Weak and Strong Convergence Theorems for Strictly Pseudononspreading Mappings and Equilibrium Problem in Hilbert Spaces

Definition 1. Let T : C → C be a mapping.

• (1)

  T is said to be nonexpansive, if ∥Tx − Ty∥ ≤ ∥x − y∥, ∀x, y ∈ C.

• (2)

  T is said to be quasinonexpansive, if F(T) is nonempty and

  $$ ()

• (3)

  T is said to be nonspreading [1, 2], if

  $$ ()

It is easy to prove that T : C → C is nonspreading if and only if

• (4)

  T : C → H is said to be k-strictly pseudononspreading in the terminology of Browder-Petryshyn [3], if there exists k ∈ [0,1) such that

  $$ ()

Remark 2. (1) If T : C → C is a nonspreading mapping with F(T) ≠ ∅, then T is quasinonexpansive and F(T) is closed and convex.

(2) Clearly every nonspreading mapping is k-strictly pseudononspreading with k = 0, but the inverse is not true. This can be seen from the following example.

Example 3. Let ℛ denote the set of all real numbers. Let T : ℛ → ℛ be a mapping defined by

It is easy to see that T is a k-strictly pseudononspreading mapping with k ∈ [0,1), but it is not nonspreading (see, [4]).

Definition 4. (1) Let T : H → H be a mapping. I − T is said to be demiclosed at 0, if for any sequence {x_n} ⊂ H with x_n⇀x^* and ∥(I − T)x_n∥ → 0, we have x^* = Tx^*.

(2) A Banach space E is said to have Opial′s property, if for any sequence {x_n} ⊂ E with x_n⇀x^*, we have

It is well known that each Hilbert space processes opial property.

(3) A mapping S : C → C is said to be semicompact, if for any bounded sequence {x_n} ⊂ C with lim _n→∞∥x_n − Sx_n∥ = 0, then there exists a subsequence $$ such that $$ converges strongly to some point x^* ∈ C.

Lemma 5 (see [5].)Let E be a uniformly convex Banach space and let B_r(0): = {x ∈ E : ∥x∥ ≤ r} be a closed ball with center 0 and radius r > 0. For any given sequence {x₁, x₂, …, x_n, …} ⊂ B_r(0) and any given number sequence {λ₁, λ₂, …, λ_n, …} with λ_i ≥ 0, $$, there exists a continuous strictly increasing and convex function g : [0,2r)→[0, ∞) with g(0) = 0 such that for any i, j ∈ 𝒩, i < j the following holds:

Lemma 6. Let H be a real Hilbert space, C be a nonempty and closed convex subset of H, and let T : C → C be a k-strictly pseudononspreading mapping.

• (i)

  If F(T) ≠ ∅, then it is closed and convex.

• (ii)

  (I − T) is demiclosed at origin.

Lemma 7. Let T : C → C be a k-strictly pseudononspreading mapping with k ∈ [0,1). Denote by T_β : = βI + (1 − β)T, where β ∈ [k, 1), then

• (i)

  F(T) = F(T_β);

• (ii)

  the following inequality holds:

  $$ ()

• (iii)

  T_β is a quasinonexpansive mapping, that is,

  $$ ()

Proof. The conclusion (i) is obvious. Now we prove the conclusion (ii). Since T is k-strictly pseudononspreading, for any x, y ∈ C we have

Take y ∈ F(T) in (8), then y ∈ F(T_β). Hence, conclusion (iii) is proved.

This completes the proof.

Lemma 8 (see [6], [7].)Let C be a nonempty and closed convex subset of a Hilbert space H and let ϕ : C × C → ℛ be a bi-function satisfying conditions: (A1), (A2), (A3), and (A4). Then, for any r > 0 and x ∈ C, there exists z ∈ C such that

Furthermore, if for given r > 0, we define a mapping T_r : C → C by then the following hold:

• (1)

  T_r is single-valued;

• (2)

  T_r is firmly nonexpansive, that is, $$;

• (3)

  F(T_r) = Ω, where Ω is the set of solutions of the equilibrium problem (11);

• (4)

  Ω is a closed and convex subset of C.

Theorem 9. Let H, C, {S_i}, k, β, {S_i,β}, ϕ, T_r, and Ω be the same as above. Let {x_n} and {u_n} be the sequences defined by

where {α_i,n}⊂(0,1) and {r_n} satisfy the following conditions:

• (a)

  $$, for each n ≥ 1;

• (b)

  for each i ≥ 1,  lim inf _n→∞α_0,nα_i,n > 0;

• (c)

  {r_n}⊂(0, ∞) and liminf _n→∞r_n > 0.

• (I)

  If $$, then both {x_n} and {u_n} converge weakly to some point x^* ∈ ℱ;

• (II)

  in addition, if there exists some positive integer m such that S_m is semicompact, then both {x_n} and {u_n} converge strongly to x^* ∈ ℱ.

Proof. First, we prove the conclusion (I). The proof is divided into three steps.

Step  1. We prove that the sequences {x_n}, {u_n}, {S_i,βu_n}, and {S_i,βx_n}, i ≥ 1 all are bounded, and for each p ∈ ℱ the limits lim _n→∞∥x_n − p∥, lim _n→∞∥u_n − p∥ exist and

In fact, it follows from Lemma 8 that $$, $$, and

Since p ∈ ℱ, by Lemma 7(i), $$. Hence, it follows from (17) and (9) that This implies that for each p ∈ ℱ, the limits lim _n→∞∥x_n − p∥ and lim _n→∞∥u_n − p∥ exist. And so {x_n} and {u_n} are bounded and (16) holds.

Furthermore, by (9), it is easy to see that for each i ≥ 1, {S_i,βu_n} and {S_i,βx_n} are also bounded.

Step  2. Next we prove that for each i ≥ 1 the following holds:

In fact, by Lemma 5 for any positive integer i ≥ 1 and p ∈ ℱ, we have This shows that Since g is a continuous and strictly increasing function with g(0) = 0. By condition (b), it yields that Therefore, we have On the other hand, it follows from Lemma 8 that $$ and for each p ∈ ℱ This shows that In view of (20) and (25) that is, In view of (27), (22), (14), and noting that {x_n − S_i,βx_n} is bounded, we have Therefore, we have

The conclusion is proved.

Step  3. Next we prove that the weak-accumulation point set W_ω(x_n) of the sequence {x_n} is a singleton and W_ω(x_n) ⊂ ℱ.

In fact, for any w ∈ W_ω(x_n), their exists a subsequence $$ such that $$. It follows from (27) that $$. Since $$, from (15) and condition (A2) we have

Since $$ and $$, it follows from condition (A4) that For any t ∈ (0,1), y ∈ C, letting y_t = ty + (1 − t)w, then y_t ∈ C. By condition (A1) and (A4), we have This implies that ϕ(y_t, y) ≥ 0. Letting t → 0, by condition (A3) we have This shows that w ∈ C is a solution to the equilibrium (11), that is, w ∈ Ω.

On the other hand, by Lemma 6, for each i ≥ 1, I − S_i is demiclosed at 0. In view of (19), we know that w ∈ ℱ. Due to the arbitrariness of w ∈ W_ω(x_n), we have W_ω(x_n) ⊂ ℱ.

Now we prove that W_ω(x_n) is a singleton. Suppose to the contrary that there exist x^*, y^* ∈ W_ω(x_n) with x^* ≠ y^*. Therefore, there exist subsequences $$ and $$ in {x_n} such that $$ and $$. Since x^*, y^* ∈ ℱ, by (16), the limits lim _n→∞∥x_n − x^*∥ and lim _n→∞∥x_n − y^*∥ exist. By using the opial property of H, we have

This is a contradiction. Therefore, W_ω(x_n) is a singleton. Without loss of generality, we can assume that W_ω(x_n) = {x^*} and x_n⇀x^*. By using (15) and (19), we have u_n⇀x^*.

This completes the proof of the conclusion (I).

Next we prove the conclusion (II).

Without loss of generality, we can assume that S₁ is semicompact. From (19) we have that

Therefore, there exists a subsequence of $$ such that $$. Since $$, we have x^* = u^* and so $$. By virtue of (16), we have

This completes the proof of Theorem 9.

Theorem 10. Let H, C, {S_i}, k, β and {S_i,β} be the same as in Theorem 9. Let {x_n} be the sequences defined by

where {α_i,n}⊂(0,1) satisfies the following conditions:

• (a)

  $$, for each n ≥ 1;

• (b)

  for each i ≥ 1,  lim inf _n→∞α_0,nα_i,n > 0.

• (I)

  If $$, then both {x_n} converge weakly to some point x^* ∈ ℱ;

• (II)

  in addition, if there exists some positive integer m such that S_m is semicompact, then {x_n} converge strongly to x^* ∈ ℱ.

Remark 11. Theorems 9 and 10 improve and extend the corresponding recent results of [4, 8–11].