Open Access

Another weak convergence theorems for accretive mappings in banach spaces

Fixed Point Theory and Applications20112011:26

DOI: 10.1186/1687-1812-2011-26

Received: 15 November 2010

Accepted: 8 August 2011

Published: 8 August 2011

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 space

1. 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
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equa_HTML.gif
In this paper, we assume that E is smooth, that is, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq1_HTML.gif exists for all x, y E with ||x|| = ||y|| = 1. This implies that J is single-valued 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 : CE is the problem of finding an element u C such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equb_HTML.gif

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-10 := {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 N .

Theorem 1.1. Let α > 0, and let A : N N be an α-inverse strongly monotone mapping, that is, 〈Ax - Ay, × - y〉 ≥ α||Ax - Ay||2for all x, y N . Suppose that {x n } is a sequence in N defined iteratively by x1 N and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equc_HTML.gif

where {λ n } [a, b] (0, 2α). If A-1 0 ≠ , then {x n } converges to some element of A-10.

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 2-uniformly 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 2-uniformly 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 : CE be an α-inverse strongly accretive mapping with S(C, A). Suppose that {x n } is iteratively defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equd_HTML.gif

where {α n } [b, c] (0, 1) and {λ n } [a, α/K2] (0, α/K2]. 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 α-inverse 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq2_HTML.gif of {x n } such that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq3_HTML.gif }. 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}
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Eque_HTML.gif

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
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equf_HTML.gif

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
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equ1_HTML.gif
(2.1)

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 : CE is said to be α-inverse strongly accretive if
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equ2_HTML.gif
(2.2)
for all x, y C. It follows from (2.2) that A is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq4_HTML.gif -Lipschitzian, that is,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equg_HTML.gif
A Banach space E is 2-uniformly smooth if there is a constant c > 0 such that 〉 E (τ) ≤ 2 for all τ > 0 where
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equh_HTML.gif
In this case, we say that a real number K > 0 is a 2-uniform smoothness constant of E if the following inequality holds for all x, y E:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equi_HTML.gif

Note that every 2-uniformly 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 2-uniformly smooth Banach space E with a 2-uniform smoothness constant K. Suppose that A : CE is an α-inverse strongly accretive mapping. Then, the following inequality holds for all x, y C and λ :
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equj_HTML.gif

where I is the identity mapping. In particular, if https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq5_HTML.gif , 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 : EC is said to be:
  1. (i)

    sunny if Q(Qx + t(x - Qx)) = Qx for all t ≥ 0;

     
  2. (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 〈x-Qx, J(Qx-y)〉 ≥ 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 : CE, 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 : CE be a mapping. Then, for each λ > 0,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equk_HTML.gif
The space E is said to satisfy Opial's condition if
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equl_HTML.gif

whenever x n x E and y E satisfy xy. 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq6_HTML.gif is a sequence of nonexpansive mappings of C into itself with https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq7_HTML.gif . Let x C and S n = T n Tn-1· · · T1for all n ≥ 1. Then, the set
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equm_HTML.gif

consists of at most one element, where https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq8_HTML.gif 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 infn→∞α n ≤ lim supn→∞α n < 1. Suppose that xn+1= α n x n + (1 - α n )y n for all n ≥ 1. If lim supn→∞(||yn+1- y n || - ||xn+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 zn+1= α n z n + (1 - α n )w n for all n ≥ 1. If the following properties are satisfied:
  1. (i)

    https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq9_HTML.gif and lim infn→∞ α n > 0;

     
  2. (ii)

    limn→∞||z n || = d and lim supn→∞||w n || ≤ d;

     
  3. (iii)

    the sequence https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq10_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq11_HTML.gif and an+1a n + b n for all n ≥ 1, then limn→∞a n exists.

Lemma 2.9 ([13]). Let {a n } and {b n } be two sequences of nonnegative real numbers. If https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq12_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq13_HTML.gif , then lim infn→∞b n = 0.

3. Main results

From now on, we assume that

  • E is 2-uniformly smooth Banach space with a 2-uniform smoothness constant K;

  • C is a nonempty closed convex subset of E;

  • QC is a sunny nonexpansive retraction from E onto C;

  • A : CE is an α-inverse strongly accretive mapping with S(C, A) ≠ and α > 0.

Suppose that {x n } is iteratively defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equn_HTML.gif

where {α n } [0, 1] and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq14_HTML.gif . 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 limn→∞||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
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equo_HTML.gif
for all n ≥ 1. This implies that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equp_HTML.gif

for all n ≥ 1. Therefore, limn→∞||x n - p|| exists, and hence, the sequence {x n } is bounded. Since A is https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq4_HTML.gif -Lipschitzian, we have {Ax n } is bounded. The proof is finished.

Proposition 3.2. The following inequality holds:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equq_HTML.gif

for all n ≥ 1.

Proof. Since Q C (I - λn+1A) and Q C are nonexpansive, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equr_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq15_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq16_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq17_HTML.gif . We observe that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equs_HTML.gif

This implies that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq18_HTML.gif . 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 : CC by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equt_HTML.gif
We get that xn+1= T n Tn-1· · · T1x1 for all n ≥ 1. It follows from Lemma 2.3 that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq19_HTML.gif . Applying Lemma 2.5, since every 2-uniformly smooth Banach space has Fréchet differentiable norm, gives
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equu_HTML.gif
consists of at most one element. But we know that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equv_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq20_HTML.gif of {x n } such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equw_HTML.gif
Since p and q also belong to S(C, A), both limits limn→∞||x n -p|| and limn→∞||x n -q|| exist. Consequently, by Opial's condition,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equx_HTML.gif

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 = 2 with the norm ||(x, y)|| = |x| + |y| for all (x, y) 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 λn+1- λ n → 0. Then, x n - y n → 0.

Proof. We will apply Lemma 2.6. Let us rewritten the iteration as
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equy_HTML.gif
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 λn+1- λ n → 0, it is a consequence of Proposition 3.2 that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equz_HTML.gif

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:
  1. (i)

    {a n } [c, 1) (0, 1) and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq9_HTML.gif ;

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq22_HTML.gif .

     

Then, x n - y n → 0.

Proof. We will apply Lemma 2.7. From the iteration, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equaa_HTML.gif
where z n x n - y n and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq23_HTML.gif . Using Proposition 3.2, we obtain
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equab_HTML.gif
It follows from https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq24_HTML.gif and Lemma 2.8 that d := limn→∞||z n || exists. We next prove that lim supn→∞||w n || ≤ d. Again, by Proposition 3.2, we get
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equac_HTML.gif
Finally, for all n ≥ 1, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equad_HTML.gif

Hence, the sequence https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq25_HTML.gif 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 2-uniformly 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 α-inverse strongly accretive mapping with S(C, A) and α > 0. Suppose that {x n } is iteratively defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equae_HTML.gif
where {α n } [0, 1] and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq15_HTML.gif satisfy one of the following conditions:
  1. (i)

    {α n } [c, d] (0, 1) and λ n+1- λ n → 0;

     
  2. (ii)

    {α n } [c, 1) (0, 1), https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq22_HTML.gif , and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq21_HTML.gif .

     

Then, {x n } converges weakly to an element in S(C, A).

Proof. Note that every 2-uniformly 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. (1)

    If https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq26_HTML.gif and {λ n } is a sequence in https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq27_HTML.gif such that λ n - λ n+1→ 0 and 0 < lim infn→∞ λ n < lim supn→∞ λ n < 1, then {α n } and {λ n } satisfy condition (i) but fail condition (ii).

     
  2. (2)

    If https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq28_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq29_HTML.gif , 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:
  1. (i)

    {λ n } [a, α/K 2] (0, α/K 2];

     
  2. (ii)

    https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq31_HTML.gif .

     

Then, x n - y n → 0.

Proof. Let p S(C, A). Note that limn→∞||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
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equaf_HTML.gif
for all n ≥ 1. Hence, for each m ≥ 1, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equag_HTML.gif
In particular, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq32_HTML.gif . It follows from https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq30_HTML.gif and Lemma 2.9 that lim infn→∞g(||x n - y n ||) = 0. By the properties of the function g, we get that lim infn→∞||x n - y n || = 0. Finally, we show that limn→∞||x n - y n || actually exists. To see this, we consider the following estimate obtained directly from Proposition 3.2:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equah_HTML.gif

The assertion follows since https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq33_HTML.gif 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/K2). 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
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equai_HTML.gif
In particular, λ n Ax n - λ n Ay n → 0 and hence
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equaj_HTML.gif

Proposition 3.12. Assume that E is a uniformly convex Banach space. Suppose that α n ≡ 0 and {λ n } (0, b] (0, α/K2). Then, x n - y n → 0.

Proof. Let us rewritten the iteration as follows:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equak_HTML.gif
Let p S(C, A). Notice that p = Q C (I -λ n A)p for all n ≥ 1. Then, limn→∞||x n -p|| exists, and hence,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equal_HTML.gif
It follows from the preceding proposition that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equam_HTML.gif

We now obtain the following weak convergence theorems in uniformly convex spaces.

Theorem 3.13. Let E be a uniformly convex and 2-uniformly 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 : CE be an α-inverse strongly accretive mapping with S(C, A) ≠ and α > 0. Suppose that {x n } is iteratively defined by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_Equan_HTML.gif
where {α n } [0, 1] and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq15_HTML.gif satisfy one of the following conditions:
  1. (i)

    https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-26/MediaObjects/13663_2010_Article_29_IEq31_HTML.gif ;

     
  2. (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/K2) is simpler than the one in Theorem 1.2.

Declarations

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 RGJ--Ph.D. scholarship. Finally, the authors thank Professor M. de la Sen and the referees for their comments and suggestions.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Khon kaen University

References

  1. Kinderlehrer D, Stampacchia G: An introduction to variational inequalities and their applications. In Pure and Applied Mathematics. Volume 88. Academic Press, Inc., New York; 1980. xiv+313Google Scholar
  2. Lions J-L, Stampacchia G: Variational inequalities. Comm Pure Appl Math 1967, 20: 493–519. 10.1002/cpa.3160200302MathSciNetView ArticleGoogle Scholar
  3. Gol'shteĭn EG, Tret'yakov NV: Modified Lagrangians in convex programming and their generalizations. Point-to-set maps and mathematical programming. Math Programming Stud 1979, 10: 86–97.View ArticleGoogle Scholar
  4. Iiduka H, Takahashi W, Toyoda M: Approximation of solutions of variational inequalities for monotone mappings. Panamer Math J 2004, 14: 49–61.MathSciNetGoogle Scholar
  5. Aoyama K, Iiduka H, Takahashi W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl Art 2006, 2006: 13. ID 35390Google Scholar
  6. Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-KMathSciNetView ArticleGoogle Scholar
  7. Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Nonlinear functional analysis. In Proceedings of Symposia in Pure Mathematics, vol. XVIII, Part 2, Chicago, IL, 1968, pp. 1–308. American Mathematical Society, Providence; 1976.Google Scholar
  8. Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleGoogle Scholar
  9. Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. J Math Anal Appl 1979, 67: 274–276. 10.1016/0022-247X(79)90024-6MathSciNetView ArticleGoogle Scholar
  10. Suzuki T: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl 2005, 2005: 103–123. 10.1155/FPTA.2005.103View ArticleGoogle Scholar
  11. Deng L: Convergence of the Ishikawa iteration process for nonexpansive mappings. J Math Anal Appl 1996, 199: 769–775. 10.1006/jmaa.1996.0174MathSciNetView ArticleGoogle Scholar
  12. Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J Math Anal Appl 1993, 178: 301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleGoogle Scholar
  13. Ofoedu EU: Strong convergence theorem for uniformly L -Lipschitzian asymptotically pseudo-contractive mapping in real Banach space. J Math Anal Appl 2006, 321: 722–728. 10.1016/j.jmaa.2005.08.076MathSciNetView ArticleGoogle Scholar
  14. Aoyama K, Kimura Y, Takahashi W, Toyoda M: On a strongly nonexpansive sequence in Hilbert spaces. J Nonlinear Convex Anal 2007, 8: 471–489.MathSciNetGoogle Scholar
  15. Bruck RE, Reich S: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J Math 1977, 3: 459–470.MathSciNetGoogle Scholar
  16. Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topol Methods Nonlinear Anal 1993, 2: 333–342.MathSciNetGoogle Scholar
  17. Reich S: A limit theorem for projections. Linear Multilinear Algebra 1983, 13: 281–290. 10.1080/03081088308817526View ArticleGoogle Scholar

Copyright

© Saejung et al; licensee Springer. 2011

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.