Open Access

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 Applications20122012:30

https://doi.org/10.1186/1687-1812-2012-30

Received: 26 September 2011

Accepted: 28 February 2012

Published: 28 February 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.

Keywords

κ-strict pseudo contraction mappingα-inverse strongly monotonegeneralized equilibrium problemvariational inequalitythe S-mapping

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 - Tyx - 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
T x - T y 2 x - y 2 + κ ( I - T ) x - ( I - T ) y 2 x , y D ( T ) .
(1.1)
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 x - T y , x - y x - y 2 - 1 - κ 2 ( I - T ) x - ( I - T ) y 2 x , y D ( T ) .
(1.2)
T is pseudo-contraction if and only if
T x - T y , x - y x - y 2 x , y D ( T ) .
T is strong pseudo-contraction if there exists a positive constant λ (0, 1)
T x - T y , x - y ( 1 - λ ) x - y 2 x , y D ( T )

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
x - y , A x - A y α A x - A y 2

for all x, y C.

The equilibrium problem for G is to determine its equilibrium points, i.e., the set
E P ( G ) = { x G : G ( x , y ) 0 , y C } .
(1.3)
Given a mapping T : CH, 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 : CH be a nonlinear mapping. The variational inequality problem is to find a u C such that
v - u , A u 0
(1.4)

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 : HC by
S r ( x ) = { z C : G ( z , y ) + 1 r y - z , z - x 0 y C } .
(1.5)

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., [516])

Let CB(H) be the family of all nonempty closed bounded subsets of H and ( . , . ) be the Hausdorff metric on CB(H) defined as
( U , V ) = max sup u U d ( u , V ) , sup v V d ( U , v ) , U , V C B ( H ) ,

where d(u, V) = infvVd(u, v), d(U, v) = infuUd(u, v), and d(u, v) = u - v.

Let C be a nonempty closed convex subset of H. Let φ : C be a real-valued function, T : CCB(H) a multivalued mapping and Φ : H × C × C 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 : CCB(H).

(H 1) For each fixed v C, (ω, u) Φ(ω, u, v) is an upper semicontinuous function from H × C to , that is, for (ω, u) H × C, whenever ω n ω and u n u as n → ∞,
lim sup n Φ ( ω n , u n , v ) Φ ( ω , u , v ) ;

(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:
( GEP ) Find u C and w T ( u ) such that Φ ( w , u , v ) + φ ( v ) - φ ( u ) 0 , v C .
(1.6)

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
w n T ( x n ) , w n - w n + 1 1 + 1 n ( T ( x n ) , T ( x n + 1 ) ) , Φ w n , u n , v + φ ( v ) - φ ( u n ) + 1 r n u n - x n , v - u n 0 , u C , x n + 1 = α n f ( x n ) + ( 1 - α n ) S u n , n = 1 , 2 , . . . .
(1.7)

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 be a lower semicontinuous and convex functional. Let T : CCB(H) be -Lipschitz continuous with constant μ, Φ : H × C × C be an equilibrium-like function satisfying (H1)-(H3) and S be a nonexpansive mapping of C into itself such that F ( S ) ( G E P ) s ( Φ , φ ) . 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
lim n α n = 0 , n = 1 α n = , n = 1 α n + 1 - α n < , lim  inf n r n > 0 a n d n = 1 r n + 1 - r n < .
If there exists a constant λ > 0 such that
Φ ( w 1 , T r 1 ( x 1 ) , T r 2 ( x 2 ) ) + Φ ( w 2 , T r 2 ( x 2 ) , T r 1 ( x 1 ) ) - λ T r 1 ( x 1 ) - T r 2 ( x 2 ) 2
(1.8)
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 x ^ = P F ( S ) ( G E P ) s ( Φ , φ ) f ( x ^ ) , there exists w ^ T ( x ^ ) such that ( x ^ , w ^ ) is a solution of (GEP) and
x n x ^ , w n w ^ a n d u n x ^ a s n .

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 satisfying (A1)-(A4), respectively. Let A : CH be a α-inverse strongly monotone mapping and let B : CH be a β-inverse strongly monotone mapping. Let T : CC be a κ-strict pseudo-contraction mapping with F = F ( T ) E P ( F , A ) E P ( G , B ) . Let {x n } be a sequence generated by x1 C = C1 and
F ( u n , u ) + ( A x n , u - u n ) + 1 r n u - u n , u n - x n 0 , u C , G ( v n , v ) + ( B x n , v - v n ) + 1 s n v - v n , v n - x n 0 , v C , z n = δ n u n + ( 1 - δ n ) v n y n = α n z n + ( 1 - α n ) T z n C n + 1 = z C n : y n - z x n - z , x n + 1 = P C n + 1 x 1 , n 1 ,
(1.9)
where { α n } n = 0 is sequence in [0,1], r n [a, b] (0, 2α) and s n [c, d] (0, 2β) satisfy the following condition:
( i ) lim n δ n = δ ( 0 , 1 ) ( i i ) 0 κ α n < 1 , n 1

Then x n converges strongly to P F x 1 .

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 α 1 ( n ) , α 2 ( n ) , . . . , α N ( n ) where α j ( n ) = α 1 n , j , α 2 n , j , α 3 n , j I × I × I , I = [ 0 , 1 ] , α 1 n , j + α 2 n , j + α 3 n , j = 1 and κ < a α 1 n , j , α 3 n , j b < 1 for all j = 1 , 2 , . . . , N - 1 , κ < c α 1 n , N 1 , κ α 3 n , N d < 1 , κ α 2 n , j e < 1 for all j = 1,2,...,N. Let x1 C = C1 and w 1 1 T ( x 1 ) , w 1 2 D ( x 1 ) , there exists sequence { w n 1 } , { w n 2 } H and {x n }, {u n }, {v n } C such that
w n 1 T ( x n ) , w n 1 - w n + 1 1 1 + 1 n ( T ( x n ) , T ( x n + 1 ) ) , w n 2 D ( x n ) , w n 2 - w n + 1 2 1 + 1 n ( D ( x n ) , D ( x n + 1 ) ) , Φ 1 w n 1 , u n , u + φ 1 ( u ) - φ 1 ( u n ) + 1 r n u n - x n , u - u n 0 , u C , Φ 2 w n 2 , v n , v + φ 2 ( v ) - φ 2 ( v n ) + 1 s n v n - x n , v - v n 0 , v C , z n = δ n P C ( I - λ A ) u n + ( 1 - δ n ) P C ( I - η B ) v n , y n = α n z n + ( 1 - α n ) S n z n , C n + 1 = z C n : y n - z x n - z , x n + 1 = P C n + 1 x 1 , n 1 .
(1.10)

where D, T : CCB(H) are -Lipschitz continuous with constant μ1, μ2, respectively, Φ1, Φ2 : H × C × C are equilibrium-like functions satisfying (H 1)-(H 3), A : CH is a α-inverse strongly monotone mapping and B : CH 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 P F x 1 where F = i = 1 N F ( T i ) ( G E P ) s ( Φ 1 , φ 1 ) ( G E P ) s ( Φ 2 , φ 2 ) F ( G 1 ) F ( G 2 ) , G1, G2 : CC are defined by G1(x) = P C (x - λAx), G2(x) = P C (x - ηBx), x C and P F x 1 is solution of the following system of variational inequality:
A x * , x - x * 0 , B x * , x - x * 0 .

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
x - P C x x - y , for all y C .

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
x - y , y - z 0 z C .
Lemma 2.2. (See [20] ) Let C be a closed convex subset of a strictly convex Banach space E. Let {T n : n } be a sequence of nonexpansive mappings on C. Suppose n = 1 F ( T n ) is nonempty. Let {λ n } be a sequence of positive numbers with n = 1 λ n = 1 . Then a mapping S on C defined by
S ( x ) = n = 1 λ n T n x

for x C is well defined, nonexpansive and F ( S ) = n = 1 F ( T n ) 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 : CC 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 { T i } i = 1 N be a finite family of κ i -strict pseudo-contractions of C into itself. For each j = 1,2,..., N, let α j = ( a 1 j , α 2 j , α 3 j ) I × I × I where I [0,1] and α 1 j + α 2 j + α 3 j = 1 . We define the mapping S : CC as follows:
U 0 = I U 1 = α 1 1 T 1 U 0 + α 2 1 U 0 + α 3 1 I U 2 = α 1 2 T 2 U 1 + α 2 2 U 1 + α 3 2 I U 3 = α 1 3 T 3 U 2 + α 2 3 U 2 + α 3 3 I U N - 1 = α 1 N - 1 T N - 1 U N - 2 + α 2 N - 1 U N - 2 + α 3 N - 1 I S = U N = α 1 N T N U N - 1 + α 2 N U N - 1 + α 3 N I .
(2.1)

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 { T i } i = 1 N be a finite family of κ-strict pseudo contraction mapping of C into C with i = 1 N F ( T i ) and κ = max{κ i : i = 1, 2,..., N} and let α j = ( a 1 j , α 2 j , α 3 j ) I × I × I , j = 1,2,3,...,N, where I = [ 0 , 1 ] , α 1 j + α 2 j + α 3 j = 1 , α 1 j , α 3 j ( κ , 1 ) for all j = 1,2,...,N - 1 and α 1 N ( κ , 1 ] , α 3 N [ κ , 1 ) α 2 j [ κ , 1 ) for all j = 1,2,..., N. Let S be the mapping generated by T1,....,T N and α1, α2,...,α N . Then F ( S ) = i = 1 N F ( T i ) 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 : CC be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition
S x - S y 1 + κ 1 - κ x - y , x , y C .

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 α j ( n ) = ( α 1 n , j , α 2 n , j , α 3 n , j ) , α j = ( α 1 j , α 2 j , α 3 j ) I × I × I , where I = [ 0 , 1 ] , α 1 n , j + α 2 n , j + α 3 n , j = 1 and α 1 j + α 2 j + α 3 j = 1 such that α i n , j α i j [ 0 , 1 ] as n → ∞ for i = 1, 3 and j = 1,2,3,..., N. For every 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 α 1 ( n ) , α 2 ( n ) , , α N ( n ) , 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 α 1 ( n ) , α 2 ( n ) , , α N ( n ) , respectively. For each n , we have
U n , 1 x n - U 1 x n = α 1 n , 1 T 1 x n + ( 1 - α 1 n , 1 ) x n - α 1 1 T 1 x n - ( 1 - α 1 1 ) x n = α 1 n , 1 T 1 x n - α 1 n , 1 x n - α 1 1 T 1 x n + α 1 1 x n = ( α 1 n , 1 - α 1 1 ) T 1 x n - ( α 1 n , 1 - α 1 1 ) x n = a 1 n , 1 - α 1 1 T 1 x n - x n
(2.2)
and for k {2, 3,..., N}, by using Lemma 2.5, we obtain
U n , k x n - U k x n = α 1 n , k T k U n , k - 1 x n + α 2 n , k U n , k - 1 x n + α 3 n , k x n - α 1 k T k U k - 1 x n - α 2 k U k - 1 x n - α 3 k x n = α 1 n , k T k U n , k - 1 x n + α 3 n , k x n - α 1 k T k U k - 1 x n - α 3 k x n + α 2 n , k U n , k - 1 x n - α 2 k U k - 1 x n = α 1 n , k T k U n , k - 1 x n - α 1 n , k T k U k - 1 x n + α 1 n , k T k U k - 1 x n - α 1 k T k U k - 1 x n + ( α 3 n , k - α 3 k ) x n + α 2 n , k U n , k - 1 x n - α 2 k U k - 1 x n = α 1 n , k ( T k U n , k - 1 x n - T k U k - 1 x n ) + ( α 1 n , k - α 1 k ) T k U k - 1 x n + ( α 3 n , k - α 3 k ) x n + α 2 n , k U n , k - 1 x n - α 2 k U k - 1 x n = α 1 n , k ( T k U n , k - 1 x n - T k U k - 1 x n ) + ( α 1 n , k - α 1 k ) T k U k - 1 x n + ( α 3 n , k - α 3 k ) x n + α 2 n , k U n , k - 1 x n - α 2 n , k U k - 1 x n + α 2 n , k U k - 1 x n - α 2 k U k - 1 x n = α 1 n , k ( T k U n , k - 1 x n - T k U k - 1 x n ) + ( α 1 n , k - α 1 k ) T k U k - 1 x n + ( α 3 n , k - α 3 k ) x n + α 2 n , k ( U n , k - 1 x n - U k - 1 x n ) + ( α 2 n , k - α 2 k ) U k - 1 x n α 1 n , k T k U n , k - 1 x n - T k U k - 1 x n + α 1 n , k - α 1 k T k U k - 1 x n + α 3 n , k - α 3 k x n + α 2 n , k U n , k - 1 x n - U k - 1 x n + α 2 n , k - α 2 k U k - 1 x n = α 1 n , k T k U n , k - 1 x n - T k U k - 1 x n + α 1 n , k - α 1 k T k U k - 1 x n + α 2 n , k U n , k - 1 x n - U k - 1 x n + 1 - α 1 n , k - α 3 n , k - 1 + α 1 k + α 3 k U k - 1 x n + α 3 n , k - α 3 k x n α 1 n , k 1 + κ 1 - κ U n , k - 1 x n - U k - 1 x n + α 1 n , k - α 1 k T k U k - 1 x n + α 2 n , k U n , k - 1 x n - U k - 1 x n + α 1 k - α 1 n , k + α 3 n , k - α 3 k U k - 1 x n + α 3 n , k - α 3 k x n 1 + κ 1 - κ U n , k - 1 x n - U k - 1 x n + α 1 n , k - α 1 k T k U k - 1 x n + 1 - κ 1 - κ U n , k - 1 x n - U k - 1 x n + α 1 k - α 1 n , k + α 3 n , k - α 3 k U k - 1 x n + α 3 n , k - α 3 k x n 2 1 - κ U n , k - 1 x n - U k - 1 x n + α 1 n , k - α 1 k T k U k - 1 x n + U k - 1 x n + α 3 n , k - α 3 k U k - 1 x n + x n .
(2.3)
By (2.2) and (2.3), we have
S n x n - S x n = U n , N x n - U N x n 2 1 - κ U n , N - 1 x n - U N - 1 x n + α 1 n , N - α 1 N T N U N - 1 x n + U N - 1 x n + α 3 n , N - α 3 N U N - 1 x n + x n 2 1 - κ 2 1 - κ U n , N - 2 x n - U N - 2 x n + α 1 n , N - 1 - α 1 N - 1 T N - 1 U N - 2 x n + U N - 2 x n + α 3 n , N - 1 - α 3 N - 1 U N - 2 x n + x n + α 1 n , N - α 1 N T N U N - 1 x n + U N - 1 x n + α 3 n , N - α 3 N U N - 1 x n + x n = 2 1 - κ 2 U n , N - 2 x n - U N - 2 x n + j = N - 1 N 2 1 - κ N - j α 1 n , j - α 1 j T j U j - 1 x n + U j - 1 x n + j = N - 1 N 2 1 - κ N - j α 3 n , j - α 3 j U j - 1 x n + x n 2 1 - κ N - 1 U n , 1 x n - U 1 x n + j = 2 N 2 1 - κ N - j α 1 n , j - α 1 j T j U j - 1 x n + U j - 1 x n + j = 2 N ( 2 1 - κ ) N - j α 3 n , j - α 3 j U j - 1 x n + x n = 2 1 - κ N - 1 α 1 n , 1 - α 1 1 T 1 x n - x n + j = 2 N 2 1 - κ N - j α 1 n , j - α 1 j T j U j - 1 x n + U j - 1 x n + j = 2 N 2 1 - κ N - j α 3 n , j - α 3 j U j - 1 x n + x n .
This together with the assumption α i n , j α i j as n → ∞ (i = 1, 3, j = 1,2,..., N), we can conclude that
lim n S n x n - S x n = 0 .

Lemma 2.7. (See [23]) Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S : CC 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
x n - u u - q , n .

Then x n q, as n → ∞.

Definition 2.2. A multivalued map T : CCB(H) is say to be -Lipschitz continuous if there exists a constant μ > 0 such that
( T ( u ) - T ( v ) ) μ u - v , u , v C ,

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
u - v ( 1 + ε ) ( U , V ) .

Let C be a nonempty closed convex subset of a real Hilbert space H. Let φ: CH be a real-valued function, T : CCB(H) be a multivalued map and Φ : H × C × C be an equilibrium-like function.

To solve the GEP, let us assume that the equilibrium-like function Φ : H × C × C satisfies the following conditions with respect to the multivalued map T: CCB(H).

(H 1) For each fixed v C, (ω, u) Φ(ω, u, v) is an upper semicontinuous function from H × C to , that is, for (ω, u) H × C, whenever ω n ω and u n u as n → ∞,
lim sup n Φ ( ω n , u n , v ) Φ ( ω , u , v ) ;

(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 be a lower semicontinuous and convex functional. Let T : CCB(H) be -Lipschitz continuous with constant μ, and Φ : H × C × C 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 : CC as follows:
T r ( x ) = u C : Φ ( w x , u , v ) + φ ( v ) - φ ( u ) + 1 r u - x , v - u 0 , v C .

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 v2 ≤ 〈T r u-T r v, u-v〉) if
Φ ( w 1 , T r ( x 1 ) , T r ( x 2 ) ) + Φ ( w 2 , T r ( x 2 ) , T r ( x 1 ) ) 0 ,

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 : CC be defined by
G ( x ) = P C ( x - λ A x ) , x C ,

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 P F x 1 .

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 : CCB(H) be -Lipschitz continuous with constant μ1, μ2, respectively, Φ12: H × C × C be equilibrium-like functions satisfying (H 1) - (H3). Let A: CH be a α-inverse strongly monotone mapping and B : CH 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 F = i = 1 N F ( T i ) ( G E P ) s ( Φ 1 , φ 1 ) ( G E P ) s ( Φ 2 , φ 2 ) F ( G 1 ) F ( G 2 ) , where G1, G2 : CC 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 α 1 ( n ) , α 2 ( n ) , . . . , α N ( n ) where α j ( n ) = α 1 n , j , α 2 n , j , α 3 n , j I × I × I , I = [ 0 , 1 ] , α 1 n , j + α 2 n , j + α 3 n , j = 1 and κ < a α 1 n , j , α 3 n , j b < 1 for all j = 1 , 2 , . . . , N - 1 , κ < c α 1 n , N 1 , κ α 3 n , N d < 1 , κ α 2 n , j e < 1 for all j = 1,2,...,N and let {x n }, {u n }, {v n }, { w n 1 } , and { w n 2 } be sequences generated by (1.10), where {α n } is a sequence in [0,1], r n , λ [a, b] (0, ) and s n , η [c, d] (0, 2β), for every n and suppose the following conditions hold:

(i) lim n δ n = δ ( 0 , 1 ) ,

(ii) 0 ≤ κα n < 1, n ≥ 1,

(iii) n = 1 α 1 n + 1 , j - α 1 n , j < , n = 1 α 3 n + 1 , j - α 3 n , j < , for all j {1,2,3,...,N}.

(iv) There exists λ1, λ2 such that
Φ 1 ( w 1 1 , T r 1 ( x 1 ) , T r 2 ( x 2 ) ) + Φ 1 ( w 2 1 , T r 2 ( x 2 ) , T r 1 ( x 1 ) ) - λ 1 T r 1 ( x 1 ) - T r 2 ( x 2 ) 2 a n d Φ 2 ( w 1 2 , T s 1 ( x 1 ) , T s 2 ( x 2 ) ) + Φ 2 ( w 2 2 , T s 2 ( x 2 ) , T s 1 ( x 1 ) ) - λ 2 T s 1 ( x 1 ) - T s 2 ( x 2 ) 2
(3.1)
for all ( r 1 , r 2 ) Θ × Θ , ( s 1 , s 2 ) Ξ × Ξ , w i 1 T ( x i ) and w i 2 D ( x i ) , for i = 1,2 where Θ = {r n : n ≥ 1} and Ξ = {s n : n ≥ 1}. Then {x n } converges strongly to P F x 1 which is a solution of (3.2):
A x * , x - x * 0 , B x * , x - x * 0 .
(3.2)
Proof. From (3.1) for every r Θ, we have
Φ 1 ( w 1 1 , T r ( x 1 ) , T r ( x 2 ) ) + Φ 1 ( w 2 1 , T r ( x 2 ) , T r ( x 1 ) ) - λ 1 T r ( x 1 ) - T r ( x 2 ) 2 0 ,
(3.3)

for all (x1, x2) C × C and w i 1 T ( x i ) , i = 1 , 2 .

Similarly, for every s Ξ, we have
Φ 2 ( w 1 2 , T s ( x 1 ) , T s ( x 2 ) ) + Φ 2 ( w 2 2 , T s ( x 2 ) , T s ( x 1 ) ) - λ 2 T s ( x 1 ) - T s ( x 2 ) 2 0 .
(3.4)

for all (x1, x2) C × C and w i 2 D ( x i ) , i = 1 , 2 . 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
( I - λ A ) x - ( I - λ A ) y 2 = x - y - λ ( A x - A y ) 2 = x - y 2 - 2 λ x - y , A x - A y + λ 2 A x - A y 2 x - y 2 - 2 α λ A x - A y 2 + λ 2 A x - A y 2 = x - y 2 + λ ( λ - 2 α ) A x - A y 2 x - y 2 .
Thus (I - λA) is nonexpansive, so is I - ηB. Since
Φ 1 ( w n 1 , u n , u ) + φ 1 ( u ) - φ 1 ( u n ) + 1 r n u n - x n , u - u n 0 , u C ,
and Theorem 2.10, we have u n = T r n x n . Since
Φ 2 ( w n 2 , v n , v ) + φ 2 ( v ) - φ 2 ( v n ) + 1 s n v n - x n , v - v n 0 , v C ,
and Theorem 2.10, we have v n = T s n x n . Let z F , again by Theorem 2.10, we have z = T r n z = T s n z = P C ( I - λ A ) z = P C ( I - η B ) z . From nonexpansiveness of { T r n } , { T s n } , { I - λ A } , and {I - ηB}, we have
z n - z = δ n ( P C ( I - λ A ) u n - z ) + ( 1 - δ n ) ( P C ( I - η B ) v n - z ) δ n P C ( I - λ A ) u n - z + ( 1 - δ n ) P C ( I - η B ) v n - z δ n T r n x n - z + ( 1 - δ n ) T s n x n - z x n - z .
(3.5)
By (3.5), we have
y n - z = α n ( z n - z ) + ( 1 - α n ) ( S n z n - z ) α n z n - z + ( 1 - α n ) S n z n - z z n - z x n - z .
(3.6)
Next, we show that C n is closed and convex for every n . It is obvious that C n is closed. In fact, we know that, for z C n ,
y n - z x n - z is equivalent to y n - x n 2 + 2 y n - x n , x n - z 0 .
So, we have that z1, z2 C n and t (0,1), it follows that
y n - x n 2 + 2 y n - x n , x n - ( t z 1 + ( 1 - t ) z 2 ) = t ( 2 y n - x n , x n - z 1 + y n - x n 2 ) + ( 1 - t ) ( 2 y n - x n , x n - z 2 + y n - x n 2 ) 0 ,
then, we have C n is convex. By Theorem 2.10 and Lemma 2.3, we conclude that F is closed and convex. This implies that P F is well defined. Next, we show that F C n for every n . Putting q F , by (3.6), it is easy to see that q C n , then we have F C n for all n . Since x n = P C n x 1 , for every w C n , we have
x n - x 1 w - x 1 , n .
(3.7)
In particular, we have
x n - x 1 P F x 1 - x 1 .
(3.8)
Since C is bounded, we have {x n } is bounded, so are {u n }, {v n }, {z n }, and {y n }. Since x n + 1 = P C n + 1 x 1 C n + 1 C n and x n = P C n x 1 , we have
0 x 1 - x n , x n - x n + 1 = x 1 - x n , x n - x 1 + x 1 - x n + 1 - x n - x 1 2 + x n - x 1 x 1 - x n + 1 ,
it implies that
x n - x 1 x n + 1 - x 1 .
Hence, we have limn→∞x n - x1 exists. Since
x n - x n + 1 2 = x n - x 1 + x 1 - x n + 1 2 = x n - x 1 2 + 2 x n - x 1 , x 1 - x n + 1 + x 1 - x n + 1 2 = x n - x 1 2 + 2 x n - x 1 , x 1 - x n + x n - x n + 1 + x 1 - x n + 1 2 = x n - x 1 2 - 2 x n - x 1 2 + 2 x n - x 1 , x n - x n + 1 + x 1 - x n + 1 2 x 1 - x n + 1 2 - x n - x 1 2 ,
(3.9)
it implies that
lim n x n - x n + 1 = 0 .
(3.10)
Since x n + 1 = P C n + 1 x 1 C n + 1 , we have
y n - x n + 1 x n - x n + 1 ,
by (3.10), we have
lim n y n - x n + 1 = 0 .
(3.11)
Since
y n - x n y n - x n + 1 + x n + 1 - x n ,
by (3.10) and (3.11), we have
lim n y n - x n = 0 .
(3.12)
Next, we show that
lim n z n - S n z n = 0 .
(3.13)
By definition of y n , we have
y n - z n = ( 1 - α n ) ( S n z n - z n ) .
(3.14)
Claim that
lim n z n - x n = 0 .
(3.15)
Putting M n = P C (I - λA)u n and N n = P C (I - ηB)v n , we have
z n - x n δ n M n - x n + ( 1 - δ n ) N n - x n .
(3.16)
Let z F . Since T r n is firmly nonexpansive mapping and T r n x n = u n , we have
z - u n 2 = T r n z - T r n x n 2 T r n z - T r n x n , z - x n = 1 2 ( u n - z 2 + x n - z 2 - u n - x n 2 ) .
Hence
u n - z 2 x n - z 2 - u n - x n 2 .
(3.17)
Since T r n is firmly nonexpansive mapping and T s n x n = v n , by using the same method as (3.17), we have
v n - z 2 x n - z 2 - v n - x n 2 .
(3.18)
By nonexpansiveness of S n and (3.17), (3.18), we have
y n - z 2 z n - z 2 δ n u n - z 2 + ( 1 - δ n ) v n - z 2 δ n ( x n - z 2 - u n - x n 2 ) + ( 1 - δ n ) ( x n - z 2 - v n - x n 2 ) = x n - z 2 - δ n u n - x n 2 - ( 1 - δ n ) v n - x n 2 ,
it implies that
δ n u n - x n 2 x n - z 2 - y n - z 2 - ( 1 - δ n ) v n - x n 2 x n - z 2 - y n - z 2 ( x n - z + y n - z ) x n - y n ,
by (3.12) and condition (i), we have
lim n u n - x n = 0 .
(3.19)
By using the same method as (3.19), we have
lim n v n - x n = 0 .
(3.20)
Since
y n - z 2 α n z n - z 2 + ( 1 - α n ) z n - z 2 α n x n - z 2 + ( 1 - α n ) z n - z 2 α n x n - z 2 + ( 1 - α n ) δ n M n - z 2 + ( 1 - δ ) N n - z 2
(3.21)
Claim that
lim n A u n - A z = lim n B v n - B z = 0 .
By nonexpansiveness of P C , we have
y n z 2 z n z 2 δ n P C ( I λ A ) u n P C ( I λ A ) z 2 + ( 1 δ n ) P C ( I η B ) v n P C ( I η B ) z 2 δ n ( I λ A ) u n ( I λ A ) z 2 + ( 1 δ n ) ( I η B ) v n ( I η B ) z 2 δ n ( u n λ A u n ( z λ A z ) 2 + ( 1 δ n ) ( v n η B v n ( z η B z ) 2 = δ n ( u n z ) λ ( A u n A z ) 2 + ( 1 δ n ) ( v n z ) η ( B v n B z ) 2 = δ n ( u n z 2 + λ 2 A u n A z 2 2 λ u n z , A u n A z ) + ( 1 δ n ) ( v n z 2 + η 2 B v n B z 2 2 η v n z , B v n B z ) δ n ( u n z 2 + λ 2 A u n A z 2 2 λ α A u n A z 2 ) + ( 1 δ n ) ( v n z 2 + η 2 B v n B z 2 2 η β B v n B z 2 ) δ n ( x n z 2 + λ ( λ 2 α ) A u n A z 2 ) + ( 1 δ n ) ( x n z 2 + η ( η 2 β ) B v n B z 2 ) = x n z 2 δ n λ ( 2 α λ ) A u n A z 2 ( 1 δ n ) η ( 2 β η ) B v n B z 2 ,
it follows that
δ n λ ( 2 α - λ ) A u n - A z 2 x n - z 2 - y n - z 2 - ( 1 - δ n ) η ( 2 β - η ) B v n - B z 2 ( x n - z + y n - z ) y n - x n ,
(3.22)
by conditions (i), (ii), λ (0, 2α) and (3.12), it implies that
lim n A u n - A z = 0 .
(3.23)
By using the same method as (3.23), we have
lim n B v n - B z = 0 .
(3.24)
By nonexpansiveness of T r n , we have
M n z 2 = P C ( u n λ A u n ) P C ( z λ A z ) 2 ( u n λ A u n ) ( z λ A z ) , M n z = 1 2 ( ( u n λ A u n ) ( z λ A z ) 2 + M n z 2 ( u n λ A u n ) ( z λ A z ) ( M n z ) 2 ) 1 2 ( u n z 2 + M n z 2 ( u n M n ) λ ( A u n A z ) 2 ) = 1 2 ( T r n x n T r n z 2 + M n z 2 u n M n 2 + 2 λ u n M n , A u n A z λ 2 A u n A z 2 ) 1 2 ( x n z 2 + M n z 2 u n M n 2 + 2 λ u n M n , A u n A z λ 2 A u n A z 2 ) .
Hence, we have
M n - z 2 x n - z 2 - u n - M n 2 + 2 λ u n - M n , A u n - A z - λ 2 A u n - A z 2 .
(3.25)
By using the same method as (3.25), we have
N n - z 2 x n - z 2 - v n - N n 2 + 2 η v n - N n , B v n - B z - η 2 B v n - B z 2 .
(3.26)
Substitute (3.25) and (3.26) in (3.21), we have
y n z 2 α n x n z 2 + ( 1 α n ) ( δ n M n z 2 + ( 1 δ n ) N n z 2 ) α n x n z 2 + ( 1 α n ) ( δ n ( x n z 2 u n M n 2 + 2 λ u n M n , A u n A z λ 2 A u n A z 2 ) + ( 1 + δ n ) ( x n z 2 v n N n 2 + 2 η v n N n , B v n B z η 2 B v n B z 2 ) ) α n x n z 2 + ( 1 α n ) ( δ n x n z 2 δ n u n M n 2 + 2 λ δ n u n M n , A u n A z + ( 1 δ n ) x n z 2 ( 1 δ n ) v n N n 2 + 2 η ( 1 δ n ) v n N n , B v n B z ) = α n x n z 2 + ( 1 α n ) ( x n z 2 δ n u n M n 2 + 2 λ δ n u n M n , A u n A z ( 1 δ n ) v n N n 2 + 2 η ( 1 δ n ) v n N n , B v n B z ) = α n x n z 2 + ( 1 α n ) x n z 2 ( 1 α n ) δ n u n M n 2 + 2 ( 1 α n ) λ δ n u n M n , A u n A z ( 1 δ n ) ( 1 α n ) v n N n 2 + 2 η ( 1 δ n ) ( 1 α n ) v n N n , B v n B z = x n z 2 ( 1 α n ) δ n u n M n 2 + 2 ( 1 α n ) λ δ n u n M n , A u n A z ( 1 δ n ) ( 1 α n ) v n N n 2 + 2 η ( 1 δ n ) ( 1 α n ) v n N n , B v n B z ,
it implies that
( 1