Viscosity approximation methods for nonexpansive nonself-mappings without boundary conditions
© Zhou and Wang; licensee Springer. 2014
Received: 12 August 2013
Accepted: 25 February 2014
Published: 12 March 2014
In this paper, we introduce several types of viscosity approximation methods for nonexpansive nonself-mappings in certain Banach spaces. Using new analysis techniques, we prove several strong convergence theorems for nonexpansive nonself-mappings in certain Banach spaces without boundary conditions.
MSC: 47H05, 47H09, 47H10.
Let C be a nonempty closed convex subset of a real Banach space E and let T be a nonexpansive mapping defined on C, that is, for all . Denote by the set of fixed points of T; that is, .
In general, the domain of T is a proper subset of E. In this case, some iterative algorithms associated with T may not be well defined. Therefore, it is interesting to find some conditions under which the algorithms considered are well defined.
Recently, the problem of approximating fixed points for nonexpansive nonself-mappings has been paid much attention to by many authors, see [1–4]. Moreover, in , Song and Chen introduced two iterative methods (one is implicit and the other is explicit) and established the strong convergence of such two methods in certain Banach spaces, if T satisfies the weakly inward condition and . Further, in , Matsushita and Takahashi introduced a new condition (2.2). They proved the following. (i) If T satisfies the weakly inward condition, then T satisfies the condition (2.2). (ii) If E is a strictly convex Banach space, is a nonexpansive mapping such that and C is a sunny nonexpansive retract of E, then T satisfies the condition (2.2). (iii) If T satisfies the condition (2.2), then , where Q is a sunny nonexpansive retraction from E onto C. Using these results, they proved two strong convergence theorems for nonexpansive nonself-mappings in certain Banach spaces without any boundary conditions.
We remark in the passing that the convergence theorems of Song and Chen  are not applicable to and for any . Although the convergence theorems of Matsushita and Takahashi  work in the Banach spaces of and for all , they could not be used to find the minimum-norm fixed point of the underlying mappings in a Hilbert space. Moreover, the proof lines of Theorem 4.2 in  are really long.
Can one present an indirect and simple method for proving Theorem 4.2 of Matsushita and Takahashi ?
Can one extend Theorem 4.2 to both the more general Banach spaces and the more broad Meir-Keeler contractions?
Can one improve the main results of  so that new convergence theorems hold true under less assumptions?
The purpose of this paper is to study and solve all problems mentioned above by improving and generalizing several recent results. Several new iterative schemes are introduced and several strong convergence theorems are established by using new analysis techniques.
for each and .
If E is smooth, then J is single valued.
If E is smooth, then J is norm-to-weak ∗ continuous.
If E has a uniformly Gâteaux differentiable norm, then J is norm-to-weak ∗ uniformly continuous on bounded sets of E.
If the norm of E is (uniformly) Fréchet differentiable, then it is (uniformly) Gâteaux differentiable.
If Banach space E is uniformly smooth, then J is norm-to-norm uniformly continuous on bounded sets of E.
and denotes the closure of the set .
Remark 2.1 (i) for all ;
(ii) if x is an interior of C, then ;
(iv) if both and satisfy the weakly inward condition, then so does a convex combination of f and g.
Let D be a subset of C and let Q be a mapping from C to D. Then Q is said to be sunny if whenever for and . A mapping Q from C into itself is said to be a retraction if . A set D is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C into D [7, 8]. It is very well known that if E is a smooth Banach space and C is a nonempty closed convex subset of E, then there exists at most one sunny nonexpansive retraction Q from E onto C. In a Hilbert space H, there exists a unique sunny nonexpansive retraction .
Q is both sunny and nonexpansive;
for all and ;
for all .
for all , where denotes the complementary set of . If E is a Hilbert space, then the MT condition is equivalent to the nowhere-normal outward condition introduced by Matsushita and Kuroiwa .
Matsushita and Takahashi  proved the following interesting results.
Proposition 2.3 Let C be a closed convex subset of a smooth Banach space E and let T be a mapping from C into E. Suppose that C is a sunny nonexpansive retract of E. If T satisfies the weakly inward condition, then T satisfies the MT condition.
Proposition 2.4 Let C be a closed convex subset of a strictly convex Banach space E and let T be a mapping from C into E. Suppose that C is a sunny nonexpansive retract of E. If , then T satisfies the MT condition.
where Q is the unique sunny nonexpansive retraction from E onto C.
By a careful observation, we have the following more rich and nice result.
where Q is the unique sunny nonexpansive retraction form E onto C.
which shows that . In view of Proposition 2.5, , thus, we get , which turns out that . The converse inclusion relation that ‘’ is obvious, consequently, . The proof is completed. □
Remark 2.2 The fact that is very important. By virtue of this fact, we can invent and create some novel argument methods for strong convergence of the some kinds of iterative algorithms.
In , Meir and Keeler proved the following interesting fixed point theorem.
Theorem MK Let be a complete metric space and let Φ be a MKC on X. Then Φ has a unique fixed point in X.
Remark 2.3 If C is a nonempty closed (convex) subset of a complete metric space , then the conclusion of Theorem MK is still true.
The following results can be found in .
for all with .
T Φ and are all Meir-Keeler contractions;
- (ii), define a mapping by
then is a MKC.
Remark 2.4 When Φ is not a self-mapping, then conclusion of (i) is still true, provided that T Φ or ΦT is well defined. The conclusion of (ii) holds true for nonself-mappings T and Φ.
- (i), there exists a unique continuous path such that(2.5)further, as , converges strongly to a fixed point z of T, which solves the following variational inequality:
- (ii)for arbitrary initial data , a sequence is generated by(2.6)
where is a real sequence satisfying conditions:
(C1) ; (C2) ; and (C3) or ().
Then converges strongly to the same point z as in (i).
where is a real sequence that satisfies certain conditions.
where is a real sequence that satisfies less restriction. We shall prove that the path defined by (2.7) and the sequences defined by (2.8) and (2.9), respectively, converge strongly to a fixed point of T which solves variational inequality (VI).
In order to achieve the objective above, we cite the following known results; see .
where is a real sequence satisfying . Then ().
where is a real sequence in satisfying conditions (i) and (ii) . Then ().
3 Main results
equivalently, , where is the unique sunny nonexpansive retraction from E onto .
It follows from Proposition 2.4 that T satisfies the MT condition.
which is equivalent to by Proposition 2.2. □
In view of Lemma 2.1, we know that , which implies that and hence , which yields .
The proof is completed. □
Remark 3.1 Theorem 3.1 improves and generalizes Theorem 2.2 due to Song and Chen ; Theorem 3.2 improves and generalizes Theorem 2.4 due to Song and Chen  and Theorem 4.2 due to Matsushita and Takahashi . We point out that our method of argumentation is much simpler than the ones used by Song and Chen , and Matsushita and Takahashi .
Proof We split the proof into four steps.
Step 1. is bounded.
for all , hence is bounded, so is .
Step 2. ().
for some positive constant .
By virtue of Proposition 2.9, we claim that ().
Step 3. , where , and is the unique sunny nonexpansive retraction from C onto , which is guaranteed by Theorem 3.1.
Step 4. ().
where satisfies condition (i) ; and (ii) . By Proposition 2.10, we conclude that (), i.e., (). This completes the proof. □
Now we prove the strong convergence of (2.9).
Proof By Proposition 2.7(i), we know that is a MKC on C. It follows from Theorem MK, has a unique fixed point z in C, i.e., .
We shall prove that as .
for all .
for all .
We now consider two possible cases.
which contradicts with (3.14). Consequently, , and hence which solves the variational inequality (VI).
Case 2. for all .
for all .
in view of Proposition 2.10, and hence , a contradiction, thus, case 2 is impossible. The proof is completed. □
Remark 3.2 We do not know whether the conclusion of Theorem 3.4 holds true when is a nonself-MKC. However, in the case where E is a Hilbert space, it is true.
Applying Theorems 3.1-3.4 to a Hilbert space H, we can obtain new strong convergence theorems, which improve and generalize the corresponding results by Matsushita and Kuroiwa , and Zhou et al. .
Theorem 3.5 Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, let T be a nonexpansive mapping from C into H with and let be a MKC. Then the conclusions of Theorems 3.1-3.4 hold true.
4 Concluding remarks
This work contains our contribution dedicated to developing and improving the viscosity approximation methods for finding fixed points of nonexpansive nonself-mappings. A novel and remarkable finding is contained in Lemma 2.1, that is, , which makes it possible to invent a novel and simple method of argumentation for establishing strong convergence theorems. We have introduced our modified viscosity approximation methods for finding fixed points of nonexpansive nonself-mappings. Applying our main results to a Hilbert space, we have drawn the corresponding conclusions announced by some authors.
The first author was supported in part by NSFC 11071053. The authors wish to thank the referee for the valuable suggestions to improve the writing of this paper.
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