Definition 1. A mapping T : H⟶H is called

• (i)

  k-Lipschitz continuous, if ‖Tx − Ty‖ ≤ k‖x − y‖ for all x, y ∈ H, where k is a positive number.

• (ii)

  Nonexpansive, if (i) holds with k = 1.

• (iii)

  η-strongly monotone, if η‖x − y‖² ≤ 〈Tx − Ty, x − y〉 for all x, y ∈ H, where η is a positive number.

• (iv)

  Firmly nonexpansive, if ‖Tx − Ty‖² ≤ 〈Tx − Ty, x − y〉 for all x, y ∈ H.

• (v)

  α-Averaged, if T = (1 − α)I + αN for some fixed α ∈ (0,1) and a nonexpansive mapping N.

In [5], we know that the metric projection P_C : H⟶C is firmly nonexpansive and (1/2)-averaged.

We collect some basic properties of averaged mappings in the following results.

Lemma 1 (see [16].)We have

• (i)

  The composite of finitely many averaged mappings is averaged. In particular, if T_i is α_i-averaged, where α_i ∈ (0,1) for i = 1,2, then the composite T₁T₂ is α-averaged, where α = α₁ + α₂ − α₁α₂.

• (ii)

  If the mappings $$ are averaged and have a common fixed point, then

Proposition 1 (see [19].)Let D be a nonempty subset of H, m ≥ 2 be an integer, and ϕ; (0,1)^m⟶(0,1) be defined by

Lemma 2 (see [20].)Assume that H₁ and H₂ are Hilbert spaces. Let A : H₁⟶H₂ be a linear bounded mapping such that A ≠ 0 and let T : H₂⟶H₂ be a nonexpansive mapping. Then, for 0 ≤ γ < 1/‖A‖², I − γA^∗(I − T)A is γ‖A‖²-averaged.

Proposition 2 (see [19].)Let C be a nonempty subset of H, and let $$ be a finite family of nonexpansive mappings from C to H. Assume that $$ and $$ such that ∑_i∈Iδ_i = 1. Suppose that, for every i ∈ I, T_i is $$-averaged; then, T = ∑_i∈Iδ_iT_i is α-averaged, where $$.

The following results play a crucial role in the next section.

Lemma 3 (see [14].)Let t be a real number in (0,1]. Let F : H⟶H be an η-strongly monotone and k-Lipschitz continuous mapping. The mapping I − tμF, for each fixed point μ ∈ (0, (2η/k²)), is contractive with constant 1 − tτ, i.e.,

Theorem 1 (see [21].)Let F be a k-Lipschitz continuous and η-strongly monotone self-mapping of H. Assume that $$ is a finite family of nonexpansive mappings from H to H such that $$. Then, the sequence {x_n} defined by the following algorithm converges strongly to the unique solution x^∗ of the variational inequality (4):

• (i)

  t_n ∈ (0,1),  t_n⟶0 as n⟶∞ and $$.

• (ii)

  $$, for some α, β ∈ (0,1), and $$ as n⟶∞,  (i = 0, …, N).

Theorem 2 (see [22].)Let F, C, $$, {t_n}, and $$ be as in Theorem 1. Then, the sequence {x_n} defined by the following algorithm:

Theorem 3. Let C and Q be two closed convex subsets in H₁ and H₂, respectively. Then, as n⟶∞, the sequence {x_n} defined by (15), where the sequences {t_n} and {α_n} satisfy conditions (C1) and (C2), respectively, converges strongly to the solution of (6).

Proof. From Lemma 2, we have that I − γA^∗(I − P_Q)A is γ‖A‖²-averaged. Since T = P_C(I − γA^∗(I − P_Q)A), by Lemma 1 (i), we get that T is λ-averaged where λ = (1 + γ‖A‖²/2). Moreover, we obtain that z ∈ Γ if and only if z ∈ Fix(T). It follows from Definition 1 (iv) that T = (1 − λ)I + λS, where S is nonexpansive. Then, iterative algorithm (15) can be rewritten as follows:

In [23], Miao and Li showed the weak convergence results of the sequence {x_n} converging to the element of Fix(T) where {x_n} is generated by the following algorithm:

Theorem 4. Let C and Q be two closed convex subsets in H₁ and H₂, respectively. Then, as n⟶∞, the sequence {x_n} defined by (17), where the sequence {t_n} satisfies condition (C1) and {β_n} and {α_n} satisfy condition (C2), converges strongly to the solution of (6).

Proof. In the proof of Theorem 3, one can rewrite iterative algorithm (17) as follows:

Moreover, we obtain the following results which are solving the common solution of variational inequality problem and multiple-sets split feasibility problem, i.e., find a point

Theorem 5. Let $$ and $$ be two finite families of closed convex subsets in H₁ and H₂, respectively. Assume that γ ∈ (0, 1/‖A‖²), {t_n} and {α_n} satisfy conditions (C1) and (C2), respectively, and the parameters {δ_n} and {ζ_n} satisfy the following conditions:

• (a)

  δ_i > 0 for 1 ≤ i ≤ N such that $$.

• (b)

  ζ_j > 0 for 1 ≤ j ≤ M such that $$.

Proof. Let T = P₁(I − γA^∗(I − P₂)A). We will show that T is averaged.

In the case of (A1), $$ and $$. Since $$ is (1/2)-averaged for all i = 1, …, N, by Proposition 1, we get that P₁ is λ₁-averaged, where λ₁ = N/(N + 1). Similarly, we have that P₂ is also averaged and so P₂ is nonexpansive. By using Lemma 2, we deduce that I − γA^∗(I − P₂)A is λ₂-averaged, where λ₂ = γ‖A‖². It follows from Lemma 1 (i) that T is λ-averaged with λ = N/(N + 1) + γ‖A‖² − (N/(N + 1))γ‖A‖².

If $$ and $$, then by using Proposition 2 and condition (a), we obtain that P₁ is (1/2)-averaged. From condition (b) and taking into account that $$ is nonexpansive, for all j = 1, …, M, we have that P₂ is also nonexpansive. It follows from Lemma 2 that I − γA^∗(I − P₂)A is γ‖A‖²-averaged. Thus, T is λ-averaged with λ = (1 + γ‖A‖²)/2.

Cases (A3) and (A4) are similar. This implies that T : = (1 − λ)I + λS, where S is nonexpansive. Moreover, by Lemma 1, we get that

Then, iterative algorithm (20) can be rewritten as follows:

Theorem 6. Let $$, $$, and {ζ_n} be as in Theorem 5. Then, as n⟶∞, the sequence {x_n}, defined by

Proof. In the proof of Theorem 5, one can rewrite iterative algorithm (23) as follows:

Example 1. We consider test problem (25), where N = M = 1, n = m = 2, φ(x) = (1 − a)‖x‖²/2 for some fixed a ∈ (0,1), and

So, we have that F : = φ^′ = (1 − a)I is a k-Lipschitz continuous and η-strongly monotone mapping with k = η = (1 − a). For each algorithm, we set aⁱ = (1/i, −1), b_i = 0, for all i = 1, …, N, and a^j = (1/j, 0), R_j = 1, for all j = 1, …, M. Taking a = 0.5, γ = 0.3, the stopping criterion is defined by E_n = ‖x_n+1 − x_n‖ < ε where ε = 10⁻⁴ and 10⁻⁶. The numerical results are listed in Table 1 with different initial points x¹, where n is the number of iterations and s is the CPU time in seconds. In Figures 1 and 2, we present the graphs illustrating the number of iterations for both methods using the stopping criterion defined as above with the different initial points shown in Table 1.

Remark 1. From the numerical analysis of our results in Table 1 and Figures 1 and 2, we get that algorithm (15) (new method) has less number of iterations and faster convergence than algorithm (8) (Buong method).

Example 2. In this example, we consider algorithm (23) for solving test problem (25), where N = 5 and M = 4. Let $$, $$, φ, a, and A be as in Example 1. In the numerical experiment, we take the stopping criterion E_n < 10⁻⁴. The numerical results are listed in Table 2 with different cases of P₁ and P₂. In Figures 3 and 4, we present the graphs illustrating the number of iterations for all cases of P₁ and P₂ using the stopping criterion as above with the different initial points appeared in Table 2. Moreover, Table 3 shows the effect of different choices of γ.

Remark 2. We observe from the numerical analysis of Table 2 that algorithm (23) has the fastest convergence when P₁ and P₂ satisfy (A4) and the slowest convergence when P₁ and P₂ satisfy (A3). Moreover, we require less iteration steps and CPU times for convergence when γ is chosen very small and close to zero.