A New Iterative Method for Solving a System of Generalized Mixed Equilibrium Problems for a Countable Family of Generalized Quasi-ϕ-Asymptotically Nonexpansive Mappings

Definition 1. (1) [1] A mapping T : C → C is said to be generalized quasi-ϕ-asymptotically nonexpansive in the light of [1], if F(T) ≠ ∅, and there exist nonnegative real sequences {ν_n} and {μ_n} with ν_n, μ_n → 0 (as n → ∞) such that

where ϕ : E × E → ℝ denotes the Lyapunov functional defined by It is obvious from the definition of ϕ that

(2) A mapping T : C → C is said to be uniformly L-Lipschitz continuous, if there exists a constant L > 0 such that

Example 2. Let C be a unit ball in a real Hilbert space l², and let T : C → C be a mapping defined by

where {a_i} is a sequence in (0,1) satisfying $$. It is shown by Goebel and Kirk [2] that where ϕ(x, y) = ∥x − y∥², $$, for all n ≥ 1, and {μ_n} is a nonnegative real sequence with μ_n → 0 as n → ∞. This shows that the mapping T defined earlier is a generalized quasi-ϕ-asymptotically nonexpansive mapping.

Let θ : C × C → ℝ be a bifunction, ψ : C → ℝ a real valued function, and B : C → E^* a nonlinear mapping. The so-called generalized mixed equilibrium problem (GMEP) is to find a u ∈ C such that

whose set of solutions is denoted by Ω(θ, B, ψ).

Lemma 3 (see [17].)Let E be a smooth, strictly convex, and reflexive Banach space, and let C be a nonempty closed convex subset of E. Then, the following conclusions hold:

• (1)

  ϕ(x, Π_Cy) + ϕ(Π_Cy, y) ≤ ϕ(x, y) for all x ∈ C and y ∈ E;

• (2)

  if x ∈ E and z ∈ C, then z = Π_Cx⇔〈z − y, Jx − Jz〉≥0, for all y ∈ C;

• (3)

  for x, y ∈ E, ϕ(x, y) = 0 if and only if x = y.

Remark 4. The following basic properties for a Banach space E can be found in Cioranescu [18].

• (i)

  If E is uniformly smooth, then J is uniformly continuous on each bounded subset of E.

• (ii)

  If E is reflexive and strictly convex, then J⁻¹ is norm-weak continuous.

• (iii)

  If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping $$ is single valued, one-to-one, and onto.

• (iv)

  A Banach space E is uniformly smooth if and only if E^* is uniformly convex.

• (v)

  Each uniformly convex Banach space E has the Kadec-Klee property; that is, for any sequence {x_n} ⊂ E, if x_n⇀x ∈ E and ∥x_n∥→∥x∥, then x_n → x, where x_n⇀x denotes that {x_n} converges weakly to x.

Lemma 5 (see [19].)Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, and let C be a nonempty closed convex subset of E. Let {x_n} and {y_n} be two sequences in C such that x_n → p and ϕ(x_n, y_n) → 0, where ϕ is the function defined by (3); then, y_n → p.

Lemma 6 (see [1].)Let E and C be the same as those in Lemma 5. Let T : C → C be a closed and generalized quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences {ν_n} and {μ_n}; then, the fixed point set F(T) of T is a closed and convex subset of C.

Lemma 7 (see [20].)Let E be a real uniformly convex Banach space, and let B_r(0) be the closed ball of E with center at the origin and radius r > 0. Then, there exists a continuous strictly increasing convex function g : [0, ∞)→[0, ∞) with g(0) = 0 such that

for all x, y, ∈B_r(0), and α, β ∈ [0,1] with α + β = 1.

Theorem 8. Let E be a real uniformly smooth and strictly convex Banach space with Kadec-Klee property, C a nonempty closed convex subset of E, and T_i : C → C, i = 1,2, … a countable family of closed and generalized quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences $$ and $$ satisfying $$ and $$ (as n → ∞ and for each i ≥ 1), and each T_i is uniformly L_i-Lipschitz continuous. Let {α_n} be a sequence in [0, ϵ] for some ϵ ∈ (0,1), and let {β_n} be a sequence in [0,1] satisfying 0 < liminf _n→∞β_n(1 − β_n). Let {x_n} be the sequence generated by

where $$, $$ is the generalized projection of E onto C_n+1, and i_n and m_n satisfy the positive integer equation: n = i + (m − 1)m/2, m ≥ i (m ≥ i, n = 1,2, …); that is, for each n ≥ 1, there exist unique i_n and m_n such that If $$ is nonempty and bounded, then {x_n} converges strongly to Π_Fx₁.

Proof. We divide the proof into several steps.

• (I)

  F and C_n (for all n ≥ 1) both are closed and convex subsets in C.

In fact, it follows from Lemma 6 that each F(T_i) is a closed and convex subset of C, so is F. In addition, with C₁ (=C) being closed and convex, we may assume that C_n is closed and convex for some n ≥ 2. In view of the definition of ϕ, we have that

where φ(v) = 2〈v, Jx_n − Jy_n〉, and a = ∥x_n∥² − ∥y_n∥² + ξ_n. This shows that C_n+1 is closed and convex.

• (II)

  F is a subset of $$.

It is obvious that F ⊂ C₁. Suppose that F ⊂ C_n for some n ≥ 2. Since E is uniformly smooth, E^* is uniformly convex. Then, for any p ∈ F ⊂ C_n, we have that

Furthermore, it follows from Lemma 7 that for any p ∈ F, we have that Substituting (18) into (17) and simplifying it, we have that This implies that p ∈ C_n+1, and so F ⊂ C_n+1.

• (III)

  x_n → x^* ∈ C as n → ∞.

In fact, since $$, from Lemma 3 (2), we have that 〈x_n − y, Jx₁ − Jx_n〉 ≥ 0, for all y ∈ C_n. Again, since $$, we have that 〈x_n − p, Jx₁ − Jx_n〉 ≥ 0, for all p ∈ F. It follows from Lemma 3 (1) that for each p ∈ F and for each n ≥ 1,

which implies that {ϕ(x_n, x₁)} is bounded, so is {x_n}. Since for all n ≥ 1, $$ and $$, we have ϕ(x_n, x₁) ≤ ϕ(x_n+1, x₁). This implies that {ϕ(x_n, x₁)} is nondecreasing; hence, the limit Since E is reflexive, there exists a subsequence $$ of {x_n} such that $$ as i → ∞. Since C_n is closed and convex and C_n+1 ⊂ C_n, this implies that C_n is weakly, closed and x^* ∈ C_n for each n ≥ 1. In view of $$, we have that Since the norm ∥·∥ is weakly lower semicontinuous, we have that and so This implies that $$, and so $$ as i → ∞. Since $$, by virtue of Kadec-Klee property of E, we obtain that Since {ϕ(x_n, x₁)} is convergent, this, together with $$, shows that lim _n→∞ϕ(x_n, x₁) = ϕ(x^*, x₁). If there exists some subsequence $$ of {x_n} such that $$ as j → ∞, then, from Lemma 3 (1), we have that that is, x^* = y, and so

• (IV)

  x^* is some member of F.

Set 𝒦_i = {k ≥ 1 : k = i + (m − 1)m/2,   m ≥ i,   m ∈ ℕ} for each i ≥ 1. Note that $$ and $$ whenever k ∈ 𝒦_i for each i ≥ 1. For example, by the definition of 𝒦₁, we have that 𝒦₁ = {1,2, 4,7, 11,16, …}, and i₁ = i₂ = i₄ = i₇ = i₁₁ = i₁₆ = ··· = 1. Then, we have that

Note that $$; that is, m_k↑∞ as 𝒦_i∋k → ∞. It follows from (27) and (28) that Since x_n+1 ∈ C_n+1, it follows from (14), (27), and (29) that as 𝒦_i∋k → ∞. Since x_k → x^* as 𝒦_i∋k → ∞, it follows from (30) and Lemma 5 that Note that $$ whenever k ∈ 𝒦_i for each i ≥ 1. From (17) and (18), for any p ∈ F, we have that that is, This, together with assumption conditions imposed on the sequences {α_n} and {β_n}, shows that $$. In view of property of g, we have that In addition, Jx_k → Jx^* implies that $$. From Remark 4 (ii), it yields that, as 𝒦_i∋k → ∞, Again, since for each i ≥ 1, as 𝒦_i∋k → ∞, This, together with (35) and the Kadec-Klee property of E, shows that On the other hand, by the assumptions that for each i ≥ 1, T_i is uniformly L_i-Lipschitz continuous, and noting again that $$, that is, m_k+1 − 1 = m_k for all k ∈ 𝒦_i, we then have From (37) and x_k → x^*, we have that $$, and $$; that is, $$. It then follows that, for each i ≥ 1, In view of the closeness of T_i, it follows from (37) that T_ix^* = x^*, namely, for each i ≥ 1, x^* ∈ F(T_i), and, hence, x^* ∈ F.

• (V)

  x^* = Π_Fx₁, and so x_n → Π_Fx₁ as n → ∞.

Put u = Π_Fx₁. Since u ∈ F ⊂ C_n and $$, we have that ϕ(x_n, x₁) ≤ ϕ(u, x₁), for all n ≥ 1. Then,

which implies that x^* = u since u = Π_Fx₁, and, hence, x_n → x^* = Π_Fx₁.

This completes the proof.

Theorem 9. Let E be the same as that in Theorem 8, and let C be a nonempty closed convex subset of E. Let $$ be a sequence of mappings defined by (42) with $$. Let {α_n} be a sequence in [0, ϵ] for some ϵ ∈ (0,1), and let {β_n} be a sequence in [0,1] satisfying 0 < liminf _n→∞β_n(1 − β_n). Let {x_n} be the sequence generated by

where i_n satisfies the positive integer equation: n = i + (m − 1)m/2, and m ≥ i (m ≥ i, n = 1,2, …). Then, {x_n} converges strongly to Π_Fx₁ which is some solution to the system of generalized mixed equilibrium problems for $$.

Proof. Note that $$ are quasi-ϕ-nonexpansive mappings; so, they are obviously generalized quasi-ϕ-asymptotically nonexpansive. Therefore, this conclusion can be obtained immediately from Theorem 8.