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J. Nonlinear Sci. Appl. 8 (2015), 354–362
Research Article
Attractive points and convergence theorems of
generalized hybrid mapping
Yuchun Zheng∗
School of Mathematics and Information Science, Henan Normal University, XinXiang, P.R. China, 453007
Abstract
In this paper, by means of the concept of attractive points of a nonlinear mapping, we prove strong convergence theorem of the Ishikawa iteration for an (α, β)−generalized hybrid mapping in a uniformly convex
Banach space, and obtain weak convergence theorem of the Ishikawa iteration for such a mapping in a
Hilbert space.
Keywords: Attractive points, generalized hybrid mapping, Ishikawa iteration, Mann iteration, Xu’s
inequality.
2010 MSC: 47H10, 54H25, 49J40, 47H05, 47H04, 65J15, 47H10.
1. Introduction
Let E be a Banach space with the norm k · k and let K be a nonempty subset of E. In 2010, Kocourek,
Takahashi and Yao [5] firstly introduced the concept of the generalized hybrid mapping, which contains the
classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings. A mapping T : K → K
is called (α, β)−generalized hybrid if there exist α, β ∈ R such that
αkT x − T yk2 + (1 − α)kx − T yk2 ≤ βkT x − yk2 + (1 − β)kx − yk2
(1.1)
for all x, y ∈ K, where R is the set of real numbers. T is said to be nonexpansive if T is (1, 0)−generalized
hybrid; T is called hybrid (Takahashi [10]) if T is ( 23 , 12 )−generalized hybrid, i.e.
3kT x − T yk2 ≤ kx − T yk2 + kT x − yk2 + kx − yk2 ∀x, y ∈ K;
T is called nonspreading (Kohsaka and Takahashi [6]) if T is (2, 1)−generalized hybrid, i.e.
2kT x − T yk2 ≤ kx − T yk2 + kT x − yk2 ∀x, y ∈ K.
∗
Corresponding author
Email address: zhengyuchun1@yeah.net (Yuchun Zheng)
Received 2015-1-16
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
355
Baillon [1] proved the first nonlinear ergodic theorem: suppose that K is a nonempty closed convex
subset of Hilbert space E and T : K → K is nonexpansive mapping such that F (T ) 6= ∅, then ∀x ∈ K, the
Cesàro means
n
1 X i
Tn x =
T x
n+1
i=0
weakly converges to a fixed point of T .
Bruck [2, 3] studied the property of Cesàro means for nonexpansive mapping in uniformly convex Banach
space. Takahashi and Yao [11] proved the nonlinear ergodic theorem for both hybrid and nonspreading
mappings in a Hilbert space. Kocourek, Takahashi and Yao [5] showed that both the nonlinear ergodic
theorem and the weak convergence theorem of the Mann iteration for (α, β)−generalized hybrid mapping.
The Mann iteration is the original definition of Mann [7] for a nonexpansive mapping T ,
xn+1 = αn xn + (1 − αn )T xn {αn } ⊂ (0, 1), x1 ∈ K.
Takahashi and Takeuchi [12] obtained the nonlinear ergodic theorem without convexity for (α, β)−generalized
hybrid mappings. Hojo and Takahashi [4] showed the strong convergence of the Halpern iteration of Cesàro
means for (α, β)−generalized hybrid mapping T under some proper conditions,
n
xn+1
1 X i
= αn u + (1 − αn )
T xn {αn } ⊂ (0, 1), u, x1 ∈ K.
n+1
i=0
In this paper, we will deal with strong and weak convergence of the Ishikawa iteration for finding
attractive points of (α, β)−generalized hybrid mappings under some conditions on the sequences {αn } and
{βn } in (0, 1),
(
xn+1 = αn xn + (1 − αn )T yn
(1.2)
yn
= βn xn + (1 − βn )T xn
Our results obviously develop and complement the corresponding ones of Kocourek, Takahashi and Yao
[5], Takahashi and Yao [11] , Takahashi and Takeuchi [12], Takahashi [10] and others.
2. Preliminaries and basic results
Let N and R be the sets of positive integers and real numbers, respectively. Let K be a nonempty subset
of a Banach space E with the norm k · k and let T be a mapping T from K to E. A point y ∈ E is called
an attractive point of T if for all x ∈ K
kT x − yk ≤ kx − yk.
We denote by A(T ) the set of all attractive points of T , i.e.,
A(T ) = {y ∈ E; kT x − yk ≤ kx − yk ∀x ∈ K}.
Takahashi and Takeuchi [12] used this concept and proved the closed and convex property of A(T ) in a
Hilbert space H. For more details, see Takahashi and Takeuchi [12].
A Banach space E is said to be uniformly convex if for all
ε ∈ [0, 2],
∃δε > 0
such that
kx + yk
< 1 − δε whenever kx − yk ≥ ε.
2
The following lemmas are well-known which can be found in [13].
kxk = kyk = 1 implies
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
356
Lemma 2.1. (Xu [13, Theorem 2]) Let q > 1 and r > 0 be two fixed real numbers. Then a Banach space is
uniformly convex if and only if there exists a continuous strictly increasing convex function g : [0, +∞) →
[0, +∞) with g(0) = 0 such that
kλx + (1 − λ)ykq ≤ λkxkq + (1 − λ)kykq − ωq (λ)g(kx − yk),
(2.1)
for all x, y ∈ Br (0) = {x ∈ E; kxk ≤ r} and λ ∈ [0, 1], where ωq (λ) = λq (1 − λ) + λ(1 − λ)q .
Note that the inequality in Lemma 2.1 is known as Xu’s inequality.
Let H be a real Hilbert space with the norm k · k and the inner produce h·, ·i. Obviously, the Xu’s
inequality is replaced by the following equality in a Hilbert space H, for x, y ∈ H and t ∈ R,
ktx + (1 − t)yk2 = tkxk2 + (1 − t)kyk2 − t(1 − t)kx − yk2 .
(2.2)
Lemma 2.2. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E
and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn }
is defined by the Ishikawa iteration
(
xn+1 = αn xn + (1 − αn )T yn
(2.3)
yn
= βn xn + (1 − βn )T xn
where the sequences {αn } and {βn } in (0, 1) such that
lim inf βn (1 − αn )(1 − βn ) > 0.
n→∞
(2.4)
Then (i) the sequence {xn } is bounded;
(ii) the limit lim kxn − uk exists for each u ∈ A(T );
n→∞
(iii) lim kxn − T xn k = 0.
n→∞
Proof. Take u ∈ A(T ).
By the definition of the attractive point, we have
kxn+1 − uk = kαn (xn − u) + (1 − αn )(T yn − u)k
≤ αn kxn − uk + (1 − αn )kT yn − uk
≤ αn kxn − uk + (1 − αn )kyn − uk
≤ αn kxn − uk + (1 − αn )(βn kxn − uk + (1 − βn )kT xn − uk)
≤ (αn + (1 − αn )(βn + (1 − βn )))kxn − uk
≤ kxn − uk
..
.
≤ kx1 − uk.
So the sequence {xn } is bounded and the sequence {kxn − uk} is monotone non-increasing, and hence the
limit lim kxn − uk exists for each u ∈ A(T ).
n→∞
Now we show (iii).
Let
r ≥ max kxn − uk.
n∈N
Then
kT yn − uk ≤ kyn − uk ≤ kxn − uk ≤ r and kT xn − uk ≤ kxn − uk ≤ r.
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
357
It follows from Lemma 2.1(q = 2) that
kxn+1 − uk2
= kαn (xn − u) + (1 − αn )(T yn − u)k2
≤ αn kxn − uk2 + (1 − αn )kT yn − uk2
≤ αn kxn − uk2 + (1 − αn )kyn − uk2
= αn kxn − uk2 + (1 − αn )kβn (xn − u) + (1 − βn )(T xn − u)k2
≤ αn kxn − uk2 + (1 − αn )(βn kxn − uk2 + (1 − βn )kT xn − uk2
−βn (1 − βn )g(kT xn − xn k))
≤ αn kxn −
uk2
+ (1 − αn )(βn kxn − uk2 + (1 − βn )kxn − uk2
−βn (1 − βn )g(kT xn − xn k))
≤ kxn −
uk2
− (1 − αn )βn (1 − βn )g(kT xn − xn k).
(2.5)
Then we have
(1 − αn )βn (1 − βn )g(kT xn − xn k) ≤ kxn − uk2 − kxn+1 − uk2 ,
and so,
∞
X
(1 − αn )βn (1 − βn )g(kT xn − xn k) ≤ kx1 − uk2 < +∞.
n=1
From the condition ((2.4)), it follows that
lim g(kT xn − xn k) = 0.
n→∞
By the property of the function g, we have
lim kT xn − xn k = 0.
n→∞
This completes the proof.
When αn = 0 for all n, the following conclusions hold obviously.
Corollary 2.3. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E
and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn }
is defined by the following iteration
xn+1 = T (βn xn + (1 − βn )T xn )
(2.6)
where the sequence {βn } in (0, 1) such that
lim inf βn (1 − βn ) > 0.
n→∞
(2.7)
Then (i) the sequence {xn } is bounded;
(ii) the limit lim kxn − uk exists for each u ∈ A(T );
n→∞
(iii) lim kxn − T xn k = 0.
n→∞
3. Strongly Convergent Theorems
Let K be a nonempty subset of a Banach space E. A mapping T : K → K is said to satisfy Condition
I if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for r ∈ (0, ∞) such that
kx − T xk ≥ f (d(x, A(T ))) for all x ∈ K,
where d(x, A(T )) = inf{kx − yk; y ∈ A(T )}. This concept was introduced by Senter and Dotson [9] and the
examples of mappings that satisfy Condition I was given.
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
358
Theorem 3.1. Let K be a nonempty closed and convex subset of a uniformly convex Banach space E and
and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅ and satisfying Condition I.
Suppose that the sequence {xn } is defined by the Ishikawa iteration
(
xn+1 = αn xn + (1 − αn )T yn
(3.1)
yn
= βn xn + (1 − βn )T xn ,
where the sequences {αn } and {βn } in (0, 1) such that
lim inf βn (1 − αn )(1 − βn ) > 0.
n→∞
(3.2)
Then the sequence {xn } converges strongly to an attractive point z of T .
Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and
kxn+1 − uk ≤ kxn − uk for each u ∈ A(T ) and lim kxn − T xn k = 0.
n→∞
(3.3)
Then Condition I implies lim f (d(xn , A(T ))) = 0, and hence
n→∞
lim d(xn , A(T )) = 0.
n→∞
(3.4)
Next we show that the sequence {xn } is a Cauchy sequence of E. In fact, for any n, m ∈ N, without loss
of generality, we may set m > n, then kxm − uk ≤ kxn − uk for each u ∈ A(T ) by ((3.3)), and so
kxn − xm k ≤ kxn − uk + ku − xm k ≤ 2kxn − uk.
(3.5)
Since u is arbitrary, then we may take the infimum for u in ((3.5)),
kxn − xm k ≤ 2 inf{kxn − uk; u ∈ A(T )} = 2d(xn , A(T )).
From ((3.4)), it follows that as lim kxn − xm k = 0, which means that {xn } is a Cauchy sequence. So there
n→∞
exists z ∈ E such that
lim kxn − zk = 0.
n→∞
By ((3.3)), we have
lim kT xn − zk = 0.
n→∞
Now we prove z ∈ A(T ). In fact, it follows from the definition of (α, β)−generalized hybrid mapping
that for all x ∈ K,
αkT xn − T xk2 + (1 − α)kxn − T xk2 ≤ βkT xn − xk2 + (1 − β)kxn − xk2 .
(3.6)
Let n → ∞ in ((3.6)). Then by the continuity of the norm k · k and the function g(t) = t2 , we have
αkz − T xk2 + (1 − α)kz − T xk2 ≤ βkz − xk2 + (1 − β)kz − xk2 ,
and hence
kz − T xk ≤ kz − xk for all x ∈ K.
So z ∈ A(T ) and lim kxn − zk = 0. The proof is completed.
n→∞
A mapping T : K → E is said to be demicompact (Petryshyn [8]) provided whenever a sequence
{xn } ⊂ K is bounded and the sequence {xn − T xn } strongly converges, then there is a subsequence {xnk }
which strongly converges.
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
359
Theorem 3.2. Let K be a nonempty closed and convex subset of a uniformly convex Banach space E and
and let T : K → K be (α, β)−generalized hybrid and demicompact with A(T ) 6= ∅. Suppose that the sequence
{xn } is defined by the Ishikawa iteration ( (3.1)) and the sequences {αn } and {βn } satisfy ( (3.2)). Then the
sequence {xn } converges strongly to an attractive point z of T .
Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and
lim kxn − T xn k = 0.
n→∞
(3.7)
Then the demicompactness of T implies there is a subsequence {xnk } ⊂ {xn } and z ∈ E such that
lim kxnk − zk = 0.
k→∞
(3.8)
By ((3.7)), we also have
lim kT xnk − zk = 0.
k→∞
From the definition of (α, β)−generalized hybrid mapping, it follows that for all x ∈ K,
αkT xnk − T xk2 + (1 − α)kxnk − T xk2 ≤ βkT xnk − xk2 + (1 − β)kxnk − xk2 .
(3.9)
Let k → ∞ in ((3.9)). Then by the continuity of the norm k · k and the function g(t) = t2 , we have
kz − T xk ≤ kz − xk for all x ∈ K.
So z ∈ A(T ). Since lim kxn − uk exists for each u ∈ A(T ) by Lemma 2.2 (ii), then we have
n→∞
lim kxn − zk = 0.
n→∞
The proof is completed.
The condition (3.2) contains αn ≡ 0 and lim inf βn (1 − βn ) > 0 as special cases. So the following result
n→∞
is obtained easily.
Corollary 3.3. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E
and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn }
is defined by the following iteration
xn+1 = T (βn xn + (1 − βn )T xn )
(3.10)
where the sequence {βn } in (0, 1) such that
lim inf βn (1 − βn ) > 0.
n→∞
(3.11)
Assume that T either satisfies Condition I or is demicompact. Then the sequence {xn } converges strongly
to an attractive point z of T .
4. Weakly Convergent Theorems
Let {xn } is a sequence in E, then xn * x will denote weak convergence of the sequence {xn } to x.
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
360
Theorem 4.1. Let K be a nonempty closed and convex subset of a Hilbert space H and let T : K → K
be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the
Ishikawa iteration
(
xn+1 = αn xn + (1 − αn )T yn
(4.1)
yn
= βn xn + (1 − βn )T xn ,
where the sequences {αn } and {βn } in (0, 1) such that
lim inf βn (1 − αn )(1 − βn ) > 0.
n→∞
(4.2)
Then the sequence {xn } converges weakly to an attractive point z of T .
Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and
lim kxn − uk exists for each u ∈ A(T ) and lim kxn − T xn k = 0.
n→∞
n→∞
(4.3)
Then there exists a subsequence {xnk } of {xn } and z ∈ H such that xnk * z. We claim z ∈ A(T ). In fact,
it follows from the definition of (α, β)−generalized hybrid mapping that for all x ∈ K,
αkT xnk − T xk2 + (1 − α)kxnk − T xk2 ≤ βkT xnk − xk2 + (1 − β)kxnk − xk2 .
(4.4)
Then we have
α(kxnk − T xk2 + 2hxnk − T x, T xnk − xnk i + kT xnk − xnk k2 ) + (1 − α)kxnk − T xk2
= αkT xnk − T xk2 + (1 − α)kxnk − T xk2
≤ βkT xnk − xk2 + (1 − β)kxnk − xk2
≤ β(kT xnk − xnk k2 + 2hT xnk − xnk , xnk − xi + kxnk − xk2 ) + (1 − β)kxnk − xk2 .
Let k → ∞. Then by ((4.3)) ( lim kxnk − T xnk k = 0), we have
k→∞
lim sup kxnk − T xk2 ≤ lim sup kxnk − xk2 .
k→∞
k→∞
Since xnk * z and
kxnk − xk2 = kxnk − T xk2 + 2hxnk − T x, T x − xi + kT x − xk2 ,
then
lim sup kxnk − xk2 = lim sup kxnk − T xk2 + 2hz − T x, T x − xi + kT x − xk2 .
k→∞
Since kz −
xk2
k→∞
= kz −
T xk2
+ 2hz − T x, T x − xi + kT x − xk2 , we have
2hz − T x, T x − xi + kT x − xk2 = kz − xk2 − kz − T xk2 ,
and hence
lim sup kxnk − xk2 = lim sup kxnk − T xk2 + kz − xk2 − kz − T xk2 .
k→∞
k→∞
From ((4.5)), it follows that
lim sup kxnk − T xk2 ≤ lim sup kxnk − xk2
k→∞
k→∞
= lim sup kxnk − T xk2 + kz − xk2 − kz − T xk2 ,
k→∞
(4.5)
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
361
and so
kz − T xk ≤ kz − xk for all x ∈ K.
That is, z ∈ A(T ).
Now we prove {xn } converges weakly to z. Suppose not, then there exists another subsequence {xni } of
{xn } which weakly converges to some y 6= z. Again in the same way, we have y ∈ A(T ).
From ((4.3)), it follows that both lim kxn − zk and lim kxn − yk exist. By the elementary properties
n→∞
n→∞
in Hilbert space, we easily obtain
lim kxn − zk2 = lim kxnk − zk2
n→∞
k→∞
= lim (kxnk − yk2 + 2hxnk − y, y − zi + ky − zk2 )
k→∞
= lim kxnk − yk2 + 2hz − y, y − zi + ky − zk2
k→∞
= lim kxnk − yk2 − ky − zk2
k→∞
= lim kxn − yk2 − ky − zk2
n→∞
= lim kxni − yk2 − ky − zk2
i→∞
= lim (kxni − zk2 + 2hxni − z, z − yi + kz − yk2 ) − ky − zk2
i→∞
= lim kxni − zk2 − 2kz − yk2
i→∞
= lim kxn − zk2 − 2kz − yk2 ,
n→∞
which implies z = y, a contradiction. Thus, {xn } converges weakly to an attractive point z of T .
Take αn ≡ 0. We also obtained easily the following.
Corollary 4.2. Let K be a nonempty closed and convex subset of a Hilbert space H and let T : K → K
be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the
following iteration
xn+1 = T (βn xn + (1 − βn )T xn )
(4.6)
where the sequence {βn } in (0, 1) such that
lim inf βn (1 − βn ) > 0.
n→∞
(4.7)
Then the sequence {xn } converges weakly to an attractive point z of T .
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