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Convergence of the modified Mann's iterative method for asymptotically κ-strictly pseudocontractive mappings
Fixed Point Theory and Applications volume 2011, Article number: 100 (2011)
Abstract
Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and K a nonempty, closed, and convex subset of E. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping with a nonempty fixed point set. We prove that (I - T) is demiclosed at 0 and obtain a weak convergence theorem of the modified Mann's algorithm for T under suitable control conditions. Moreover, we also elicit a necessary and sufficient condition that guarantees strong convergence of the modified Mann's iterative sequence to a fixed point of T in a real Banach spaces with the Fréchet differentiable norm.
2000 AMS Subject Classification: 47H09; 47H10.
1 Introduction
Let E and E* be a real Banach space and the dual space of E, respectively. Let K be a nonempty subset of E. Let J denote the normalized duality mapping from E into 2E* given by J(x) = {f ∈ E* : 〈x, f〉 = ||x||2 = ||f||2}, for all x ∈ E, where 〈·,·〉 denotes the duality pairing between E and E*. In the sequel, we will denote the set of fixed points of a mapping T : K → K by F (T) = {x ∈ K : Tx = x}.
A mapping T : K → K is called asymptotically κ-strictly pseudocontractive with sequence such that lim n→∞ κ n = 1 (see, e.g., [1–3]) if for all x, y ∈ K, there exist a constant κ ∈ [0, 1) and j(x - y) ∈ J(x - y) such that
If I denotes the identity operator, then (1) can be written as
The class of asymptotically κ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces, j is the identity and it is shown by Osilike et al. [2] that (1) (and hence (2)) is equivalent to the inequality
where lim n→∞ λ n = lim n→∞ [1 + 2(κ n - 1)] = 1, λ = (1 - 2κ) ∈ [0, 1).
A mapping T with domain D(T) and range R(T) in E is called strictly pseudocontractive of Browder-Petryshyn type [4], if for all x, y ∈ D(T), there exists κ ∈ [0, 1) and j(x - y) ∈ J(x - y) such that
If I denotes the identity operator, then (3) can be written as
In Hilbert spaces, (3) (and hence (4)) is equivalent to the inequality
It is shown in [5] that the class of asymptotically κ-strictly pseudocontractive mappings and the class of κ-strictly pseudocontractive mappings are independent.
A mapping T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that
for all x, y ∈ K and is said to be demiclosed at a point p if whenever {x n } ⊂ D(T) such that {x n } converges weakly to x ∈ D(T ) and {Tx n } converges strongly to p, then Tx = p.
Kim and Xu [6] studied weak and strong convergence theorems for the class of asymptotically κ-strictly pseudocontractive mappings in Hilbert space. They obtained a weak convergence theorem of modified Mann iterative processes for this class of mappings. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection method. They proved the following.
Theorem KX [6] Let K be a closed and convex subset of a Hilbert space H. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence {κ n } ⊂ [1, ∞) such thatand F(T ) ≠ ∅. Letbe a sequence generated by the modified Mann's iteration method:
Assume that the control sequence is chosen in such a way that κ + λ ≤ α n ≤ 1 - λ for all n, where λ ∈ (0, 1) is a small enough constant. Then, {x n } converges weakly to a fixed point of T.
The modified Mann's iteration scheme was introduced by Schu [7, 8] and has been used by several authors (see, for example, [1–3, 9–11]). One question is raised naturally: is the result in Theorem KX true in the framework of the much general Banach space?
Osilike et al. [5] proved the convergence theorems of modified Mann iteration method in the framework of q-uniformly smooth Banach spaces which are also uniformly convex. They also obtained that a modified Mann iterative process {x n } converges weakly to a fixed point of T under suitable control conditions. However, the control sequence {α n } ⊂ [0,1] depended on the Lipschizian constant L and excluded the natural choice These are motivations for us to improve the results. We prove the demiclosedness principle and weak convergence theorem of the modified Mann's algorithm for T in the framework of uniformly convex Banach spaces which have the Fréchet differentiable norm. Moreover, we also elicit a necessary and sufficient condition that guarantees strong convergence of the modified Mann's iterative sequence to a fixed point of T in a real Banach spaces with the Fréchet differentiable norm.
We will use the notation:
-
1.
⇀ for weak convergence.
-
2.
denotes the weak ω-limit set of {x n }.
2 Preliminaries
Let E be a real Banach space. The space E is called uniformly convex if for each ε > 0, there exists a δ > 0 such that for x, y ∈ E with ||x|| ≤ 1, ||y|| ≤ 1, ||x - y|| ≥ ε, we have . The modulus of convexity of E is defined by
for all ε ∈ [0,2]. E is uniformly convex if δ E (0) = 0 and δ E (ε) > 0 for all ε ∈ (0, 2]. The modulus of smoothness of E is the function ρ E : [0, ∞) ∈ [0, ∞) defined by
E is uniformly smooth if and only if
E is said to have a Fréchet differentiable norm if for all x ∈ U = {x ∈ E : ||x|| = 1}
exists and is attained uniformly in y ∈ U. In this case, there exists an increasing function b : [0, ∞) → [0, ∞) with such that for all x, h ∈ E
It is well known (see, for example, [[12], p. 107]) that uniformly smooth Banach space has a Fréchet differentiable norm.
Lemma 2.1 [2, p. 80] Let , , be nonnegative sequences of real numbers satisfying the following inequality
If and , then lim n →∞ a n exists. If in addition has a subsequence which converges strongly to zero, then lim n →∞ a n = 0.
Lemma 2.2 [2, p. 78] Let E be a real Banach space, K a nonempty subset of E, and T : K → K an asymptotically κ-strictly pseudocontractive mapping. Then, T is uniformly L-Lipschitzian.
Lemma 2.3 [[13], p. 29] Let K be a nonempty, closed, convex, and bounded subset of a uniformly convex Banach space E, and let T : K → E be a nonexpansive mappings. Let {x n } be a sequence in K such that {x n } converges weakly to some point x ∈ K. Then, there exists an increasing continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter of K such that
Lemma 2.4 [[14], p. 9] Let E be a real Banach space with the Fréchet differentiable norm.
For x ∈ E, let β*(t) be defined for 0 < t < ∞ by
Then, lim t →0 + β*(t) = 0 and
Remark 2.5 In a real Hilbert space, we can see that β*(t) = t for t > 0. In our more general setting, throughout this article we will still assume that
where β* is a function appearing in (6).
Then, we prove the demiclosedness principle of T in the uniformly convex Banach space which has the Fréchet differentiable norm.
Lemma 2.6 Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm. Let K be a nonempty, closed, and convex subset of E and T : K → K an asymptotically κ-strictly pseudocontractive mapping with F(T) ≠ ∅. Then, (I - T) is demiclosed at 0.
Proof. Let {x n } be a sequence in K which converges weakly to p ∈ K and {x n - Tx n } converges strongly to 0. We prove that (I - T)(p) = 0. Let x* ∈ F(T). Then, there exists a constant r > 0 such that ||x n - x*|| ≤ r, ∀n ≥ 1. Let , and let . Then, C is nonempty, closed, convex, and bounded, and {x n } ⊆ C. Choose any α ∈ (0, κ) and let T α,n : K → K be defined for all x ∈ K by
Then for all x, y ∈ K,
where . (In fact, in (7) the domain of β*(·) requires ||x - y - (Tnx-Tny)|| ≠ 0. But when ||x - y - (Tnx-Tny)|| = 0, we have ||T α,n x-T α,n y||2 = ||x - y||2, which still satisfies the inequality . So we do not specially emphasize the situation that the argument of β*(·) equals 0 in this inequality and the following proof of Theorem 3.1.) Define G α,m : K → E by
Then, G α,m is nonexpansive and it follows from Lemma 2.3 that there exists an increasing continuous function h : [0, ∞) → [0, ∞) with h(0) = 0 depending on the diameter of K such that
Observe that
and as n → ∞
Thus, it follows from (9) and (10) that
so that (8) implies that
Observe that
so that
Since T is continuous, we have (I - T)(p) = 0, completing the proof of Lemma 2.6. □
Lemma 2.7 Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping with F(T) ≠ ∅. Let be the sequence satisfying the following conditions:
-
(a)
exists for every p ∈ F(T );
-
(b)
;
-
(c)
exists for all t ∈ [0, 1] and for all p1, p2 ∈ F (T ).
Then, the sequence {x n } converges weakly to a fixed point of T.
Proof. Since lim n →∞ ||x n - p|| exists, then {x n } is bounded. By (b) and Lemma 2.6, we have . Assume that and that and are subsequences of {x n } such that and , respectively. Since E has the Fréchet differentiable norm, by setting x = p1 - p2, h = t(x n - p1) in (5) we obtain
where b is an increasing function. Since ||x n - p1|| ≤ M, ∀n ≥ 1, for some M > 0, then
Therefore,
Hence, . Since , then lim n →∞ 〈x n - p1, j(p1 - p2)〉 exists. Since lim n →∞ 〈x n - p1, j(p1 - p2)〉 = 〈p - p1, j(p1 - p2)〉, for all . Set p = p2. We have 〈p2 - p1, j(p1 - p2)〉 = 0, that is, p2 = p1. Hence, is singleton, so that {x n } converges weakly to a fixed point of T. □
3 Main results
Theorem 3.1 Let E be a real uniformly convex Banach space which has the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence , such that and let F(T) ≠ ∅. Assume that the control sequence is chosen so that
-
(i*)
0 < α n < κ, n ≥ 1;
-
(ii*)
. (11)
Given x1 ∈ K, then the sequence is generated by the modified Mann's algorithm:
converges weakly to a fixed point of T.
Proof. Pick a p ∈ F(T). We firstly show that lim n →∞ ||x n - p|| exists. To see this, using (2) and (6), we obtain
Obviously,
Let δ n = 1 + 2α n (κ n - 1). Since , we have
then (14) implies lim n →∞ ||x n - p|| exists by Lemma 2.1 (and hence the sequence {||x n - p||} is bounded, that is, there exists a constant M > 0 such that ||x n - p|| < M ).
Then, we prove lim n →∞ ||x n - Tx n || = 0. In fact, it follows from (13) that
Then,
Since , then (15) implies that lim inf n →∞ ||x n - Tnx n || = 0. Thus lim n →∞ ||x n - Tnx n || = 0.
By Lemma 2.2 we know that T is uniformly L-Lipschitzian, then there exists a constant L > 0, such that
Hence, lim n →∞ ||x n - Tx n || = 0.
Now we prove that for all p1, p2 ∈ F(T), lim n →∞ ||tx n + (1 - t)p1 - p2|| exists for all t ∈ [0, 1]. Let σ n (t) = ||tx n + (1 - t)p1 - p2||. It is obvious that lim n →∞ σ n (0) = ||p1 - p2|| and lim n →∞ σ n (1) = lim n →∞ ||x n - p2|| exist. So, we only need to consider the case of t ∈ (0, 1).
Define T n : K → K by
Then for all x, y ∈ K,
By the choice of α n , we have ||T n x - T n y||2 ≤ [1 + 2α n (κ n - 1)]||x - y||2. For the convenience of the following discussing, set , then ||T n x - T n y|| ≤ λ n ||x - y||.
Set S n,m = T n + m -1T n + m -2 ··· T n , m ≥ 1. We have
and
Set b n,m = ||S n,m (tx n + (1 - t)p1) - tS n,m x n - (1 - t)S n,m p1||. If ||x n - p1|| = 0 for some n0, then x n = p1 for any n ≥ n0 so that lim n →∞ ||x n - p1|| = 0, in fact {x n } converges strongly to p1 ∈ F(T). Thus, we may assume ||x n - p1|| > 0 for any n ≥ 1. Let δ denote the modulus of convexity of E. It is well known (see, for example, [[15], p. 108]) that
for all t ∈ [0, 1] and for all x, y ∈ E such that ||x|| ≤ 1, ||y|| ≤ 1. Set
Then, ||w n , m || ≤ 1 and ||z n , m || ≤ 1 so that it follows from (16) that
Observe that
and
it follows from (17) that
Since E is uniformly convex, then is nondecreasing, and since , hence it follows from (18) that
Since and since δ(0) = 0 and lim n →∞ ||x n - p1|| exists, then the continuity of δ yields lim n →∞ b n,m = 0 uniformly for all m ≥ 1. Observe that
Hence, lim sup n →∞ σ n (t) ≤ lim inf n →∞ σ n (t), this ensures that lim n →∞ σ n (t) exists for all t ∈ (0, 1).
Now, apply Lemma 2.7 to conclude that {x n } converges weakly to a fixed point of T. □
Theorem 3.2 Let E be a real Banach space with the Fréchet differentiable norm, and let K be a nonempty, closed, and convex subset of E. Let T : K → K be an asymptotically κ-strictly pseudocontractive mapping for some 0 ≤ κ < 1 with sequence {κ n } ⊂ [1, ∞) such that , let F(T) ≠ ∅. Let {α n } be a real sequence satisfying the condition (11). Given x1 ∈ K, let be the sequence generated by the modified Mann's algorithm (12). Then, the sequence {x n } converges strongly to a fixed point of T if and only if
where d(x n , F(T)) = inf p ∈ F ( T )||x n - p||.
Proof. In the real Banach space E with the Fréchet differentiable norm, we still have
as we have already proved in Theorem 3.1. Thus, [d(x n +1 - p)]2 ≤ δ n [d(x n - p)]2 and it follows from Lemma 2.1 that lim n -∞ d(x n , F(T )) exists.
Now if {x n } converges strongly to a fixed point p of T, then lim n →∞ ||x n - p|| = 0. Since
we have lim inf n →∞ d(x n , F(T )) = 0.
Conversely, suppose lim inf n →∞ d(x n , F(T)) = 0, then the existence of lim n →∞ d(x n , F (T)) implies that lim n →∞ d(x n , F(T)) = 0. Thus, for arbitrary ε > 0 there exists a positive integer n0 such that for any n ≥ n0.
From (19), we have
and for some M > 0, ||x n - p|| < M. Now, an induction yields
Since , then there exists a positive integer n1 such that , ∀n ≥ n1. Choose N = max{n0, n1}, then for all n, m ≥ N + 1 and for all p ∈ F (T ) we have
Taking infimum over all p ∈ F(T), we obtain
Thus, is Cauchy. We can also prove lim n →∞ ||x n - Tx n || = 0 as we have done in Theorem 3.1. Suppose lim n →∞ x n = u. Then,
Thus, u ∈ F(T). □
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This study was supported by the Youth Teacher Foundation of North China Electric Power University.
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Zhang, Y., Xie, Z. Convergence of the modified Mann's iterative method for asymptotically κ-strictly pseudocontractive mappings. Fixed Point Theory Appl 2011, 100 (2011). https://doi.org/10.1186/1687-1812-2011-100
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DOI: https://doi.org/10.1186/1687-1812-2011-100