A New Hybrid Algorithm for a Pair of Quasi-ϕ-Asymptotically Nonexpansive Mappings and Generalized Mixed Equilibrium Problems in Banach Spaces

Remark 2.1. The following basic properties can be found in Cioranescu [10].

• (i)

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

• (ii)

  If E is a reflexive and strictly convex Banach space, then J⁻¹ is hemicontinuous.

• (iii)

  If E is a smooth, strictly convex, and reflexive Banach space, then J is singlevalued, 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.

Lemma 2.2 (see [11], [12].)Let E be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of E. Then, the following conclusions hold:

• (a)

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

• (b)

  if x ∈ E and z ∈ C, then

• (c)

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

Remark 2.3. If E is a real Hilbert space H, then ϕ(x, y) = | | x − y | |² and Π_C is the metric projection P_C of H onto C.

Definition 2.4 (see [13].)(1) A mapping T : C → C is said to be relatively nonexpansive if F(T) ≠ ∅, $$, and

(2) A mapping T : C → C is said to be closed if, for any sequence {x_n} ⊂ C with x_n → x and Tx_n → y, Tx = y.

Definition 2.5 (see [14].)(1) A mapping T : C → C is said to be quasi-ϕ-nonexpansive if F(T) ≠ ∅ and

(2) A mapping T : C → C is said to be quasi-ϕ-asymptotically nonexpansive if F(T) ≠ ∅ and there exists a real sequence {k_n}⊂[1, ∞) with k_n → 1 such that

(3) A pair of mappings T₁, T₂ : C → C is said to be uniformly quasi-ϕ-asymptotically nonexpansive if F(T₁) ⋂ F(T₂) ≠ ∅ and there exists a real sequence {k_n}⊂[1, ∞) with k_n → 1 such that for i = 1,2

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

Remark 2.6. (1) From the definition, it is easy to know that each relatively nonexpansive mapping is closed.

(2) The class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, and the class of quasi-ϕ-nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse may be not true.

Lemma 2.7 (see [15].)Let E be a uniformly convex Banach space, r > 0 a positive number, and B_r(0) a closed ball of E. Then, for any given subset {x₁, x₂, …, x_N} ⊂ B_r(0) and for any positive numbers {λ₁, λ₂, …, λ_N} with $$, there exists a continuous, strictly increasing, and convex function g : [0,2r)→[0, ∞) with g(0) = 0 such that, for any i, j ∈ {1,2, …, N} with i < j,

Lemma 2.8 (see [15].)Let E be a real uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let T : C → C be a closed and quasi-ϕ-asymptotically nonexpansive mapping with a sequence {k_n}⊂[1, ∞), k_n → 1. Then F(T) is a closed convex subset of C.

Lemma 2.9. Let E be a smooth, strictly convex, and reflexive Banach space and C a nonempty closed convex subset of E. Let Θ : C × C → ℝ a bifunction satisfying the conditions (A₁)–(A₄). Let r > 0 and x ∈ E. Then, the followings hold.

• (i)

  (Blum and Oettli [3]) there exists z ∈ C such that

  $$ (2.13)

• (ii)

  (Takahashi and Zembayashi [8]) Define a mapping T_r : E → C by

  $$ (2.14)

Then, the following conclusions hold:

• (a)

   T_r is single-valued,

• (b)

   T_r is a firmly nonexpansive-type mapping, that is, ∀z, y ∈ E,

  $$ (2.15)

• (c)

   $$,

• (d)

   EP (Θ) is closed and convex,

• (e)

   ϕ(q, T_rx) + ϕ(T_rx, x) ≤ ϕ(q, x),   ∀ q ∈ F(T_r).

Lemma 2.10 (see [16].)Let E be a smooth, strictly convex, and reflexive Banach space, and C a nonempty closed convex subset of E. Let A : C → E^* be a continuous and monotone mapping, ψ : C → ℝ a lower semicontinuous and convex function, and Θ : C × C → ℝ a bifunction satisfying conditions (A₁)–(A₄). Let r > 0 be any given number and x ∈ E any given point. Then, the following hold.

• (i)

  There exists u ∈ C such that

  $$ (2.16)

• (ii)

  If we define a mapping K_r : C → C by

Then, the mapping K_r has the following properties:

• (a)

   K_r is single valued,

• (b)

   K_r is a firmly nonexpansive-type mapping, that is,

  $$ (2.18)

• (c)

   $$,

• (d)

   Ω is closed and convex,

• (e)

   

  $$ (2.19)

Remark 2.11. It follows from Lemma 2.9 that the mapping K_r is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-nonexpansive.

Theorem 3.1. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A : C → E^* be a continuous and monotone mapping, ψ : C → ℝ a lower semicontinuous and convex, function, and Θ : C × C → ℝ a bifunction satisfying conditions (A₁)–(A₄). Let S,   T : C → C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that S and T are uniformly L-Lipschitz continuous and that G = F(T) ⋂ F(S) ⋂ Ω is a nonempty and bounded subset in C. Let {x_n} be the sequence generated by

where J : E → E^* is the normalized duality mapping, {α_n} and {β_n} are sequences in [0,1] and {r_n}⊂[a, ∞) for some a > 0, ξ_n = sup _u∈G(k_n − 1)ϕ(u, x_n). Suppose that the following conditions are satisfied:

• (i)

   $$,

• (ii)

   $$.

Then {x_n} converges strongly to Π_{F(S)∩F(T)∩Ω}x₀, where Π_{F(S)∩F(T)∩Ω} is the generalized projection of E onto F(S)∩F(T)∩Ω.

Proof. Firstly, we define two functions H : C × C → ℝ and K_r : C → C by

By Lemma 2.10, we know that the function H satisfies conditions (A₁)–(A₄) and K_r has properties (a)–(e). Therefore, (3.1) is equivalent to We divide the proof of Theorem 3.1 into five steps.

(I) First we prove that C_n and Q_n are both closed and convex subsets of C for all n ≥ 0.

In fact, it is obvious that Q_n is closed and convex for all n ≥ 0. Again we have that

Hence C_n, ∀n ≥ 0, is closed and convex, and so C_n∩Q_n is closed and convex for all n ≥ 0.

(II) Next we prove that F(T)∩F(S)∩Ω ⊂ C_n∩Q_n, ∀n ≥ 0.

Putting $$, ∀n ≥ 0, by Lemma 2.10 and Remark 2.11, $$ is relatively nonexpansive. Again since S and T are quasi-ϕ-asymptotically nonexpansive, for any given u ∈ F(S)∩F(T)∩Ω, we have that

From (3.5) we have that This implies that u ∈ C_n, ∀n ≥ 0, and so F(T)∩F(S)∩Ω ⊂ C_n, ∀n ≥ 0.

Now we prove that F(T)∩F(S)∩Ω ⊂ C_n∩Q_n, ∀n ≥ 0.

In fact, from Q₀ = C, we have that F(T)∩F(S)∩Ω ⊂ C₀∩Q₀. Suppose that F(T)∩F(S)∩Ω ⊂ C_k∩Q_k, for some k ≥ 0. Now we prove that F(T)∩F(S)∩Ω ⊂ C_k+1∩Q_k+1. In fact, since $$, we have that

Since F(T)∩F(S)∩Ω ⊂ C_k∩Q_k, for any z ∈ F(T)∩F(S)∩Ω, we have that This shows that z ∈ Q_k+1, and so F(T)∩F(S)∩Ω ⊂ Q_k+1. The conclusion is proved.

(III) Now we prove that {x_n} is bounded.

From the definition of Q_n, we have that $$, ∀n ≥ 0. Hence, from Lemma 2.2(1),

This implies that {ϕ(x_n, x₀)} is bounded. By virtue of (2.4), {x_n} is bounded. Denote Since $$ and $$, from the definition of $$, we have that This implies that {ϕ(x_n, x₀)} is nondecreasing, and so the limit lim _n→∞ϕ(x_n, x₀) exists. Without loss of generality, we can assume that By the way, from the definition of {ξ_n}, (2.4), and (3.10), it is easy to see that

(IV) Now, we prove that {x_n} converges strongly to some point p ∈ G = F(T)∩F(S)∩Ω.

In fact, since {x_n} is bounded in C and E is reflexive, there exists a subsequence $$ such that $$. Again since Q_n is closed and convex for each n ≥ 0, it is weakly closed, and so p ∈ Q_n for each n ≥ 0. Since $$, from the defintion of $$, we have that

Since we have that This implies that $$, that is, $$. In view of the Kadec-Klee property of E, we obtain that $$.

Now we first prove that x_n → p  (n → ∞). In fact, if there exists a subsequence $$ such that $$, then we have that

Therefore we have that p = q. This implies that

Now we first prove that p ∈ F(T)∩F(S). In fact, by the construction of Q_n, we have that $$. Therefore, by Lemma 2.2(a) we have that

In view of x_n+1 ∈ C_n∩Q_n ⊂ C_n and noting the construction of C_n we obtain From (3.13) and (3.19), we have that

From (2.4) it yields that (| | x_n+1||−| | u_n| | ) ² → 0 and (| | x_n+1||−| | z_n| | ) ² → 0. Since | | x_n+1||→| | p||, we have that

Hence, we have that

This implies that {Jz_n} is bounded in E^*. Since E is reflexive, and so E^* is reflexive, there exists a subsequence $$ such that $$. In view of the reflexiveness of E, we see that J(E) = E^*. Hence, there exists x ∈ E such that Jx = p₀. Since

taking liminf _n→∞ on both sides of the equality above and in view of the weak lower semicontinuity of norm ||·||, it yields that that is, p = x. This implies that p₀ = Jp, and so Jz_n⇀Jp. It follows from (3.23) and the Kadec-Klee property of E^* that $$ (as n → ∞). Noting that J⁻¹ : E^* → E is hemicontinuous, it yields that $$. It follows from (3.22) and the Kadec-Klee property of E that $$.

By the same way as given in the proof of (3.18), we can also prove that

From (3.18) and (3.26), we have that

Since J is uniformly continuous on any bounded subset of E, we have that For any u ∈ F(T)⋂F(S)⋂Ω, it follows from (3.5) that Since From (3.27) and (3.28), it follows that

In view of condition (i) and liminf _n→∞α_n(1 − α_n) > 0, we see that

It follows from the property of g that Since x_n → p and J is uniformly continuous, it yields that Jx_n → Jp. Hence from (3.33) we have that Since J⁻¹ : E^* → E is hemicontinuous, it follows that On the other hand, we have that This together with (3.35) shows that

Furthermore, by the assumption that T is uniformly L-Lipschitz continuous, we have that

This together with (3.18) and (3.37), yields | | Tⁿ⁺¹x_n − Tⁿx_n|| → 0 (as n → ∞). Hence from (3.37) we have that Tⁿ⁺¹x_n → p, that is, TTⁿx_n → p. In view of (3.37) and the closeness of T, it yields that Tp = p. This implies that p ∈ F(T).

By the same way as given in the proof of (3.23) to (3.31), we can also prove that

Since $$, from (2.19), (3.6), (3.13), and (3.39), we have that

From (2.4) it yields that (| | u_n||−| | y_n| | ) ² → 0. Since | | u_n||→| | p||, we have that

Hence we have that

By the same way as given in the proof of (3.26), we can also prove that From (3.39) and (3.43) we have that Since J is uniformly continuous on any bounded subset of E, we have that For any u ∈ F(T)⋂F(S)⋂Ω, it follows from (3.6), (3.13), and (3.39) that In view of condition (ii) and liminf _n→∞β_n(1 − β_n) > 0, we see that It follows from the property of g that Since x_n → p and J is uniformly continuous, it yields, Jx_n → Jp. Hence from (3.48) we have that Since J⁻¹ : E^* → E is hemicontinuous, it follows that On the other hand, we have that This together with (3.50) shows that

Furthermore, by the assumption that S is uniformly L-Lipschitz continuous, we have that

This together with (3.26) and (3.52), yields that | | Sⁿ⁺¹z_n − Sⁿz_n|| → 0 (as n → ∞). Hence from (3.52) we have that Sⁿ⁺¹z_n → p, that is, SSⁿz_n → p. In view of (3.52) and the closeness of T, it yields that Sp = p. This implies that p ∈ F(S).

Next we prove that p ∈ Ω. From (3.45) and the assumption that r_n ≥ a, ∀n ≥ 0, we have that

Since $$, we have that Replacing n by n_k in (3.55), from condition (A₂), we have that By the assumption that y ↦ H(x, y) is convex and lower semicontinuous, it is also weakly lower semicontinuous. Letting n_k → ∞ in (3.55), from (3.54) and condition (A₄), we have that H(y, p) ≤ 0, ∀y ∈ C.

For t ∈ (0,1] and y ∈ C, letting y_t = ty + (1 − t)p, there are y_t ∈ C and H(y_t, p) ≤ 0. By conditions (A₁) and (A₄), we have that

Dividing both sides of the above equation by t, we have that H(y_t, y) ≥ 0, ∀y ∈ C. Letting t ↓ 0, from condition (A₃), we have that H(p, y) ≥ 0, ∀y ∈ C, that is, Θ(p, y)+〈Ap, y − p〉+ψ(y) − ψ(p) ≥ 0, ∀y ∈ C. Therefore p ∈ Ω, and so p ∈ F(T)⋂F(S)⋂Ω.

(V) Finally, we prove that $$.

Let $$. From w ∈ F(T)⋂F(S)⋂Ω ⊂ C_n∩Q_n, and $$, we have that

Since the norm is weakly lower semicontinuous, this implies that It follows from the definition of $$ and (3.59) that we have p = w. Therefore, $$. This completes the proof of Theorem 3.1.

Remark 3.2. Theorem 3.1 improves and extends the corresponding results in [7–9].

• (a)

  For the framework of spaces, we extend the space from a uniformly smooth and uniformly convex Banach space to a uniformly smooth and strictly convex Banach space with the Kadec-Klee property(note that each uniformly convex Banach space must have the Kadec-Klee property).

• (b)

  For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, or weak relatively nonexpansive mappings to a pair of quasi-ϕ-asymptotically nonexpansive mappings.

• (c)

  For the equilibrium problem, we extend the generalized equilibrium problem to the generalized mixed equilibrium problem.

Theorem 3.3. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A : C → E^* be a continuous and monotone mapping and Θ : C × C → ℝ a bifunction satisfying conditions (A₁)–(A₄). Let S,   T : C → C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that S and T are uniformly L-Lipschitz continuous and that G = F(T)⋂F(S)⋂GEP  is a nonempty and bounded subset in C. Let {x_n} be the sequence generated by

where J : E → E^* is the normalized duality mapping, {α_n} and {β_n} are sequences in [0,1], and {r_n}⊂[a, ∞) for some a > 0, ξ_n = sup _u∈G(k_n − 1)ϕ(u, x_n). If {α_n} and {β_n} satisfy conditions (i)-(ii) in Theorem 3.1, then {x_n} converges strongly to Π_{F(S)∩F(T)∩GEP}x₀, where GEP  is the set for the solutions of generalized equilibrium problem (1.3).

Proof. Putting ψ = 0 in Theorem 3.1, the conclusion of Theorem 3.3 can be obtained from Theorem 3.1.

Theorem 3.4. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let ψ : C → ℝ be a lower semicontinuous and convex function and Θ : C × C → ℝ a bifunction satisfying conditions (A₁)–(A₄). Let S, T : C → C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that S and T are uniformly L-Lipschitz continuous and that G = F(T)⋂F(S)⋂MEP  is a nonempty and bounded subset in C. Let {x_n} be the sequence generated by

where J : E → E^* is the normalized duality mapping, {α_n} and {β_n} are sequences in [0,1], and {r_n}⊂[a, ∞) for some a > 0,   ξ_n = sup _u∈G(k_n − 1)ϕ(u, x_n). If {α_n} and {β_n} satisfy conditions (i)-(ii) in Theorem 3.1, then {x_n} converges strongly to Π_{F(S)∩F(T)∩MEP}x₀, where MEP is the set of solutions for mixed equilibrium problem (1.4).

Proof. Putting A = 0 in Theorem 3.1, the conclusion of Theorem 3.4 can be obtained from Theorem 3.1.

Theorem 3.5. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A : C → E^* be a continuous and monotone mapping and ψ : C → ℝ a lower semicontinuous and convex function. Let S, T : C → C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that S and T are uniformly L-Lipschitz continuous and that G = F(T)⋂F(S)⋂VI (C, A, ψ) is a nonempty and bounded subset in C. Let {x_n} be the sequence generated by

where J : E → E^* is the normalized duality mapping, {α_n} and {β_n} are sequences in [0,1], and {r_n}⊂[a, ∞) for some a > 0,   ξ_n = sup _u∈G(k_n − 1)ϕ(u, x_n). If {α_n} and {β_n} satisfy conditions (i)-(ii) in Theorem 3.1, then {x_n} converges strongly to Π_{F(S)∩F(T)∩VI (C,A,ψ)}x₀, where VI (C, A, ψ) is the set of solutions for the mixed variational inequality (1.5).

Proof. Putting Θ = 0 in Theorem 3.1, the conclusion of Theorem 3.5 can be obtained from Theorem 3.1.

Theorem 3.6. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let Θ : C × C → ℝ be a bifunction satisfying conditions (A₁)–(A₄). Let S, T : C → C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that S and T are uniformly L -Lipschitz continuous and that G = F(T)⋂F(S)⋂EP (Θ) is a nonempty and bounded subset in C. Let {x_n} be the sequence generated by

where J : E → E^* is the normalized duality mapping, {α_n} and {β_n} are sequences in [0,1], and {r_n}⊂[a, ∞) for some a > 0, ξ_n = sup _u∈G(k_n − 1)ϕ(u, x_n). If {α_n} and {β_n} satisfy conditions (i)-(ii) in Theorem 3.1, then {x_n} converges strongly to Π_{F(S)∩F(T)∩EP (Θ)}x₀, where EP (Θ) is the set of solutions for the equilibrium problem (1.6).

Proof. Putting ψ = 0 and A = 0 in Theorem 3.1, the conclusion of Theorem 3.6 can be obtained from Theorem 3.1.

Theorem 3.7. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A : C → E^* be a continuous and monotone mapping and S,   T : C → C two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that S and T are uniformly L-Lipschitz continuous and that G = F(T)⋂F(S)⋂VI (C, A) is a nonempty and bounded subset in C. Let {x_n} be the sequence generated by

where J : E → E^* is the normalized duality mapping, {α_n} and {β_n} are sequences in [0,1], and {r_n}⊂[a, ∞) for some a > 0, ξ_n = sup _u∈G(k_n − 1)ϕ(u, x_n). If {α_n} and {β_n} satisfy conditions (i)-(ii) in Theorem 3.1, then {x_n} converges strongly to Π_{F(S)∩F(T)∩VI (C,A)}x₀, where VI (C, A) is the set of solutions for the variational inequality (1.7)

Proof. Putting ψ = 0 and Θ = 0 in Theorem 3.1, the conclusion of Theorem 3.7 can be obtained from Theorem 3.1.

Theorem 3.8. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A : C → E^* be a continuous and monotone mapping, ψ : C → ℝ a lower semicontinuous and convex function, and Θ : C × C → ℝ a bifunction satisfying conditions (A₁)–(A₄). Let S : C → C be a closed and quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that S is uniformly L-Lipschitz continuous and that F(S)⋂Ω is a nonempty and bounded subset in C. Let {x_n} be the sequence generated by

where J : E → E^* is the normalized duality mapping, {α_n} and {β_n} are sequences in [0,1], and {r_n}⊂[a, ∞) for some a > 0,   ξ_n = sup _{u∈F(S)∩Ω}(k_n − 1)ϕ(u, x_n). If {β_n} satisfy condition (ii) in Theorem 3.1, then {x_n} converges strongly to Π_F(S)∩Ωx₀.

Proof. Taking T = I in Theorem 3.1, we have that z_n = x_n, ∀n ≥ 0. Hence, the conclusion of Theorem 3.8 is obtained.

Theorem 3.9. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let A : C → E^* be a continuous and monotone mapping, ψ : C → ℝ a lower semicontinuous and convex function, and Θ : C × C → ℝ a bifunction satisfying conditions (A₁)–(A₄). Suppose that Ω is a nonempty subset in C. Let {x_n} be the sequence generated by

where {r_n}⊂[a, ∞) for some a > 0. Then {x_n} converges strongly to Π_Ωx₀.

Proof. Taking T = S = I in Theorem 3.1, the conclusion is obtained.

Theorem 3.10. Let E be a uniformly smooth and strictly convex Banach space with the Kadec-Klee property and C a nonempty closed convex subset of E. Let S, T : C → C be two closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that S and T are uniformly L-Lipschitz continuous and that F(T)⋂F(S) is a nonempty and bounded subset in C. Let {x_n} be the sequence generated by

where J : E → E^* is the normalized duality mapping, {α_n} and {β_n} are sequences in [0,1], and ξ_n = sup _{u∈F(S)∩F(T)}(k_n − 1)ϕ(u, x_n). If {α_n} and {β_n} satisfy conditions (i)-(ii) in Theorem 3.1, then {x_n} converges strongly to Π_F(S)∩F(T)x₀.

Proof. Taking A = Θ = 0 and r_n = 1, ∀n ≥ 0 in Theorem 3.1, the conclusion of Theorem 3.10 is obtained.