Abstract
In this paper, an iterative sequence for relatively nonexpansive multi-valued mappings by using the notion of generalized projection is introduced, and then weak and strong convergence theorems are proved.
2000 Mathematics Subject Classification: 47H09; 47H10; 47J25.
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1 Introduction and preliminaries
Let D be a nonempty closed convex subset of a real Banach space X. A single-valued mapping T : D → D is called nonexpansive if ||T(x) - T(y)|| ≤ ||x - y|| for all x, y ∈ D. Let N(D) and CB(D) denote the family of nonempty subsets and nonempty closed bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by
for A1, A2 ∈ CB(D), where d(x, A1) = inf {||x - y||; y ∈ A1}. The multi-valued mapping T : D → CB(D) is called nonexpansive if H(T(x), T(y)) ≤ ||x - y|| for all x, y ∈ D. An element p ∈ D is called a fixed point of T : D → N(D) (respectively, T : D → D) if p ∈ F(T) (respectively, T(p) = p). The set of fixed points of T is represented by F(T).
Let X be a real Banach space with dual X*. We denote by J the normalized duality mapping from X to defined by
where 〈.,.〉 denotes the generalized duality pairing.
The Banach space X is strictly convex if ||(x + y)/2|| < 1 for all x, y ∈ X with ||x|| = ||y|| = 1 and x ≠ y. The Banach space X is uniformly convex if lim n →∞ ||x n - y n || = 0 for any two sequences {x n }, {y n } ⊆ X with ||x n || = ||y n || = 1 for all n ∈ ℕ and lim n →∞ ||(x n + y n )/2|| = 1.
Lemma 1.1. [1]Let X be a uniformly convex Banach space and B r = {x ∈ X : ||x|| ≤ r}, r > 0. Then, there exists a continuous, strictly increasing, and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that
for all x, y ∈ B r and all α, β ∈ [0, 1] with α + β = 1.
The norm of Banach space X is said to be Gâteaux differentiable if for each x, y ∈ S(X):= {x ∈ X : ||x|| = 1} the limit
exists. In this case, X is called smooth. The norm of Banach space X is said to be Fréchet differentiable if for each x ∈ S(X), limit (1.1) is attained uniformly for y ∈ S(X) and the norm is uniformly Fréchet differentiable if limit (1.1) is attained uniformly for x, y ∈ S(X). In this case, X is said to be uniformly smooth. The following properties of J are well known [2]:
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1.
X (X*, resp.) is uniformly convex if and only if X* (X, resp.) is uniformly smooth;
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2.
If X is smooth, then J is single-valued and norm-to-weak* continuous;
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3.
If X is reflexive, then J is onto;
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4.
If X is strictly convex, then J(x) ∩ J(y) = ∅ for all x ≠ y;
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5.
If X has a Fréchet differentiable norm, then J is norm-to-norm continuous;
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6.
If X is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of X.
The normalized duality mapping J of a smooth Banach space X is called weakly sequentially continuous if x n ⇀ x implies that , where ⇀ denotes the weak convergence and denotes the weak* convergence.
Let X be a smooth Banach space. The function ϕ : X × X → ℝ is defined by
It is obvious from the definition of the function ϕ that
In addition, the function ϕ has the following property:
Lemma 1.2. [3, Remark 2.1] Let X be a strictly convex and smooth Banach space, then ϕ(x, y) = 0 if and only if x = y.
Lemma 1.3. [4]Let X be a uniformly convex and smooth Banach space and r > 0. Then
for all y, z ∈ B r = {x ∈ X; ||x|| ≤ r}, where g : [0, ∞) → [0, ∞) is a continuous, strictly increasing and convex function with g(0) = 0.
Let D be a nonempty closed convex subset of a smooth Banach space X. A point p ∈ D is called an asymptotic fixed point of T : D → D[5], if there exists a sequence {x n } in D which converges weakly to p and lim n →∞ ||x n - T(x n )|| = 0. The set of asymptotic fixed points of T is represented by . A mapping T : D → D is called relatively nonexpansive [3, 6–8], if the following conditions are satisfied:
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1.
F(T) is nonempty;
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2.
ϕ(p, T(x)) ≤ ϕ(p, x), ∀x ∈ D, p ∈ F(T);
3..
Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X. It is known that [4, 9] for any x ∈ X, there exists a unique point x0 ∈ D such that
Following Alber [9], we denote such an element x0 by Π D x. The mapping Π D is called the generalized projection from X onto D. If X is a Hilbert space, then ϕ(y, x) = ||y - x||2 and Π D is the metric projection of X onto D.
Lemma 1.4. [4, 9]Let D be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach space X. Then
Lemma 1.5. [4, 9]Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space X. Let x ∈ X and z ∈ D, then
In 2004, Matsushita and Takahashi [10] introduced the following iterative sequence for finding a fixed point of relatively nonexpansive mapping T : D → D. Given x1 ∈ D,
where D is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space X, Π D is the generalized projection onto D and {α n } is a sequence in [0, 1].
They proved weak and strong convergence theorems in uniformly convex and uniformly smooth Banach space X.
Iterative methods for approximating fixed points of multi-valued mappings in Banach spaces have been studied by some authors, see for instance [11–14].
Let D be a nonempty closed convex subset of a smooth Banach space X. We define an asymptotic fixed point for a multi-valued mapping as follows.
Definition 1.6. A point p ∈ D is called an asymptotic fixed point of T : D → N(D), if there exists a sequence {x n } in D which converges weakly to p and lim n →∞d(x n , T(x n )) = 0.
Moreover, we define a relatively nonexpansive multi-valued mapping as follows.
Definition 1.7. A multi-valued mapping T : D → N(D) is called relatively nonexpansive, if the following conditions are satisfied:
1. F(T) is nonempty;
2. ϕ(p, z) ≤ ϕ(p, x), ∀x ∈ D, z ∈ T(x), p ∈ F(T);
3.,
whereis the set of asymptotic fixed points of T.
There exist relatively nonexpansive multi-valued mappings that are not nonexpansive.
Example 1.8. Let I = [0,1], X = Lp (I), 1 < p < ∞ and D = {f ∈ X; f(x) ≥ 0, ∀x ∈ I}. Let T : D → CB(D) be defined by
It is clear that F(T) = {0}. Let . Then, there exists a sequence {f n } in D which converges weakly to h, and z n = d(f n , T(f n )) → 0. Let n ∈ ℕ, we have
Since z n → 0, we have ||f n || p → 0. Therefore, f n → 0. Hence, h = 0. Therefore,. Let f ∈ D such that f(x) > 1 for all x ∈ I, and g ∈ T(f), then
Next, let f ∈ D such that there exists x ∈ I such that f(x) ≤ 1, then
Hence, T is relatively nonexpansive. However, if f(x) = 2 and g(x) = 1 for all x ∈ I, we get . Then, H(T(f), T(g)) > ||f - g|| p = 1. Hence, T is not nonexpansive.
In this article, inspired by Matsushita and Takahashi [10], we introduce the following iterative sequence for finding a fixed point of relatively nonexpansive multi-valued mapping T : D → N(D). Given x1 ∈ D,
where z n ∈ T(x n ) for all n ∈ ℕ, D is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space X, Π D is the generalized projection onto D and {α n } is a sequence in [0, 1]. We prove weak and strong convergence theorems in uniformly convex and uniformly smooth Banach space X.
2 Main results
In this section, at first, concerning the fixed point set of a relatively nonexpansive multi-valued mapping, we prove the following proposition.
Proposition 2.1. Let X be a strictly convex and smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : D → N(D) is a relatively nonexpansive multi-valued mapping. Then, F(T) is closed and convex.
Proof. First, we show F(T) is closed. Let {x n } be a sequence in F(T) such that x n → x*. Since T is relatively nonexpansive, we have
for all z ∈ T(x*) and for all n ∈ ℕ. Therefore,
By Lemma 1.2, we obtain x* = z. Hence, T(x*) = {x*}. So, we have x* ∈ F(T). Next, we show F(T) is convex. Let x, y ∈ F(T) and t ∈ (0, 1), put p = tx + (1 - t)y. We show p ∈ F(T). Let w ∈ T(p), we have
By Lemma 1.2, we obtain p = w. Hence, T(p) = {p}. So, we have p ∈ F(T). Therefore, F(T) is convex. □
Remark 2.2. Let X be a strictly convex and smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : D → N(D) is a relatively nonexpansive multi-valued mapping. If p ∈ F(T), then T(p) = {p}.
Proposition 2.3. Let X be a uniformly convex and smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : D → N(D) is a relatively nonexpansive multi-valued mapping. Let {α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 for all n ∈ ℕ. For a given x1 ∈ D, let {x n } be the iterative sequence defined by (1.5). Then, {Π F (T)x n } converges strongly to a fixed point of T, where Π F (T)is the generalized projection from D onto F(T).
Proof. By Proposition 2.1, F(T) is closed and convex. So, we can define the generalized projection Π F (T)onto F(T). Let p ∈ F(T). From Lemma 1.4, we have
Hence, lim n → ∞ϕ(p, x n ) exists. So, {ϕ(p, x n )} is bounded. Then, by (1.2) we have {x n } is bounded, and hence, {z n } is bounded. Let u n = Π F (T)x n , for all n ∈ ℕ. Then, we have
Therefore
for all m ∈ ℕ. From Lemma 1.4, we obtain
By (2.4) and (2.6) we have
It follows that {ϕ(u n , x n )} converges. From u n + m = Π F (T)x n + m and Lemma 1.4, we have
Hence, by (2.5) we obtain
for all m, n ∈ ℕ. Let r = sup n ∈ℕ ||u n ||. From Lemma 1.3, there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that
for all m, n ∈ ℕ, n > m. Therefore, {u n } is a Cauchy sequence. Since X is complete and F(T) is closed, there exists q ∈ F(T) such that {u n } converges strongly to q. □
If the duality mapping J is weakly sequentially continuous, we have the following weak convergence theorem.
Theorem 2.4. Let X be a uniformly convex and uniformly smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : D → N(D) is a relatively nonexpansive multi-valued mapping. Let {α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 for all n ∈ ℕ and lim inf n →∞α n (1 - α n ) > 0. For a given x1 ∈ D, let {x n } be the iterative sequence defined by (1.5). If J is weakly sequentially continuous, then {x n } converges weakly to a fixed point of T.
Proof. As in the proof of Proposition 2.3, {x n } and {z n } are bounded. So, there exists r > 0 such that x n , z n ∈ B r for all n ∈ ℕ. Since X is a uniformly smooth Banach space, X* is a uniformly convex Banach space. Let p ∈ F(T). By Lemma 1.1, there exists a continuous, strictly increasing and convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that
Hence
Since lim n →∞ϕ(p, x n ) exists and lim inf n →∞α n (1 - α n ) > 0, we obtain
Therefore,
Since J-1 is uniformly norm-to-norm continuous on bounded sets, we have
Since d(x n , T(x n )) ≤ ||x n - z n ||, we obtain
Let u n = Π F ( T ) x n . By Lemma 1.5, we have
for each w ∈ F(T). From Proposition 2.3, there exists p ∈ F(T) such that {u n } converges strongly to p. Let be a subsequence of {x n } such that converges weakly to q. Then, by (2.11) we have q ∈ F(T). It follows from (2.12) that
Let j → ∞ in inequality (2.13), since J is weakly sequentially continuous we have
Since J is monotone, we have
It follows from (2.14) and (2.15) that
Since X is strictly convex, we have p = q. Therefore, {x n } converges weakly to p. The proof is complete. □
Theorem 2.5. Let X be a uniformly convex and uniformly smooth Banach space, and D a nonempty closed convex subset of X. Suppose T : D → N(D) is a relatively nonexpansive multi-valued mapping. Let {α n } be a sequence of real numbers such that 0 ≤ α n ≤ 1 for all n ∈ ℕ and lim inf n →∞α n (1 - α n ) > 0. For a given x1 ∈ D, let {x n } be the iterative sequence defined by (1.5). If the interior of F(T) is nonempty, then {x n } converges strongly to a fixed point of T.
Proof. Since the interior of F(T) is nonempty, there exists p ∈ F(T) and r > 0 such that p + rh ∈ F(T), whenever ||h|| ≤ 1. By (1.3) for any q ∈ F(T) we have
Therefore,
Since p + rh ∈ F(T), as in the proof of Proposition 2.3, we have
It follows from (2.18) and (2.19) that
Then, by (2.18) and (2.20) we have
whenever ||h|| ≤ 1. Therefore, we obtain
It follows that
for all m, n ∈ ℕ, n > m. As in the proof of Proposition 2.3, {ϕ(p, x n )} converges. Hence, {J(x n )} is a Cauchy sequence. Since X* is complete, {J(x n )} converges strongly to a point in X*. Since X* has a Fréchet differentiable norm, then J-1 is norm-to-norm continuous on X*. Hence, {x n } converges strongly to some point u in D. As in the proof of Theorem 2.4, lim n →∞d(x n , T(x n )) = 0. Hence, we have u ∈ F(T), where u = lim n →∞ Π F ( T )x n . □
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Acknowledgements
AR would like to thank the School of Mathematics of the Institute for Research in Fundamental Sciences, Tehran, Iran, for supporting this research (Grant No. 90470122).
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Homaeipour, S., Razani, A. Weak and strong convergence theorems for relatively nonexpansive multi-valued mappings in Banach spaces. Fixed Point Theory Appl 2011, 73 (2011). https://doi.org/10.1186/1687-1812-2011-73
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DOI: https://doi.org/10.1186/1687-1812-2011-73