Implicit Iterative Scheme for a Countable Family of Nonexpansive Mappings in 2-Uniformly Smooth Banach Spaces

Definition 1. A mapping T is said to be

• (1)

  η-strongly accretive if for each x, y ∈ X, there exists a j(x − y) ∈ J(x − y) and η > 0, such that

  $$ ()

• (2)

  L-Lipschitzian continuous if for each x, y ∈ X, there exists a constant L > 0, such that

  $$ ()

In particular, T is called nonexpansive if L = 1; it is said to be contractive if L < 1.

Yamada [3] introduced the hybrid steepest descent method:

where T is a nonexpansive mapping in Hilbert spaces. Under some appropriate conditions, Yamada [3] proved that the sequence {x_n} generated by (4) converges strongly to the unique solution x^* ∈ Fix (T) of the variational inequality: VIP(F, X):〈Fx^*, x − x^*〉≥0, for all x ∈ Fix (T).

Moudafi [4] introduced the classical viscosity approximation method for nonexpansive mappings and defined a sequence {x_n} by

where {α_n} is a sequence in (0,1). Xu [6] proved that under certain appropriate conditions on {α_n}, the sequence {x_n} generated by (5) converges strongly to the unique solution x^* ∈ C of the variational inequality: 〈(I − f)x^*, x − x^*〉≥0, for all x ∈ Fix (T), (where C = Fix (T)) in Hilbert spaces as well as in some Banach spaces.

Marino and Xu [7] considered the following general iterative method in Hilbert spaces:

where A is a strongly positive bounded linear operator. It is proved that if the sequence {α_n} satisfies appropriate conditions, the sequence {x_n} generated by (6) converges strongly to the unique solution $$ of the variational inequality: $$, for all x ∈ C, where C is the fixed points set of a nonexpansive mapping T.

Tian [8] considered the following general iterative algorithm (GIA) in Hilbert spaces:

It is proved that if the sequence {α_n} satisfies appropriate conditions, the sequence {x_n} generated by (7) converges strongly to the unique solution $$ of the variational inequality: $$, for all x ∈ Fix (T).

In 2001, Soltuz [9] introduced the following backward Mann scheme iteration:

where T is a nonexpansive mapping and got strong convergence in Hilbert spaces.

In order to find a common fixed point of a finite family of nonexpansive mappings {T_i : i ∈ J}, where J stands for {1,2, 3, …, N}, in 2001, Xu and Ori [10] introduced the following implicit process:

where T_n = T_n(modN), and a weak convergence is obtained in real Hilbert spaces.

Ceng et al. [11] introduced an iterative algorithm to find a common fixed point of a finite family of nonexpansive semigroups in reflexive Banach spaces with a weak sequentially continuous duality mapping, which satisfy the uniformly asymptotical regularity condition:

Under some appropriate conditions one the parameter sequences {α_n}, {β_n}, {γ_n}, and the sequence {x_n} generated by (10) converges strongly to the approximate solution of a variational inequality problem.

In order to find a common element of the solution set of a general system of variational inequalities and the fixed-point set of the mapping S, Ceng et al. [12] constructed a new relaxed extragradient iterative method:

Under mild assumptions, they obtained a strong convergence theorem.

Yao et al. [13] introduced the following Halpern-type implicit iterative method where T is a continuous pseudocontraction:

and obtained a strong convergence theorem in Banach spaces.

Hu [14] introduced an iteration for a nonexpansive mapping in Banach spaces, which guarantee a uniformly Gêteaux differentiable norm as follows:

and several strong convergent theorems are obtained.

Very recently, Jung [15] proposed an iterative process in the frame of Hilbert spaces as follows:

where S is a mapping defined by Sx = kx + (1 − k)Tx and T is a k-strictly pseudocontraction. Strong convergence theorems are established.

Motivated and inspired by Soltuz [9], Xu and Ori [10], Ceng et al. [11], Ceng et al. [12], Yao et al. [13], Hu [14], and Jung [15], we consider the following new implicit iteration in real 2-uniformly smooth Banach spaces:

where {α_n} and {β_n} are real sequences in (0,1), V is an L-Lipschitzian continuous with Lipschitzian constant L > 0, F is an η-strongly accretive and κ-Lipschitzian continuous mapping with κ > 0 and η > 0, and $$ is a countable family of nonexpansive mappings.

In this paper, we prove that the implicit iterative process (15) has strong convergence and find the unique solution $$ of variational inequality:

Our results improve and extend the corresponding conclusions announced by many others.

Lemma 2 (see [17].)Let X be a real q-uniformly smooth Banach space for some q > 1, then there exists some positive constant d_q, such that

in particular, if X is a real 2-uniformly smooth Banach space, then there exists a best smooth constant K > 0, such that

Lemma 3 (see [18].)Assume that a Banach space X has a weakly continuous duality mapping J_q:

• (i)

  for all x, y ∈ X, the following inequality holds:

  $$ ()

•   

  in particular, for all x, y ∈ X, there holds:

  $$ ()

• (ii)

  assume that a sequence {x_n} ⊂ X converges weakly to a point x ∈ X, then the following equation holds:

  $$ ()

Lemma 4 (see [19].)Assume that {a_n} is a sequence of nonnegative real numbers, such that

where {γ_n} is a sequence in (0,1) and {δ_n} is a sequence in ℝ, such that

• (a)

  $$;

• (b)

  limsup _n→∞ δ_n ≤ 0 or $$.

Then lim _n→∞ a_n = 0.

Lemma 5. Let X be a real 2-uniformly smooth Banach space. Let T be a nonexpansive mapping over X, and let F : X → X be an η-strongly accretive and κ-Lipschitzian continuous mapping with κ > 0 and η > 0. For 0 < t < σ ≤ 1 and μ ∈ (0, min {1, η/K²κ²}), set τ = μ(η − μK²κ²), and define a mapping T^t : X → X by T^t : = σI − tμF. Then T^t is a contraction on X; that is, ∥T^tx − T^ty∥≤(σ − tτ)∥x − y∥.

Proof. From 0 < μ < η/K²κ², we have η − μK²κ² > 0. Setting η < 1/2, we have 0 < 2(η − μK²κ²) < 1. For each x, y ∈ X, by Lemma 2, we have

Hence, it implies that

This completes the proof.

Definition 6 (see [20].)Let X be a real Banach space, let C be a nonempty subset of X, and let $$ be a countable family of mappings of C with $$. Then {T_n} is said to satisfy the AKTT  condition, if for any bounded subset D of C, the following inequality holds:

Lemma 7 (see [20].)Let X be a Banach space, let C be a nonempty closed subset of X, and let {T_n} be a family of self-mappings of C satisfying the AKTT condition. Then for each x ∈ C, {T_nx} converges strongly to a point in C. Moreover, let the mapping T be defined by

Then for any bounded subset D of C, the following equality holds:

Lemma 8 (see [1].)Suppose that q > 1. Then the following inequality holds:

for arbitrary positive real numbers a and b.

Proposition 9 (the path convergence). Let T : X → X be a nonexpansive mapping with Fix (T) ≠ ∅, and let V : X → X be an L-Lipschitzian continuous mapping with Lipschitzian constant L > 0. F : X → X is an η-strongly accretive and κ-Lipschitzian continuous mapping with κ > 0 and η > 0. For t ∈ (0,1), let μ ∈ (0, min {1, η/K²κ²}), and set τ = μ(η − μK²κ²) and 0 < γ < τ/L. Then assume that {x_t} is defined by

Then {x_t} converges strongly as t → 0⁺ to a fixed point $$ of  T, which is the unique solution of the variational inequality VIP:

Proof. Consider a mapping S_t on X defined by

It is easy to see that S_t is a contraction. Indeed, for any x, y ∈ X, by Lemma 5, we have Hence, by the Banach contraction mapping principle, S_t has a unique fixed point, denoted by x_t, which uniquely solves the fixed point equation (33).

We divided the proof into several steps.

Step 1. We show the uniqueness of the solution of the variational inequality (34). Assume that both x₁ ∈ Fix (T) and x₂ ∈ Fix (T) are solutions of the variational inequality (34), then we have

Adding up (37) yields Indeed, from the given conditions μ ∈ (0, min {1, η/K²κ²}), and τ = μ(η − μK²κ²), 0 < γ < τ/L, we have

Thus, we conclude that x₁ = x₂. So the uniqueness of the variational inequality (35) is guaranteed.

Step 2. We show that {x_t} is bounded. Taking p ∈ Fix (T), it follows from Lemma 5 that

It follows that Hence {x_t} is bounded, so are {V(x_t)} and {FTx_t)}.

Step 3. Next, we will show that {x_t} has a subsequence converging strongly to x^* ∈ Fix (T).

Assume t_n → 0, and set $$. By the definition of {x_n}, we have

Since {x_n} is bounded, there exists a subsequence $$ of {x_n} converging weakly to x^* ∈ X as k → ∞.

Set $$. Define a mapping B : X → ℝ by

Again, J is weakly continuous, by Lemma 3, and it follows that From (42), we have and we also note that so, we obtain This implies that Tx^* = x^*; that is, x^* ∈ Fix (T).

By Lemma 5, we have

This implies that Since {x_n} is bounded, there exists a subsequence $$ of {x_n} satisfying Since the mapping J is single-valued and weakly continuous, it follows from (50) that $$ as k → ∞. Thus, there exists a subsequence, such that $$.

Step 4. Finally, we show that x^* is the unique solution of variational inequality (34).

Since x_t = tγV(x_t)+(I − tμF)Tx_t, we can derive that

It follows that, for any x ∈ Fix (T),

Since T is a nonexpansive mapping, for all x, y ∈ X, we conclude that

Now replacing t in (52) with t_n and letting n → ∞, from (42), we have that $$, thus we can conclude that

So, x^* is a solution of (34). Hence, $$ by uniqueness. Therefore, $$ as t → 0⁺. This completes the proof.

Proposition 10 (the demiclosed result). Let T : X → X be a nonexpansive mapping with Fix (T) ≠ ∅, and let V be an L-Lipschitzian continuous self-mapping on X with Lipschitzian constant L > 0. F : X → X is an η-strongly accretive and κ-Lipschitzian continuous mapping with κ > 0 and η > 0. Assume that the net {x_t} is defined as Proposition 9 which converges strongly as t → 0⁺ to $$. Suppose that the sequence {x_n} ⊂ X is bounded and satisfies the condition lim _n→∞∥x_n − Tx_n∥ = 0 (the so-called demiclosed property). Then the following inequality VIP holds:

Proof. Set a_n(t) = ∥Tx_n − x_n∥∥x_t − x_n∥. From the given condition and the boundness of {x_t} and {x_n}, it is obvious that a_n(t) → 0 when n → ∞.

From (33) and the fact that T is a nonexpansive mapping, we obtain that

which implies that It follows that Taking the limsup  as t → 0 and recalling (42) and the continuity of F, we conclude that

On the other hand, since X is a real 2-uniformly smooth Banach space, and J is single-valued and strong-weak* uniformly continuous on X, as t → 0⁺, we have

Thus, from (59) and (60), we obtain So (55) is valid. This completes the proof.

Theorem 11. Let $$ be a countable family of self-nonexpansive mappings on X, such that $$. Let V be an L-Lipschitzian continuous self-mapping on X with Lipschitzian constant L > 0. F : X → X is an η-strongly accretive and κ-Lipschitzian continuous mapping with κ > 0 and η > 0. Suppose that the sequences {α_n} and {β_n} satisfy the controlling conditions (C1)-(C2). Let μ ∈ (0, min {1, η/K²κ²}), and set τ = μ(η − μK²κ²) and (τ − 1)/L < γ < τ/L. Assume that ({T_n}, T) satisfies the AKTT condition. Then {x_n} defined by (62) converges strongly to a common fixed point $$ of  $$ which equivalently solves the following variational inequality:

Proof. First we show that {x_n} is well defined. Consider a mapping S_n on X defined by

It is easy to see that S_n is a contraction. Indeed, for any x, y ∈ X, by Lemma 5, we have

Hence, S_n is a contraction. By the Banach contraction mapping principle, we conclude that S_n has a unique fixed point, denoted by x_n. So (62) is well defined.

Then we show that {x_n} is bounded. Taking any p ∈ S, we have

which implies that By induction, it follows that Hence {x_n} is bounded, so are the {V(x_n)} and {F(x_n)}.

Next, we show that

From (C1) and the definition of {x_n}, we observe that By Lemma 7 and (70), we have

Let x_t be defined by (33), from Propositions 9 and 10, and we have that {x_t} converges strongly to $$ and

As required, finally we show that $$. As a matter of fact, by Lemmas 5 and 8, we have

which implies that It is easily to see that Thus, (C2) yields that $$. Applying Lemma 4 and (72) to (74), we conclude that $$.

This completes the proof.

Remark 12. Our result in Proposition 9 extends Theorem 3.1 of Tian [8] from real Hilbert spaces to real 2-uniformly smooth Banach spaces. If we set β_n = 0, our result in Theorem 11 extends Theorem 3.2 of Tian [8] from real Hilbert spaces to real 2-uniformly smooth Banach spaces as well as from a single nonexpansive mapping to a countable family of nonexpansive mappings.

Remark 13. In 2008, Hu [14] introduced a modified Halpern-type iteration for a single nonexpansive mapping in Banach spaces which have a uniformly Gêteaux differentiable norm as follows:

Under some appropriate assumptions, he proved that the sequence {x_n} defined by the iteration process (76) converges strongly to the fixed point of T.

Corollary 14. If we take γ = 1, F = I, μ = 1, and β_n = 0 in (62), we extend the classical viscosity approximation [4] under a mild assumption: the contraction mapping f is replaced by an L-Lipschitzian continuous mapping V. Our proving process needs no Banach limit and is different from the proving process given by Xu [6] in some aspects.

Remark 15. Ceng et al. [11] introduced the following iterative algorithm to find a common fixed point of a finite family of nonexpansive semigroups in reflexive Banach spaces:

Under some appropriate conditions one the parameter sequences {α_n}, {β_n}, {γ_n}, and the sequence {x_n} converges strongly to the approximate solution of a variational inequality problem.

If we set α_n + β_n = 1 and γ_n = 0, the algorithm is simplified into viscosity-form iterative schemes for a finite family of nonexpansive semigroups. Our algorithms are considered in full space and avoid the generalized projections or sunny nonexpansive retractions in Banach space. For further improving our works, in order to obtain more general results, we should take the results given by Ceng et al. in [11] into account.