On the Strong Convergence of an Algorithm about Firmly Pseudo-Demicontractive Mappings for the Split Common Fixed-Point Problem

Definition 2.1. We say that T is demicontractive [6] means that there exists constant β < 1 such that

T is pseudo-demicontractive [9] means that there exists constant α > 1 such that

Definition 2.2. We say that T is firmly pseudo-demicontractive means that there exist constants α > 1, β > 1 such that

The inequality (2.5) is equivalent to

An operator satisfying (2.5) will be referred to as a α, β firmly pseudo-demicontractive mapping. It is worth noting that the class of firmly pseudo-demicontractive maps contains important operators such as the demicontractive maps, quasi-nonexpansive maps, and the strictly pseudo-contractive maps with fixed points.

Next, let us recall several concepts:

• (1)

  a mapping T : H → H is said to be nonexpansive if

  $$ ()

• (2)

  a mapping T : H → H is said to be quasi-nonexpansive if

  $$ ()

• (3)

  a mapping T : H → H is said to be strictly pseudo-contractive if

  $$ ()

Obviously, the nonexpansive operators are both quasi-nonexpansive and strictly pseudo-contractive maps and are well known for being demiclosed.

Lemma 2.3 (see [5].)An operator T is said to be closed at a point y ∈ Rⁿ if for every x ∈ Rⁿ and every sequence x_k in Rⁿ, such that, x_k → x(k → ∞) and T(x_k) → y, we have Tx = y.

In what follows, only the particular case of closed at zero will be used, which is the particular case when y = 0.

Lemma 2.4 (see [12].)Let {a_n},{b_n} and {δ_n} be sequences of nonnegative real numbers satisfying the inequality

where {δ_n} ⊂ l¹, {b_n} ⊂ l¹, then lim _n→∞ a_n exists.

Algorithm 3.1. Initialization: let x₀ ∈ H₁ be arbitrary.

Iterative step: for k ∈ N set u_k = x_k + γA^*(T − I)Ax_k and let

where (1 − β)/λ < γ < 0 with λ being the spectral radius of the operator A^*A, t_k > 0.

Theorem 3.2. Let H₁, H₂ are two Hilbert spaces, A is a bounded linear operator, U : H₁ → H₁,   T : H₂ → H₂ are two firmly pseudo-demicontractive operators with Fix (U) = C and Fix (T) = Q, α, β, and μ, θ are two firmly pseudo-demicontractive coefficients of U, T, respectively. Let $$, δ_k = t_k(μ − 1) and {b_k} ⊂ l¹, {δ_k} ⊂ l¹. T − I is closed at the origin, λ the spectral radius of the operator A^*A, then the sequence generated by the modified algorithm (3.4) converges strongly to the solution of (3.1).

Proof. Taking y ∈ Γ, that is, y ∈ Fix (U), Ay ∈ Fix (T), and using (2.6), we obtain that

Using the expression of u_k in Algorithm 3.1, we also have From the definition of λ, it follows that

Now, by setting θ = 2γ〈x_k − y, A^*(T − I)Ax_k〉 and using the fact (2.5) and its equivalent form (2.6), we infer that

Substituting (3.8), (3.7), and (3.6) into (3.5), we get the following inequality: Since ((1 − β)/λ) < γ < 0, t_k > 0 and α > 1, β > 1, μ > 1, θ > 1, we obtain that −t_k(1 − μ) > 0, (1 − t_k(1 − μ))γ(α − 1) < 0, (1 − t_k(1 − μ))γ(β − 1 + γλ) < 0, and t_k(θ + t_k − 1) > 0, thus, Setting $$ and δ_k = t_k(μ − 1), (3.10) can be formulated as We also can denote that $$, $$, thus (3.11) can be rewritten as

Obviously, {a_k}, {b_k}, and {δ_k} are sequences of nonnegative real numbers. Since {b_k} ⊂ l¹, {δ_k} ⊂ l¹, that is, $$, $$ from Lemma 2.4, $$ exists.

Since λ being the spectral radius of the operator A^*A, (3.9) also can be reformulated as the following:

Taking limits from both side of (3.13), (1 − t_k(1 − μ))γ(α − 1)λ < 0, t_k(μ − 1) → 0, (1 − t_k(1 − μ))γ(β − 1 + γλ) < 0, $$, we obtain that If we take limits from both sides of (3.9), we can get the following: Because T − I is closed at the origin, from (3.15) and (3.16), using Lemma 2.3, we have

The sequence generated by modified algorithm (3.1) converges strongly to the solution of SCFP. The proof is completed.

Corollary 3.3. Let H₁, H₂ are two Hilbert spaces, A is a bounded linear operator, U : H₁ → H₁,   T : H₂ → H₂ are two pseudo-demicontractive operators with Fix (U) = C and Fix (T) = Q, α and μ are two pseudo-demicontractive coefficients of U, T, respectively. Let $$, δ_k = t_k(μ − 1), and {b_k} ⊂ l¹, {δ_k} ⊂ l¹. T − I is closed at the origin, and λ is the spectral radius of the operator A^*A, then the sequence generated by the modified algorithm (3.4) converges strongly to the solution of (3.1).

Proof. Taking β = θ = 1, here (3.9) as the following:

Because from (3.18), we get The same lines as the proof of Thereom 3.2, we obtain the desired result.