Existence and Strong Convergence Theorems for Generalized Mixed Equilibrium Problems of a Finite Family of Asymptotically Nonexpansive Mappings in Banach Spaces

Example 1.1 (see [2].)Let B_H be the closed unit ball in the Hilbert space H = l₂ and S : B_H → B_H a mapping defined by

where {a_n} is a sequence of real numbers such that 0 < a_i < 1 and $$. Then That is, S is Lipschitzian but not nonexpansive. Observe that Here $$ as n → ∞. Therefore, S is asymptotically nonexpansive but not nonexpansive.

Special Cases (1) If T is monotone that is T is relaxed η-ξ monotone with η(x, y) = x − y for all x, y ∈ C and ξ = 0, (1.8) is reduced to the following generalized equilibrium problem (GEP).

The solution set of (1.10) is denoted by GEP (f), that is,

(2) In the case of T ≡ 0 and φ ≡ 0, (1.8) is reduced to the following classical equilibrium problem

The set of all solution of (1.12) is denoted by EP (f), that is,

(3) In the case of f ≡ 0, (1.8) is reduced to the following variational-like inequality problem [3].

Theorem 2.1. Let C be a nonempty convex subset of a smooth Banach space E and let x ∈ E and y ∈ C. Then the following are equivalent:

• (a)

  yis a best approximation to x : y = P_Cx,

• (b)

  y is a solution of the variational inequality:

  $$ ()
  where J is a duality mapping and P_C is the metric projection from E onto C.

Lemma 2.2 (see [25].)Let E be a uniformly convex Banach space, let {α_n} be a sequence of real numbers such that 0 < b ≤ α_n ≤ c < 1 for all n ≥ 1, and let {x_n} and {y_n} be sequences in E such that limsup _n→∞∥x_n∥≤d, limsup _n→∞∥y_n∥≤d and lim _n→∞∥α_nx_n + (1 − α_n)y_n∥ = d. Then lim _n→∞∥x_n − y_n∥ = 0.

Theorem 2.3 (see [23].)Let C be a bounded, closed, and convex subset of a uniformly convex Banach space E. Then there exists a strictly increasing, convex, and continuous function γ : [0, ∞)→[0, ∞) such that γ(0) = 0 and

for any asymptotically nonexpansive mapping S of C into C with {k_n}, any elements x₁, x₂, …, x_n ∈ C, any numbers λ₁, λ₂, …, λ_n ≥ 0 with $$ and each m ≥ 1.

Lemma 2.4 (see [26], Lemma 1.6.)Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : C → C be an asymptotically nonexpansive mapping. Then (I − S) is demiclosed at 0, that is, if x_n⇀x and (I − S)x_n → 0, then x ∈ F(S).

Lemma 2.5 (see [7], Lemma 3.2.)Let C be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, let T : C → E^* be an η-hemicontinuous and relaxed η − ξ monotone mapping. Let f be a bifunction from C × C to ℛ satisfying (A1), (A3), and (A4) and let φ be a lower semicontinuous and convex function from C to ℛ. Let r > 0 and z ∈ C. Assume that

• (i)

  η(x, y) + η(y, x) = 0 for all x, y ∈ C;

• (ii)

  for any fixed u, v ∈ C, the mapping x ↦ 〈Tv, η(x, u)〉 is convex and lower semicontinuous;

• (iii)

  ξ : E → ℛ is weakly lower semicontinuous, that is, for any net {x_β}, x_β converges to x in σ(E, E^*) which implies that ξ(x) ≤ liminf ξ(x_β).

Then there exists x₀ ∈ C such that

Lemma 2.6 (see [7], Lemma 3.3.)Let C be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, let T : C → E^* be an η-hemicontinuous and relaxed η-ξ monotone mapping. Let f be a bifunction from C × C to ℛ satisfying (A1)–(A4) and let φ be a lower semicontinuous and convex function from C to ℛ. Let r > 0 and define a mapping Φ_r : E → C as follows:

for all x ∈ E. Assume that

• (i)

  η(x, y) + η(y, x) = 0, for all x, y ∈ C;

• (ii)

  for any fixed u, v ∈ C, the mapping x ↦ 〈Tv, η(x, u)〉 is convex and lower semicontinuous and the mapping x ↦ 〈Tu, η(v, x)〉 is lower semicontinuous;

• (iii)

  ξ : E → ℛ is weakly lower semicontinuous;

• (iv)

  for any x, y ∈ C, ξ(x − y) + ξ(y − x) ≥ 0.

Then, the following holds:

• (1)

  Φ_r is single valued;

• (2)

  〈Φ_rx − Φ_ry, J(Φ_rx − x)〉≤〈Φ_rx − Φ_ry, J(Φ_ry − y)〉 for all x, y ∈ E;

• (3)

  F(Φ_r) = EP (f, T);

• (4)

  EP (f, T) is nonempty closed and convex.

Theorem 3.1. Let C be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E, let T : C → E^* be an η-hemicontinuous and relaxed η-ξ monotone mapping. Let f be a bifunction from C × C to ℝ satisfying the following conditions (A1)–(A4):

• (A1)

  f(x, x) = 0 for all x ∈ C;

• (A2)

  f(x, y) + f(y, x) ≤ min {ξ(x − y), ξ(y − x)} for all x, y ∈ C;

• (A3)

  for all y ∈ C, f(·, y) is weakly upper semicontinuous;

• (A4)

  for all x ∈ C, f(x, ·) is convex.

For any r > 0 and x ∈ E, define a mapping Φ_r : E → C as follows: where φ is a lower semicontinuous and convex function from C to ℝ. Assume that

• (i)

  η(x, y) + η(y, x) = 0, for all x, y ∈ C;

• (ii)

  for any fixed u, v ∈ C, the mapping x ↦ 〈Tv, η(x, u)〉 is convex and lower semicontinuous and the mapping x ↦ 〈Tu, η(v, x)〉 is lower semicontinuous;

• (iii)

  ξ : E → ℝ is weakly lower semicontinuous.

Then, the following holds:

• (1)

  Φ_r is single valued;

• (2)

  〈Φ_rx − Φ_ry, J(Φ_rx − x)〉≤〈Φ_rx − Φ_ry, J(Φ_ry − y)〉 for all x, y ∈ E;

• (3)

  F(Φ_r) = GMEP (f, T);

• (4)

  GMEP (f, T) is nonempty closed and convex.

Proof. For each x ∈ E. It follows from Lemma 2.5 that Φ_r(x) is nonempty.

(1) We prove that Φ_r is single valued. Indeed, for x ∈ E and r > 0, let z₁,  z₂ ∈ Φ_rx. Then

Adding the two inequalities, from (i) we have Setting Δ : = min {ξ(z₁ − z₂), ξ(z₂ − z₁)} and using (A2), we have that is, Since T is relaxed η-ξ monotone and r > 0, one has In (3.5) exchanging the position of z₁ and z₂, we get that is, Now, adding the inequalities (3.6) and (3.8), we have Hence, Since J is monotone and E is strictly convex, we obtain that z₁ − x = z₂ − x and hence z₁ = z₂. Therefore S_r is single valued.

(2) For x, y ∈ C, we have

Setting Λ_x,y : = min {ξ(Φ_rx − Φ_ry), ξ(Φ_ry − Φ_rx)} and applying (A2), we get that is, In (3.13) exchanging the position of Φ_rx and Φ_ry, we get Adding the inequalities (3.13) and (3.14), we have It follows that Hence The conclusions (3), (4) follow from Lemma 2.6.

Example 3.2. Define ξ : ℝ → ℝ and f : ℝ × ℝ → ℝ by

It is easy to see that f satisfies (A1), (A3), (A4), and (A2): f(x, y) + f(y, x) ≤ min {ξ(x − y), ξ(x − y)}, for  all (x, y) ∈ ℝ × ℝ.

Remark 3.3. Theorem 3.1 generalizes and improves [7, Lemma 3.3] in the following manners.

• (1)

  The condition f(x, y) + f(y, x) ≤ 0 has been weakened by (A2) that is f(x, y) + f(y, x) ≤ min {ξ(x − y), ξ(y − x)} for all x, y ∈ C.

• (2)

  The control condition ξ(x − y) + ξ(y − x) ≥ 0 imposed on the mapping ξ in [7, Lemma 3.3] can be removed.

If T is monotone that is T is relaxed η-ξ monotone with η(x, y) = x − y for all x, y ∈ C and ξ = 0, we have the following results.

Corollary 3.4. Let C be a nonempty, bounded, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E. Let T : C → E^* be a monotone mapping and f be a bifunction from C × C to ℝ satisfying the following conditions (i)–(iv):

• (i)

  f(x, x) = 0 for all x ∈ C;

• (ii)

  f(x, y) + f(y, x) ≤ 0 for all x, y ∈ C;

• (iii)

  for all y ∈ C, f(·, y) is weakly upper semicontinuous;

• (iv)

  for all x ∈ C, f(x, ·) is convex.

For any r > 0 and x ∈ E, define a mapping Φ_r : E → C as follows: where φ is a lower semicontinuous and convex function from C to ℝ. Then, the following holds:

• (1)

  Φ_r is single valued;

• (2)

  〈Φ_rx − Φ_ry, J(Φ_rx − x)〉≤〈Φ_rx − Φ_ry, J(Φ_ry − y)〉 for all x, y ∈ E;

• (3)

  F(Φ_r) = GEP (f);

• (4)

  GEP (f) is nonempty closed and convex.

Theorem 4.1. Let E be a uniformly convex and smooth Banach space and let C be a nonempty, bounded, closed, and convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4). Let T : C → E^* be an η-hemicontinuous and relaxed η-ξ monotone mapping and φ a lower semicontinuous and convex function from C to ℝ. Let, for each 1 ≤ i ≤ N, S_i : C → C be an asymptotically nonexpansive mapping with a sequence $$, respectively, such that k_n,i → 1 as n → ∞. Assume that $$ is nonempty. Let {x_n} be a sequence generated by (1.20), where {t_n} and {r_n} are real sequences in (0,1) satisfying lim _n→∞t_n = 0 and liminf _n→∞r_n > 0. Then {x_n} converges strongly, as n → ∞, to P_Ωx₀, where P_Ω is the metric projection of E onto Ω.

Proof. First, define the sequence {k_n} by k_n : = max {k_n,i : 1 ≤ i ≤ N} and so k_n → 1 as n → ∞ and

where h(n) = j + 1 if jN < n ≤ (j + 1)N, j = 1,2 … , N and n = jN + i(n); i(n)∈{1,2, …, N}. Next, we rewrite the algorithm (1.20) as the following relation: where Φ_r is the mapping defined by (3.19). We show that the sequence {x_n} is well defined. It is easy to verify that C_n∩D_n is closed and convex and Ω ⊂ C_n for all n ≥ 0. Next, we prove that Ω ⊂ C_n ∩ D_n. Indeed, since D₀ = C, we also have Ω ⊂ C₀ ∩ D₀. Assume that Ω ⊂ C_k−1 ∩ D_k−1 for k ≥ 2. Utilizing Theorem 3.1 (2), we obtain which gives that hence Ω ⊂ D_k. By the mathematical induction, we get that Ω ⊂ C_n∩D_n for each n ≥ 0 and hence {x_n} is well defined. Now, we show that Put w = P_Ωx₀, since Ω ⊂ C_n∩D_n and $$, we have Since x_n+2 ∈ D_n+1 ⊂ D_n and $$, we have Hence the sequence {∥x_n − x₀∥} is bounded and monotone increasing and hence there exists a constant d such that Moreover, by the convexity of D_n, we also have 1/2(x_n+1 + x_n+2) ∈ D_n and hence This implies that By Lemma 2.2, we have Furthermore, we can easily see that Next, we show that Fix κ ∈ {1,2, …, N} and put m = n − κ. Since $$, we have x_n ∈ C_n−1⊆⋯⊆C_m. Since t_m > 0, there exists y₁, …, y_P ∈ C and a nonnegative number λ₁, …, λ_P with λ₁ + ⋯+λ_P = 1 such that By the boundedness of C and {k_n}, we can put the following: This together with (4.14) implies that for all i ∈ {1, …, N}. Therefore, for each i ∈ {1, …, P}, we get Moreover, since each S_i, i ∈ {1,2, …, N}, is asymptotically nonexpansive, we can obtain that It follows from Theorem 2.3 and the inequalities (4.17)–(4.19) that Since lim _n→∞k_n = 1 and lim _n→∞t_n = 0, it follows from the above inequality that Hence (4.13) is proved. Next, we show that From the construction of C_n, one can easily see that The boundedness of C and lim _n→∞t_n = 0 implies that On the other hand, since for any positive integer n > N, n = (n − N)(mod N) and n = (h(n) − 1)N + i(n), we have that is Thus, Applying the facts (4.11), (4.13), and (4.24) to the above inequality, we obtain Therefore, for any j = 1,2, …, N, we have which gives that as required. Since {x_n} is bounded, there exists a subsequence $$ of {x_n} such that $$. It follows from Lemma 2.4 that $$ for all l = 1,2, …, N. That is $$.

Next, we show that $$. By the construction of D_n, we see from Theorem 2.1 that $$. Since x_n+1 ∈ D_n, we get

Furthermore, since liminf _n→∞r_n > 0, we have as n → ∞. By (4.32), we also have $$. By the definition of $$, for each y ∈ C, we obtain By (A3), (4.32), (ii), the weakly lower semicontinuity of φ and η-hemicontinuity of T, we have Hence, This shows that $$ and hence $$.

Finally, we show that x_n → w as n → ∞, where w : = P_Ωx₀. By the weakly lower semicontinuity of the norm, it follows from (4.6) that

This shows that and $$. Since E is uniformly convex, we obtain that $$. It follows that $$. So we have x_n → w as n → ∞. This completes the proof.

Theorem 5.1. Let E be a uniformly convex and smooth Banach space and let C be a nonempty, bounded, closed, and convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4). Let T : C → E^* be an η-hemicontinuous and relaxed η-ξ monotone mapping and φ a lower semicontinuous and convex function from C to ℝ. Let S be an asymptotically nonexpansive mapping with a sequence {k_n}, such that k_n → 1 as n → ∞. Assume that Ω : = F(S)∩GMEP (f, T) is nonempty. Let {x_n} be a sequence generated by

where {t_n} and {r_n} are real sequences in (0,1) satisfying lim _n→∞t_n = 0 and liminf _n→∞r_n > 0. Then {x_n} converges strongly, as n → ∞, to P_Ωx₀, where P_Ω is the metric projection of E onto Ω.

Theorem 5.2. Let E be a uniformly convex and smooth Banach space and let C be a nonempty, bounded, closed, and convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4). Let T : C → E^* be an η-hemicontinuous and relaxed η-ξ monotone mapping and φ a lower semicontinuous and convex function from C to ℝ. Let S be a nonexpansive mapping of C into itself such that Ω : = F(S)∩GMEP (f, T) ≠ ∅. Let {x_n} be the sequence in C generated by

where {t_n} and {r_n} are real sequences in (0,1) satisfying lim _n→∞t_n = 0 and liminf _n→∞r_n > 0. Then, the sequence {x_n} converges strongly to P_Ωx₀.

If one takes T ≡ 0 and φ ≡ 0 in Theorem 4.1, then one obtains the following result concerning an equilibrium problem in a Banach space setting.

Theorem 5.3. Let E be a uniformly convex and smooth Banach space and let C be a nonempty, bounded, closed, and convex subset of E. Let f be a bifunction from C × C to ℝ satisfying (A1)–(A4) and let S be an asymptotically nonexpansive mapping of C into itself such that $$. Let {x_n} be the sequence in C generated by

where {t_n} and {r_n} are real sequences in (0,1) satisfying lim _n→∞t_n = 0 and liminf _n→∞r_n > 0. Then the sequence {x_n} converges strongly to P_Ωx₀.

Theorem 5.4. Let E be a uniformly convex and smooth Banach space, C a nonempty, bounded, closed, and convex subset of E and S an asymptotically nonexpansive mapping of C into itself such that $$. Let {x_n} be the sequence in C generated by

If {t_n}⊂(0,1) and lim _n→∞t_n = 0, then {x_n} converges strongly to P_Ωx₀.