Shrinking Projection Method of Fixed Point Problems for Asymptotically Pseudocontractive Mapping in the Intermediate Sense and Mixed Equilibrium Problems in Hilbert Spaces

Theorem NT. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive mapping of C into itself such that F(T) ≠ ∅. Suppose that x₁ = x ∈ C chosen arbitrarily and {x_n} the sequence defined by

where 0 ≤ α_n ≤ α < 1. Then {x_n} converges strongly to P_F(T)(x₁).

Theorem S. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let T : C → C be a completely continuous uniformly L-Lipschitzian such that L > 0 and asymptotically pseudocontractive mapping defined as in (1.9) with the sequence {k_n}⊂[1, ∞) such that lim _n→∞  k_n = 1. Let q_n = 2k_n − 1 for all n ∈ ℕ. Suppose that x₁ = x ∈ C chosen arbitrarily and {x_n} the sequence defined by

where {α_n}, {β_n}⊂(0,1) such that ϵ ≤ α_n ≤ β_n ≤ b for some ϵ > 0 and $$ and $$. Then {x_n} converges strongly to some fixed point of T.

Theorem Z. Let C be a nonempty bounded closed convex subset of a real Hilbert space H. Let T : C → C be a uniformly L-Lipschitzian such that L > 0 and asymptotically pseudocontractive mapping with a fixed point defined as in (1.8) with the sequence {k_n}⊂[1, ∞) such that lim _n→∞  k_n = 1. Suppose that x₁ = x ∈ C chosen arbitrarily and {x_n} the sequence defined by

where {α_n}⊂[a, b] such that 0 < a < b < 1/(1 + L). Then {x_n} converges strongly to P_F(T)(x₁).

Lemma 2.1 (see [15].)Let H be a Hilbert space. For any x, y ∈ H and λ ∈ ℝ, we have

Lemma 2.2 (see [3].)Let C be a nonempty closed convex subset of a Hilbert space H. Then the following inequality holds:

Lemma 2.3 (see [16].)Let C be a nonempty closed convex subset of a Hilbert space H, Φ : C × C → ℝ satisfying the conditions (A1)–(A5), and let φ : C → ℝ ∪ {+∞} be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r > 0, define a mapping S_r : C → C as follows:

for all x ∈ C. Then, the following statement hold:

• (1)

  for each x ∈ C, S_r(x) ≠ ∅;

• (2)

  S_r is single-valued;

• (3)

  S_r is firmly nonexpansive; that is, for any x, y ∈ C,

  $$ ()

• (4)

  F(S_r) = MEP (Φ, φ);

• (5)

  MEP (Φ, φ) is closed and convex.

Lemma 2.4 (see [3].)Every Hilbert space H has Radon-Riesz property or Kadec-Klee property, that is, for a sequence {x_n} ⊂ H with x_n⇀x and ∥x_n∥→∥x∥ then x_n → x.

Lemma 2.5 (see [7].)Let C be a nonempty closed convex of a real Hilbert space H, and let T : C → C be a uniformly L-Lipschitz and asymptotically pseudocontractive mapping in the intermediate sense such that F(T) is nonempty. Then I − T is demiclosed at zero. That is, whenever {x_n} is a sequence in C weakly converging to some x ∈ C and the sequence {(I − T)x_n} strongly converges to some y, it follows that (I − T)x = y.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert space H, Φ a bifunction from C × C into ℝ satisfying the conditions (A1)–(A5), and φ : C → ℝ ∪ {+∞} a proper lower semicontinuous and convex function with either (B1) or (B2) holds. Let T : C → C be a uniformly L -Lipschitzian such that L > 0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {k_n}⊂[1, ∞) and {τ_n}⊂[0, ∞) such that lim _n→∞  k_n = 1 and lim _n→∞  τ_n = 0. Let q_n = 2k_n − 1 for all n ∈ ℕ. Assume that Ω∶ = F(T)∩MEP (Φ, φ) be a nonempty bounded subset of C. For x₁ = x ∈ C chosen arbitrarily, suppose that {x_n}, {y_n}, {z_n}, and {u_n} are generated iteratively by

where $$ and Δ_n = sup {∥x_n − z∥:z ∈ Ω} < ∞ satisfying the following conditions:

• (C1)

  {α_n}, {β_n}⊂(0,1) such that a ≤ α_n ≤ β_n ≤ b for some a > 0 and $$;

• (C2)

  {r_n}⊂[r, ∞) for some r > 0;

• (C3)

  $$.

Then the sequences {x_n}, {y_n}, {z_n}, and {u_n} converge strongly to w = P_Ω(x₁).

Proof. Pick p ∈ Ω. Therefore, by (3.1) and the definition of $$ in Lemma 2.3, we have

and by Tp = p, and Lemma 2.3 (4), we have By (3.2), (3.3), and the nonexpansiveness of $$, we have By (3.3), Lemma 2.1, the uniformly L-Lipschitzian of T, and the asymptotically pseudocontractive mapping in the intermediate sense of T, we have Substituting (3.6) and (3.7) into (3.5), and by the condition (C1) and (3.4), we have where $$ and Δ_n∶ = sup {∥x_n − z∥:z ∈ Ω}.

Firstly, we prove that C_n∩Q_n is closed and convex for all n ∈ ℕ. It is obvious that C₁∩Q₁ is closed, and by mathematical induction that C_n∩Q_n is closed for all n ≥ 2, that is C_n∩Q_n is closed for all n ∈ ℕ. Let $$. Since for any z ∈ C, $$ is equivalent to

for all n ∈ ℕ. Therefore, for any z₁, z₂ ∈ C_n+1∩Q_n+1 ⊂ C_n∩Q_n and ϵ ∈ (0,1), we have for all n ∈ ℕ, and we have for all n ∈ ℕ. Since C₁∩Q₁ is convex, and by putting n = 1 in (3.9), (3.10), and (3.11), we have C₂∩Q₂ is convex. Suppose that x_k is given and C_k∩Q_k is convex for some k ≥ 2. It follows by putting n = k in (3.9), (3.10), and (3.11) that C_k+1∩Q_k+1 is convex. Therefore, by mathematical induction, we have C_n∩Q_n is convex for all n ≥ 2, that is, C_n∩Q_n is convex for all n ∈ ℕ. Hence, we obtain that C_n∩Q_n is closed and convex for all n ∈ ℕ.

Next, we prove that Ω ⊂ C_n∩Q_n for all n ∈ ℕ. It is obvious that p ∈ Ω ⊂ C = C₁∩Q₁. Therefore, by (3.1) and (3.8), we have p ∈ C₂, and note that p ∈ C = Q₂, and so p ∈ C₂∩Q₂. Hence, we have Ω ⊂ C₂∩Q₂. Since C₂∩Q₂ is a nonempty closed convex subset of C, there exists a unique element x₂ ∈ C₂∩Q₂ such that $$. Suppose that x_k ∈ C_k∩Q_k is given such that $$, and p ∈ Ω ⊂ C_k∩Q_k for some k ≥ 2. Therefore, by (3.1) and (3.8), we have p ∈ C_k+1. Since $$, therefore, by Lemma 2.2, we have

for all z ∈ C_k∩Q_k. Thus, by (3.1), we have p ∈ Q_k+1, and so p ∈ C_k+1∩Q_k+1. Hence, we have Ω ⊂ C_k+1∩Q_k+1. Since C_k+1∩Q_k+1 is a nonempty closed convex subset of C, there exists a unique element x_k+1 ∈ C_k+1∩Q_k+1 such that $$. Therefore, by mathematical induction, we obtain Ω ⊂ C_n∩Q_n for all n ≥ 2, and so Ω ⊂ C_n∩Q_n for all n ∈ ℕ, and we can define $$ for all n ∈ ℕ. Hence, we obtain that the iteration (3.1) is well defined.

Next, we prove that {x_n} is bounded. Since $$ for all n ∈ ℕ, we have

for all z ∈ C_n∩Q_n. It follows by p ∈ Ω ⊂ C_n∩Q_n that ∥x_n − x₁∥≤∥p − x₁∥ for all n ∈ ℕ. This implies that {x_n} is bounded, and so are {y_n}, {z_n}, and {u_n}.

Next, we prove that ∥x_n − x_n+1∥→0 and ∥u_n − u_n+1∥→0 as n → ∞. Since $$, therefore, by (3.13), we have ∥x_n − x₁∥≤∥x_n+1 − x₁∥ for all n ∈ ℕ. This implies that {∥x_n − x₁∥} is a bounded nondecreasing sequence; there exists the limit of ∥x_n − x₁∥, that is

for some m ≥ 0. Since x_n+1 ∈ Q_n+1, therefore, by (3.1), we have It follows by (3.15) that Therefore, by (3.14), we obtain Indeed, from (3.1) we have substituting y = u_n+1 into (3.18) and y = u_n into (3.19), we have Therefore, by the condition (A2), we get It follows that Thus, we have It follows by the condition (C2) that where M = sup _n≥1∥u_n − x_n∥<∞. Therefore, by the condition (C3) and (3.17), we obtain

Next, we prove that ∥Tⁿu_n − u_n∥→0 and ∥Tu_n − u_n∥→0 as n → ∞. Since x_n+1 ∈ C_n+1, by (3.1), we have

it follows by the condition (C1) that Since lim _n→∞  q_n = 1 and the condition (C1), we have lim _n→∞(1 − bq_n − b²L² − b) = 1 − 2b − b²L² > 0. Therefore, from (3.27) by (3.17) and lim _n→∞  θ_n = 0, we obtain By the uniformly L-Lipschitzian of T, we have Therefore, by (3.25) and (3.28), we obtain

Next, we prove that ∥y_n − x_n∥→0, ∥u_n − z_n∥→0 and ∥u_n − x_n∥→0 as n → ∞. From (3.26), by the condition (C1), we have

it follows that Therefore, by (3.17), lim _n→∞  θ_n = 0, and lim _n→∞(1 − bq_n − b²L² − b) = 1 − 2b − b²L² > 0, we obtain From (3.1), we have ∥u_n − z_n∥ = β_n∥u_n − Tⁿu_n∥; therefore, by (3.28), we obtain By (3.2), (3.3), and the firmly nonexpansiveness of $$, we have it follows that Therefore, from (3.8), we have it follows by the condition (C1) that Therefore, by (3.33), lim _n→∞  θ_n = 0, and lim _n→∞(1 − bq_n − b²L² − b) = 1 − 2b − b²L² > 0, we obtain Since {u_n} is bounded, there exists a subsequence $$ of {u_n} which converges weakly to $$. Next, we prove that $$. From $$ and $$ as i → ∞ by (3.30), therefore, by Lemma 2.5, we obtain $$. From (3.1), we have it follows by the condition (A2) that Hence, Therefore, from (3.39) and by $$ as i → ∞, we obtain For a constant t with 0 < t < 1 and y ∈ C, let $$. Since $$, thus, y_t ∈ C. So, from (3.43), we have By (3.44), the conditions (A1) and (A4), and the convexity of φ, we have it follows that Therefore, by the condition (A3) and the weakly lower semicontinuity of φ, we have $$ as t → 0 for all y ∈ C, and hence, we obtain $$, and so $$.

Since Ω is a nonempty closed convex subset of C, there exists a unique w ∈ Ω such that w = P_Ω(x₁). Next, we prove that x_n → w as n → ∞. Since w = P_Ω(x₁), we have ∥x₁ − w∥≤∥x₁ − z∥ for all z ∈ Ω; it follows that

Since w ∈ Ω ⊂ C_n∩Q_n, therefore, by (3.13), we have Since $$ by (3.39) and $$, we have $$ as i → ∞. Therefore, by (3.47), (3.48) and the weak lower semicontinuity of norm, we have It follows that Since $$ as i → ∞, therefore, we have Hence, from (3.50), (3.51), Kadec-Klee property, and the uniqueness of w = P_Ω(x₁), we obtain It follows that {x_n} converges strongly to w and so are {y_n}, {z_n}, and {u_n}. This completes the proof.

Remark 3.2. The iteration (3.1) is the difference with the iterative scheme of Qin et al. [7] as follows.

• (1)

  The sequence {x_n} is a projection sequence of x₁ onto C_n∩Q_n for all n ∈ ℕ such that

  $$ ()

• (2)

  A solving of a common element of the set of fixed point for an asymptotically pseudocontractive mapping in the intermediate sense and the set of solutions of the mixed equilibrium problems by iteration is obtained.

We define the condition (B3) as the condition (B1) such that φ = 0. If φ = 0, then Theorem 3.1 is reduced immediately to the following result.

Corollary 3.3. Let C be a nonempty closed convex subset of a real Hilbert space H, and let Φ a bifunction from C × C into ℝ satisfying the conditions (A1)–(A5) with either (B2) or (B3) holds. Let T : C → C be a uniformly L-Lipschitzian such that L > 0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {k_n}⊂[1, ∞) and {τ_n}⊂[0, ∞) such that lim _n→∞  k_n = 1 and lim _n→∞  τ_n = 0. Let q_n = 2k_n − 1 for all n ∈ ℕ. Assume that Ω : = F(T)∩EP (Φ) be a nonempty bounded subset of C. For x₁ = x ∈ C chosen arbitrarily, suppose that {x_n}, {y_n}, {z_n}, and {u_n} are generated iteratively by

where $$ and Δ_n = sup {∥x_n − z∥:z ∈ Ω} < ∞ satisfying the following conditions:

•  

  (C1) {α_n}, {β_n}⊂(0,1) such that a ≤ α_n ≤ β_n ≤ b for some a > 0 and $$;

•  

  (C2) {r_n}⊂[r, ∞) for some r > 0;

•  

  (C3) $$.

Then the sequences {x_n}, {y_n}, {z_n}, and {u_n} converge strongly to w = P_Ω(x₁).

Corollary 3.4. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C → C be a uniformly L-Lipschitzian such that L > 0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {k_n}⊂[1, ∞) and {τ_n}⊂[0, ∞) such that lim _n→∞  k_n = 1 and lim _n→∞  τ_n = 0. Let q_n = 2k_n − 1 for all n ∈ ℕ. Assume that F(T) be a nonempty bounded subset of C. For x₁ = x ∈ C chosen arbitrarily, suppose that {x_n}, {y_n}, and {z_n} are generated iteratively by

where $$, Δ_n = sup {∥x_n − z∥:z ∈ F(T)} < ∞ and {α_n}, {β_n}⊂(0,1) such that a ≤ α_n ≤ β_n ≤ b for some a > 0 and $$. Then the sequences {x_n}, {y_n} and {z_n} converge strongly to w = P_F(T)(x₁).

Corollary 3.5. Let C is a nonempty closed convex subset of a real Hilbert space H, and ζ : C → ℝ ∪ {+∞} be a proper lower semicontinuous and convex function with either (B2) or (B4) holds. Let T : C → C be a uniformly L-Lipschitzian such that L > 0 and asymptotically pseudocontractive mapping in the intermediate sense defined as in (1.13) with the sequences {k_n}⊂[1, ∞) and {τ_n}⊂[0, ∞) such that lim _n→∞  k_n = 1 and lim _n→∞  τ_n = 0. Let q_n = 2k_n − 1 for all n ∈ ℕ. Assume that Ω : = F(T)∩Argmin(ζ) be a nonempty bounded subset of C. For x₁ = x ∈ C chosen arbitrarily, suppose that {x_n}, {y_n}, {z_n} and {u_n} are generated iteratively by

where $$ and Δ_n = sup {∥x_n − z∥:z ∈ Ω} < ∞ satisfying the following conditions:

•  

  (C1) {α_n}, {β_n}⊂(0,1) such that a ≤ α_n ≤ β_n ≤ b for some a > 0 and $$;

•  

  (C2) {r_n}⊂[r, ∞) for some r > 0;

•  

  (C3) $$.

Then the sequences {x_n}, {y_n}, {z_n}, and {u_n} converge strongly to w = P_Ω(x₁).