1. Introduction
Let E be a real Banach space with the dual space E*. Let C be a nonempty closed convex subset of E. Let T : C β C be a nonlinear mapping. We denote by F(T) the set of fixed points of T.
A mapping T : C β C is said to be nonexpansive if
Three classical iteration processes are often used to approximate a fixed point of nonexpansive mapping. The first one is introduced by Halpern [3] and is defined as follows: Take an initial point x0 β C arbitrarily and define {xn} recursively by
where is a sequence in the interval [0, 1]. The second iteration process is now known as Mannβs iteration process[6] which is defined as
where the initial point x1 is taken in C arbitrarily and the sequence is in the interval [0,1]. The third iteration process is referred to as Ishikawaβs iteration process [5] which is defined recursively by
where the initial point x1 is taken in C arbitrarily, are sequences in the interval [0, 1].
In general not much is known regarding the convergence of the iteration processes (1.1)-(1.3) unless the underlying space E has elegant properties which we briefly mention here.
Recently, Matsushita and Takahashi [7] proved strong convergence theorems for approximation of fixed points of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. More precisely, they proved the following theorem.
Theorem 1.1. Let E be a uniformly convex and uniformly smooth Banach space, let C be a nonempty closed convex subset of E, let T be a relatively nonexpansive mapping from C into itself and let {Ξ±n} be a sequence of real numbers such that 0 β€ Ξ±n < 1 and lim supnββ Ξ±n < 1. Suppose that {xn} is given by
where J is the duality mapping on E. If F(T) is nonempty, then {xn} converges strongly to Ξ F(T)x0, where Ξ F(T) is the generalized projection from C onto F(T).
In [4], Hao introduced the following iterative scheme for approximating a fixed point of asymptotically quasi-Ο-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach space: x0 β E, C1 = C, x1 = Ξ C1x0,
where ΞΎn = max{0, suppβF(T),xβC(Ο(p, Tnx) β Ο(p, x))}.
Motivated by the fact above, the purpose of this paper is to prove a strong convergence theorem for finding a fixed point of asymptotically quasi-Ο-nonexpansive mappings in the intermediate sense in a reflexive, strictly convex and smooth Banach space, which has the Kadec-Klee property.
2. Preliminaries
Let E be a real Banach space and let E* be the dual space of E. The duality mapping J : E β 2E* is defined by
By Hahn-Banach theorem, J(x) is nonempty.
The modulus of convexity of E is the function Ξ΄E : (0, 2] β [0, 1] defined by
E is said to be uniformly convex if βΞ΅ β (0, 2], there exists a Ξ΄ = Ξ΄(Ξ΅) > 0 such that for x, y β E with β₯xβ₯ β€ 1, β₯yβ₯ β€ 1 and β₯x β yβ₯ β₯ Ξ΅, then Equivalently, E is uniformly convex if and only if Ξ΄E(Ξ΅) > 0, βΞ΅ β (0, 2]. E is strictly convex if for all x, y β E, x β y, β₯xβ₯ = β₯yβ₯ = 1, we have β₯Ξ»x+(1βΞ»)yβ₯ < 1, βΞ» β (0, 1). The space E is said to be smooth if the limit
exists for all x, y β S(E) = {z β E : β₯zβ₯ = 1}. It is also said to be uniformly smooth if the limit exists uniformly in x, y β S(E).
It is well known that if E is uniformly smooth, then J is norm-to-norm uniformly continuous on each bounded subset of E. If E is smooth, then J is single-valued.
Recall that a Banach space E has the Kadec-Klee property if for any sequence {xn} β E and x β E with xn β x and β₯xnβ₯ β β₯xβ₯, then β₯xn β xβ₯ β 0 as n β β. It is well known that if E is a uniformly convex Banach space, then E has the Kadec-Klee property.
In what follows, we always use Ο : E Γ E β β to denote the Lyapunov functional defined by
It follows from the definition of Ο that
and
Following Alber [1], the generalized projection Ξ C : E β C is defined by
The existence and uniqueness of the operator Ξ C follows from the properties of the function Ο(x, y) and strict monotonicity of mapping J (see [1,2,10]).
Lemma 2.1 ([1]). Let E be a reflexive, strictly convex and smooth Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:
Remark 2.1. If E is a real Hilbert space, then Ο(x, y) = β₯x β yβ₯2 and Ξ C is the metric projection PC of E onto C.
Definition 2.2. Let C be a nonempty closed convex subset of E and let T be a mapping from C into itself. A point p β C is said to be an asymptotic fixed point of T if C contains a sequence {xn}, which converges weakly to p and limnββ β₯xn β Txnβ₯ = 0.
The set of asymptotic fixed points of T is denoted by
Definition 2.3. A mapping T : C β C is said to be
(1) relatively nonexpansive if and
for all x β C and p β F(T);
(2) quasi-Ο-nonexpansive if F(T)β Ο and
for all x β C and p β F(T);
(3) asymptotically quasi-Ο-nonexpansive if F(T)β Ο and there exists a sequence {kn} β [0,β) with kn β 1 as n β β such that
for all x β C, p β F(T) and n β₯ 1;
(4) asymptotically quasi-Ο-nonexpansive in the intermediate sense if F(T)β Ο and
Put
Remark 2.2. From the definition, it is obvious that ΞΎn β 0 as n β β and
Remark 2.3. (1) It is easy to see that the class of quasi-Ο-nonexpansive mappings contains the class of relatively nonexpansive mappings.
(2) The class of asymptotically quasi-Ο-nonexpansive mappings is more general than the class of relatively asymptotically nonexpansive mappings.
(3) The class of asymptotically quasi-Ο-nonexpansive mappings in the intermediate sense is a generalization of the class of asymptotically quasi-nonexpansive mappings in the intermediate sense in the framework.
Recall that T is said to be asymptotically regular on C if for any bounded subset K of C,
Definition 2.4. A mapping T : C β C is said to be closed if for any sequence {xn} β C with xn β x and Txn β y, Tn = y.
Lemma 2.5 ([4]). Let E be a reflexive, strictly convex and smooth Banach space such that both E and E* have the Kadec-Klee property. Let C be a nonempty closed and convex subset of E. Let T : C β C be a closed and asymptotically quasi-Ο-nonexpansive mapping in the intermediate sense. Then F(T) is a closed convex subset of C.
3. Main results
Theorem 3.1. Let E be a reflexive, strictly convex and smooth Banach space such that both E and E* have the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let T : C β C be a closed, asymptotically regu-lar and asymptotically quasi-Ο-nonexpansive mapping in the intermediate sense. Let {Ξ±n} be a sequence in [0, 1] and {Ξ²n} be a sequence in (0, 1) satisfying the following conditions:
Let {xn} be a sequence generated by
where ΞΎn = max{0, suppβF(T);xβC(Ο(p, Tnx) β Ο(p, x))}, Ξ Cn+1 is the general-ized projection of E onto Cn+1. If F(T) is bounded in C, then {xn} converges strongly to Ξ F(T)x1.
Proof. It follows from Lemma 2.2 that F(T) is a closed convex subset of C, so that Ξ F(T)x is well defined for any x β C.
We split the proof into six steps.
Step 1. We first show that Cn, n β₯ 1, is nonempty, closed and convex.
It is obvious that C1 = C is closed and convex. Suppose that Cn is closed and convex for some n β₯ 2. For z1, z2 β Cn+1, we see that z1, z2 β Cn. It follows that z = tz1 + (1 β t)z2 β Cn, where t β (0, 1). Notice that
and
These are equivalent to
and
Multiplying t and 1 β t on both sides of (3.2) and (3.3), respectively, we obtain that
That is,
Therefore, we have
This implies that Cn+1 is closed and convex for all n β₯ 1.
Step 2. We show that F(T) β Cn, βn β₯ 1.
For n = 1, we have F(T) β C1 = C. Now, assume that F(T) β Cn for some n β₯ 2. Put wn = Jβ1(Ξ²nJxn + (1βΞ²n)JTnxn). For each x* β F(T), we obtain from (2.2) and (2.3) that
and
Therefore, we have
So, x* β Cn+1. It implies that F(T) β Cn+1.
Step 3. We prove that {xn} is bounded and limnββ Ο(xn, x1) exists.
Since xn = Ξ Cnx1, we have from Lemma 2.1 that
Again, since F(T) β Cn, we have
It follows from Lemma 2.1 that for each u β F(T) and for each n β₯ 1,
Therefore, {Ο(xn, x1)} is bounded. By virtue of (2.1), {xn} is also bounded. Again, since xn = Ξ Cnx1, xn+1 = Ξ Cn+1x1 and xn+1 β Cn+1 β Cn for all n β₯ 1, we have
This implies that {Ο(xn, x1)} is nondecreasing and bounded. Hence, limnββ Ο(xn, x1) exists.
Step 4. Next, we prove that where is some point in C.
Now, since {xn} is bounded and the space E is reflexive, we may assume that there exists a subsequence {xni} of {xn} such that Since Cn is closed and convex, it is easy to see that for each n β₯ 1. This implies that
On the other hand, it follows from the weak lower semicontinuity of the norm that
which implies that as ni β β. Hence, as ni β β. In view of the Kadec Klee property of E, we see that as ni β β. If there exists another subsequence such that we have
which implies This shows that
Step 5. Now we prove that
Since and limnββ Ο(xn, x1) exists, we see that
Hence, we have
Since xn+1 β Cn+1, xn β and Ξ±n β 0, it follows from (3.1) and Remark 2.2 that
as n β β. This implies that
Therefore we obtain
and so
This shows that {Jyn} is bounded. Since E is reflexive, E* is reflexive. Without loss of generality, we can assume that In view of reflexivity of E, we see that J(E) = E*. Hence, there exists y β E such that This implies that J(yn) β Jy. And
Taking lim infnββ for both sides of (3.7), we have from (3.4) that
which shows that and so
It follows from (3.6) and the Kadec-Klee property of E* that Since Jβ1 is norm-weak-continuous, we have
It follows from (3.5),(3.8) and the Kadec-Klee property of E that we have
On the other hand, since {xn} is bounded and T is asymptotically quasi-Ο-nonexapnsive in the intermediate sense, for any given p β F(T), we have from (2.3) that
This implies that {Tnxn} is bounded. Since
it implies that {wn} is also bounded. From (3.1), we have
It follows from (3.9) that as n β β. Since Jβ1 is norm-weaklycontinuous, this implies that
as n β β. Note that
This together with (3.10) shows that
as n β β. Since we have Since
By condition (ii) and (3.11), we have that
Since Jβ1 is norm-weakly-continuous, this implies that
It follows from (3.12) that
This together with (3.13) and the Kadec-Klee property of E shows that
as n β β. Again, by the asymptotic regularity of T, we have
as n β β. That is, It follows from the closedness of T that
Step 6. Finally, we prove that
Let w = Ξ F(T)x1. Since w β F(T) β Cn and xn = Ξ Cnx1, we have
This implies that
From the definition of and (3.14), we see that This completes the proof.
Remark 3.1. If we take Ξ±n = 0 for all n β β, then the iterative scheme (3.1) reduces to following scheme:
where
which is (1.2) and an improvement to (1.1).
In the framework of Hilbert spaces, Theorem 3.1 is reduced to the following.
Corollary 3.2. Let E be a Hilbert space. Let C be a nonempty closed convex subset of H. Let T : C β C be a closed, asymptotically regular and asymptot-ically quasi-Ο-nonexpansive mapping in the intermediate sense. Let {Ξ±n} be a sequence in [0, 1] and {Ξ²n} be a sequence in (0, 1) satisfying the following condi-tions:
Let {xn} be a sequence generated by
where
is the metric projection from E onto Cn+1. If F(T) is bounded in C, then {xn} converges strongly to PF(T)x1.
Proof. . If E is a Hilbert space, then J = I (the identity mapping) and Ο(x, y) = β₯x β yβ₯2. We can obtain the desired conclusion easily from Theorem 3.1. This completes the proof.
If T is quasi-Ο- nonexpansive, then Theorem 3.1 is reduced to the following without involving boundedness of F(T) and asymptotically regularity on C.
Corollary 3.3. Let E be a reflexive, strictly convex and smooth Banach space such that both E and E* have the Kadec-Klee property. Let C be a nonempty closed convex subset of E. Let T : C β C be a closed, quasi-Ο-nonexpansive mapping with F(T) β Ο. Let {Ξ±n} be a sequence in [0, 1] and {Ξ²n} be a sequence in (0, 1) satisfying the following conditions:
Let {xn} be a sequence generated by
where is the generalized projection of E onto Cn+1. Then {xn} converges strongly to Ξ F(T)x1.
Remark 3.2. (1) By Remark 3.1, Theorem 3.1 extends Theorem 2.1 of Hao [4].
(2) Theorem 3.1 generelized Theorem 3.1 of Matsushita and Takahashi [7] in the following respects:
(3) Corollary 3.1 generalized and improves Corollary 2.5 of Hao [4], Theorem 3.4 of Nakajo and Takahashi [8] and Theorem 2.1 of Su and Qin [9] in the following aspects:
μ°Έκ³ λ¬Έν
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