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Hybrid iterative scheme for a generalized equilibrium problems, variational inequality problems and fixed point problem of a finite family of κ i -strictly pseudocontractive mappings
Fixed Point Theory and Applications volume 2012, Article number: 30 (2012)
Abstract
In this article, by using the S-mapping and hybrid method we prove a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of generalized equilibrium defined by Ceng et al., which is a solution of two sets of variational inequality problems. Moreover, by using our main result we have a strong convergence theorem for finding a common element of the set of fixed point problems of a finite family of κ i -strictly pseudocontractive mappings and the set of solution of a finite family of generalized equilibrium defined by Ceng et al., which is a solution of a finite family of variational inequality problems.
1 Introduction
Let H be a real Hilbert space and let C be a nonempty closed convex subset of H. A mapping T of H into itself is called nonexpansive if ∥Tx - Ty∥ ≤ ∥x - y∥ for all x, y ∈ H. We denote by F(T) the set of fixed points of T (i.e., F(T) = {x ∈ H : Tx = x}) Goebel and Kirk [1] showed that F(T) is always closed convex, and also nonempty provided T has a bounded trajectory.
Recall the mapping T is said to be κ-strict pseudo-contration if there exist κ ∈ [0, 1) such that
Note that the class of κ-strict pseudo-contractions strictly includes the class of nonexpansive mappings, that is T is nonexpansive if and only if T is 0-strict pseudo-contractive. If κ = 1, T is said to be pseudo-contraction mapping. T is strong pseudo-contraction if there exists a positive constant λ ∈ (0, 1) such that T + λI is pseudo-contraction. In a real Hilbert space H (1.1) is equivalent to
T is pseudo-contraction if and only if
T is strong pseudo-contraction if there exists a positive constant λ ∈ (0, 1)
The class of κ-strict pseudo-contractions falls into the one between classes of nonexpansive mappings and pseudo-contraction mappings and class of strong pseudo-contraction mappings is independent of the class of κ-strict pseudo-contraction.
A mapping A of C into H is called inverse-strongly monotone, see [2] if there exists a positive real number α such that
for all x, y ∈ C.
The equilibrium problem for G is to determine its equilibrium points, i.e., the set
Given a mapping T : C → H, let G(x, y) = 〈Tx, y - x〉 for all x, y ∈ C. Then, z ∈ EP(F) if and only if 〈Tz, y - z〉 ≥ 0 for all y ∈ C, i.e., z is a solution of the variational inequality. Let A : C → H be a nonlinear mapping. The variational inequality problem is to find a u ∈ C such that
for all v ∈ C. The set of solutions of the variational inequality is denoted by VI(C, A).
In 2005, Combettes and Hirstoaga [3] introduced some iterative schemes of finding the best approximation to the initial data when EP(G) is nonempty and proved strong convergence theorem.
Also in [3] Combettes and Hiratoaga, following [4] define S r : H → C by
hey proved that under suitable hypotheses G, S r is single-valued and firmly nonexpansive with F(S r ) = EP(G).
Numerous problems in physics, optimization, and economics reduce to find a element of EP(G) (see, e.g., [5–16])
Let CB(H) be the family of all nonempty closed bounded subsets of H and be the Hausdorff metric on CB(H) defined as
where d(u, V) = infv∈Vd(u, v), d(U, v) = infu∈Ud(u, v), and d(u, v) = ∥u - v∥.
Let C be a nonempty closed convex subset of H. Let φ : C → ℝ be a real-valued function, T : C → CB(H) a multivalued mapping and Φ : H × C × C → ℝ an equilibrium-like function, that is, Φ(w, u, v) + Φ(w, v, u) = 0 for all (w, u, v) ∈ H × C × C which satisfies the following conditions with respect to the multivalued map T : C → CB(H).
(H 1) For each fixed v ∈ C, (ω, u) ↦ Φ(ω, u, v) is an upper semicontinuous function from H × C to ℝ, that is, for (ω, u) ∈ H × C, whenever ω n → ω and u n → u as n → ∞,
(H 2) For each fixed (w, v) ∈ H × C, u ↦ Φ(w, u, v) is a concave function;
(H 3) For each fixed (w, u) ∈ H × C, v ↦ Φ(w, u, v) is a convex function.
In 2009, Ceng et al. [17] introduced the following generalized equilibrium problem (GEP) as follows:
The set of such solutions u ∈ C of (GEP) is denote by (GEP) s (Φ, φ).
In the case of φ ≡ 0 and Φ(w, u, v) ≡ G(u, v), then (GEP) s (Φ, φ) is denoted by EP(G). By using Nadler's theorem they introduced the following algorithm:
Let x1 ∈ C and w1 ∈ T(x1), there exists sequences {w n } ⊆ H and {x n }, {u n } ⊆ C such that
They proved a strong convergence theorem of the sequence {x n } generated by (1.7) as follows:
Theorem 1.1. (See [17] ) Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H and let φ : C → ℝ be a lower semicontinuous and convex functional. Let T : C → CB(H) be -Lipschitz continuous with constant μ, Φ : H × C × C → ℝ be an equilibrium-like function satisfying (H1)-(H3) and S be a nonexpansive mapping of C into itself such that . Let f be a contraction of C into itself and let {x n }, {w n }, and {u n } be sequences generated by (1.7), where {α n } ⊆ [0,1] and {r n } ⊆ (0, ∞) satisfy
If there exists a constant λ > 0 such that
for all (r1, r2) ∈ Ξ × Ξ,(x1, x2) ∈ C × C and w i ∈ T(x i ), i = 1, 2, where Ξ = {r n : n ≥ 1}, then for , there exists such that is a solution of (GEP) and
In 2011, Kangtunyakarn [18] proved the following theorem for strict pseudocontractive mapping in Hilbert space by using hybrid method as follows:
Theorem 1.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let F and G be bifunctions from C × C into ℝ satisfying (A1)-(A4), respectively. Let A : C → H be a α-inverse strongly monotone mapping and let B : C → H be a β-inverse strongly monotone mapping. Let T : C → C be a κ-strict pseudo-contraction mapping with . Let {x n } be a sequence generated by x1 ∈ C = C1 and
where is sequence in [0,1], r n ∈ [a, b] ⊂ (0, 2α) and s n ⊂ [c, d] ⊂ (0, 2β) satisfy the following condition:
Then x n converges strongly to .
From motivation of (1.7) and (1.9), we define the following algorithm as follows:
Algorithm 1.3. Let T i , i = 1,2,...,N, be κ i -pseudo contraction mappings of C into itself and κ = max{κ i : i = 1,2,..., N} and let S n be the S-mappings generated by T1, T2, ..., T N and where and for all for all j = 1,2,...,N. Let x1 ∈ C = C1 and , there exists sequence and {x n }, {u n }, {v n } ⊆ C such that
where D, T : C → CB(H) are -Lipschitz continuous with constant μ1, μ2, respectively, Φ1, Φ2 : H × C × C → ℝ are equilibrium-like functions satisfying (H 1)-(H 3), A : C → H is a α-inverse strongly monotone mapping and B : C → H is a β-inverse strongly monotone mapping.
In this article, we prove under some control conditions on {δ n }, {α n }, {s n }, and {r n } that the sequence {x n } generated by (1.7) converges strongly to where , G1, G2 : C → C are defined by G1(x) = P C (x - λAx), G2(x) = P C (x - ηBx), ∀x ∈ C and is solution of the following system of variational inequality:
2 Preliminaries
In this section, we need the following lemmas and definition to prove our main result.
Let C be a nonempty closed convex subset of H. Then for any x ∈ H, there exists a unique nearest point in C, denoted by P C x, such that
The following lemma is a property of P C .
Lemma 2.1. (See [19].) Given x ∈ H and y ∈ C. Then P C x = y if and only if there holds the inequality
Lemma 2.2. (See [20] ) Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : n ∈ ℕ} be a sequence of nonexpansive mappings on C. Suppose is nonempty. Let {λ n } be a sequence of positive numbers with . Then a mapping S on C defined by
for x ∈ C is well defined, nonexpansive and hold.
The following lemma is well known.
Lemma 2.3. Let H be Hilbert space, C be a nonempty closed convex subset of H. Let T : C → C be a κ-strictly pseudo-contractive, then the fixed point set F(T) of T is closed and convex so that the projection PF(T)is well defined.
In 2009, Kangtunyakarn and Suantai [21] introduced the S-mapping generated by a finite family of κ-strictly pseudo contractive mappings and real numbers as follows:
Definition 2.1. Let C be a nonempty convex subset of real Hilbert space. Let be a finite family of κ i -strict pseudo-contractions of C into itself. For each j = 1,2,..., N, let where I ∈ [0,1] and . We define the mapping S : C → C as follows:
This mapping is called S-mapping generated by T1, ..., T N and α1, α2, ..., α N .
Lemma 2.4. (See [21] ) Let C be a nonempty closed convex subset of real Hilbert space. Let be a finite family of κ-strict pseudo contraction mapping of C into C with and κ = max{κ i : i = 1, 2,..., N} and let , j = 1,2,3,...,N, where for all j = 1,2,...,N - 1 and for all j = 1,2,..., N. Let S be the mapping generated by T1,....,T N and α1, α2,...,α N . Then and S is a nonexpansive mapping.
Lemma 2.5. (See [22] ) Let C be a nonempty closed convex subset of a real Hilbert space H and S : C → C be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition
We prove the following lemma by using the concept of the S-mapping as follows:
Lemma 2.6. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T i , i = 1,2,...,N be κ i strictly pseudo-contraction mappings of C into itself and κ = max{κ i : i = 1,2,...,N} and let , where and such that as n → ∞ for i = 1, 3 and j = 1,2,3,..., N. For every n ∈ ℕ, let S and S n be the S-mapping generated by T1, T2,..., T N and α1, α2,...,α N and T1, T2,..., T N and , respectively. Then limn→∞∥S n x n - Sx n ∥ = 0 for every bounded sequence {x n } in C.
Proof. Let {x n } be bounded sequence in C, U k and Un,kbe generated by T1,T2,...,T N and α1,α2,...,α N and T1,T2,...,T N and , respectively. For each n ∈ ℕ, we have
and for k ∈ {2, 3,..., N}, by using Lemma 2.5, we obtain
By (2.2) and (2.3), we have
This together with the assumption as n → ∞ (i = 1, 3, j = 1,2,..., N), we can conclude that
Lemma 2.7. (See [23]) Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : C → C be a nonexpansive mapping. Then I - S is demi-closed at zero.
Lemma 2.8. (See [24]) Let C be a closed convex subset of H. Let {x n } be a sequence in H and u ∈ H. Let q = P C u, if {x n } is such the ω(x n ) ⊂ C and satisfy the condition
Then x n → q, as n → ∞.
Definition 2.2. A multivalued map T : C → CB(H) is say to be -Lipschitz continuous if there exists a constant μ > 0 such that
where is the Hausdorff metric on CB(H).
Lemma 2.9. (Nadler's theorem, see [25]) Let (X, ∥ ⋅ ∥) be a normed vector space and is the Hausdorff metric on CB(H). If U, V ∈ CB(X), then for any given ϵ > 0 and u ∈ U, there exists v ∈ V such that
Let C be a nonempty closed convex subset of a real Hilbert space H. Let φ: C → H be a real-valued function, T : C → CB(H) be a multivalued map and Φ : H × C × C → ℝ be an equilibrium-like function.
To solve the GEP, let us assume that the equilibrium-like function Φ : H × C × C → ℝ satisfies the following conditions with respect to the multivalued map T: C → CB(H).
(H 1) For each fixed v ∈ C, (ω, u) ↦ Φ(ω, u, v) is an upper semicontinuous function from H × C to ℝ, that is, for (ω, u) ∈ H × C, whenever ω n → ω and u n → u as n → ∞,
(H 2) For each fixed (w, v) ∈ H × C, u ↦ Φ(w, u, v) is a concave function;
(H 3) For each fixed (w, u) ∈ H × C, v ↦ Φ(w, u, v) is a convex function.
Theorem 2.10. (See [17]) Let C be a nonempty, bounded, closed, and convex subset of a real Hilbert space H, and let φ : C → ℝ be a lower semicontinuous and convex functional. Let T : C → CB(H) be -Lipschitz continuous with constant μ, and Φ : H × C × C → ℝ be an equilibrium-like function satisfying (H 1)-(H 3). Let r > 0 be a constant. For each x ∈ C, take w x ∈ T(x) arbitrarily and define a mapping T r : C → C as follows:
Then, there hold the following:
(a) T r is single-valued;
(b) T r is firmly nonexpansive (that is, for any u, v ∈ C, ∥T r u - T r v∥2 ≤ 〈T r u-T r v, u-v〉) if
for all (x1, x2) ∈ C × C and all w i ∈ T(xi), i = 1,2;
(c) F(T r ) = (GEP) s (Φ, φ)
(d) (GEP) s (Φ, φ) is closed and convex.
Lemma 2.11. (See [26]) Let C be a nonempty closed convex subset of a Hilbert space H and let G : C → C be defined by
with ∀λ > 0. Then x* ∈ VI (C, A) if and only if x* ∈ F(G).
3 Main results
In this section, we prove a strong convergence theorem of the sequence {x n } generated by (1.10) to .
Theorem 3.1. Let C be a nonempty bounded, closed, and convex subset of Hilbert space H and let φ1, φ2 : be a lower semicontinuous and convex function. Let D, T : C → CB(H) be -Lipschitz continuous with constant μ1, μ2, respectively, Φ1,Φ2: H × C × C→ ℝ be equilibrium-like functions satisfying (H 1) - (H3). Let A: C → H be a α-inverse strongly monotone mapping and B : C → H be a β-inverse strongly monotone mapping, let T i , i = 1,2,...,N, be κ i -pseudo contraction mappings of C into itself and κ = max{κ i :i = 1,2,..., N} with , where G1, G2 : C → C are defined by G1(x) P C (x-λAx), G2(x) = P C (x-ηBx), ∀x ∈ C. Let S n be the S-mappings generated by T1,T2,...,T N and where and for all for all j = 1,2,...,N and let {x n }, {u n }, {v n }, , and be sequences generated by (1.10), where {α n } is a sequence in [0,1], r n , λ ∈ [a, b] ⊂ (0, 2α) and s n , η ∈ [c, d] ⊂ (0, 2β), for every n ∈ ℕ and suppose the following conditions hold:
(i) ,
(ii) 0 ≤ κ ≤ α n < 1, ∀n ≥ 1,
(iii) , for all j ∈ {1,2,3,...,N}.
(iv) There exists λ1, λ2 such that
for all and , for i = 1,2 where Θ = {r n : n ≥ 1} and Ξ = {s n : n ≥ 1}. Then {x n } converges strongly to which is a solution of (3.2):
Proof. From (3.1) for every r ∈ Θ, we have
for all (x1, x2) ∈ C × C and .
Similarly, for every s ∈ Ξ, we have
for all (x1, x2) ∈ C × C and . From (3.3) and (3.4), we have Theorem 2.10 hold.
It is easy to see that I - λ A and I - η B are nonexpansive mapping. Indeed, since A is a α-inverse strongly monotone mapping with λ ∈ (0, 2α), we have
Thus (I - λA) is nonexpansive, so is I - ηB. Since
and Theorem 2.10, we have . Since
and Theorem 2.10, we have . Let , again by Theorem 2.10, we have . From nonexpansiveness of , and {I - ηB}, we have
By (3.5), we have
Next, we show that C n is closed and convex for every n ∈ ℕ. It is obvious that C n is closed. In fact, we know that, for z ∈ C n ,
So, we have that ∀z1, z2 ∈ C n and t ∈ (0,1), it follows that
then, we have C n is convex. By Theorem 2.10 and Lemma 2.3, we conclude that is closed and convex. This implies that is well defined. Next, we show that for every n ∈ ℕ. Putting , by (3.6), it is easy to see that q ∈ C n , then we have for all n ∈ ℕ. Since , for every w ∈ C n , we have
In particular, we have
Since C is bounded, we have {x n } is bounded, so are {u n }, {v n }, {z n }, and {y n }. Since and , we have
it implies that
Hence, we have limn→∞∥x n - x1∥ exists. Since
it implies that
Since , we have
by (3.10), we have
Since
by (3.10) and (3.11), we have
Next, we show that
By definition of y n , we have
Claim that
Putting M n = P C (I - λA)u n and N n = P C (I - ηB)v n , we have
Let . Since is firmly nonexpansive mapping and , we have
Hence
Since is firmly nonexpansive mapping and , by using the same method as (3.17), we have
By nonexpansiveness of S n and (3.17), (3.18), we have
it implies that
by (3.12) and condition (i), we have
By using the same method as (3.19), we have
Since
Claim that
By nonexpansiveness of P C , we have
it follows that
by conditions (i), (ii), λ ∈ (0, 2α) and (3.12), it implies that
By using the same method as (3.23), we have
By nonexpansiveness of , we have
Hence, we have
By using the same method as (3.25), we have
Substitute (3.25) and (3.26) in (3.21), we have
it implies that
from (3.12), (3.23), (3.24) and conditions (i) and (ii), we have
By using the same method as (3.28), we have
By (3.19) and (3.28), we have
By (3.20) and (3.29), we have
From (3.16), (3.30) and (3.31), we have
From (3.12) and (3.32), we have
From (3.14), (3.33) and condition (i), we have (3.13).
Let a ∈ (0,1), by (3.10) there exists N0 ∈ ℕ such
Thus, for any number n, p ∈ ℕ, p > 0, we have
Since a ∈ (0,1), we have limn→∞an= 0. By (3.35), we have {x n } is Cauchy sequence in Hilbert space. Let limn→∞x n = x*. Since T : C → CB(H) be -Lipschitz continuous with constant μ1 and (1.10), we have
By (3.34) and for any number n, p ∈ ℕ, p > 0, we have
Since a ∈ (0,1), we have limn→∞an= 0. By (3.37), we have is cauchy sequence in Hilbert space. Let . Next, we will prove that . Since , we have
Since
by limn→∞x n = x* and , we have , this implies that . By using the same method as above, we have and .
Let ω(x n ) be the set of all weakly ω-limit of {x n }. We shall show that . Since {x n } is bounded, then . Let q ∈ ω(x n ), there exists a subsequence of {x n } converse weakly to q. Since {x n } is a Cauchy sequence in Hilbert space, we have as {i → ∞}, it implies that x n → q as n → ∞. Since limn→ ∞x n = x* and limn→∞x n = q, we have x* = q, then we have and . From (3.19) and x n → q as n → ∞, we have u n → q as n → ∞.
By , we have
by (3.19), (H 1) and lower semicontinuity of φ1, we have
then, we have
By using the same method as (3.39), we have
Since for all and for all j = 1,2,..., N. Without loss of generality, we may assume as i → ∞, and as i → ∞, ∀j = 1,2,...,N.
Let S be S-mapping generated by T1, T2 ..., T N . and β1, β2,...,β N , where . By Lemma 2.4, we have S is nonexpansive and .
By Lemma 2.6, we have
By (3.13) and (3.41), we have
Since as i → ∞ and (3.32), we have as i → ∞. By as i → ∞, (3.42) and Lemma 2.7, we have
Next, we define Q : C → C by
By Lemma 2.2, we have
From (3.44), we have
by condition (i), (3.19), (3.20), and (3.32), we have
Since as i → ∞ and Lemma 2.7, we have
From (3.39), (3.40), (3.43), and (3.47), we have . Hence . Hence, by Lemma 2.8 and (3.8), it implies that {x n } converges strongly to . This completes the proof.
Remark 3.2. If we take and φ1 = φ2, then the Algorithm 1.3 reduces to the following algorithm:
under the same conditions of Theorem 3.1, we have the sequence {x n } generated by algorithm (3.48) converges strongly to where , where G1 : C → C is defined by G 1(x) = P C (x - λA x) ∀x ∈ C and is a solution of 〈Ax*, x -x*〉 ≥ 0
4 Application
In this section, by using our main result we prove a strong convergence theorem of the sequence {x n } generated by Algorithm 4.1 as follows:
Algorithm 4.1. Let T i , i = 1,2,...,N, be κ i -pseudo contraction mappings of C into itself and κ = max{κ i : i = 1,2,..., N} and let S n be the S-mappings generated by T1, T2,..., T N and where and for all for all j = 1,2,...,N. Let x1 ∈ C = C1 and , there exists sequence and such that
The following result can be obtained from Theorem 3.1. We, therefore, omit the proof.
Theorem 4.2. Let C be a nonempty bounded, closed, and convex subset of Hilbert space H and let φ i : be a lower semicontinuous and convex function, for all i = 1,2,..., N. Let Ti: C → CB(H) be -Lipschitz continuous with constant μ i , Φ i : H × C × C → ℝ be equilibrium-like functions satisfying (H 1)-(H3) and A i : C → H be a α i -inverse strongly monotone mappings ∀i = 1,2,..., N and let T i , i = 1,2,...,N, be κ i -pseudo contraction mappings of C into itself and κ = max{κ i : i = 1,2,..., N} with , where G i : C → C is defined by G i (x) = P C (x - λ i A i x) ∀x ∈ C, i = 1,2,...,N. Let S n be the S-mappings generated by T1, T2,..., T N and where and for all for all j = 1,2,...,N and let , be sequences generated by (4.1), where {α n } is a sequence in [0,1], and n ∈ ℕ and suppose the following conditions hold:
(i) .
(ii) 0 ≤ κ ≤ α n < 1, ∀n ≥ 1,
(iii) , for all j ∈ {1,2,...,N}.
(iv) There exists λi, ∀i = 1,2,..., N such that
for all for k = 1,2 where . Then {x n } converges strongly to and is a solutions of (4.3):
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Kangtunyakarn, A. Hybrid iterative scheme for a generalized equilibrium problems, variational inequality problems and fixed point problem of a finite family of κ i -strictly pseudocontractive mappings. Fixed Point Theory Appl 2012, 30 (2012). https://doi.org/10.1186/1687-1812-2012-30
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DOI: https://doi.org/10.1186/1687-1812-2012-30