Strong and Weak Convergence Theorems for an Infinite Family of Lipschitzian Pseudocontraction Mappings in Banach Spaces

Definition 1.1. (1) A mapping T : C → C is said to be pseudocontraction [1], if for any x, y ∈ C, there exists j(x − y) ∈ J(x − y) such that

It is well known that [1] the condition (1.3) is equivalent to the following:

(2) T : C → C is said to be strongly pseudocontractive, if there exists k ∈ (0,1) such that

(3) T : C → C is said to be strictly pseudocontractive in the terminology of Browder and Petryshyn [1], if there exists λ > 0 such that

In this case, we say T is a λ-strictly pseudocontractive mapping.

(4) T : C → C is said to be L-Lipschitzian, if there exists L > 0 such that

Remark 1.2. It is easy to see that if T : C → C is a λ-strictly pseudocontractive mapping, then it is a (1 + λ)/λ-Lipschitzian mapping.

In fact, it follows from (1.6) that for any x, y ∈ C,

Lemma 1.3 (see [2], Theorem 13.1 or [3].)Let E be a real Banach space, C be a nonempty closed convex subset of E, and T : C → C be a continuous strongly pseudocontractive mapping. Then T has a unique fixed point in C.

Remark 1.4. Let E be a real Banach space, C be a nonempty closed convex subset of E and T : C → C be a Lipschitzian pseudocontraction mapping. For every given u ∈ C and s ∈ (0,1), define a mapping U_s : C → C by

The concept of pseudocontractive mappings is closely related to accretive operators. It is known that T is pseudocontractive if and only if I − T is accretive, where I is the identity mapping. The importance of accretive mappings is from their connection with theory of solutions for nonlinear evolution equations in Banach spaces. Many kinds of equations, for example, Heat, wave, or Schrödinger equations can be modeled in terms of an initial value problem:

In order to approximate a fixed point of Lipschitzian pseudocontractive mapping, in 1974, Ishikawa introduced a new iteration (it is called Ishikawa iteration). Since then, a question of whether or not the Ishikawa iteration can be replaced by the simpler Mann iteration has remained open. Recently Chidume and Mutangadura [4] solved this problem by constructing an example of a Lipschitzian pseudocontractive mapping with a unique fixed point for which every Mann-type iteration fails to converge.

Inspired by the implicit iteration introduced by Xu and Ori [5] for a finite family of nonexpansive mappings in a Hilbert space, Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9] proposed and studied convergence theorems for an implicit iteration process for a finite or infinite family of continuous pseudocontractive mappings.

The purpose of this paper is to study the strong and weak convergence problems of the implicit iteration processes for an infinite family of Lipschitzian pseudocontractive mappings in Banach spaces. The results presented in this paper extend and improve some recent results of Xu and Ori [5], Osilike [6], Chen et al. [7], Zhou [8] and Boonchari and Saejung [9].

For this purpose, we first recall some concepts and conclusions.

A Banach space E is said to be uniformly convex, if for each ɛ > 0, there exists a δ > 0 such that for any x, y ∈ E with ∥x∥, ∥y∥≤1 and ∥x − y∥≥ɛ, ∥x + y∥≤2(1 − δ) holds. The modulus of convexity of E is defined by

Concerning the modulus of convexity of E, Goebel and Kirk [10] proved the following result.

Lemma 1.5 (see [10], Lemma 10.1.)Let E be a uniformly convex Banach space with a modulus of convexity δ_E. Then δ_E : [0,2]→[0,1] is continuous, increasing, δ_E(0) = 0, δ_E(t) > 0 for t ∈ (0,2] and

A Banach space E is said to satisfy the Opial condition, if for any sequence {x_n} ⊂ E with x_n⇀x, then the following inequality holds:

Lemma 1.6 (Zhou [8]). Let E be a real reflexive Banach space with Opial condition. Let C be a nonempty closed convex subset of E and T : C → C be a continuous pseudocontractive mapping. Then I − T is demiclosed at zero, that is, for any sequence {x_n} ⊂ E, if x_n⇀y and ∥(I − T)x_n∥→0, then (I − T)y = 0.

Lemma 1.7 (Chang [11]). Let $$ be the normalized duality mapping, then for any x, y ∈ E,

Definition 1.8 (see [12].)Let {T_n} : C → E be a family of mappings with $$. We say {T_n} satisfies the AKTT-condition, if for each bounded subset B of C the following holds:

Lemma 1.9 (see [12].)Suppose that the family of mappings {T_n} : C → C satisfies the AKTT-condition. Then for each y ∈ C, {T_ny} converges strongly to a point in C. Moreover, let T : C → C be the mapping defined by

Theorem 2.1. Let E be a uniformly convex Banach space with a modulus of convexity δ_E, and C be a nonempty closed convex subset of E. Let {T_n} : C → C be a family of L_n-Lipschitzian and pseudocontractive mappings with L : = sup _n≥1L_n < ∞ and ℱ : = ⋂_n≥1F(T_n) ≠ ∅. Let {x_n} be the sequence defined by

• (i)

  limsup _n→∞α_n < 1;

• (ii)

  there exists a compact subset K ⊂ E such that $$;

• (iii)

  {T_n} satisfies the AKTT-condition, and F(T) ⊂ ℱ, where T : C → C is the mapping defined by (1.19).

Proof. First, we note that, by Remark 1.4, the method is well defined. So, we can divide the proof in three steps.

• (I)

  For each p ∈ ℱ the limit lim _n→∞∥x_n − p∥ exists.

•  

  In fact, since {T_n} is pseudocontractive, for each p ∈ ℱ, we have

  $$ (2.2)
  Simplifying, we have that
  $$ (2.3)
  Consequently, the limit lim _n→∞∥x_n − p∥ exists, and so the sequence {x_n} is bounded.

• (II)

  Now, we prove that lim _n→∞∥x_n − T_nx_n∥ = 0.

•  

  In fact, by virtue of (2.1) and (1.4), we have

  $$ (2.4)
  Letting u = (x_n−1 − p)/∥x_n−1 − p∥ and v = (x_n − p)/∥x_n−1 − p∥, from (2.3), we know that ∥u∥ = 1, ∥v∥≤1. It follows from (2.4) and Lemma 1.5 that
  $$ (2.5)
  Simplifying, we have that
  $$ (2.6)
  This implies that
  $$ (2.7)
  Letting lim _n→∞∥x_n − p∥ = r, if r = 0, the conclusion of Theorem 2.1 is proved. If r > 0, it follows from the property of modulus of convexity δ_E that ∥x_n−1 − x_n∥→0    (n → ∞). Therefore, from (2.1) and the condition (i), we have that
  $$ (2.8)
  In view of (2.1) and (2.8), we have
  $$ (2.9)

• (III)

  Now, we prove that {x_n} converges strongly to some point in ℱ.

•  

  In fact, it follows from (2.9) and condition (ii) that there exists a subsequence $$ such that $$ (as n_i → ∞), $$ and $$ (some point in C). Furthermore, by Lemma 1.9, we have $$. consequently, we have

  $$ (2.10)
  This implies that p = Tp, that is, p ∈ F(T) ⊂ ℱ. Since $$ and the limit lim _n→∞∥x_n − p∥ exists, we have x_n → p.

This completes the proof of Theorem 2.1.

Theorem 2.2. Let E be a uniformly convex Banach space satisfying the Opial condition. Let C be a nonempty closed convex subset of E and {T_n} : C → C be a family of L_n-Lipschitzian pseudocontractive mappings with L : = sup _n≥1L_n < ∞ and ℱ : = ⋂_n≥1F(T_n) ≠ ∅. Let {x_n} be the sequence defined by (2.1) and {α_n} be a sequence in (0, 1). If the following conditions are satisfied:

• (i)

  limsup _n→∞α_n < 1,

• (ii)

  for any bounded subset B of C

  $$ (2.11)

Proof. By the same method as given in the proof of Theorem 2.1, we can prove that the sequence {x_n} is bounded and

Now, we prove that

Indeed, for each m ≥ 1, we have

Finally, we prove that {x_n} converges weakly to some point u ∈ ℱ.

In fact, since E is uniformly convex, and so it is reflexive. Again since {x_n} ⊂ C is bounded, there exists a subsequence $$ such that $$. Hence from (2.13), for any m > 1, we have

Next, we prove that W_ω(x_n) is a singleton. Let us suppose, to the contrary, that if there exists a subsequence $$ such that $$ and q ≠ u. By the same method as given above we can also prove that q ∈ ℱ : = ⋂_n≥1F(T_n)∩W_ω(x_n). Taking p = u and p = q in (2.12). We know that the following limits

This completes the proof of Theorem 2.2.

Lemma 2.3 (see [12] or [9].)Let E be a smooth Banach space, C be a closed convex subset of E. Let {S_n} : C → C be a family of λ_n-strictly pseudocontractive mappings with $$ and λ : = inf _n≥1λ_n > 0. For each n ≥ 1 define a mapping T_n : C → C by:

• (i)

  $$, for all n ≥ 1;

• (ii)

  $$, for all k ≥ 1;

• (iii)

  $$.

• (1)

  each T_n, n ≥ 1 is a λ-strictly pseudocontractive mapping;

• (2)

  {T_n} satisfies the AKTT-condition;

• (3)

  if T : C → C is the mapping defined by

  $$ (2.22)

Theorem 2.4. Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E. Let {S_n} : C → C be a family of λ_n-strictly pseudocontractive mappings with $$ and λ : = inf _n≥1λ_n > 0. For each n ≥ 1 define a mapping T_n : C → C by

• (i)

  $$, for all n ≥ 1;

• (ii)

  $$, for all k ≥ 1;

• (iii)

  $$.

Let {x_n} be the sequence defined by

• (i)

  limsup _n→∞α_n < 1;

• (ii)

  there exists a compact subset K ⊂ E such that $$. Then, {x_n} converges strongly to some point p ∈ ℱ.

Proof. Since {S_n} : C → C is a family of λ_n-strictly pseudocontractive mappings with λ : = inf _n≥1λ_n > 0. Therefore, {S_n} is a family of λ-strictly pseudocontractive mappings. By Remark 1.2, {S_n} is a family of (1 + λ)/λ-Lipschitzian and strictly pseudocontractive mappings. Hence, by Lemma 2.3, {T_n} defined by (2.21) is a family of (1 + λ)/λ-Lipschitzian, strictly pseudocontractive mappings with $$ and it has also the following properties:

• (1)

  {T_n} satisfies the AKTT-condition;

• (2)

  if T : C → C is the mapping defined by (2.22), then Tx = lim _n→∞T_nx, x ∈ C and $$. Hence, by Definition 1.1, {T_n} is also a family of (1 + λ)/λ-Lipschitzian and pseudocontractive mappings having the properties (1) and (2) and $$. Therefore, {T_n} satisfies all the conditions in Theorem 2.1. By Theorem 2.1, the sequence {x_n} converges strongly to some point $$.

This completes the proof of Theorem 2.4.