Strong Convergence of Mann’s Iteration Process in Banach Spaces

Mann’s iteration process for finding a fixed point of a nonexpansive mapping in a Banach space is considered. This process is known to converge weakly in some class of infinite-dimensional Banach spaces (e.g., uniformly convex Banach spaces with a Fréchet differentiable norm), but not strongly even in a Hilbert space. Strong convergence is therefore a nontrivial problem. In this paper we provide certain conditions either on the underlying space or on the mapping under investigation so as to guarantee the strong convergence of Mann’s iteration process and its variants.


Introduction
Let X be a real Banach space with norm · , let C be a nonempty closed convex subset of X, and let T : C → C be a nonexpansive mapping (i.e., Tx − Ty ≤ x − y for x, y ∈ C). We use Fix(T) to denote the set of fixed points of T; i.e., Fix(T) = {x ∈ C : Tx = x}. It is known that Fix(T) is nonempty if X is uniformly convex and C is bounded. Mann's iteration process [1], an averaged iterative scheme, is used to find a point in Fix(T). This process generates a sequence {x n } via the recursive process: x n+1 = (1 − t n )x n + t n Tx n , n = 0, 1, · · · , where the initial point x 0 ∈ C is arbitrary and (t n ) ⊂ [0, 1]. It is known [2] that if X is uniformly convex with a Fréchet differentiable norm, if Fix(T) is nonempty, and if ∑ ∞ n=0 t n (1 − t n ) = ∞, then the sequence (x n ) generated by (1) converges weakly to a point in Fix(T). However, the counterexample in [3] shows that the algorithm (1) fails, in general, to converge strongly even in Hilbert space unless C is compact [4]. Therefore, efforts have been made to study sufficient conditions to guarantee strong convergence of Mann's algorithm (1) without assuming compactness of C. For instance, Gwinner [5] imposed the ϕ-accretiveness condition on I − T to prove strong convergence of (1) in a uniformly convex Banach space. Some authors have made modifications of Mann's iteration process in order to get strong convergence (see, e.g., [6][7][8]).
In this paper we continue to study the strong convergence of Mann's iteration process. We improve Gwinner's strongly convergent result ( [5], Theorem 1) by removing the condition t n ≤ b with b ∈ (0, 1). We also prove strong convergence of Mann's iteration process in a reflexive Banach space with Opial's and Kadec-Klee properties when I − T is ϕ-accretive. A regularization method is introduced to approximate a fixed point of T. This method implicitly yields a sequence of approximate solutions and we shall prove (in Theorem 4) its strong convergence to a solution of a variational inequality. Combining this regularization method with Mann's method, we obtain a new iteration process (see (22)) when the regularizer R = I − f with f a contraction. We will prove that this process converges strongly in a Banach space whenever the sequence of approximate solutions of the implicit regularization converges strongly.
We use the notation: • "x n x" stands for the weak convergence of (x n ) to x, • "x n → x" stands for the strong convergence of (x n ) to x, • ω w (x n ) := {x : ∃x n k x} is the set of all weak accumulation points of the sequence (x n ).

Uniform Convexity
Let X be a real Banach space with norm · . Recall that the modulus of convexity of X is defined as (cf. [9]) We say that X is uniformly convex if Examples of uniformly convex Banach spaces include Hilbert spaces H and l p (and also L p ) spaces for 1 < p < ∞. As a matter of fact, the moduli of convexity of these spaces are The following inequality characterization of uniform convexity is convenient in application.
Recall that a Banach space X is said to satisfy the Kadec-Klee property (also known as property (H)) if the (sequential) weak and strong topologies on the unit sphere coincide; equivalently, given any sequence {u n } and a point u in X, the following implication holds: (u n → u weakly and u n → u ) =⇒ u n → u strongly.
Every uniformly convex Banach space satisfies the Kadec-Klee property.

Duality Maps
Let X be a real Banach space with norm · . The notion of general duality maps on X was introduced by Browder [11][12][13][14]. By a gauge we mean a function µ : R + → R + satisfying the properties: Associated with a gauge µ is the duality map J µ : X → X * ( [12]) defined by A special case is given by choosing the gauge µ(t) = t p−1 for t ≥ 0, where 1 < p < ∞. In this case the corresponding duality map, which is denoted by J p and referred to as the generalized duality map of order p of X, is given by In particular, we denote J for J 2 and call it the normalized duality map. In more detail, J is defined by For a Hilbert space H, its normalized duality map is identified with the identity map I (with the dual space H * being identified with H through the Riesz canonical embedding).
Recall that a Banach space X is said to have a (sequentially) weakly continuous duality map if, for some gauge µ, the duality map J µ : X → X * is (sequentially) continuous when X is endowed with the weak topology and the dual space X * endowed with the weak-star topology. It is known that, for 1 < p < ∞ (p = 2), the sequence space p has a sequentially weakly continuous duality map J p ; while the function space L p does not [15]. However, the normalized duality map J of p (p = 2) is not weakly continuous [16], Proposition 3.2.

Proposition 2.
[17] Assume a Banach space X has a weakly continuous duality map J µ for some gauge µ. If (u n ) is a sequence in X weakly convergent to a point u, then we have where Ψ is defined by Ψ(t) = t 0 µ(s)ds, t ≥ 0. In particular, X satisfies the Opial property ( [15]): whenever (u n ) is a sequence in X weakly convergent to u.

Demiclosedness Principle for Nonexpansive Mappings
Let C be a nonempty closed convex subset of a Banach space X and let T : where X w is endowed with the weak topology. Equivalently, I − T is demiclosed if and only if the implication below holds: This is called the demiclosedness principle for nonexpansive mappings, which holds in the following Banach spaces: • Uniformly convex Banach spaces [12]; • Banach spaces satisfying Opial's property [15], in particular, Banach spaces with a weakly continuous duality map J µ for some gauge µ.

Accretive Operators
Let C be a nonempty closed convex subset of a Banach space X. Recall that an operator S : C → X is said to be accretive (cf. [18]) if, for each x, y ∈ C, Here J : X → X * is the normalized duality map (or a selection in the case of multivalued). Definition 1 ([5,19]). Let ϕ : [0, ∞) → [0, ∞) be a continuous, strictly increasing function with ϕ(0) = 0 and lim t→∞ ϕ(t) = ∞. Then we say that an operator S : for all x, y ∈ C, or respectively, for all x, y ∈ C, where J is the normalized duality map of X.
Notice that if S is ϕ-accretive, then the set of zeros of S, This impliesx =x.

A Useful Lemma
The lemma below is helpful in proving strong convergence of a sequence (x n ) to a point x in a Banach space by proving convergence to zero of the real sequence ( x n − x ).
where (λ n ) and (δ n ) are sequences in (0,1) and (β n ) is a sequence in R. Assume Then lim n→∞ s n = 0.

Strong Convergence of Mann's Algorithm
Let X be a real Banach space, let C be a nonempty closed convex subset of X, and let T : C → C be a nonexpansive mapping with a fixed point. We use Fix(T) to note the set of fixed points of T. It is known that if X is strictly convex, then Fix(T) is convex. For finding a fixed point of T, Mann's iterative algorithm [1] (see also [22]) is often used. This algorithm generates a sequence (x n ) as follows: where the initial point x 0 ∈ C is arbitrary, and t n ∈ [0, 1] for all n ≥ 0.

Lemma 2. [23]
In a Banach space X, let T : C → C be a nonexpansive mapping and let (x n ) be a sequence generated by Mann's method (10). Then Consequently, both lim n→∞ x n − z and lim n→∞ x n − Tx n exist.
We begin with a strong convergence result on the Mann algorithm (10), which improves ( [5], Theorem 1) by removing the restriction of t n ≤ b with b ∈ (0, 1). Theorem 1. Let X be a real uniformly convex Banach space, let C be a nonempty closed convex subset of X, and let T be a nonexpansive self-mapping of C with Fix(T) = ∅. Suppose S := I − T is ϕ-accretive and (t n ) satisfies the divergence condition Then (x n ) converges in norm to a fixed point of T.
Proof. First observe that the ϕ-accretiveness of S = I − T implies that T has a unique fixed point, i.e., Fix(T) = {z} is singleton. Set r = x 0 − z . By Lemma 2(i), x n − z ≤ r for all n.
Applying Proposition 1(i), we get It turns out that Consequently, The divergence condition (11) together with Lemma 2(ii) implies that lim n→∞ x n − Tx n = 0. By the demiclosedness principle of S = I − T, we have that ω w (x n ) ⊂ Fix(T) = {z}. This proves that x n z. Now using the ϕ-accretiveness of S = I − T, we obtain It turns out that x n → z . Therefore, the Kadec-Klee property of the uniform convexity of X implies that x n → z. The proof is complete.
Then (x n ) converges in norm to a fixed point of T.
Proof. We sketch the proof here. Under the condition (13), we have by ( [24], Lemma 2) that x n − Tx n → 0 strongly. Now Opial's property together with the fact that lim n→∞ x n − z exists for every z ∈ Fix(T) [Lemma 2(i)] implies that the sequence {x n } converges weakly to a point z in Fix(T). In the meanwhile, we still have (12) which implies x n → z . Now the strong convergence of {x n } to z follows from the Kadec-Klee property of X.

Remark 2.
It is unclear if, in Theorem 2, the divergence condition (13) can be weakened to the divergence condition (11). Alternatively, it is unknown whether x n − Tx n → 0 holds true in a Banach space with Opial's condition, without the assumption that sup n≥0 t n < 1. (Note: Ishikawa ([24], Lemma 2) requires this assumption.)

Regularization
Finding a fixed point of a nonexpansive mapping T is equivalent to finding a zero of the accretive operator S = I − T. Thus regularization methods may apply. In [5], Gwinner introduced the regularization operator S ε := (1 − ε)(I − T) + εR, where ε ∈ (0, 1) is a regularization parameter and R : C → X is a mapping which is referred to as regularizer. The following result on this regularization was proved.

Theorem 3 ([5] Theorem 2)
. Let X be a uniformly convex Banach space, C a nonempty closed convex subset of X, and T a nonexpansive self-mapping of C with Fix(T) = ∅. Assume R : C → X is a continuous, bounded, uniformly ϕ-accretive operator. Assume the normalized duality map J : X → X * is weakly sequentially continuous. Choose positive real numbers δ n and ε n ∈ (0, 1) such that If the approximate solutionsỹ n ∈ C satisfy then the sequence {ỹ n } converges strongly to a fixed pointẑ of T, which is uniquely determined by the variational inequality: Remark 3. The condition in Theorem 3 that the normalized duality map J : X → X * be weakly continuous is quite restrictive because this condition rules out the applicability of Theorem 3 to the uniformly convex sequence space p (1 < p < ∞, p = 2) since the normalized duality map J of p (1 < p < ∞, p = 2) fails to be weakly continuous ( [16], Proposition 3.2).Below we use the condition that X admits a weakly continuous duality map J µ for some gauge µ; as a result, our theorem applies to every p space (1 < p < ∞) which has a weakly continuous duality map J µ with gauge µ(t) = t p−1 for t ≥ 0.
Our next result improves Theorem 3 (i.e., [5], Theorem 2) twofold: first we remove the uniform convexity of the space X and secondly, we replace the normalized duality map J with a general duality map J µ for some gauge µ, the latter being more flexible in applications (such as p spaces for 1 < p < ∞). Theorem 4. Assume X is a reflexive Banach space with a weakly sequentially continuous duality map J µ : X → X * for some gauge µ, C a nonempty closed convex subset of X, and T : C → C a nonexpansive mapping with Fix(T) = ∅. Assume R : C → X is a continuous, bounded, uniformly ϕ-accretive operator. Assume δ n and ε n satisfy the conditions (i) and (ii) in Theorem 3. Then the sequence {ỹ n } of approximate solutions determined by (14) converges strongly to the unique solutionẑ of the variational inequality (15).
Adding them up and making use of the uniform ϕ-accretiveness of R yields It follows immediately thatẑ =z and the solution of the variational inequality (15) is unique. Hence, we have verified that the sequence {ỹ n } converges strongly to the unique solution of (15).
Take R to be of the form where f : C → C is an α-contraction, with α ∈ [0, 1). Namely, Then it is easy to find that R is uniformly ϕ-accretive with ϕ(t) = (1 − α)t for t ≥ 0 (namely, I − T is strongly accretive): Consequently, the following result is a direct consequence of Theorem 4.

Corollary 1.
Assume X is a reflexive Banach space with a weakly sequentially continuous duality map J µ : X → X * for some gauge µ, C a nonempty closed convex subset of X, and T : C → C a nonexpansive mapping with Fix(T) = ∅. Assume f : C → C is an α-contraction for some α ∈ [0, 1). Assume δ n and ε n satisfy the conditions (i) and (ii) in Theorem 3. Then the sequence {ỹ n } of approximate solutions determined by converges strongly to the unique solutionẑ of the variational inequality: Remark 4. Theorem 4 and Corollary 1 both are applicable to p spaces for each 1 < p < ∞ since the generalized duality map J µ of p with gauge µ(t) = t p−1 is weakly continuous; Gwinner's theorem ( [5], Theorem 2) however is not applicable since the normalized duality map J of p for 1 < p < ∞, p = 2 is not weakly continuous.
Applying Mann's iteration method to the regularized operator (1 − ε)(I − T) + εR with the regularizer R of form (19) leads to the iteration process [5], which we call regularized Mann iteration method: where the initial point z 0 ∈ C is arbitrary, f : C → C is an α-contraction with α ∈ [0, 1), and α n , ε n ∈ (0, 1) for n ≥ 0. The convergence of this algorithm depends on the convergence of the approximate scheme (20) as shown below.
Theorem 5. Assume X is a Banach space, C a nonempty closed convex subset of X, and T : C → C a nonexpansive mapping with Fix(T) = ∅. Assume R : C → C is an α-contraction with α ∈ [0, 1). Assume (α n ), (ε n ), and (δ n ) satisfy the conditions: Assume the sequence {ỹ n } determined by the approximate equation (20) converges strongly to the unique solutionẑ of the variational inequality (21). Then the sequence {z n } generated by the iteration process (22) also converges strongly to the same solutionẑ of (21).
The conditions (C2)-(C4) assert that γ n = o(α n ε n ). Noticing condition (C1), we can apply Lemma 1 to Equation (26) to conclude z n+1 −ỹ n → 0. Now since it is assumed thatỹ n →z, the unique solution of the variational inequality (21), it is also that z n →z. This finishes the proof.

Remark 6.
To the best of our knowledge, it looks to be the first time in the literature to use the approximate solutionsỹ n to study strong convergence of the regularized Mann's iteration process (22). Previously, exact solutions y n were used (see, e.g., [5]). Here y n is the exact fixed point of the contraction (1 − ε n )T(·) + ε n f (·): y n = (1 − ε n )T(y n ) + ε n f (y n ).
A key assumption in Theorem 5 is that the sequence {ỹ n } be strongly convergent, which has been proved in a reflexive Banach space with a weakly continuous duality map in Theorem 4. Consequently, we have the following result. Corollary 2. Assume X is a reflexive Banach space with a weakly sequentially continuous duality map J µ : X → X * for some gauge µ (in particular, X = p for 1 < p < ∞), C a nonempty closed convex subset of X, and T : C → C a nonexpansive mapping with Fix(T) = ∅. Assume f : C → C is an α-contraction for some α ∈ [0, 1). Assume the sequence {ỹ n } satisfies (20) and let {z n } be generated by the regularized iteration process (22). Suppose α n , ε n , δ n satisfy the conditions (C1)-(C4) in Theorem 5. Then {z n } converges strongly to the unique solutionẑ of the variational inequality (21).