In this section, we shall prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem (1.2), set of variational inequalities for an α-inverse strongly monotone mapping and the set of common fixed points for a pair of asymptotically quasi-ϕ-nonexpansive mappings in Banach spaces.
Theorem 3.1. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ ∅. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1) - (A 4), and φ : C → ℝ be a lower semi-continuous and convex function. Let T : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequence
such that
as n → ∞ and S : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequence
such that
as n → ∞. Assume that T and S are uniformly asymptotically regular on C and Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B, φ.) ≠ ∅.
Let {x
n
} be the sequence defined by x0 ∈ E and
where
as n → ∞,
for each n ≥ 1, M
n
= sup{ϕ(z, x
n
) : z ∈ Ω } for each n ≥ 1, {α
n
} and {β
n
} are sequences in [0, 1], {λ
n
} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where
is the 2-uniformly convexity constant of E and {r
n
} ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied: lim infn→∞(1 -α
n
) > 0 and lim infn→∞(1 -β
n
) > 0. Then, the sequence {x
n
} converges strongly to ΠΩx0, where ΠΩ is generalized projection of E onto Ω.
Proof. We have several steps to prove this theorem as follows:
Step 1. We first show that Cn+1is closed and convex for each n ≥ 1. Indeed, it is obvious that C1 = C is closed and convex. Suppose that C
i
is closed and convex for each i ∈ ℕ. Next, we prove that Ci+1is closed and convex. For any z ∈ Ci+1, we know that ϕ(z, u
i
) ≤ ϕ (z, x
i
) + θ
i
is equivalent to
where
and M
i
= sup{ϕ(z, x
i
) : z ∈ Ω} for each i ≥ 1. Hence, Ci+1is closed and convex. Then, for each n ≥ 1, we see that C
n
is closed and convex. Hence,
is well defined.
By the same argument as in the proof of [[43], Lemma 2.4], one can show that F(T) ∩ F(S) is closed and convex. We also know that VI(A, C) = U-10 is closed and convex, and hence from Lemma 2.12(d), Ω := F(S) ∩ F(T) ∩ VI(A, C) ∩ GMEP(f, B, φ) is a nonempty, closed and convex subset of C. Consequently, ΠΩ is well defined.
Step 2. We show that the sequence {x
n
} is well defined. Next, we prove that Ω ⊂ C
n
for each n ≥ 1. If n = 1, Ω ⊂ C1 = C is obvious. Suppose that Ω ⊂ C
i
for some positive integer i. For every q ∈ Ω, we obtain from the assumption that q ∈ C
i
. It follows, from Lemma 2.1 and Lemma 2.8, that
Thus, q ∈ VI(A, C) and A is α-inverse-strongly monotone, we have
From Lemma 2.2 and ||Ay|| ≤ ||Ay - Au|| for all y ∈ C and q ∈ Ω, we obtain
Substituting (3.3) and (3.4) into (3.2), we have
As Ti is asymptotically quasi-ϕ-nonexpansive mapping, we also have
It follows that
This shows that q ∈ Ci+1. This implies that Ω ⊂ C
n
for each n ≥ 1.
From
, we see that
Since Ω ⊂ C
n
for each n ≥ 1, we arrive at
Hence, the sequence {x
n
} is well defined.
Step 3. Now, we prove that {x
n
} is bounded.
In view of Lemma 2.1, we see that
for each q ∈ C
n
. Therefore, we obtain that the sequence ϕ(x
n
, x0) is bounded, and so are {x
n
}, {w
n
}, {y
n
}, {z
n
}, {Tnw
n
} and {Snx
n
}.
Step 4. We show that {x
n
} is a Cauchy sequence.
Since
and
, we have
This implies that {ϕ(x
n
, x0)} is nondecreasing, and limn →∞ϕ(x
n
, x0) exists.
For m > n and from Lemma 2.1, we have
Letting m, n → ∞ in (3.9), we see that ϕ(x
m
, x
n
) → 0. It follows from Lemma 2.6 that ||x
m
- x
n
|| → 0 as m, n → ∞. Hence, {x
n
} is a Cauchy sequence. Since E is a Banach space and C is closed and convex, we can assume that p ∈ C such that x
n
→ p as n → ∞.
Step 5. We will show that p ∈ Ω:= F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B, φ).
(a) First, we show that p ∈ F(T) ∩ F(S).
By taking m = n + 1 in (3.9), we obtain that
Since
, from definition of Cn+1, we have
and from (3.5) and (3.6), we also have
Since E is uniformly smooth and uniformly convex, from (3.10)-(3.12), θ
n
→ 0 as n → ∞ and
Lemma 2.6, it follows that
and by using triangle inequality, we have
Since J is uniformly norm-to-norm continuous, we also have
and
Since
, and from (3.7), we have
Since ||x
n
- u
n
|| → 0 and J is uniformly continuous, we have
Since {x
n
} and {u
n
} are bounded, it follows from (3.14) and (3.15) that ϕ(y
n
, u
n
) → 0 as n → ∞. Since E is smooth and uniformly convex, from Lemma 2.6, we have
Since J is uniformly norm-to-norm continuous, we also have
Again from (3.1) and (3.16), we have
This implies that ||JTnw
n
- Jx
n
|| → 0. Again since J-1 is uniformly norm-to-norm continuous, we also have
For p ∈ Ω, we note that
It follows from (3.22) and x
n
→ p as n → ∞, that
On other hand, we have
Since T is uniformly asymptotically regular and from (3.24), we obtain that
Thai is, TTnwn → p as n → ∞. From the closedness of T, we see that p ∈ F(T). Furthermore, For q ∈ Ω, from (3.7) and (3.18) that
and hence
From (3.18) and lim infn→∞(1 -β
n
) > 0, obtain that
From Lemma 2.1, Lemma 2.8 and (3.4), we compute
Applying Lemma 2.6 and (3.27) that
Since J is uniformly norm-to-norm continuous on bounded sets, by (3.28), we have
From(3.1), (3.20) and (ii), we have
Since J-1 is uniformly norm-to-norm continuous on bounded sets
We observe that
It follows from (3.31) and x
n
→ p as n → ∞, we obtain
On other hand, we have
Since S is uniformly asymptotically regular and (3.33), we obtain that
that is, SSnz
n
→ p as n → ∞. From the closedness of S, we see that p ∈ F(S). Hence, p ∈ F(T) ∩ F(S).
(b) We show that p ∈ GMEP(f, B, φ). From (A2), we have
and hence
For t with 0 < t ≤ 1 and y ∈ C, let y
t
= t
y
+ (1 - t)p. Then, we get y
t
∈ C. From (3.35), it follows that
we know that y
n
, u
n
→ p as n → ∞, and
as n → ∞. Since B is monotone, we know that 〈By
t
- Bu
n
, y
t
- u
n
〉 ≥ 0. Thus, it follows from (A4) that
Based on the conditions (A1), (A4) and convexity of φ, we have
and hence
From (A3) and the weakly lower semicontinuity of φ, and letting t → 0, we also have
This implies that p ∈ GMEP(f, B, φ).
(c) We show that p ∈ VI(A, C). Indeed, define a set-valued U : E ⇉ E* by Lemma 2.14, U is maximal monotone and U-10 = VI(A, C). Let (v, w) ∈ G(U). Since w ∈ Uv = Av + N
C
(v), we get w - Av ∈ N
C
(v).
From w
n
∈ C, we have
On the other hand, since
. Then from Lemma 2.7, we have
and thus
It follows from (3.36) and (3.37) that
where M = supn≥1||v - w
n
||. Takeing the limit as n → ∞, (3.28) and (3.29), we obtain 〈v - p, w〉 ≥ 0. Based on the maximality of U, we have p ∈ U-10 and hence p ∈ VI(A, C). Hence, by (a), (b) and (c), we obtain p ∈ Ω.
Step 5. Finally, we prove that p = ΠΩx0. Taking the limit as n → ∞ in (3.8), we obtain that
and hence, p = ΠΩx0 by Lemma 2.1. This completes the proof.
The following Theorems can readily be derived from Theorem 3.1.
Corollary 3.2. Let E be a uniformly smooth and 2-uniformly convex Banach space, and C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø,. Let f : C × C → ℝ be a bifunction satisfying the conditions (A 1) - (A 4), and φ : C → ℝ be a lower semi-continuous and convex function. Let T : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequence
such that
as n → ∞ and S : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequence
such that
as n → ∞. Assume that T and S are uniformly asymptotically regular on C and Ω:= F(T) ∩ F(S) ∩ VI(A, C) ∩ MEP(f, φ) ≠ ∅. Let {x
n
} be the sequence defined by x0 ∈ E and
where
as n → ∞,
for each n ≥ 1, M
n
= sup{ϕ(z, x
n
) : z ∈ Ω} for each n ≥ 1, {α
n
} and {β
n
} are sequences in [0, 1], {λ
n
} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where
is the 2-uniformly convexity constant of E and {r
n
} ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:
-
(i)
lim infn→∞(1 -α
n
) > 0,
-
(ii)
lim infn→∞(1 -β
n
) > 0.
Then, the sequence {x
n
} converges strongly to ΠΩx0, where ΠΩ is generalized projection of E onto Ω.
Proof. Putting B ≡ 0 in Theorem 3.1, the conclusion of Theorem 3.2 can be obtained.
Corollary 3.3. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and φ : C → ℝ be a lower semi-continuous and convex function. Let T : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequence
such that
as n → ∞ and S : C → C be a closed and asymptotically quasi-ϕ-nonexpansive mapping with the sequence
such that
as n → ∞. Assume that T and S are uniformly asymptotically regular on C and Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ MVI(B, C) ≠ ∅. Let {x
n
} be the sequence defined by x0 ∈ E and
where
as n → ∞,
for each n ≥ 1, M
n
= sup{ϕ(z, x
n
) : z ∈ Ω} for each n ≥ 1, {α
n
} and {β
n
} are sequences in [0, 1], {λ
n
} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where
is the 2-uniformly convexity constant of E and {r
n
} ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:
-
(i)
lim infn→∞(1 -α
n
) > 0;
-
(ii)
lim infn→∞(1 -β
n
) > 0.
Then, the sequence {x
n
} converges strongly to ΠΩx0, where ΠΩ is generalized projection of E onto Ω.
Proof. Putting f ≡ 0 in Theorem 3.1, the conclusion of Theorem 3.2 can be obtained.
Since every closed relatively asymptotically nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive, we obtain the following corollary.
Corollary 3.4. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1) - (A 4), and φ. C → ℝ be a lower semi-continuous and convex function. Let T. C → C be a closed and relatively asymptotically nonexpansive mapping with the sequence
such that
as n → ∞ and S. C →C be a closed and relatively asymptotically nonexpansive mapping with the sequence
such that
as n → ∞. Assume that T and S are uniformly asymptotically regular on C and Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ ∅. Let {x
n
} be the sequence defined by x0 ∈ E and
where
as n → ∞,
for each n ≥ 1, M
n
= sup{ϕ (z, x
n
). z ∈ Ω} for each n ≥ 1, {α
n
} and {β
n
} are sequences in [0, 1], {λ
n
} ⊂ [a, b] for some a, b with 0 < a < b < c2α/ 2, where
is the 2-uniformly convexity constant of E and {r
n
} ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:
-
(i)
lim infn→∞(1 - α
n
) > 0;
-
(ii)
lim infn→∞(1 - β
n
) > 0.
Then, the sequence {x
n
} converges strongly to ΠΩx0, where ΠΩis generalized projection of E onto Ω.
Since every closed relatively nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive, we obtain the following corollary.
Corollary 3.5. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1) - (A 4), and φ : C → ℝ be a lower semi-continuous and convex function. Let T, S : C → C be closed relatively nonexpansive mappings such that Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ ∅. Let {x
n
} be the sequence defined by x0 ∈ E and
where {α
n
} and {β
n
} are sequences in [0, 1], {λ
n
} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where
is the 2-uniformly convexity constant of E and {r
n
} ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:
-
(i)
lim infn→∞(1 - α
n
) > 0,
-
(ii)
lim infn→∞(1 - β
n
) > 0.
Then, the sequence {x
n
} converges strongly to ΠΩx0, where ΠΩis generalized projection of E onto Ω.
Corollary 3.6. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1) - (A 4), and φ : C → ℝ be a lower semi-continuous and convex function. Let T, S : C → C be a closed quasi-ϕ-nonexpansive mappings Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ ∅. Let {x
n
} be the sequence defined by x0 ∈ E and
where {α
n
} and {β
n
} are sequences in [0, 1], {λ
n
} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where
is the 2-uniformly convexity constant of E and {r
n
} ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:
-
(i)
lim infn→∞(1 - α
n
) > 0;
-
(ii)
lim infn→∞(1 - β
n
) > 0.
Then, the sequence {x
n
} converges strongly to ΠΩx0, where ΠΩ is generalized projection of E onto Ω
Proof. Since every closed quasi-ϕ-nonexpansive mapping is asymptotically quasi-ϕ-nonexpansive, the result is implied by Theorem 3.1.
Corollary 3.7. Let E be a uniformly smooth and 2-uniformly convex Banach space, C be a nonempty closed convex subset of E. Let A be an α-inverse-strongly monotone mapping of C into E* satisfying ||Ay|| ≤ ||Ay - Au||, ∀y ∈ C and u ∈ VI(A, C) ≠ Ø. Let B : C → E* be a continuous and monotone mapping and f : C × C → ℝ be a bifunction satisfying the conditions (A 1) - (A 4), and φ : C → ℝ be a lower semi-continuous and convex function. Let T, S : C → C be closed relatively nonexpansive mappings such that Ω := F(T) ∩ F(S) ∩ VI(A, C) ∩ GMEP(f, B,φ) ≠ Ø. Let {x
n
} be the sequence defined by x0 ∈ E and
where {α
n
} and {β
n
} are sequences in [0, 1], {λ
n
} ⊂ [a, b] for some a, b with 0 < a < b < c2α/2, where
is the 2-uniformly convexity constant of E and {r
n
} ⊂ [d, ∞) for some d > 0. Suppose that the following conditions are satisfied:
-
(i)
lim infn→∞(1 - α
n
) > 0;
-
(ii)
lim infn→∞(1 - β
n
) > 0.
Then, the sequence {x
n
} converges strongly to ΠΩx0, where ΠΩis generalized projection of E onto Ω.
Proof. Since every closed relatively nonexpansive mapping is quasi-ϕ-nonexpansive, the result is implied by Theorem 3.1.
Remark 3.8. Corollaries 3.7, 3.6 and 3.7 improve and extend the corresponding results of Saewan et al. [[51], Theorem 3.1] in the sense of changing the closed relatively quasi-nonexpansive mappings to be the more general than the closed and asymptotically quasi-ϕ-nonexpansive mappings and adjusting a problem from the classical equilibrium problem to be the generalized equilibrium problem.