Implicit Mann Type Iteration Method Involving Strictly Hemicontractive Mappings in Banach Spaces

Remark 1.1. (1) X is called uniformly smooth if X^* is uniformly convex.

(2) In a uniformly smooth Banach space, J is uniformly continuous on bounded subsets of X.

Definition 1.2. The mapping T is called Lipshitz if there exists a constant L > 0 such that

for all x, y ∈ D(T). If L = 1, then T is called nonexpansive and if 0 ≤ L < 1, then T is called contractive.

Definition 1.3 (see [1], [2].)(1) The mapping T is said to be pseudocontractive if

for all x, y ∈ D(T) and r > 0.

(2) The mapping T is said to be strongly pseudocontractive if there exists a constant t > 1 such that

for all x, y ∈ D(T) and r > 0.

(3) The mapping T is said to be local strongly pseudocontractive if for each x ∈ D(T) there exists a constant t > 1 such that

for all y ∈ D(T) and r > 0.

(4) The mapping T is said to be strictly hemicontractive if F(T) ≠ ∅ and if there exists a constant t > 1 such that

for all x ∈ D(T),  q ∈ F(T) and r > 0.

Theorem 1.4. Let X be a real Banach space and let K be a nonempty closed convex subset of X. Let {T_i : i ∈ I} be N strictly pseudocontractive mappings from K to K with $$. Let {α_n} be a real sequence satisfying the following conditions:

• (i)

  0 < α_n < 1,

• (ii)

  $$,

• (iii)

  $$.

From arbitrary x₀ ∈ K, define the sequence {x_n} by the implicit iteration process (1.8). Then {x_n} converges strongly to a common fixed point of the mappings {T_i : i ∈ I} if and only if lim  inf _n→∞d(x_n, ℱ) = 0.

Remark 1.5. One can easily see that for α_n = 1 − 1/n^1/2,  $$. Hence the results of Osilike [11] are needed to be improved.

Lemma 2.1 (see [4].)Let X be a smooth Banach space. Suppose that one of the following holds:

• (a)

  J is uniformly continuous on any bounded subsets of X,

• (b)

  〈x − y, j(x) − j(y)〉≤∥x−y∥² for all x, y in X,

• (c)

  for any bounded subset D of X, there is a function c : [0, ∞)→[0, ∞) such that

for all x, y ∈ D, where c satisfies $$.

Then for any ϵ > 0 and any bounded subset K, there exists δ > 0 such that

for all x, y ∈ K and s ∈ [0, δ].

Remark 2.2. (1) If X is uniformly smooth, then (a) in Lemma 2.1 holds.

(2) If X is a Hilbert space, then (b) in Lemma 2.1 holds.

Lemma 2.3 (see [8].)Let T : D(T) ⊂ X → X be a mapping with F(T) ≠ ∅. Then T is strictly hemicontractive if and only if there exists a constant t > 1 such that for all x ∈ D(T) and q ∈ F(T), there exists j(x − q) ∈ J(x − q) satisfying

Lemma 2.4 (see [10].)Let X be an arbitrary normed linear space and let T : D(T) ⊂ X → X be a mapping.

• (a)

  If T is a local strongly pseudocontractive mapping and F(T) ≠ ∅, then F(T) is a singleton and T is strictly hemicontractive.

• (b)

  If T is strictly hemicontractive, then F(T) is a singleton.

Lemma 2.5 (see [10].)Let {θ_n},{σ_n}, and {ω_n} be nonnegative real sequences and let ϵ^′ > 0 be a constant satisfying

where $$,  θ_n ≤ 1 for all n ≥ 1 and $$. Then limsup _n→∞σ_n ≤ ϵ^′.

Lemma 3.1. Let X be a smooth Banach space. Suppose that one of the following holds:

• (a)

  J is uniformly continuous on any bounded subsets of X,

• (b)

  〈x − y, j(x) − j(y)〉≤∥x−y∥² for all x,y in X,

• (c)

  for any bounded subset D of X, there is a function c : [0, ∞)→[0, ∞) such that

for all x, y ∈ D, where c satisfies $$.

Then for any ϵ > 0 and any bounded subset K, there exists δ > 0 such that

for all x, y, z ∈ K and α, β, γ ∈ [0, δ];α + β + γ = 1.

Theorem 3.2. Let X  be a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let K be a nonempty closed bounded convex subset of X and let T : K → K be a continuous strictly hemicontractive mapping. Let {α_n},{β_n} and {γ_n} be real sequences in [0,1] satisfying conditions

• (iv)

  α_n + β_n + γ_n = 1, for  all  n ≥ 1,

• (v)

  $$,

• (vi)

  $$ and $$.

For a sequence {v_n} in K, suppose that {x_n} is the sequence generated from an arbitrary x₀ ∈ K by

satisfying $$.

Then the sequence {x_n} converges strongly to a unique fixed point q of  T.

Proof. By [2, Corollary 1], T has a unique fixed point q in K. It follows from Lemma 2.4 that F(T) is a singleton. That is, F(T) = {q} for some q ∈ K.

Set M = 1 + diam  K. It is easy to verify that

Also Consider where the first inequality holds by the convexity of ∥·∥².

Now we put k = 1/t, where t satisfies (2.3). Using (3.4) and Lemma 3.1, we infer that

where Also, we have implies as n → ∞, and consequently as n → ∞. Since J is uniformly continuous on any bounded subsets of X, we have For any given ϵ > 0 and the bounded subset K, there exists a δ > 0 satisfying (2.2). Note that (3.13) and (vi) ensure that there exists an N such that Now substituting (3.6) in (3.8) to obtain by using (3.7), implies for all n ≥ N.

Put

and we have from (3.16) For k < 1/2, set δ = 1/2(1 − k) < 1. Because α_n ≤ δ, we imply 1 − α_n ≥ 1 − δ and 2(1 − k)α_n ≤ 1. Now observe that $$,θ_n ≤ 1 for all n ≥ 1 and $$. It follows from Lemma 2.5 that Letting ϵ^′ → 0⁺, we obtain that $$, which implies that x_n → q as n → ∞. This completes the proof.

Corollary 3.3. Let X  be a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let K be a nonempty closed bounded convex subset of X and let T : K → K be a Lipschitz strictly hemicontractive mapping. Let {α_n},{β_n} and {γ_n} be real sequences in [0,1] satisfying the conditions (iv)–(vi).

From arbitrary x₀ ∈ K, define the sequence {x_n} by the implicit iteration process (3.4). Then the sequence {x_n} converges strongly to a unique fixed point q of T.

Corollary 3.4. Let X  be a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let K be a nonempty closed bounded convex subset of X and let T : K → K be a continuous strictly hemicontractive mapping. Suppose that {α_n} be a real sequence in [0,1] satisfying the conditions (v) and lim _n→∞α_n = 0.

From arbitrary x₀ ∈ K, define the sequence {x_n} by the implicit iteration process (1.8). Then the sequence {x_n} converges strongly to a unique fixed point q of T.

Corollary 3.5. Let X be a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let K be a nonempty closed bounded convex subset of X and let T : K → K be a Lipschitz strictly hemicontractive mapping. Suppose that {α_n} be a real sequence in [0,1] satisfying the conditions (v) and lim_n→∞α_n = 0.

From arbitrary x₀ ∈ K, define the sequence {x_n} by the implicit iteration process (1.8). Then the sequence {x_n} converges strongly to a unique fixed point q of T.

Remark 3.6. Similar results can be found for the iteration processes involved error terms; we omit the details.

Remark 3.7. Theorem 3.2 and Corollary 3.3 extend and improve Theorem 1.4 in the following directions.

We do not need the assumption liminf _n→∞d(x_n, ℱ) = 0 as in Theorem 1.4.

Algorithm 4.1. For a given x₀ ∈ K, compute the sequence {x_n} by the implicit iteration process of arbitrary fixed order p ≥ 2:

which is called the multistep implicit iteration process, where {α_n}, {β_n},  {γ_n}, and $$,i = 1,2, …, p − 1 are real sequences in [0,1] and α_n + β_n + γ_n = 1, for  all  n ≥ 1.

Algorithm 4.2. For a given x₀ ∈ K, compute the sequence {x_n} by the iteration process

where {α_n}, {β_n},  {γ_n}, $$ and $$ are real sequences in [0,1] satisfying some certain conditions.

For p = 2, we obtain the following two-step implicit iteration process.

Algorithm 4.3. For a given x₀ ∈ K, compute the sequence {x_n} by the iteration process

where {α_n}, {β_n},  {γ_n} and $$ are real sequences in [0,1] satisfying some certain conditions.

If T₁ = T,  T₂ = I and $$ in (4.3), we obtain the following implicit Mann iteration process.

Algorithm 4.4. For any given x₀ ∈ K, compute the sequence {x_n} by the iteration process

where {α_n} is a real sequence in [0,1] satisfying some certain conditions.

Theorem 4.5. Let X  be a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let K be a nonempty closed bounded convex subset of X and let T, T₁, T₂, …, T_p : K → K(p ≥ 2) be p + 1 mappings. Let T, T₁ be continuous strictly hemicontractive mappings. Let {α_n}, {β_n},  {γ_n} and $$,i = 1,2, …, p − 1 be real sequences in [0,1] satisfying the conditions (iv)–(vi) and $$. For arbitrary x₀ ∈ K, define the sequence {x_n} by (4.1). Then {x_n} converges strongly to the common fixed point of $$.

Proof. By applying Theorem 3.2 under assumption that T and T₁ are continuous strictly hemicontractive mappings, we obtain Theorem 4.5 which proves strong convergence of the iteration process defined by (4.1). Consider by taking T₁ = T and $$,

From (4.5) and the condition $$, we obtain This completes the proof.

Corollary 4.6. Let X  be a smooth Banach space satisfying any one of the Axioms (a)–(c) of Lemma 3.1. Let K be a nonempty closed bounded convex subset of X and let T, T₁, T₂, …, T_p : K → K(p ≥ 2) be p + 1 mappings. Let T, T₁ be Lipschitz strictly hemicontractive mappings. Let {α_n}, {β_n},   {γ_n} and $$,   i = 1,2, …, p − 1 be real sequences in [0,1] satisfying the conditions (iv)-(vi) and $$. For arbitrary x₀ ∈ K, define the sequence {x_n} by (4.1). Then {x_n} converges strongly to the common fixed point of $$.