arrow] E is called 2-generalized nonspreading [29] if there exist [[alpha].sub.1],[[alpha].sub.2], [[beta].sub.1], [[beta].sub.2], [[gamma].sub.1], [[gamma].sub.2], [[delta].sub.1], [[delta].sub.2] [member of] R such that

(3.1) [mathematical expression not reproducible]

for all x,y [member of] C; see also [30]. Such a mapping is called ([[alpha].sub.1], [[alpha].sub.2], [[beta].sub.1], [[beta].sub.1], [[gamma].sub.1], [[gamma].sub.2], [[delta].sub.1], [[delta].sub.1])-generalized nonspreading. We know that a (0, [[alpha].sub.2], 0, [[beta].sub.2], 0, [[gamma].sub.2], 0, [[delta].sub.2])- generalized nonspreading mapping is generalized nonspreading in the sense of [14]. We also know that a (0,1, 0,1, 0,1, 0, 0)-generalized nonspreading mapping is nonspreading in the sense of [18].

Now we prove an attractive point theorem for commutative 2-generalized non- spreading mappings in a Banach space. Before proving it, we prove the following result.

Lemma 3.1. Let E be a smooth, strictly convex and reflexive Banach space with the duality mapping J and let C be a nonempty subset of E. Let S and T be mappings of C into itself. Let {[x.sub.n]} be a bounded sequence of E and let [mu] be a mean on [l.sup.[infinity]]. Suppose that

[mathematical expression not reproducible]

for all y [member of] C. Then A(S) [intersection] A(T) is nonemppty. Additionally, if C is closed and convex and {[x.sub.n]} [subset] C, then F(S) [intersection] F(T) is nonempty.

Proof. Using a mean [mu] and a bounded sequence {[x.sub.n]}, we define a function g : [E.sup.*] [right arrow] R as follows:

g([x.sup.*]) = [[mu].sub.n]<[x.sub.n], [x.sup.*]>

for all [x.sup.*] [member of] [E.sup.*]. Since [mu] is linear, g is also linear. Furthermore, we have

[mathematical expression not reproducible]

for all [x.sup.*] [member of] [E.sup.*]. Then g is a linear and bounded real- valued function on [E.sup.*]. Since E is reflexive, there exists a unique element z of E such that

g([x.sup.*]) = [[mu].sub.n] <[x.sub.n], [x.sup.*]> = <z, [x.sup.*]>

for all [x.sup.*] [member of] [E.sup.*]. From (2.3) we have that for y [member of] C and n [member of] N,

[phi]([x.sub.n], y) = [phi]([x.sub.n], Sy) + [phi](Sy, y) + 2<[x.sub.n] - Sy, JSy - Jy>.

So, we have that for y [member of] C,

[mathematical expression not reproducible].

Since, by assumption, [[mu].sub.n][phi]([x.sub.n], Sy) [less than or equal to] [[mu].sub.n][phi]([x.sub.n], y) for all y [member of] C, we have

[[mu].sub.n] [phi]([x.sub.n], y) [less than or equal to] [[mu].sub.n][phi]([x.sub.n], y) + 0(Sy, y) + 2<z - Sy, JSy - Jy)

This implies that

0 [less than or equal to] [phi](Sy, y) + 2<z - Sy, JSy - Jy>.

Using (2.4), we have that

0 [less than or equal to] [phi](Sy, y) + [phi](z, y) + [phi](Sy, Sy) - [phi](z Sy) - 0(Sy, y)

and hence [phi](z, Sy) [less than or equal to] [phi](z, y). This implies that z is an element of A(S). Similarly, we have that [phi](z, Ty) [less than or equal to] [phi](z, y) and hence z [member of] A(T). Therefore we have z [member of] A(S) [intersection] A(T). Additionally, if C is closed and convex and {[x.sub.n]} [subset] C, we have that z [member of] [bar.co]{[x.sub.n]: n [member of] N} [subset] C. In fact, if z [not member of] C, then there exists [y.sup.*] [member of] [E.sup.*] by the separation theorem [23] such that <z, [y.sup.*]> < [inf.sub.y [member of] C] <y,[y.sup.*]>. So, from {[x.sub.n]} [subset] C we have

[mathematical expression not reproducible].

This is a contradiction. Then we have z [member of] C. Since z [member of] A(S) [intersection] A(T) and z [member of] C, we have that

[phi](z, Sz) [less than or equal to] 0(z, z) = 0 and [phi](z, Tz) [less than or equal to] [phi](z, z) = 0

and hence [phi](z, Sz) = 0 and [phi](z, Tz) = 0. Since E is strictly convex, we have z [member of] F(S) [intersection] F(T). This completes the proof.

Using Lemma 3.1, we prove an attractive point theorem for commutative 2- generalized nonspreading mappings in a Banach space.

Theorem 3.2. Let C be a nonempty subset of a smooth, strictly convex and reflexive Banach space E and let S and T be commutative 2-generalized nonspreading mappings of C into itself. Suppose that there exists an element z [member of] C such that {[S.sup.k][T.sup.l]z: k, l [member of] N [union] {0}} is bounded. Then A(S) [intersection] A(T) is nonempty. Additionally, if C is closed and convex, then F(S) [intersection] F(T) is nonempty.

Proof. Since S is a 2-generalized nonspreading mapping of C into itself, there exist [mathematical expression not reproducible] such that for all x,y [member of] C,

(3.2) [mathematical expression not reproducible].

By assumption, we can take z [member of] C such that {[S.sup.k][T.sup.l]z : k,l [member of] N U {0}} is bounded. Replacing x by [S.sup.k][T.sup.l]z in (3.2), we have that for any y [member of] C and k, l [member of] N U {0},

[mathematical expression not reproducible].

This implies that

[mathematical expression not reproducible].

Summing up these inequalities with respect to k = 0,1, ..., n, we have

[mathematical expression not reproducible].

Furthermore, summing up these inequalities with respect to l = 0,1, ..., n, we have

[mathematical expression not reproducible].

Dividing by [(n + 1).sup.2], we have

[mathematical expression not reproducible],

where [mathematical expression not reproducible]. Since {[S.sup.k][T.sup.l]z} is bounded by assumption, {[S.sub.n]z} is bounded. Taking a Banach limit [mu] to both sides of this inequality, we have that

0 [less than or equal to] [phi](Sy, y) + 2[[mu].sub.n]<[S.sub.n]Z - Sy, JSy - Jy>

and hence

0 [less than or equal to] [phi](Sy, y) + [[mu].sub.n][phi]([S.sub.n]z, y) + [phi](Sy, Sy) - [[mu].sub.n][phi]([S.sub.n]z, Sy) - [phi](Sy, y)

Thus, we have

[[mu].sub.n][phi] ([S.sub.n] z, Sy) [less than or equal to] [[mu].sub.n] [phi] ([S.sub.n]z, y).

Similarly, replacing S and T by T and S, respectively, we have

[[mu].sub.n][phi]([S.sub.n]z, Ty) [less than or equal to] [[mu].sub.n][phi]([S.sub.n]z, y).

Using Lemma 3.1, we have that A(S) [intersection] A(T) is nonempty. Additionally, if C is closed and convex, then F(S) [intersection] F(T) is nonempty.

Since commutative 2-generalized hybrid mappings in a Hilbert space are commutative 2-generalized nonspreading mappings in a Banach space, as a direct sequence of Theorem 3.2, we have the following theorem proved by Hojo, Takahashi and Takahashi [6] in a Hilbert space.

Theorem 3.3 ([6]). Let H be a Hilbert space, let C be a nonempty subset of H and let S and T be commutative 2-generalized hybrid mappings of C into itself. Suppose that there exists an element z G C such that {[S.sup.k][T.sup.l]z: k,l [member of] NU{0}} is bounded. Then A(S) [intersection] A(T) is nonempty. Additionally, if C is closed and convex, then F(S) if F(T) is nonempty.

4. NONLINEAR ERGODIC THEOREMS

Let E be a smooth Banach space, let C be a nonempty subset of E and let J be the duality mapping from E into [E.sup.*]. Observe that if T : C [right arrow] E is a 2-generalized nonspreading mapping and F (T) [not equal to] 0, then

[phi](u,Ty) [less than or equal to] [phi](u, y)

for all u [member of] F(T) and y [member of] C. Indeed, putting x = u [member of] F(T) in (3.1), we obtain that

[mathematical expression not reproducible].

So, we have that

(4.1) [phi](u, Ty) [less than or equal to] [phi](u, y)

for all u [member of] F(T) and y [member of] C. Similarly, putting y = u [member of] F(T) in (3.1), we obtain that for x [member of] C,

[mathematical expression not reproducible]

and hence

[mathematical expression not reproducible].

If [mathematical expression not reproducible], then we have from (4.1) that

[mathematical expression not reproducible].

So, we have that

(4.2) [phi](Tx,u) [less than or equal to] [phi](x,u)

for all x [member of] C and u [member of] F(T). This implies that T is generalized nonexpansive in the sense of [8].

Now using the technique developed by [22] and [28], we can prove a mean convergence theorem without convexity for commutative 2-generalized nonspreading mappings in a Banach space. For proving this result, we need the following lemmas.

Lemma 4.1. Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty subset of E. Let S and T be commutative 2-generalized nonspreading mappings of C into itself. If [[S.sup.k][T.sup.l]z : k,l [member of] N [union] {0}} for some z [member of] C is bounded and

[S.sub.n]x = 1/[(1 + n).sup.2] [n.summation over (k=0)] [n.summation over (l=0)] [S.sup.k][T.sup.l]x

for all x [member of] C and n [member of] N [union] {0}, then every weak cluster point of {[S.sub.n]x} is a point of A(S) [intersection] A(T). Additionally, if C is closed and convex, then every weak cluster point of {[S.sub.n]x} is a point of F(S) [intersection] F(T).

Proof. Since S : C [right arrow] C is 2-generalized nonspreading, we have that for all x,y [member of] C, (3.2) holds. Since there exists z [member of] C such that {[S.sup.k][T.sup.l]z : k,l [member of] N U {0}} is bounded, {[S.sup.k][T.sup.l]x : k,l [member of] NU{0}} for all x [member of] C is bounded. Then as in the proof of Theorem 3.2, we have that for any y [member of] C

[mathematical expression not reproducible].

Since {[S.sup.k][T.sup.l]x} is bounded, {[S.sub.n]x} is bounded. Thus we have a subsequence {[mathematical expression not reproducible]} of {[S.sub.n]x} such that {[mathematical expression not reproducible]} converges weakly to a point u [member of] E. Letting [n.sub.i] [right arrow] [infinity], we obtain

0 [less than or equal to] f(Sy, y) + 2<u - Sy, JSy - Jy>.

Using (2.4), we have that

0 [less than or equal to] [phi](Sy, y) + [phi](u, y) + [phi](Sy, Sy) - [phi](u, Sy) - [phi](Sy, y)

and hence

[phi](u, Sy) [less than or equal to] [phi](u, y)

This implies that u is an element of A(S). Similarly, we have that

[phi](u, Ty) [less than or equal to] [phi](u, y).

and hence u [member of] A(T). Therefore we have u [member of] A(S) [intersection] A(T). Additionally, if C is closed and convex, we have that {[S.sub.n]x} [subset] C and then

u [member of] [bar.co]{[S.sub.n]x : n [member of] N} [subset] C.

Since u [member of] A(S) [intersection] A(T) and u [member of] C, we have that

[phi](u, Su) [less than or equal to] [phi](u, u) = 0 and [phi](u, Tu) [less than or equal to] [phi](u, u) = 0

and hence

[phi](u, Su) = 0 and [phi](u,Tu) = 0.

Since E is strictly convex, we have [member of] G F(S) [intersection] F(T). This completes the proof.

Let E be a smooth Banach space. Let C be a nonempty subset of E and let T be a mapping of C into E. We denote by B(T) the set of skew-attractive points of T, i.e., B(T) = {z [member of] E: [phi](Tx, z) [less than or equal to] [phi](x, z), [for all]x [member of] C}. The following result was proved by Lin and Takahashi [20].

Lemma 4.2 ([20]). Let E be a smooth Banach space and let C be a nonempty subset of E. Let T be a mapping from C into E. Then B(T) is closed.

Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty subset of E. Let T be a mapping of C into E. Define a mapping [T.sup.*] as follows:

[T.sup.*][x.sup.*] = JT[J.sup.-1][x.sup.*], V[x.sup.*] [member of] JC,

where J is the duality mapping on E and [J.sup.-1] is the duality mapping on [E.sup.*]. A mapping [T.sup.*] is called the duality mapping of T; see also [31] and [7]. It is easy to show that if T is a mapping of C into itselt, then [T.sup.*] is a mapping of JC into itself. In fact, for [x.sup.*] [member of] JC, we have [J.sup.-1] [x.sup.*] [member of] C and hence T[J.sup.-1] [x.sup.*] [member of] C. So, we have

[T.sup.*][x.sup.*] = JT[J.sup.-1] [x.sup.*] [member of] JC.

Then, [T.sup.*] is a mapping of JC into itself. Using Lemma 2.4, we have the following result.

Lemma 4.3 ([20]). Let E be a smooth, strictly convex and reflexive Banach space and let C be a nonempty subset of E. Let T be a mapping of C into E and let [T.sup.*] be the duality mapping of T. Then, the following hold:

(1) JB(T) = A([T.sup.*]);

(2) JA(T) = B([T.sup.*]).

In particular, JB(T) is closed and convex.

Let D = {(k,l) : k,l [member of] N [union] {0}}. Then D is a directed set by the binary relation:

(k,l) [less than or equal to] (i,j) if k [less than or equal to] i and l [less than or equal to] j.

Theorem 4.4. Let E be a uniformly convex Banach space with a Frechet differentiable norm and let C be a nonempty subset of E. Let S,T : C [right arrow] C be commutative 2-generalized nonspreading mappings such that {[S.sup.k][T.sup.l]z : k,l [member of] N [union] {0}} for some z [member of] C is bounded, A(S) = B(S) and A(T) = B(T). Let R be the sunny generalized nonexpansive retraction of E onto B(S) [intersection] B(T). Then, for any x [member of] C,

[S.sub.n]x = 1/[(n + 1).sup.2] [n.summation over (k=0)][N.summation over (l=0)] [S.sup.k][T.sup.l]x

converges weakly to an element q of A(S) [intersection] A(T), where q = [lim.sub.(k,l)[member of]D] R[S.sup.k][T.sup.l]x.

Proof. We have from Theorem 3.2 that A(S) [intersection] A(T) = B(S) [intersection] B(T) is nonempty. We know from Lemmas 2.11, 4.2 and 4.3 that B(S) [intersection] B(T) is closed, and

J(B(S) [intersection] B(T)) = JB(S) [intersection] JB(T)

is closed and convex. So, from Lemma 2.5 and Lemma 2.7 there exists the sunny generalized nonexpansive retraction R of E onto B(S) [intersection] B(T). From Lemma 2.8, this retraction R is characterized by

[mathematical expression not reproducible].

We also know from Lemma 2.6 that

0 [less than or equal to] <v - Rv, JRv - Ju>, [for all]u [member of] B(S) [intersection] B(T), v [member of] C.

Adding up [phi](Rv,u) to both sides of this inequality, we have

(4.3) [mathematical expression not reproducible].

Since [phi](Sz,u) [less than or equal to] [phi](z, u) and [phi](Tz, u) [less than or equal to] [phi](z, u) for any u [member of] B(S) [intersection] B(T) and z [member of] C, it follows that for any (k, l), (i, j) [member of] D with (k, l) [less than or equal to] (i, j),

[mathematical expression not reproducible].

Hence the net [phi]([S.sup.k][T.sup.l]x, R[S.sup.k][T.sup.l]x) is nonincreasing. Putting u = R[S.sup.k][T.sup.l]x and v = [S.sup.l][T.sup.j]x with (k,l) [less than or equal to] (i, j) in (4.3), we have from Lemma 2.3 that

[mathematical expression not reproducible],

where g is a strictly increasing, continuous and convex real-valued function with g(0) = 0. From the properties of g, {R[S.sup.k][T.sup.l]x} is a Cauchy net; see [19]. Therefore {R[S.sup.k][T.sup.l]x} converges strongly to a point q [member of] B(S) [intersection] B(T). Next, consider a fixed x [member of] C and an arbitrary subsequence [mathematical expression not reproducible] which converges weakly to a point v. From the proof of Lemma 4.1, we know that v [member of] A(S) [intersection] A(T) = B(S) [intersection] B(T). Rewriting the characterization of the retraction R, we have that for any u [member of] B(S) [intersection] B (T),

0 [less than or equal to] <[S.sup.k][T.sup.l]x - R[S.sup.k][T.sup.l]x, [JRS.sup.k][T.sup.l]x - Ju>

and hence

[mathematical expression not reproducible],

where K is an upper bound for [parallel][S.sup.k][T.sup.l]x - [RS.sup.k][T.sup.l]x[parallel]. Summing up these inequalities for k = 0,1, ..., n and l = 0,1, ..., n and dividing by [(n + 1).sup.2], we arrive to

[mathematical expression not reproducible],

where [mathematical expression not reproducible]. Letting [n.sub.i] [right arrow] [infinity] and remembering that J is continuous, we get

<v - q, Ju - Jq> [less than or equal to], 0.

This holds for any u [member of] B(S) [intersection] B(T). Therefore Rv = q. But because v [member of] B(S) [intersection] B(T), we have v = q. Thus the sequence {[S.sub.n]x} converges weakly to the point q [member of] A(S) [intersection] A(T).

Using Theorem 4.4, we obtain the following theorems.

Theorem 4.5. Let E be a uniformly convex Banach space with a Frechet differentiable norm. Let S,T : E [right arrow] E be commutative [mathematical expression not reproducible]-generalized nonspreading mappings such that [[alpha].sub.1] - [[beta].sub.1] = 0, [mathematical expression not reproducible], respectively.

Assume that {[S.sup.k][T.sup.l]z : k,l [member of] N [union] {0}} for some z [member of] C is bounded. Let R be the sunny generalized nonexpansive retraction of E onto F(S) [intersection] F(T). Then, for any x [member of] E,

[mathematical expression not reproducible]

converges weakly to an element q of F(S) [intersection] F(T), where q = [lim.sub.(k,l) [member of] D] [RS.sup.k][T.sup.l]x.

Proof. Since {[S.sup.k][T.sup.l]z: k, l [member of] N [union] {0}} for some z [member of] C is bounded, we have that A(S) [intersection] A(T) = F(S) [intersection] F(T) is nonempty. We also know that [[alpha].sub.2] > [[beta].sub.2] together with [[alpha].sub.1] - [[beta].sub.1] = 0, [[gamma].sub.1] [less than or equal to] [[delta].sub.1] and [[gamma].sub.2] [less than or equal to] [[delta].sub.2] implies that

[phi](Sx, u) [less than or equal to] [phi](x, u)

for all x [member of] E and u [member of] F(S). Similarly, [[alpha]'.sub.2] > [[beta]'.sub.2] together with [[alpha]'.sub.1] - [[beta]'.sub.1] = 0, [[gamma]'.sub.1] [less than or equal to] [[delta]'.sub.1] and [[gamma]'.sub.2] [less than or equal to] [[delta]'.sub.2] implies that

[phi](Tx, v) [less than or equal to] [less than or equal to] [phi](x, v)

for all x [member of] E and v [member of] F(T). Thus, we have that F(S) = B(S) and F(T) = B(T). Therefore, we have the desired result from Theorem 4.4.

Theorem 4.6 ([6]). Let H be a Hilbert space and let C be a nonempty subset of H. Let S and T be commutative 2-generalized hybrid mappings of C into itself such that {[S.sup.k][T.sup.l]z : k, l [member of] N [union] {0}} for some z [member of] C is bounded. Let P be the metric projection of H onto A(S) if A(T). Then, for any x [member of] C,

[S.sub.n]x = 1/[(n + 1).sup.2] [n.summation over (k=0)] [n.summation over (l=0)] [S.sup.k][T.sup.l]x

converges weakly to an element q of A(S) [intersection] A(T), where q = [lim.sub.(k,l)[member of]D] P[S.sup.k][T.sup.l]x. In particular, if C is closed and convex, {[S.sub.n]x} converges weakly to an element q of F(S) f F(T).

Proof. We have from Theorem 3.2 that A(S) [intersection] A(T) is nonempty. We also have that A(S) = B(S) and A(T) = B(T). Since A(S) [intersection] A(T) is a nonempty, closed and convex subset of H, there exists the metric projection of H onto A(S) [intersection] A(T). In a Hilbert space, the metric projection of H onto A(S) [intersection] A(T) is equivalent to the sunny generalized nonexpansive retraction of H onto A(S) [intersection] A(T). On the other hand, commutative 2-generalized hybrid mappings S, T : C [right arrow] C are commutative 2-generalized nonspreading mappings. So, we have the desired result from Theorem 4.6. Furthermore, if C is closed and convex, we have that q G F(S) [intersection] F(T) and then {[S.sub.n]x} converges weakly to q [member of] F(S) [intersection] F(T).

Remark We do not know whether a mean convergence theorem of Baillon's type for nonspreading mappings in a Banach space holds or not.

Acknowledgements. The first author was partially supported by Grant-in-Aid for Scientific Research No. 15K04906 from Japan Society for the Promotion of Science. The second and the third authors were partially supported by the grant MOST 105- 2115-M-037-001 and the grant MOST 105-2115-M-039-002-MY3, respectively.

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WATARU TAKAHASHI, CHING-FENG WEN, AND JEN-CHIH YAO

Dedicated to Professor Ravi Agarwal on the occasion of his 70th birthday

Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan; Keio Research and Education Center for Natural Sciences, Keio University, Kouhoku-ku, Yokohama 223-8521, Japan; and Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan

wataru@is.titech.ac.jp; wataru@a00.itscom.net

Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan

cfwen@kmu.edu.tw

Center for General Education, China Medical University, Taichung 40402, Taiwan

yaoj c@mail.cmu.edu.tw
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