- Open Access
A new general iterative algorithm with Meir-Keeler contractions for variational inequality problems in q-uniformly smooth Banach spaces
© Guan et al.; licensee Springer. 2013
- Received: 20 June 2013
- Accepted: 23 August 2013
- Published: 9 November 2013
In this paper, we generalize the iterative scheme and extend the space studied in (Fixed Point Theory and Applications 2012:46, 2012). Further, we prove some strong convergence theorems of the new iterative scheme for variational inequality problems in q-uniformly smooth Banach spaces under very mild conditions. Our results improve and extend corresponding ones announced by many others.
Throughout this paper, we denote by X and a real Banach space and the dual space of X, respectively. Let C be a nonempty closed convex subset of X.
for all , where denotes the generalized duality pairing between X and . In particular, is called the normalized duality mapping and for . It is well known that if X is smooth, then is single-valued, which is denoted by .
Without loss of generality, we can assume that and .
Recall that if C and D are nonempty subsets of a Banach space X such that C is nonempty closed convex and , then a mapping is sunny  provided for all and , whenever . A mapping is called a retraction if for all . Furthermore, P is a sunny nonexpansive retraction from C onto D if P is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D.
Proposition 1.1 (Banach )
Let be a complete metric space, and let f be a contraction on X, then f has a unique fixed point.
Proposition 1.2 (Meir and Keeler )
Let be a complete metric space, and let ϕ be a Meir-Keeler contraction (MKC, for short) on X, that is, for every , there exists such that implies for all . Then ϕ has a unique fixed point.
This proposition is one of generalizations of Proposition 1.1, because the contractions are Meir-Keeler contractions.
Proposition 1.3 
is sunny and nonexpansive.
Variational inequality theory has emerged as a great important tool in studying a wide class of unilateral, free, obstacle, moving and equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. This field is dynamics and it is experiencing an explosive growth in both theory and applications. Several numerical methods have been developed for solving variational inequalities and related optimization problems; see [4–7] and the references therein.
where is a nonlinear mapping. The set of solutions of (1.5) is denoted by .
where T is a nonexpansive mapping of K into itself and f is a contraction on K. They obtained a strong convergence theorem under some mild restrictions on the parameters.
where T is a k-strictly pseudo-contraction, f is a contraction and A is a strong positive linear bounded operator, is the metric projection. They proved, under certain appropriate assumptions on the sequences and , that defined by (1.7) converges strongly to a fixed point of the k-strictly pseudo-contraction, which solves some variational inequality.
where is a -strictly pseudo-contraction, ϕ is an MKC contraction and is an L-Lipschitzian and η-strongly monotone mapping in a Hilbert space, is the metric projection. Under certain appropriate assumptions on the sequences , , and , the sequence defined by (1.8) converges strongly to a common fixed point of an infinite family of -strictly pseudo-contractions, which solves some variational inequality.
Question 1 Can the space in Song  be extended from a Hilbert space to a q-uniformly smooth Banach space?
Question 2 Can the projection in Song  be changed to the sunny nonexpansive retraction and be put to other place of the iteration process?
Question 3 Can we extend the iterative scheme of algorithm (1.8) to a more general iterative scheme?
Question 4 Can we remove the very strict condition which is necessary in Lemma 3.1 and Theorem 3.2 of Song ?
where is a -strictly pseudo-contraction, ϕ is an MKC contraction, is the sunny nonexpansive retraction and is an L-Lipschitzian and η-strongly accretive mapping in a q-uniformly smooth Banach space. Under some suitable assumptions on the sequences , , and , the sequence defined by (1.9) converges strongly to a common fixed point of an infinite family of -strictly pseudo-contractions, which solves some variational inequality.
A Banach space X is said to be strictly convex if whenever x and y are not collinear, then .
A Banach space X is said to be uniformly smooth if as . A Banach space X is said to be q-uniformly smooth if there exists a fixed constant such that with . A typical example of uniformly smooth Banach spaces is , where . More precisely, is -uniformly smooth for every .
exists for all x, y on the unit sphere . If, for each , the limit (2.3) is uniformly attained for , then the norm of X is said to be uniformly Gâteaux differentiable. The norm of X is said to be Fréchet differentiable if, for each , the limit (2.3) is attained uniformly for .
In order to prove our main results, we need the following lemmas.
Lemma 2.1 
Lemma 2.2 
Lemma 2.3 
. Then .
Lemma 2.4 
Let C be a nonempty convex subset of a real q-uniformly smooth Banach space X, and let be a λ-strict pseudo-contraction. For , we define . Then, as , , is nonexpansive such that .
Lemma 2.5 
for arbitrary positive real numbers a, b.
Lemma 2.6 Let F be an L-Lipschitzian and η-strongly accretive operator on a nonempty closed convex subset C of a real q-uniformly smooth Banach space X with and . Then is a contraction with contraction coefficient .
where . Hence, G is a contraction with contraction coefficient . This completes the proof. □
Lemma 2.7 (, Demiclosedness principle)
Let C be a nonempty closed convex subset of a reflexive Banach space X which satisfies Opial’s condition, and suppose that is nonexpansive. Then the mapping is demiclosed at zero, that is, , implies .
is sunny and nonexpansive.
Proof From Proposition 1.3, we have (i) ⇔ (ii) ⇔ (iii). We need only to prove (iii) ⇔ (iv).
Indeed, if , it follows from the fact that , .
If , then , . This completes the proof. □
Lemma 2.9 
converges strongly to some point in C for each ;
Furthermore, if the mapping is defined by for all , then .
Thus and the uniqueness is proved. Below, we use to denote the unique solution of (2.3).
First, we prove that is bounded.
Assume that for , fixed for each t.
Case 1. . In this case, we can see easily that is bounded.
which implies . Thus is bounded.
Then, we prove that () as .
Since X is reflexive and is bounded, there exists a subsequence of such that . Setting , we obtain .
We claim .
Since as , by (2.10) we obtain that . Hence, we have .
Now, we prove that solves the variational inequality (2.4).
Now replacing t in (2.13) with and letting , notice that for , we obtain , i.e., is a solution of (2.4). Hence by uniqueness. Thus, we have shown that every cluster point of (at ) equals , therefore as . □
Lemma 2.11 Let X be a q-uniformly smooth Banach space, and let C be a nonempty convex subset of X. Assume that is a countable family of -strict pseudo-contractions for some and such that . Assume that is a positive sequence such that . Then is a λ-strict pseudo-contraction with and .
Proof Let , where . Then is a λ-strict pseudo-contraction with .
where , which shows that is a λ-strict pseudo-contraction.
Using the same means, our proof method can easily carry over to the general finite case.
Thus, for , if with and , then strongly converges.
Thus, H is a λ-strict pseudo-contraction.
Step 3. We prove .
where . Thus, we obtain , it follows that . □
3 Main results
, , , ;
, , ;
We divide the rest of the proof into two parts.
Step 1. We will prove that the sequence is bounded.
which implies that is bounded, so are and .
Step 2. We claim that as .
This completes the proof. □
Proof The proof of the lemma will be split into three parts.
Step 3. We will prove that .
Now, we will obtain the contradiction from two cases.
Case 1. Fix (), if for some such that , and for the other such that .
It contradicts .
Put and . Applying Lemma 2.3 to (3.11), we obtain as , which contradicts . Thus . This completes the proof. □
, , , ;
, , ;
Proof Combining the proof of Lemma 3.1 with Lemma 3.2, we can obtain the conclusion. □
Theorem 3.1 improves and extends Theorem 3.2 of Song . Especially, our results extend the above results from a Hilbert space to a more general q-uniformly smooth and uniformly convex Banach space.
We change the metric projection in Song  into the sunny nonexpansive retraction and put it to the other place of the iteration process so that our iteration process is better defined.
We generalize the iteration process so that our iteration process is more general.
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