Generalized Mixed Equilibrium Problems and Fixed Point Problem for a Countable Family of Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

Special Examples

• (i)

  If A = 0, then the problem (1.2) is reduced to the mixed equilibrium problem (MEP), and the set of its solutions is denoted by

  $$ (1.4)

• (ii)

  If ψ ≡ 0, then the problem (1.2) is reduced to the generalized equilibrium problem (GEP), and the set of its solutions is denoted by

  $$ (1.5)

• (iii)

  If A = 0, ψ = 0, then the problem (1.2) is reduced to the equilibrium problem (EP), and the set of its solutions is denoted by

  $$ (1.6)

• (iv)

  If F = 0, then the problem (1.2) is reduced to the mixed variational inequality of Browder type (VI), and the set of its solutions is denoted by

  $$ (1.7)

Remark 1.1. The following basic properties for Banach space E and for the normalized duality mapping J can be found in Cioranescu [6].

• (i)

  If E is an arbitrary Banach space, then J is monotone and bounded;

• (ii)

  If E is a strictly convex Banach space, then J is strictly monotone;

• (iii)

  If E is a a smooth Banach space, then J is single-valued, and hemicontinuous; that is, J is continuous from the strong topology of E to the weak star topology of E^*;

• (iv)

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

• (v)

  If E is a reflexive and strictly convex Banach space with a strictly convex dual E^* and J^* : E^* → E is the normalized duality mapping in E^*, then $$ and J^*J = I_E;

• (vi)

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

• (vii)

  A Banach space E is uniformly smooth if and only if E^* is uniformly convex. If E is uniformly smooth, then it is smooth and reflexive.

Definition 1.2. (1) A mapping T : C → C is said to be nonexpansive if

(2) A mapping T : C → C is said to be relatively nonexpansive [8] if $$ and

(3) A mapping T : C → C is said to be weak relatively nonexpansive [9] if $$ and

(4) 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, then Tx = y.

Definition 1.3. (1) A mapping T : C → C is said to be quasi-ϕ-nonexpansive [10] if F(T) ≠ ∅ and

(2) A mapping T : C → C is said to be quasi-ϕ-asymptotically nonexpansive [11], if F(T) ≠ ∅ and there exists a real sequence {k_n}⊂[1, ∞) with k_n → 1 such that

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

Definition 1.4. (1) A mapping T : C → C is said to be total quasi-ϕ-asymptotically nonexpansive if F(T) ≠ ∅ and there exist nonnegative real sequences {ν_n}, {μ_n} with ν_n → 0, μ_n → 0 (as n → ∞) and a strictly increasing continuous function ζ : ℝ⁺ → ℝ⁺ with ζ(0) = 0 such that for all   x ∈ C, P ∈ F(T)

(2) A countable family of mappings {T_n} : C → C is said to be uniformly total quasi-ϕ-asymptotically nonexpansive, if $$ and there exist nonnegative real sequences {ν_n}, {μ_n} with ν_n → 0,   μ_n → 0 (as n → ∞) and a strictly increasing continuous function ζ : ℝ⁺ → ℝ⁺ with ζ(0) = 0 such that for all $$

Remark 1.5. From the definition, it is easy to know that

• (1)

  each relatively nonexpansive mapping is closed;

• (2)

  taking ζ(t) = t, t ≥ 0,   ν_n = (k_n − 1) and μ_n = 0, then (1.16) can be rewritten as

  $$ (1.20)
  This implies that each quasi-ϕ-asymptotically nonexpansive mapping must be a total quasi-ϕ-asymptotically nonexpansive mapping, but the converse is not true;

• (3)

  the class of quasi-ϕ-asymptotically nonexpansive mappings contains properly the class of quasi-ϕ-nonexpansive mappings as a subclass, but the converse is not true;

• (4)

  the class of quasi-ϕ-nonexpansive mappings contains properly the class of weak relatively nonexpansive mappings as a subclass, but the converse is not true;

• (5)

  the class of weak relatively nonexpansive mappings contains properly the class of relatively nonexpansive mappings as a subclass, but the converse is not true.

Remark 1.6. If A is an α-inverse strongly monotone mapping, then it is 1/α-Lipschitz continuous.

Iterative approximation of fixed points for relatively nonexpansive mappings in the setting of Banach spaces has been studied extensively by many authors. In 2005, Matsushita and Takahashi [12] obtained some weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Recently, Ofoedu and Malonza [4], Zhang [5], Su et al. [13], Zegeye and Shahzad [14], Wattanawitoon and Kumam [15], Qin et al. [16], Takahashi and Zembayashi [17], Chang et al. [18, 19], Yao et al. [20, 21], Qin et al. [22], and Cho et al. [23, 24] extend the notions from relatively nonexpansive mappings, weakly relatively nonexpansive mappings or quasi-ϕ-nonexpansive mappings to quasi-ϕ-asymptotically nonexpansive mappings and also prove some strongence theorems to approximate a common fixed point of quasi-ϕ-nonexpansive mappings or quasi-ϕ-asymptotically nonexpansive mappings.

Lemma 2.1 (see [7], [26].)Let E be a smooth, strictly convex and reflexive Banach space and C be 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

  $$ (2.1)

• (c)

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

Remark 2.2. 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.

Lemma 2.3 (see [18].)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 sequence $$ and for any given sequence $$ of positive numbers with $$, then there exists a continuous, strictly increasing and convex function g : [0,2r)→[0, ∞) with g(0) = 0 such that for any positive integers i,   j with i < j,

Lemma 2.4. 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 T : C → C be a closed and total quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences {ν_n}, {μ_n} and a strictly increasing continuous functions ζ : ℝ⁺ → ℝ⁺ such that μ₁ = 0, ν_n → 0, μ_n → 0 (as n → ∞) and ζ(0) = 0. Then F(T) is a closed convex subset of C.

Proof. Letting {p_n} be a sequence in F(T) with p_n → p (as n → ∞), we prove that p ∈ F(T). In fact, from the definition of T, we have

Therefore we have that is, p ∈ F(T).

Next we prove that F(T) is convex. For any p, q ∈ F(T),   t ∈ (0,1), putting w = tp + (1 − t)q, we prove that w ∈ F(T). Indeed, in view of the definition of ϕ(x, y), we have

Since μ_n → 0 and ν_n → 0, we have ϕ(w, Tⁿw) → 0 (as n → ∞). From (1.10) we have ∥Tⁿw∥→∥w∥. Consequently ∥JTⁿw∥→∥Jw∥. This implies that {JTⁿw} is a bounded sequence. Since E is reflexive, E^* is also reflexive. So we can assume that Again since E is reflexive, we have J(E) = E^*. Therefore there exists x ∈ E such that Jx = f₀. By virtue of the weakly lower semicontinuity of norm ∥·∥, we have that is, w = x which implies that f₀ = Jw. Hence from (2.6) we have JTⁿw⇀Jw ∈ E^*. Since ∥JTⁿw∥→∥w∥ and E^* has the Kadec-Klee property, we have JTⁿw → Jw. Since E is uniformly smooth, E^* is uniformly convex, which in turn implies that E^* is smooth. From Remark 1.1(iii) it yields that J⁻¹ : E^* → E is hemi-continuous. Therefore we have Tⁿw⇀w. Again since ∥Tⁿw∥→∥w∥, by using the Kadec-Klee property of E, we have Tⁿw → w. This implies that TTⁿw = Tⁿ⁺¹w → w. Since T is closed, we have w = Tw.

This completes the proof of Lemma 2.4.

Lemma 2.5. Let E be a smooth, strictly convex and reflexive Banach space and C be a nonempty closed convex subset of E. Let f : C × C → ℝ be a bifunction satisfying the following conditions:

•  

  (A1)    f(x, x) = 0,     for  all  x ∈ C,

•  

  (A2) f is monotone, that is, f(x, y) + f(y, x) ≤ 0,   for  all  x, y ∈ C,

•  

  (A3) lim  sup_t↓0f(x + t(z − x), y) ≤ f(x, y)  for  all  x, z, y ∈ C,

•  

  (A4) The function y ↦ f(x, y) is convex and lower semi-continuous.

Then the following conclusions hold:

•  

  (1) (Blum and Oettli [27]) for any given r > 0 and x ∈ E, there exists a unique z ∈ C such that

  $$ (2.8)

•  

  (2) (Takahashi and Zembayashi [28]) for any given r > 0 and x ∈ E, define a mapping $$ by

Then, the following conclusions hold:

•  

  (a) $$ is single-valued;

•  

  (b) $$ is firmly nonexpansive-type mapping, that is, for all  z, y ∈ E,

  $$ (2.10)

•  

  (c) $$ and $$ is quasi-ϕ-nonexpansive;

•  

  (d) EP (f) is closed and convex;

•  

  (e) $$.

For solving the generalized mixed equilibrium problem (1.2), let us assume that the following conditions are satisfied:

•  

  (1) E is a smooth, strictly convex, and reflexive Banach space and C is a nonempty closed convex subset of E;

•  

  (2) A : C → E^* is β-inverse strongly monotone mapping;

•  

  (3) F : C × C → ℝ is bifunction satisfying the conditions (A1), (A3), (A4) in Lemma 2.5 and the following condition (A2)^′:

•  

  (A2)^′ for some γ ≥ 0 with γ ≤ β

  $$ (2.11)

•  

  (4) ψ : C → ℝ is a lower semicontinuous and convex function.

Under the assumptions as above, we have the following results.

Lemma 2.6. Let E, C, A, F, ψ satisfy the above conditions (1)–(4). Denote by

For any given r > 0 and x ∈ E, define a mapping $$ by Then, the following hold:

•  

  (a) $$ is single-valued;

•  

  (b) $$ is a firmly nonexpansive-type mapping, that is, for all z, y ∈ E,

  $$ (2.14)

•  

  (c) $$;

•  

  (d) GMEP (F, A, ψ) is closed and convex;

•  

  (e)  

  $$ (2.15)

Proof. It follows from Lemma 2.5 that in order to prove the conclusions of Lemma 2.6 it is sufficient to prove that the function Γ : C × C → ℝ satisfies the conditions (A1)–(A4) in Lemma 2.5.

In fact, by the similar method as given in the proof of Lemma 2.4 in [1], we can prove that the function Γ satisfies the conditions (A1), (A3), and (A4). Now we prove that Γ also satisfies the conditions (A2).

Indeed, for any x, y ∈ C, by condition (A2)′, we have

This implies that the function Γ satisfies the conditions (A2). Therefore the conclusions of Lemma 2.6 can be obtained from Lemma 2.3 immediately.

Remark 2.7. It follows from Lemma 2.5 that the mapping $$ is a relatively nonexpansive mapping. Thus, it is quasi-ϕ-nonexpansive.

Theorem 3.1. Let $$ be the same as above. Suppose that

is a nonempty and bounded subset of C. For any given x₀ ∈ C, let {x_n} be the sequence generated by where $$ is the mapping defined by (2.13) with Γ = Γ_i, r = r_i,n, and r_k,n ∈ [d, ∞), k = 1,2, …, M, n ≥ 1 for some $$ is the generalized projection of E onto the set C_n+1,   and {α_n,i}, {α_n} are sequences in [0,1] satisfying the following conditions:

•  

  (a) $$ for all n ≥ 0;

•  

  (b) lim inf_n→∞α_n,0 · α_n,i > 0 for all i ≥ 1;

•  

  (c) 0 < α ≤ α_n < 1 for some α ∈ (0,1).

Then {x_n} converges strongly to Π_ℱx₀, where Π_ℱ is the generalized projection from E onto ℱ.

Proof. We divide the proof of Theorem 3.1 into five steps.

(i) We first prove that ℱ and C_n both are closed and convex subset of C for all n ≥ 0.

In fact, it follows from Lemmas 2.4 and 2.6 that F(S_i),   i ≥ 1 and GMEP (F_j, A_j, ψ_j)  (j = 1,2, …, M) both are closed and convex. Therefore ℱ is a closed and convex subset in C. Furthermore, it is obvious that C₀ = C is closed and convex. Suppose that C_n is closed and convex for some n ≥ 1. Since the inequality ϕ(v, u_n) ≤ ϕ(v, x_n) + η_n is equivalent to

therefore, we have

This implies that C_n+1 is closed and convex. The desired conclusions are proved. These in turn show that Π_ℱx₀ and $$ are well defined.

(ii) We prove that {x_n} and $$ for all i ≥ 1 are both bounded sequences in C.

By the definition of C_n, we have $$ for all n ≥ 0. It follows from Lemma 2.1 (a) that

This implies that {ϕ(x_n, x₀)} is bounded. By virtue of (1.10), {x_n} is bounded. Since $$ for all u ∈ ℱ and $$ is bounded for all i ≥ 1, and so {z_n}is bounded in E. Denote M by

In view of the structure of {C_n}, we have $$ and $$. This implies that x_n+1 ∈ C_n and

Therefore {ϕ(x_n, x₀)} is convergent. Without loss of generality, we can assume that (iii) Next, we prove that $$ for all n ≥ 0.

Indeed, it is obvious that ℱ ⊂ C₀ = C. Suppose that ℱ ⊂ C_n for some n ≥ 0. Since $$, by Lemma 2.6 and Remark 2.7, $$ is quasi-ϕ-nonexpansive. Again since E is uniformly smooth, E^* is uniformly convex. Hence, For any given u ∈ ℱ ⊂ C_n and for any positive integer j > 0, from Lemma 2.3 we have

Hence u ∈ C_n+1 and so ℱ ⊂ C_n for all n ≥ 0. By the way, from the definition of {η_n} and ζ and (3.10), it is easy to see that (IV) Now, we prove that {x_n} converges strongly to some point

First, we prove that {x_n} converges strongly to some point $$.

In fact, since {x_n} is bounded in C and E is reflexive, there exists a subsequence $$ such that $$. Again since C_n is closed and convex for each n ≥ 1, it is weakly closed, and so p ∈ C_n for each n ≥ 0. Since $$, from the definition of $$, we have

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

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

Therefore we have p = q. This implies that

Now we prove that $$. In fact, by the construction of C_n, we have that C_n+1 ⊂ C_n and $$. Therefore by Lemma 2.1(a) we have

In view of x_n+1 ∈ C_n and noting the construction of C_n+1 we obtain that From (1.10) it yields (∥x_n+1∥−∥u_n∥) ² → 0. Since∥x_n+1∥→∥p∥, we have Hence we have

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

taking liminf _n→∞ on the both sides of above equality and in view of the weak lower semicontinuity of norm ∥·∥, then it yields that That is p = x. This implies that p₀ = Jp, and so Ju_n⇀Jp. It follows from (3.24) and the Kadec-Klee property of E^* that $$ (as n → ∞). Note that J⁻¹ : E^* → E is hemi-continuous, it yields that $$. It follows from (3.23) and the Kadec-Klee property of E that $$.

By the similar way as given in the proof of (3.20), we can also prove that

From (3.20) and (3.27) we have that

Since J is uniformly continuous on any bounded subset of E, we have For any j ≥ 1 and any u ∈ ℱ, it follows from (3.13), (3.20), and (3.27) that Since from (3.28) and (3.29), it follows that In view of condition (b) and condition (c), we have 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 (3.34) we have Since J⁻¹ : E^* → E is hemicontinuous, it follows that On the other hand, for each j ≥ 1 we have This together with (3.36) shows that

Furthermore, by the assumption that for each j ≥ 1, S_j is uniformly L_i-Lipschitz continuous, hence we have

This together with (3.20) and (3.38), yields $$ (as n → ∞). Hence from (3.36) we have $$, that is, $$. In view of (3.38) and the closeness of S_j, it yields that S_jp = p, for  all  j ≥ 1. This implies that $$.

Next, we prove that $$. Denote that

By the similar method as in the proof of (3.13), we can prove that It follows from Lemma 2.6, (2.15), (3.32) that for any u ∈ ℱ, From (1.10) it yields $$. Since $$, we have

Hence we have

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

taking $$ on the both sides of above equality and in view of the weak lower semicontinuity of norm ∥·∥, it yields that This is, p = x. This implies that p₀ = Jp, and so $$. It follows from (3.44) and the Kadec-Klee property of E^* that $$ (as n_i → ∞). Note that J⁻¹ : E^* → E is hemicontinuous it yields that $$. It follows from (3.43) and the Kadec-Klee property of E that $$.

By the similar way as given in the proof of (3.20), we can also prove that

From (3.27) and (3.47) we have that Since J is uniformly continuous on any bounded subset of E, we have

Since

By the similar way as above, we can also prove that

From (3.51) and the assumption that r_n ≥ d,   ∀ n ≥ 0, we have In the proof of Lemma 2.6 we have proved that the function Γ_i, i = 1,2, …, M defined by (3.6) satisfies the condition (A1)–(A4) and Therefore for any y ∈ C we have This implies that for some constant M₁ > 0. Since the function y ↦ Γ_i(x, y) is convex and lower semi-continuous, letting n → ∞ in (3.55), from (3.52) and (3.55), for each i, we have Γ_i(y, p) ≤ 0,   for  all  y ∈ C.

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

Dividing both sides of the above equation by t, we have Γ_i(y_t, y) ≥ 0, for  all  y ∈ C. Letting t ↓ 0, from condition (A3), we have Γ_i(p, y) ≥ 0,   for  all  y ∈ C,   for  all  i = 1,2, …, M, that is, for each i = 1,2, …, M, we have This implies that $$. Therefore, we have that

(V) Now, we prove x_n → Π_ℱx₀.

Let w = Π_ℱx₀. From w ∈ ℱ ⊂ C_n+1 and $$, we have ϕ(x_n+1, x₀) ≤ ϕ(w, x₀), for  all  n ≥ 0. This implies that

By the definition of Π_ℱx₀ and (3.59), we have p = w. Therefore, x_n → Π_ℱx₀. This completes the proof of Theorem 3.1.

Theorem 3.2. Let $$ be the same as above. Let $$ be an infinite family of closed and uniformly quasi-ϕ-asymptotically nonexpansive mappings with a sequence {k_n}⊂[1, ∞) and k_n → 1. Suppose that for each i ≥ 1, S_i is uniformly L_i-Lipschitz continuous and that

is a nonempty and bounded subset of C. For any given x₀ ∈ C, let {x_n} be the sequence generated by where ξ_n = sup _u∈𝒢(k_n − 1)ϕ(u, x_n),   r_n ∈ [d, ∞) for some d > 0,    and for i ≥ 0,   {α_n,i},   {α_n} are sequences in [0,1] satisfying the following conditions:

•  

  (a) $$ for all n ≥ 0;

•  

  (b) lim inf_n→∞α_n,0 · α_n,i > 0 for all i ≥ 1;

•  

  (c) 0 < α ≤ α_n < 1 for some α ∈ (0,1).

Then {x_n} converges strongly to Π_𝒢x₀.

Proof. Since $$ is an infinite family of closed quasi-ϕ-asymptotically nonexpansive mappings, it is an infinite family of closed and uniformly total quasi-ϕ-asymptotically nonexpansive mappings with sequence ζ(t) = t, t ≥ 0, ν_n = k_n − 1, μ_n = 0. Hence ζ_n = ν_nsup _u∈𝒢ζ(ϕ(u, x_n)) + μ_n = sup_u∈𝒢(k_n − 1)ϕ(u, x_n) → 0. Therefore all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 is obtained from Theorem 3.1 immediately.

Remark 3.3. Theorems 3.1 and 3.2 improve and extend the corresponding results in [8, 11, 15, 16, 18–24, 28] and others in the following aspects.

•  

  (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 Kadec-Klee property).

•  

  (b) For the mappings, we extend the mappings from nonexpansive mappings, relatively nonexpansive mappings, quasi-ϕ-nonexpansive mapping or quasi-ϕ-asymptotically nonexpansive mappings to a countable family of total quasi-ϕ-asymptotically nonexpansive mappings.

•  

  (c) We extend a single generalized mixed equilibrium problem to a system of generalized mixed equilibrium problems.