Strong Convergence of the Viscosity Approximation Process for the Split Common Fixed-Point Problem of Quasi-Nonexpansive Mappings

Algorithm 1. Initialization: Let x₀ ∈ H be arbitrary.

Iterative step: Set T = U(I + γA^*(S − I)A). For k ∈ N, let

where f : H → H is a contraction of modulus ρ,   ω_k ∈ (0,1/2),   γ ∈ (0,1/λ) with λ being the spectral radius of the operator A^*A, and α_k ∈ (0,1).

This paper establishes the strong convergence of the sequence given by (1.12) to the unique solution of the variational inequality problem VIP(I − f, Γ):

Now we give a series of preliminary results needed for the convergence analysis of algorithm (1.12).

Lemma 1.1. Let H be a real Hilbert space and T : H → H a quasi-nonexpansive mapping. Then, the following properties are reached:

• (i)

  $$;

• (ii)

  $$ and $$.

Remark 1.2. Let F : = I − f, where f is the contraction defined in (1.7). It is a simple matter to see that the operator F is (1 − ρ) strongly monotone over H; that is,

Lemma 1.3 (see [18], Lemma 1.3.)Let {δ_n} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence $$ of {δ_n} which satisfies $$ for all j ≥ 0. Also consider the sequence of integers $$ defined by

Then $$ is a nondecreasing sequence verifying lim _n→∞τ(n) = ∞, and, for all n ≥ n₀, it holds that δ_τ(n) ≤ δ_τ(n)+1 and one has

Proof. Clearly, we can see that {τ(n)} is a well-defined sequence, and the fact that it is nondecreasing is obvious as well as lim _n→∞τ(n) = ∞ and δ_τ(n) ≤ δ_τ(n)+1. Let us prove (1.16). It is easily observed that τ(n) ≤ n. Consequently, we prove (1.16) by distinguishing the three cases: (c1) τ(n) = n; (c2) τ(n) = n − 1; (c3) τ(n) < n − 1. In the first case (i.e., τ(n) = n), (1.16) is immediately given by δ_τ(n) ≤ δ_τ(n)+1. In the second case (i.e., τ(n) = n − 1), (1.16) becomes obvious. In the third case (i.e., τ(n) ≤ n − 2), by (1.15) and for any integer n ≥ n₀, we easily observe that δ_j ≥ δ_j+1 for τ(n) + 1 ≤ j ≤ n − 1; namely,

which entails the desired result.

Theorem 2.1. Given a bounded linear operator A : H₁ → H₂, let U : H₁ → H₁ and S : H₂ → H₂ be quasi-nonexpansive mappings with nonempty fixed-point set F(U) = C and F(S) = Q. Assume that U − I and S − I are demiclosed at origin. Let {x_k} be the sequence given by (1.12) with γ ∈ (0,1/λ), ω_k ∈ (0,1/2) such that 0 < lim  inf _k→∞ω_k ≤ lim  sup _k→∞ω_k < 1/2 and {α_k}⊂(0,1) such that lim _k→∞α_k = 0 and ∑_kα_k = ∞. If Γ ≠ ∅, then the sequence {x_k} strongly converges to a split common fixed-point x^* ∈ Γ, verifying x^* = P_Γf(x^*) which equivalently solves the following variational inequality problem:

Proof. Set $$. Then $$.

Firstly, we prove that {x_k} is bounded. Taking y ∈ Γ, that is, y ∈ F(U), Ay ∈ F(S). We have

From the definition of $$, we get On the other hand, we have From the definition of λ, it follows that Now, by using property (ii) of Lemma 1.1, we obtain Combining (2.4)–(2.6), we have From property (i) of Lemma 1.1, we have From (2.3) and (2.8), we have Combining (2.2), (2.3), and (2.9), it follows that   It is obviously that and hence {x_k} is bounded. Let x^* = P_Γf(x^*). We have and hence By (2.9) we obtain that It follows from (2.13) that and hence Setting $$, we have so that (2.16) can be rewritten as Now using (2.12) again, we have which yields From (2.18) and (2.20), we obtain It follows from Remark 1.2 that and hence

The rest of the proof will be divided into two parts.

Theorem 2.2. Given a bounded linear operator A : H₁ → H₂, let U : H₁ → H₁ and S : H₂ → H₂ be quasi-nonexpansive mappings with nonempty fixed-point set F(U) = C and F(S) = Q. Assume that U − I and S − I are demiclosed at origin. Let x₀ ∈ H be arbitrary and {x_k} the sequence given by

where T = U(I + γA^*(S − I)A), f : H → H a contraction of modulus ρ,   γ ∈ (0,1/λ), ω ∈ (0,1/2), and {α_k}⊂(0,1) such that lim _k→∞α_k = 0 and ∑_kα_k = ∞. If Γ ≠ ∅, then the sequence {x_k} strongly converges to a split common fixed-point x^* ∈ Γ, verifying x^* = P_Γf(x^*) which equivalently solves the following variational inequality problem: