Abstract
We present two weak convergence theorems for inverse strongly accretive mappings in Banach spaces, which are supplements to the recent result of Aoyama et al. [Fixed Point Theory Appl. (2006), Art. ID 35390, 13pp.].
2000 MSC: 47H10; 47J25.
Keywords:
weak convergence theorem; accretive mapping; Banach space1. Introduction
Let E be a real Banach space with the dual space E*. We write 〈x, x* 〉 for the value of a functional x*∈ E* at x ∈ E. The normalized duality mapping is the mapping J : E → 2E* given by
In this paper, we assume that E is smooth, that is, exists for all x, y ∈ E with x = y = 1. This implies that J is singlevalued and we do consider the singleton Jx as an element in E*. For a closed convex subset C of a (smooth) Banach space E, the variational inequality problem for a mapping A : C → E is the problem of finding an element u ∈ C such that
The set of solutions of the problem above is denoted by S(C, A). It is noted that if C = E, then S(C, A) = A^{1}0 := {x ∈ E : Ax = 0}. This problem was studied by Stampacchia (see, for example, [1,2]). The applicability of the theory has been expanded to various problems from economics, finance, optimization and game theory.
Gol'shteĭn and Tret'yakov [3] proved the following result in the finite dimensional space ℝ^{N}.
Theorem 1.1. Let α > 0, and let A : ℝ^{N }→ ℝ^{N }be an αinverse strongly monotone mapping, that is, 〈Ax  Ay, ×  y〉 ≥ αAx  Ay^{2 }for all x, y ∈ ℝ^{N}. Suppose that {x_{n}} is a sequence in ℝ^{N }defined iteratively by x_{1 }∈ ℝ^{N }and
where {λ_{n}}⊂ [a, b] ⊂ (0, 2α). If A^{1 }0 ≠ ∅, then {x_{n}} converges to some element of A^{1}0.
The result above was generalized to the framework of Hilbert spaces by Iiduka et al. [4]. Note that every Hilbert space is uniformly convex and 2uniformly smooth (the related definitions will be given in the next section). Aoyama et al. [[5], Theorem 3.1] proved the following result.
Theorem 1.2. Let E be a uniformly convex and 2uniformly smooth Banach space with the uniform smoothness constant K, and let C be a nonempty closed convex subset of E. Let Q_{C }be a sunny nonexpansive retraction from E onto C, let α > 0 and let A : C → E be an αinverse strongly accretive mapping with S(C, A) ≠ ∅. Suppose that {x_{n}} is iteratively defined by
where {α_{n}} ⊂ [b, c] ⊂ (0, 1) and {λ_{n}} ⊂ [a, α/K^{2}] ⊂ (0, α/K^{2}]. Then, {x_{n}} converges weakly to some element of S(C, A).
Motivated by the result of Aoyama et al., we prove two more convergence theorems for αinverse strongly accretive mappings in a Banach space, which are supplements to Theorem 1.2 above. The first one is proved without the presence of the uniform convexity, while the last one is proved in uniformly convex space with some different control conditions on the parameters.
The paper is organized as follows: In Section 2, we collect some related definitions and known fact, which are referred in this paper. The main results are presented in Section 3. We start with some common tools in proving the main results in Section 3.1. In Section 3.2, we prove the first weak convergence theorem without the presence of uniform convexity. The second theorem is proved in uniformly convex Banach spaces in Section 3.3.
2. Definitions and related known fact
Let E be a real Banach space. If {x_{n}} is a sequence in E, we denote strong convergence of {x_{n}} to x ∈ E by x_{n }→ x and weak convergence by x_{n }⇀ x. Denote by ω_{w }({x_{n}}) the set of weakly sequential limits of the sequence {x_{n}}, that is, ω_{w }({x_{n}}) = {p : there exists a subsequence of {x_{n}} such that }. It is known that if {x_{n}} is a bounded sequence in a reflexive space, then ω_{w }({x_{n}}) = ∅.
The space E is said to be uniformly convex if for each ε ∈ (0, 2) there exists δ > 0 such that for any x, y ∈ U := {z ∈ E : z = 1}
The following result was proved by Xu.
Lemma 2.1 ([6]). Let E be a uniformly convex Banach space, and let r > 0. Then, there exists a strictly increasing, continuous and convex function g : [0, 2r] → [0, ∞) such that g(0) = 0 and
for all α ∈ [0, 1] and x, y ∈ B_{r }:= {z ∈ E : z ≤ r}.
The space E is said to be smooth if the limit
exists for all x, y ∈ U. The norm of E is said to be Fréchet differentiable if for each x ∈ U, the limit (2.1) is attained uniformly for y ∈ U.
Let C be a nonempty subset of a smooth Banach space E and α > 0. A mapping A : C → E is said to be αinverse strongly accretive if
for all x, y ∈ C. It follows from (2.2) that A is Lipschitzian, that is,
A Banach space E is 2uniformly smooth if there is a constant c > 0 such that 〉_{E }(τ) ≤ cτ^{2 }for all τ > 0 where
In this case, we say that a real number K > 0 is a 2uniform smoothness constant of E if the following inequality holds for all x, y ∈ E:
Note that every 2uniformly smooth Banach space has the Fréchet differentiable norm and hence it is reflexive.
The following observation extracted from Lemma 2.8 of [5] plays an important role in this paper.
Lemma 2.2. Let C be a nonempty closed convex subset of a 2uniformly smooth Banach space E with a 2uniform smoothness constant K. Suppose that A : C → E is an αinverse strongly accretive mapping. Then, the following inequality holds for all x, y ∈ C and λ ∈ ℝ:
where I is the identity mapping. In particular, if , then I  λA is nonex pansive, that is, (I  λA)x  (I  λA)y ≤ x  y for all x, y ∈ C.
Let C be a subset of a Banach space E. A mapping Q : E → C is said to be:
(i) sunny if Q(Qx + t(x  Qx)) = Qx for all t ≥ 0;
(ii) a retraction if Q^{2 }= Q.
It is known that a retraction Q from a smooth Banach space E onto a nonempty closed convex subset C of E is sunny and nonexpansive if and only if 〈xQx, J(Qxy)〉 ≥ 0 for all x ∈ E and y ∈ C. In this case, Q is uniquely determined. Using this result, Aoyama et al. obtained the following result. Recall that, for a mapping T : C → E, the set of fixed points of T is denoted by F (T), that is, F (T) = {x ∈ C : x = Tx}.
Lemma 2.3 ([5]). Let C be a nonempty closed convex subset of a smooth Banach space
E. Let Q_{C }be a sunny nonexpansive retraction from E onto C, and let A : C → E be a mapping. Then, for each λ > 0,
The space E is said to satisfy Opial's condition if
whenever x_{n }⇀ x ∈ E and y ∈ E satisfy x ≠ y. The following results are known from theory of nonexpansive mappings. It should be noted that Oplial's condition and the Fréchet differentiability of the norm are independent in uniformly convex space setting.
Lemma 2.4 ([7], [8]). Let C be a nonempty closed convex subset of a Banach space. E. Suppose that E is uniformly convex or satisfies Opial's condition. Suppose that T is a nonexpansive mapping of C into itself. Then, I  T is demiclosed at zero, that is, if {x_{n}} is a sequence in C such that x_{n }⇀ p and x_{n } Tx_{n }→ 0, then p = Tp.
Lemma 2.5 ([9]). Let C be a nonempty closed convex subset of a uniformly convex Banach space with a Fréchet differentiable norm. Suppose that is a sequence of nonexpansive mappings of C into itself with . Let x ∈ C and S_{n }= T_{n}T_{n1 }· · · T_{1 }for all n ≥ 1. Then, the set
consists of at most one element, where D is the closed convex hull of D.
The following two lemmas are proved in the absence of uniform convexity, and they are needed in Section 3.2.
Lemma 2.6 ([10]). Let {x_{n}} and {y_{n}} be bounded sequences in a Banach space and {α_{n}} be a real sequence in [0, 1] such that 0 < lim inf_{n→∞ }α_{n }≤ lim sup_{ n→∞ }α_{n }< 1. Suppose that x_{n+1 }= α_{n}x_{n }+ (1  α_{n})y_{n }for all n ≥ 1. If lim sup_{n→∞}(y_{n+1 } y_{n}  x_{n+1 } x_{n}) ≤ 0, then x_{n } y_{n }→ 0.
Lemma 2.7 ([11]). Let {z_{n}} and {w_{n}} be sequences in a Banach space and {α_{n}} be a real sequence in [0, 1]. Suppose that z_{n+1 }= α_{n}z_{n }+ (1  α_{n})w_{n }for all n ≥ 1. If the following properties are satisfied:
(i) and lim inf_{n→∞ }α_{n }> 0;
(ii) lim_{n→∞ }z_{n} = d and lim sup_{n→∞}w_{n} ≤ d;
(iii) the sequence is bounded;
then d = 0.
We also need the following simple but interesting results.
Lemma 2.8 ([12]). Let {a_{n}} and {b_{n}} be two sequences of nonnegative real numbers.
If and a_{n+1 }≤ a_{n }+ b_{n }for all n ≥ 1, then lim_{n→∞ }a_{n }exists.
Lemma 2.9 ([13]). Let {a_{n}} and {b_{n}} be two sequences of nonnegative real numbers. If and , then lim inf_{n→∞ }b_{n }= 0.
3. Main results
From now on, we assume that
• E is 2uniformly smooth Banach space with a 2uniform smoothness constant K;
• C is a nonempty closed convex subset of E;
• Q_{C }is a sunny nonexpansive retraction from E onto C;
• A : C → E is an αinverse strongly accretive mapping with S(C, A) ≠ ∅ and α > 0.
Suppose that {x_{n}} is iteratively defined by
where {α_{n}}⊂ [0, 1] and . For convenience, we write y_{n }≡ Q_{C }(x_{n } λ_{n }Ax_{n}).
3.1. Some properties of the sequence {x_{n}} for weak convergence theorems
We start with some propositions, which are the common tools for proving the main results in the next two subsections.
Proposition 3.1. If p ∈ S(C, A), then lim_{n→∞ }x_{n } p exists, and hence, the sequences {x_{n}} and {Ax_{n}} are both bounded.
Proof. Let p ∈ S(C, A). By the nonexpansiveness of Q_{C }(I  λ_{n }A) for all n ≥ 1 and
Lemma 2.3, we have
for all n ≥ 1. This implies that
for all n ≥ 1. Therefore, lim_{n→∞ }x_{n } p exists, and hence, the sequence {x_{n}} is bounded. Since A is Lipschitzian, we have {Ax_{n}} is bounded. The proof is finished.
Proposition 3.2. The following inequality holds:
for all n ≥ 1.
Proof. Since Q_{C }(I  λ_{n+1}A) and Q_{C }are nonexpansive, we have
□
Proposition 3.3. Suppose that E is a reflexive Banach space such that either it is uniformly convex or it satisfies Opial's condition. Suppose that {x_{n}} is a bounded sequence of C satisfying x_{n } Q_{C }(I  λ_{n}A)x_{n }→ 0 and .
Then, {x_{n}} converges weakly to some element of S(C, A).
Proof. Suppose that E is a uniformly convex Banach space or a reflexive Banach space satisfying Opial's condition. Then, ω_{w }({x_{n}}) ≠ ∅. We first prove that ω_{w}({x_{n}}) ⊂ S(C, A). To see this, let z ∈ ω_{w}({x_{n}}). Passing to a subsequence, if necessary, we assume that there exists a subsequence {n_{k}} of {n} such that and . We observe that
This implies that . By the nonexpansiveness of Q_{C }(I  λA), Lemmas 2.3 and 2.4, we obtain that z ∈ F (Q_{C }(I  λA)) = S(C, A). Hence ω_{w}({x_{n}}) ⊂ S(C, A).
We next prove that ω_{w}({x_{n}}) is exactly a singleton in the following cases.
Case 1: E is uniformly convex. We follow the idea of Aoyama et al. [5] in this case. For any n ≥ 1, we define a nonexpansive mapping T_{n }: C → C by
We get that x_{n+1 }= T_{n}T_{n1 }· · · T_{1}x_{1 }for all n ≥ 1. It follows from Lemma 2.3 that . Applying Lemma 2.5, since every 2uniformly smooth Banach space has Fréchet differentiable norm, gives
consists of at most one element. But we know that
Therefore, ω_{w}({x_{n}}) is a singleton.
Case 2: E satisfies Opial's condition. Suppose that p and q are two different elements of ω_{w}({x_{n}}). There are subsequences and of {x_{n}} such that
Since p and q also belong to S(C, A), both limits lim_{n→∞ }x_{n }p and lim_{n→∞ }x_{n }q exist. Consequently, by Opial's condition,
This is a contradiction. Hence, ω_{w}({x_{n}}) is a singleton, and the proof is finished. □
Remark 3.4. There exists a reflexive Banach space such that it satisfies Opial's condition but it is not uniformly convex. In fact, we consider E = ℝ^{2 }with the norm (x, y) = x + y for all (x, y) ∈ ℝ^{2 }. Note that E is finite dimensional, and hence it is reflexive and satisfies Opial's condition. To see that E is not uniformly convex, let x = (1, 0) and y = (0, 1), it follows that x  y = (1, 1) = 2 and x + y/2 = (1/2, 1/2) = 1 ≰ 1  δ for all δ > 0.
3.2. Convergence results without uniform convexity
In this subsection, we make use of Lemmas 2.6 and 2.7 to show that x_{n } y_{n }→ 0 under the additional restrictions on the sequences {α_{n}} and {λ_{n}}.
Proposition 3.5. Suppose that {α_{n}}⊂ [c, d] ⊂ (0, 1) and λ_{n+1 } λ_{n }→ 0. Then, x_{n } y_{n }→ 0.
Proof. We will apply Lemma 2.6. Let us rewritten the iteration as
It follows from Proposition 3.1 that {x_{n}} and {Ax_{n}} are bounded. Then, {y_{n}} = {(I  λ_{n}A) x_{n}} is bounded. Since λ_{n+1 } λ_{n }→ 0, it is a consequence of Proposition 3.2 that
Since all the requirements of Lemma 2.6 are satisfied, x_{n } y_{n }→ 0. □
Proposition 3.6. Suppose that {α_{n}} and {λ_{n}} satisfy the following properties:
(i) {a_{n}} ⊂ [c, 1) ⊂ (0, 1) and ;
(ii) and .
Then, x_{n } y_{n }→ 0.
Proof. We will apply Lemma 2.7. From the iteration, we have
where z_{n }≡ x_{n } y_{n }and . Using Proposition 3.2, we obtain
It follows from and Lemma 2.8 that d := lim_{n→∞ }z_{n} exists. We next prove that lim sup_{n→∞ }w_{n} ≤ d. Again, by Proposition 3.2, we get
Finally, for all n ≥ 1, we have
Hence, the sequence is bounded. It follows then that d = 0. □
We now have the following weak convergence theorems without uniform convexity.
Theorem 3.7. Let E be a 2uniformly smooth Banach space satisfying Opial's condition. Let C be a nonempty closed convex subset of E. Let Q_{C }be a sunny nonexpansive retraction from E onto C and A : C → E be an αinverse strongly accretive mapping with S(C, A) ≠ ∅ and α > 0. Suppose that {x_{n}} is iteratively defined by
where {α_{n}} ⊂ [0, 1] and satisfy one of the following conditions:
(i) {α_{n}} ⊂ [c, d] ⊂ (0, 1) and λ_{n+1 } λ_{n }→ 0;
(ii) {α_{n}} ⊂ [c, 1) ⊂ (0, 1), , , and .
Then, {x_{n}} converges weakly to an element in S(C, A).
Proof. Note that every 2uniformly smooth Banach space is reflexive. The result follows from Propositions 3.3, 3.5 and 3.6. □
Remark 3.8. Conditions (i) and (ii) in Theorem 3.7 are not comparable.
(1) If and {λ_{n}} is a sequence in such that λ_{n } λ_{n+1 }→ 0 and 0 < lim inf_{n→∞ }λ_{n }< lim sup_{n→∞ }λ_{n }< 1, then {α_{n}} and {λ_{n}} satisfy condition (i) but fail condition (ii).
(2) If and , then {α_{n}} and {λ_{n}} satisfy condition (ii) but fail condition (i).
Remark 3.9. Note that the Opial property and uniform convexity are independent. Theorem 3.7 is a supplementary to Theorem 3.1 of Aoyama et al. [5].
3.3. Convergence results in uniformly convex spaces
In this subsection, we prove two more convergence theorems in uniformly convex spaces, which are also a supplementary to Theorem 3.1 of Aoyama et al. [5]. Let us start with some propositions.
Proposition 3.10. Assume that E is a uniformly convex Banach space. Suppose that {α_{n}} and {λ_{n}} satisfy the following properties:
(i) {λ_{n}} ⊂ [a, α/K^{2}] ⊂ (0, α/K^{2}];
(ii) and .
Then, x_{n } y_{n }→ 0.
Proof. Let p ∈ S(C, A). Note that lim_{n→∞ }x_{n } p exists and hence both {x_{n}} and {y_{n}} are bounded. By the uniform convexity of E and Lemma 2.1, there exists a continuous and strictly increasing function g such that
for all n ≥ 1. Hence, for each m ≥ 1, we have
In particular,. It follows from and Lemma 2.9 that lim inf_{n→∞ }g(x_{n } y_{n}) = 0. By the properties of the function g, we get that lim inf_{n→∞ }x_{n } y_{n} = 0. Finally, we show that lim_{n→∞ }x_{n } y_{n} actually exists. To see this, we consider the following estimate obtained directly from Proposition 3.2:
The assertion follows since and Lemma 2.8. □
Let us recall the concept of strongly nonexpansive sequences introduced by Aoyama et al. (see [14]). A sequence of nonexpansive mappings {T_{n}} of C is called a strongly nonexpansive sequence if x_{n } y_{n } (T_{n}x_{n } T_{n}y_{n}) → 0 whenever {x_{n}} and {y_{n}} are sequences in C such that {x_{n }y_{n}} is bounded and x_{n }y_{n}T_{n}x_{n }T_{n}y_{n} → 0. It is noted that if {T_{n}} is a constant sequence, then this property reduces to the concept of strongly nonexpansive mappings studied by Bruck and Reich [15].
Proposition 3.11. Assume that E is a uniformly convex Banach space and {λ_{n}}⊂ (0, b] ⊂ (0, α/K^{2}). Then, {Q_{C }(I  λ_{n}A)} is a strongly nonexpansive sequence.
Proof. Notice first that Q_{C }is a strongly nonexpansive mapping (see [16,17]). Next, we prove that {I  λ_{n}A} is a strongly nonexpansive sequence and then the assertion follows. Let {x_{n}} and {y_{n}} be sequences in C such that {x_{n } y_{n}} is bounded and x_{n } y_{n}(I  λ_{n}A)x_{n } (I  λ_{n}A)y_{n} → 0. It follows from Lemma 2.2 that
In particular, λ_{n}Ax_{n } λ_{n}Ay_{n }→ 0 and hence
Proposition 3.12. Assume that E is a uniformly convex Banach space. Suppose that α_{n }≡ 0 and {λ_{n}} ⊂ (0, b] ⊂ (0, α/K^{2}). Then, x_{n } y_{n }→ 0.
Proof. Let us rewritten the iteration as follows:
Let p ∈ S(C, A). Notice that p = Q_{C }(I λ_{n}A)p for all n ≥ 1. Then, lim_{n→∞ }x_{n }p exists, and hence,
It follows from the preceding proposition that
□
We now obtain the following weak convergence theorems in uniformly convex spaces.
Theorem 3.13. Let E be a uniformly convex and 2uniformly smooth Banach space. Let C be a nonempty closed convex subset of E. Let Q_{C }be a sunny nonexpansive retraction from E onto C and A : C → E be an αinverse strongly accretive mapping with S(C, A) ≠ ∅ and α > 0. Suppose that {x_{n}} is iteratively defined by
where {α_{n}} ⊂ [0, 1] and satisfy one of the following conditions:
(i) and ;
(ii) α_{n }≡ 0 and {λ_{n}}⊂ [a, b] ⊂ (0, α/K^{2}).
Then, {x_{n}} converges weakly to an element in S (C, A).
Proof. The result follows from Propositions 3.3, 3.10 and 3.12. □
Remark 3.14. It is easy to see that conditions (i) and (ii) in Theorem 3.13 are not comparable.
Remark 3.15. Compare Theorem 3.13 to Theorem 1.2 of Aoyama et al., our result is a supplementary to their result. It is noted that, for example, our iteration scheme with α_{n }≡ 0 and λ_{n }≡ α/(α/K^{2}) is simpler than the one in Theorem 1.2.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contribute equally and significantly in this research work. All authors read and approved the final manuscript.
Acknowledgements
The first author is supported by the Thailand Research Fund, the Commission on Higher Education of Thailand and Khon Kaen University under Grant number 5380039. The second author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand. The third author is supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0188/2552) and Khon Kaen University under the RGJPh.D. scholarship. Finally, the authors thank Professor M. de la Sen and the referees for their comments and suggestions.
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