Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces with Applications

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1 Applied Mathematical Sciences, Vol. 7, 2013, no. 3, Iterative Method for a Finite Family of Nonexpansive Mappings in Hilbert Spaces with Applications S. Imnang Department of Mathematics, Faculty of Science Thaksin University Phatthalung Campus, Phatthalung, 93110, Thailand Centre of Excellence in Mathematics, CHE Si Ayutthaya Road, Bangkok 10400, Thailand suwicha.n@hotmail.com Abstract In this paper, we introduce a new iterative method for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove that the studied iterative method converges strongly to a common element of these three sets. Consequently, we apply our main result to the problem of approximating a zero of a finite family of maximal monotone mappings in Hilbert spaces. The theorems presented in this paper, improve and extend the corresponding results of Takahashi and Toyoda [18] and many others. Mathematics Subject Classification: 47H10, 49J40, 47H05, 47H09, 46B20 Keywords: Nonexpansive mapping; A general system of variational inequalities; Mixed equilibrium problem; Demi-closedness principle 1 Introduction Let H be a real Hilbert space with inner product.,. and C be a nonempty closed convex subset of H. A mapping T : C C is said to be nonexpansive mapping if Tx Ty x y for all x, y C. The fixed point set of T is denoted by F (T ):={x C : Tx = x}. For a given nonlinear operator A : C H, we consider the following variational inequality problem of finding x C such that Ax,x x 0, x C. (1.1)

2 104 S. Imnang The set of solutions of the variational inequality (1.1) is denoted by VI(C, A). Variational inequality theory has emerged as an important tool in studying a wide class of obstacle, unilateral, free, moving, equilibrium problems arising in several branches of pure and applied sciences in a unified and general framework. The variational inequality problem has been extensively studied in the literature, see, for example, Piri [13], Qin et al. [15], Shehu [16], Wangkeeree and Preechasilp [21], Yao et al. [23], Yao et al. [25] and the references therein. For solving the variational inequality problem in the finite-dimensional Euclidean space R n under the assumption that a set C R n is closed and convex, a mapping A of C into R n is monotone and k-lipschitz-continuous and VI(C, A) is nonempty, Korpelevich [9] introduced the following so-called extragradient method: x 0 = x C, y n = P C (x n λax n ), x n+1 = P C (x n λay n ), for every n = 0, 1, 2,..., where λ (0, 1/k) and P C is the projection of R n onto C. He showed that the sequences {x n } and {y n } generated by this iterative process converge to the same point z VI(C, A). Later on, the idea of Korpelevich was generalized and extended by many authors, see e.g. [4, 8, 13, 15, 16, 21, 23] for finding a common element of the set of fixed points and the set of solutions of the variational inequality. Let A, B : C H be two mappings. In 2008, Ceng et al. [1] considered the following problem of finding (x,y ) C C such that { λay + x y,x x 0, x C, μbx + y x,x y 0, x C, (1.2) which is called a general system of variational inequalities (in short, GSVI), where λ and μ are positive numbers. In particular, if A = B, problem GSVI (1.2) reduces to find (x,y ) C C such that { λay + x y,x x 0, x C, μax + y x,x y 0, x C, (1.3) which is defined by Verma [19], and is called the new system of variational inequalities. The set of solutions of GSVI (1.3) is denoted by Φ. Further, if we add up the requirement that x = y, then problem (1.3) reduces to the classical variational inequality VI(C, A). Ceng et al. [1] introduced and studied a relaxed extragradient method for finding a common element of the set of solutions of problem GSVI (1.2) for the α and β-inverse-strongly monotone

3 Iterative method for a finite family mappings and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Let x 1 = v C and {x n }, {y n } are given by { yn = P C (x n μbx n ), x n+1 = a n v + b n x n +(1 a n b n )SP C (y n λay n ), n 1, where λ (0, 2α),μ (0, 2β) and {a n }, {b n } [0, 1]. Then, they proved that the sequence {x n } converges strongly to a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of problem GSVI (1.2) under some control conditions. Some related works, we refer to see [2, 5, 8, 10, 20, 24]. Recently, in 2012, Ceng et al. [2] considered an iterative method for the system of GSVI (1.2) and obtained a strong convergence theorem for the two different systems of GSVI (1.2) and the set of fixed points of a strict pseudocontraction mapping in a real Hilbert space. Let ϕ : C R {+ } be a proper extended real-valued function and F be a bifunction from C C to R, where R is the set of real numbers. Ceng and Yao [3] considered the following mixed equilibrium problem (in short, MEP): Find x C such that F (x, y)+ϕ(y) ϕ(x), y C. (1.4) The set of solution of MEP (1.4) is denoted by MEP(F, ϕ). It is easy to see that x is a solution of MEP (1.4) implies that x domϕ = {x C ϕ(x) < + }. If ϕ = 0, then the MEP (1.4) becomes the following equilibrium problem: Find x C such that F (x, y) 0, y C. (1.5) The set of solution of (1.5) is denoted by EP(F ). If F = 0, then the MEP (1.4) reduces to the convex minimization problem: Find x C such that ϕ(y) ϕ(x), y C. If ϕ = 0 and F (x, y) = Ax, y x for all x, y C, where A is a mapping from C into H, then MEP (1.4) reduces to the classical variational inequality and EP(F )=VI(C, A). For solving problem MEP (1.4), Ceng and Yao [3] introduced a hybrid iterative scheme for finding a common element of the set MEP(F, ϕ) and the set of common fixed points of finite many nonexpansive mappings in a Hilbert space. Some related works, we refer to see [8, 16, 20, 23]. Motivated and inspired by the works in the literature, in this paper, we introduce a general iterative method for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a

4 106 S. Imnang mixed equilibrium problem and the set of common fixed points of a finite family of nonexpansive mappings in a real Hilbert space. Furthermore, we prove that the studied iterative method converges strongly to a common element of these three sets. Consequently, we apply our main result to the problem of approximating a zero of a finite family of maximal monotone mappings in Hilbert spaces. The theorems presented in this paper, improve and extend the corresponding results of Takahashi and Toyoda [18] and many others. 2 Preliminaries In this section, we recall the well known results and give some useful lemmas that will be used in the next section. Let C be a nonempty closed convex subset of a real Hilbert space H. For every point x H, there exists a unique nearest point in C, denoted by P C x, such that x P C x x y, y C. P C is called the metric projection of H onto C. It is well known that P C is a nonexpansive mapping of H onto C and satisfies x y, P C x P C y P C x P C y 2, x, y H. (2.1) Obviously, this immediately implies that (x y) (P C x P C y) 2 x y 2 P C x P C y 2, x, y H. (2.2) Recall that, P C x is characterized by the following properties: P C x C and x P C x, y P C x 0, x y 2 x P C x 2 + P C x y 2, (2.3) for all x H and y C; see Goebel and Kirk [6] for more details. For solving the mixed equilibrium problem, let us give the following assumptions for the bifunction F, ϕ and the set C: (A1) F (x, x) = 0 for all x C; (A2) F is monotone, i.e. F (x, y)+f (y, x) 0 for all x, y C; (A3) For each y C, x F (x, y) is weakly upper semicontinuous; (A4) For each x C, y F (x, y) is convex; (A5) For each x C, y F (x, y) is lower semicontinuous;

5 Iterative method for a finite family (B1) For each x H and r>0, there exist a bounded subset D x C and y x C such that for any z C \ D x, F (z, y x )+ϕ(y x )+ 1 r y x z, z x <ϕ(z). (B2) C is a bounded set. In the sequel we shall need to use the following lemmas. Lemma 2.1. ([12]) Let C be a nonempty closed convex subset of H. LetF be a bifunction from C C to R satisfying (A1)-(A5) and let ϕ : C R {+ } be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r>0 and x H, define a mapping T r : H C as follows. T r (x) = {z C : F (z, y)+ϕ(y)+ 1r } y z, z x ϕ(z), y C for all x H. Then the following conclusions hold: (1) For each x H, T r (x) Ø; (2) T r is single-valued; (3) T r is firmly nonexpansive, i.e. for any x, y H, T r (x) T r (y) 2 T r x T r y, x y ; (4) F (T r )=MEP(F, ϕ); (5) MEP(F, ϕ) is closed and convex. Lemma 2.2. ([22]) Assume {a n } is a sequence of nonnegative real numbers such that a n+1 (1 γ n )a n + δ n, where {γ n } is a sequence in (0, 1) and {δ n } is a sequence such that (i) n=1 γ n = ; (ii) lim sup n δ n /γ n 0or n=1 δ n <. Then lim n a n =0. Lemma 2.3. ([11]) Let (H,.,. ) be an inner product space. x, y, z H and α, β, γ [0, 1] with α + β + γ =1, we have αx + βy + γz 2 = α x 2 + β y 2 + γ z 2 αβ x y 2 αγ x z 2 βγ y z 2. Then, for all Lemma 2.4. ([17]) Let {x n } and {y n } be bounded sequences in a Banach space X and let {b n } be a sequence in [0, 1] with 0 < lim inf n b n lim sup n b n < 1. Suppose x n+1 =(1 b n )y n +b n x n for all integers n 1 and lim sup n ( y n+1 y n x n+1 x n ) 0. Then, lim n y n x n =0.

6 108 S. Imnang Lemma 2.5. ([6]) Demi-closedness principle. Assume that T is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. If T has a fixed point, then I T is demi-closed: that is, whenever {x n } is a sequence in C converging weakly to some x C (for short, x n x C), and the sequence {(I T )x n } converges strongly to some y (for short, (I T )x n y), it follows that (I T )x = y. Here I is the identity operator of H. The following lemma is an immediate consequence of an inner product. Lemma 2.6. In a real Hilbert space H, there holds the inequality x + y 2 x 2 +2 y, x + y, x, y H. In 2009, Kangtunyakarn and Suantai [7] introduced a new mapping called the S-mapping. Let {T i } N i=1 be a finite family of nonexpansive mappings of C into itself. For each n N, and j =1, 2,...,N, let α (n) j =(α n,j 1,α n,j 2,α n,j 3 )be such that α n,j 1,α n,j 2,α n,j 3 [0, 1] with α n,j 1 + α n,j 2 + α n,j 3 = 1. They defined the new mapping S n : C C as follows: U n,0 = I, U n,1 = α n,1 1 T 1 U n,0 + α n,1 2 U n,0 + α n,1 3 I, U n,2 = α n,2 1 T 2U n,1 + α n,2 2 U n,1 + α n,2 3 I, U n,3 = α n,3 1 T 3 U n,2 + α n,3 2 U n,2 + α n,3 3 I,. U n,n 1 = α n,n 1 1 T N 1 U n,n 2 + α n,n 1 2 U n,n 2 + α n,n 1 3 I, S n = U n,n = α n,n 1 T N U n,n 1 + α n,n 2 U n,n 1 + α n,n 3 I. The mapping S n is called the S-mapping generated by T 1,T 2,...,T N and α (n) α (n) 2,...,α (n) N. Nonexpansivity of each T i ensures the nonexpansivity of S n. Lemma 2.7. ([7]) Let C be a nonempty closed convex subset of a strictly convex Banach space X. Let{T i } N i=1 be a finite family of nonexpansive mappings of C into itself with N i=1 F (T i) Øand let α j =(α j 1,αj 2,αj 3 ), j =1, 2,...,N, where α j 1,αj 2,αj 3 [0, 1], αj 1 +αj 2 +αj 3 =1, αj 1 (0, 1) for all j =1, 2,...,N 1, α1 N (0, 1] and αj 2,αj 3 [0, 1) for all j =1, 2,...,N.LetS be the S-mapping generated by T 1,T 2,...,T N and α 1,α 2,...,α N. Then F (S) = N i=1 F (T i). Lemma 2.8. ([7]) Let C be a nonempty closed convex subset of a Banach space X. Let {T i } N i=1 be a finite family of nonexpansive mappings of C into itself and for all n N and all j {1, 2,...,N}, let α (n) j =(α n,j 1,α n,j 2,α n,j 3 ), α j = 1,

7 Iterative method for a finite family (α1,α j 2,α j 3) j where α n,j 1,α n,j 2,α n,j 3 [0, 1], α1,α j 2,α j j 3 [0, 1], α n,j 1 +α n,j 2 +α n,j 3 =1 and α j 1 + α j 2 + α j 3 =1. Suppose α n,j i α j i as n for all i {1, 3} and all j = 1, 2, 3,...,N. Let S and S n be the S-mappings generated by T 1,T 2,...,T N and α 1,α 2,...,α N and T 1,T 2,...,T N and α (n) 1,α (n) 2,...,α (n) N, respectively. Then lim n S n x Sx =0for every x C. Lemma 2.9. ([1]) For given x,y C, (x,y ) is a solution of problem (1.2) if and only if x is a fixed of the mapping G : C C defined by G(x) =P C [P C (x μbx) λap C (x μbx)], x C, where y = P C (x μbx ). Throughout this paper, the set of fixed points of the mapping G is denoted by GV I(C, A, B). Lemma ([21]) Let A : C H be a L A -Lipschitzian and relaxed (c, d)- cocoercive mapping and B : C H be a L B -Lipschitzian and relaxed (c,d )- cocoercive mapping. Let the mapping G : C C be defined by G(x) =P C [P C (x μbx) λap C (x μbx)], x C. If 0 <λ< 2(d cl2 A ) L 2 A and 0 <μ< 2(d c L 2 B ). Then G is nonexpansive. L 2 B 3 Main Results We are now in a position to state and prove our main results. Lemma 3.1. Let B : C H be a L-Lipschitzian and relaxed (c,d )-cocoercive mapping. If 0 <μ 2(d c L 2 ), then I μb is nonexpansive. L 2 Proof. For any x, y C, we have (I μb)x (I μb)y 2 = (x y) μ(bx By) 2 = x y 2 + μ 2 Bx By 2 2μ x y, Bx By x y 2 + μ 2 Bx By 2 2μ[ c Bx By 2 + d x y 2 ] = x y 2 + μ 2 Bx By 2 +2μc Bx By 2 2μd x y 2 x y 2 + μ 2 L 2 x y 2 +2μc L 2 x y 2 2μd x y 2 =(1+2μc L 2 2μd + μ 2 L 2 ) x y 2 x y 2,

8 110 S. Imnang hence I μb is nonexpansive. Theorem 3.2. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C C to R satisfying (A1)-(A5) and ϕ : C R {+ } be a proper lower semicontinuous and convex function. Let A : C H be a L A -Lipschitzian and relaxed (c, d)-cocoercive mapping and B : C H be a L B -Lipschitzian and relaxed (c,d )-cocoercive mapping. Let {T i } N i=1 be a finite family of nonexpansive self-mappings of C such that Ω= N i=1 F (T i) GV I(C, A, B) MEP(F, ϕ) Ø. For all j {1, 2,...,N}, let α (n) j =(α n,j 1,α n,j 2,α n,j 3 ) be such that α n,j 1,α n,j 2,α n,j 3 [0, 1], α n,j 1 + α n,j } [η N, 1] with α n,j 3 =1, {α n,j 1 }N 1 j=1 [η 1,θ 1 ] with 0 <η 1 θ 1 < 1, {α n,n 0 <η N 1 and {α n,j 2 } N j=1, {αn,j 3 } N j=1 [0,θ 2] with 0 θ 2 < 1. Let S n be the S-mappings generated by T 1,T 2,...,T N and α (n) 1,α (n) 2,...,α (n) N. Assume that either (B1) or (B2) holds and that v is an arbitrary point in C. Let x 1 C and {x n }, {u n }, {y n } be the sequences defined by F (u n,y)+ϕ(y) ϕ(u n )+ 1 r n y u n,u n x n 0, y C, y n = P C (u n μbu n ), x n+1 = a n v + b n x n +(1 a n b n )S n P C (y n λay n ), n 1, where 0 <λ< 2(d cl2 A ) L 2 A conditions hold: and 0 <μ< 2(d c L 2 B ). Suppose that the following L 2 B (C1) lim n a n =0and n=1 a n = ; (C2) 0 < lim inf n b n lim sup n b n < 1; (C3) lim inf n r n > 0 and lim n r n+1 r n =0; (C4) lim n α n+1,i 1 α n,i 1 =0for all i {1, 2,...,N} and lim n α n+1,j 3 α n,j 3 =0for all j {2, 3,...,N}. Then {x n } converges strongly to x = P Ω v and (x, y) is a solution of problem (1.2), where y = P C (x μbx). Proof. Let x Ω and {T rn } be a sequence of mappings defined as in Lemma 2.1. It follows from Lemma 2.9 that x = P C [P C (x μbx ) λap C (x μbx )]. Put y = P C (x μbx ) and t n = P C (y n λay n ), then x = P C (y λay ) and x n+1 = a n v + b n x n +(1 a n b n )S n t n.

9 Iterative method for a finite family From Lemma 3.1 and nonexpansiveness of P C and T rn, we have t n x 2 = P C (y n λay n ) P C (y λay ) 2 which, implies that y n y 2 = P C (u n μbu n ) P C (x μbx ) 2 u n x 2 = T rn x n T rn x 2 x n x 2, (3.1) x n+1 x = a n v + b n x n +(1 a n b n )S n t n x a n v x + b n x n x +(1 a n b n ) t n x a n v x + b n x n x +(1 a n b n ) x n x max{ v x, x 1 x }. Thus, {x n } is bounded. Consequently, the sequences {u n }, {y n }, {t n }, {Ay n }, {Bu n } and {S n t n } are also bounded. Also, observe that t n+1 t n = P C (y n+1 λay n+1 ) P C (y n λay n ) y n+1 y n = P C (u n+1 μbu n+1 ) P C (u n μbu n ) u n+1 u n. (3.2) On the other hand, from u n = T rn x n domϕ and u n+1 = T rn+1 x n+1 domϕ, we have F (u n,y)+ϕ(y) ϕ(u n )+ 1 r n y u n,u n x n 0, y C, (3.3) and F (u n+1,y)+ϕ(y) ϕ(u n+1 )+ 1 r n+1 y u n+1,u n+1 x n+1 0, y C. (3.4) Putting y = u n+1 in (3.3) and y = u n in (3.4), we have F (u n,u n+1 )+ϕ(u n+1 ) ϕ(u n )+ 1 r n u n+1 u n,u n x n 0, and F (u n+1,u n )+ϕ(u n ) ϕ(u n+1 )+ 1 r n+1 u n u n+1,u n+1 x n+1 0.

10 112 S. Imnang From the monotonicity of F, we obtain that u n+1 u n, u n x n u n+1 x n+1 0, r n r n+1 and hence u n+1 u n,u n u n+1 + u n+1 x n r n (u n+1 x n+1 ) 0. r n+1 Then, we have and hence u n+1 u n 2 u n+1 u n,x n+1 x n +(1 { u n+1 u n x n+1 x n + 1 r n )(u n+1 x n+1 ) r n+1 r n r n+1 u n+1 x n+1 u n+1 u n x n+1 x n + 1 r n+1 r n+1 r n u n+1 x n+1. (3.5) }, It follows from (3.2) and (3.5) that t n+1 t n x n+1 x n + 1 r n+1 r n+1 r n u n+1 x n+1. (3.6) Let x n+1 = b n x n +(1 b n )z n. Then, we obtain z n+1 z n = x n+2 b n+1 x n+1 1 b n+1 x n+1 b n x n 1 b n = a n+1v +(1 a n+1 b n+1 )S n+1 t n+1 a nv +(1 a n b n )S n t n 1 b n+1 1 b n = a n+1 (v S n+1 t n+1 )+ a n (S n t n v)+s n+1 t n+1 S n t n. 1 b n+1 1 b n (3.7) Next, we estimate S n+1 t n+1 S n t n.

11 Iterative method for a finite family For each k {2, 3,...,N}, we have U n+1,k t n U n,k t n = α n+1,k 1 T k U n+1,k 1 t n + α n+1,k 2 U n+1,k 1 t n + α n+1,k 3 t n α n,k 1 T k U n,k 1 t n α n,k 2 U n,k 1 t n α n,k 3 t n = α n+1,k 1 (T k U n+1,k 1 t n T k U n,k 1 t n ) +(α n+1,k 1 α n,k 1 )T k U n,k 1 t n +(α n+1,k 3 α n,k 3 )t n + α n+1,k 2 (U n+1,k 1 t n U n,k 1 t n )+(α n+1,k 2 α n,k 2 )U n,k 1 t n α n+1,k 1 U n+1,k 1 t n U n,k 1 t n + α n+1,k 1 α n,k 1 T k U n,k 1 t n + α n+1,k 3 α n,k 3 t n + α n+1,k 2 U n+1,k 1 t n U n,k 1 t n + α n+1,k 2 α n,k 2 U n,k 1 t n =(α n+1,k 1 + α n+1,k 2 ) U n+1,k 1 t n U n,k 1 t n + α n+1,k 1 α n,k 1 T k U n,k 1 t n + α n+1,k 3 α n,k 3 t n + α n+1,k 2 α n,k 2 U n,k 1 t n U n+1,k 1 t n U n,k 1 t n + α n+1,k 1 α n,k 1 T k U n,k 1 t n + α n+1,k 3 α n,k 3 t n + (α n,k 1 α n+1,k 1 )+(α n,k 3 α n+1,k 3 ) U n,k 1 t n U n+1,k 1 t n U n,k 1 t n + α n+1,k 1 α n,k 1 T k U n,k 1 t n + α n+1,k 3 α n,k 3 t n + α n,k 1 α n+1,k 1 U n,k 1 t n + α n,k 3 α n+1,k 3 U n,k 1 t n = U n+1,k 1 t n U n,k 1 t n + α n+1,k 1 α n,k 1 ( T k U n,k 1 t n + U n,k 1 t n ) + α n+1,k 3 α n,k 3 ( t n + U n,k 1 t n ). (3.8) It follow from (3.8) that S n+1 t n S n t n = U n+1,n t n U n,n t n N U n+1,1 t n U n,1 t n + α n+1,j 1 α n,j 1 ( T ju n,j 1 t n + U n,j 1 t n ) + N j=2 j=2 α n+1,j 3 α n,j 3 ( t n + U n,j 1 t n ) = α n+1,1 1 α n,1 1 T 1 t n t n N + α n+1,j 1 α n,j 1 ( T ju n,j 1 t n + U n,j 1 t n ) + j=2 N j=2 α n+1,j 3 α n,j 3 ( t n + U n,j 1 t n ).

12 114 S. Imnang This together with the condition (C4), we obtain lim S n+1t n S n t n =0. (3.9) n It follows from (3.6) that S n+1 t n+1 S n t n t n+1 t n + S n+1 t n S n t n x n+1 x n + 1 r n+1 r n u n+1 x n+1 r n+1 + S n+1 t n S n t n. (3.10) By (3.7) and (3.10), we have z n+1 z n x n+1 x n a n+1 v S n+1 t n+1 + a n S n t n v 1 b n+1 1 b n + S n+1 t n+1 S n t n x n+1 x n a n+1 v S n+1 t n+1 + a n S n t n v 1 b n+1 1 b n + 1 r n+1 r n u n+1 x n+1 r n+1 + S n+1 t n S n t n. This together with (C1)-(C3) and (3.9), we obtain that lim sup z n+1 z n x n+1 x n 0. n Hence, by Lemma 2.4, we get x n z n 0asn. Consequently, lim n x n+1 x n = lim n (1 b n ) z n x n =0. (3.11) From (C3), (3.2) and (3.5), we also have u n+1 u n 0, t n+1 t n 0 and y n+1 y n 0, as n. Since therefore x n+1 x n = a n (v x n )+(1 a n b n )(S n t n x n ), S n t n x n 0 as n. (3.12) Next, we prove that lim n x n u n = 0. From Lemma 2.1(3), we have u n x 2 = T rn x n T rn x 2 T rn x n T rn x,x n x = u n x,x n x = 1 2{ un x 2 + x n x 2 x n u n 2}.

13 Iterative method for a finite family Hence u n x 2 x n x 2 x n u n 2. (3.13) From Lemma 2.3, (3.1) and (3.13), we have x n+1 x 2 a n v x 2 + b n x n x 2 +(1 a n b n ) t n x 2 a n v x 2 + b n x n x 2 +(1 a n b n ) u n x 2 a n v x 2 + b n x n x 2 +(1 a n b n ) [ x n x 2 x n u n 2] a n v x 2 + x n x 2 (1 a n b n ) x n u n 2. It follows that (1 a n b n ) x n u n 2 a n v x 2 + x n x 2 x n+1 x 2 a n v x 2 +( x n x + x n+1 x ) x n+1 x n. From the conditions (C1), (C2) and (3.11), we obtain Since lim x n u n =0. (3.14) n S n t n u n S n t n x n + x n u n, it follows from (3.12) and (3.14) that lim S nt n u n =0. (3.15) n Next, we show that Ay n Ay 0 and Bu n Bx 0asn. From (3.1), we have x n+1 x 2 a n v x 2 + b n x n x 2 +(1 a n b n ) t n x 2 a n v x 2 + b n x n x 2 +(1 a n b n ) y n y 2 a n v x 2 + b n x n x 2 +(1 a n b n ) (u n μbu n ) (x μbx ) 2

14 116 S. Imnang and a n v x 2 + b n x n x 2 +(1 a n b n ) [ u n x 2 2μ u n x,bu n Bx + μ 2 Bu n Bx 2] a n v x 2 + b n x n x 2 +(1 a n b n ) [ u n x 2 +2μc Bu n Bx 2 2μd u n x 2 + μ 2 Bu n Bx 2] a n v x 2 + b n x n x 2 +(1 a n b n ) [ x n x 2 +(2μc + μ 2 2μd ) Bu L 2 n Bx 2] B a n v x 2 + x n x 2 +(1 a n b n )(2μc + μ 2 2μd ) Bu L 2 n Bx 2, B x n+1 x 2 a n v x 2 + b n x n x 2 +(1 a n b n ) t n x 2 Therefore, we have = a n v x 2 + b n x n x 2 +(1 a n b n ) P C (y n λay n ) P C (y λay ) 2 a n v x 2 + b n x n x 2 +(1 a n b n ) (y n λay n ) (y λay ) 2 a n v x 2 + b n x n x 2 +(1 a n b n ) [ x n x 2 +(2λc + λ 2 2λd ) Ay L 2 n Ay 2] A a n v x 2 + x n x 2 +(1 a n b n )(2λc + λ 2 2λd ) Ay L 2 n Ay 2. A (1 a n b n )(2μc + μ 2 2μd ) Bu L 2 n Bx 2 B a n v x 2 +( x n x + x n+1 x ) x n+1 x n, and (1 a n b n )(2λc + λ 2 2λd L 2 A ) Ay n Ay 2 a n v x 2 +( x n x + x n+1 x ) x n+1 x n.

15 Iterative method for a finite family This together with (3.11), (C1) and (C2), we obtain Ay n Ay 0 and Bu n Bx 0 as n. (3.16) Next, we prove that S n t n t n 0asn. From (2.1) and nonexpansiveness of I μb, we get y n y 2 = P C (u n μbu n ) P C (x μbx ) 2 By (3.1), we obtain Hence, (u n μbu n ) (x μbx ),y n y = 1 [ (un μbu n ) (x μbx ) 2 + y n y 2 2 (u n μbu n ) (x μbx ) (y n y ) 2] 1 [ un x 2 + y n y 2 (u n x ) (y n y ) μ (u n x ) (y n y ),Bu n Bx μ 2 Bu n Bx 2]. y n y 2 u n x 2 (u n x ) (y n y ) 2 +2μ (u n x ) (y n y ),Bu n Bx μ 2 Bu n Bx 2 x n x 2 (u n x ) (y n y ) 2 +2μ (u n x ) (y n y ),Bu n Bx μ 2 Bu n Bx 2. x n+1 x 2 a n v x 2 + b n x n x 2 +(1 a n b n ) y n y 2 which implies that a n v x 2 + b n x n x 2 +(1 a n b n ) [ x n x 2 (u n x ) (y n y ) 2 +2μ (u n x ) (y n y ),Bu n Bx μ 2 Bu n Bx 2] a n v x 2 + x n x 2 (1 a n b n ) (u n x ) (y n y ) 2 +(1 a n b n )2μ (u n x ) (y n y ) Bu n Bx, (1 a n b n ) (u n x ) (y n y ) 2 a n v x 2 +(1 a n b n )2μ (u n x ) (y n y ) Bu n Bx +( x n x + x n+1 x ) x n+1 x n. This together with (C1), (3.11) and (3.16), we obtain (u n x ) (y n y ) 0 as n. (3.17)

16 118 S. Imnang From Lemma 2.6 and (2.2), it follows that (y n t n )+(x y ) 2 = (y n λay n ) (y λay ) [ P C (y n λay n ) P C (y λay ) ] + λ(ay n Ay ) 2 (y n λay n ) (y λay ) [ P C (y n λay n ) P C (y λay ) ] 2 +2λ Ay n Ay, (y n t n )+(x y ) (y n λay n ) (y λay ) 2 P C (y n λay n ) P C (y λay ) 2 +2λ Ay n Ay (y n t n )+(x y ) (y n λay n ) (y λay ) 2 S n P C (y n λay n ) S n P C (y λay ) 2 +2λ Ay n Ay (y n t n )+(x y ) (y n λay n ) (y λay ) (S n t n x ) [ (y n λay n ) (y λay ) + S n t n x ] +2λ Ay n Ay (y n t n )+(x y ) = u n S n t n + x y (u n y n ) λ(ay n Ay ) [ (y n λay n ) (y λay ) + S n t n x ] +2λ Ay n Ay (y n t n )+(x y ). This together with (3.15), (3.17) and (3.16), we obtain (y n t n )+(x y ) 0asn. This together with (3.12), (3.14) and (3.17), we obtain that S n t n t n S n t n x n + x n u n + (u n y n ) (x y ) + (y n t n )+(x y ) 0 as n. (3.18) Next, we show that lim sup v x, x n x 0, n where x = P Ω v. Indeed, since {t n } and {S n t n } are two bounded sequences in C, we can choose a subsequence {t ni } of {t n } such that t ni z C and lim sup v x, S n t n x = lim v x, S ni t ni x. n i Since lim n S n t n t n =0, we obtain that S ni t ni zas i. Next, we show that z Ω. (a) We first show z N i=1 F (T i). We can assume that α n,j 1 α j 1 (0, 1) and α n,n 1 α1 N (0, 1] as n for all j {1, 2,...,N 1} and α n,j 3 α j 3 [0, 1) as n for j =1, 2,...,N. Let S be the S-mappings generated by T 1,T 2,...,T N and α 1,α 2,...,α N where

17 Iterative method for a finite family α j = (α1,α j 2,α j 3), j for j = 1, 2,...,N. From Lemma 2.8, we have S n t n St n 0asn. Since St n t n St n S n t n + S n t n t n, it follows by (3.18) that St n t n 0asn. Since t ni zand St n t n 0, we obtain by Lemma 2.5 and Lemma 2.7 that z F (S) = N i=1 F (T i). (b) Now, we show that z GV I(C, A, B). Since t n x n S n t n t n + S n t n x n, it follows from (3.18) and (3.12) that t n x n 0asn. Furthermore, by Lemma 2.10, we have G : C C is nonexpansive. Then, we have t n G(t n ) = P C (y n λay n ) G(t n ) [ = P C P (un μbu n ) λap (u n μbu n ) ] G(t n ) = G(u n ) G(t n ) u n t n u n x n + x n t n, which implies t n G(t n ) 0asn. Again by Lemma 2.5, we have z GV I(C, A, B). (c) We show that z MEP(F, ϕ). Since t ni zand x n t n 0, we obtain that x ni z. From u n x n 0, we also obtain that u ni z. By using the same argument as that in the proof of [12, Theorem 3.1, pp. 1825], we can show that z MEP(F, ϕ). Therefore there holds z Ω. On the other hand, it follows from (2.3) and S ni t ni zas i that lim sup n Hence, we have v x, x n x = lim sup v x, S n t n x = lim v x, S ni t ni x n i = v x, z x 0. (3.19) x n+1 x 2 = a n v + b n x n +(1 a n b n )S n t n x, x n+1 x = a n v x, x n+1 x + b n x n x, x n+1 x +(1 a n b n ) S n t n x, x n+1 x

18 120 S. Imnang a n v x, x n+1 x b n( x n x 2 + x n+1 x 2 ) (1 a n b n )( t n x 2 + x n+1 x 2 ) a n v x, x n+1 x b n( x n x 2 + x n+1 x 2 ) (1 a n b n )( x n x 2 + x n+1 x 2 ) = a n v x, x n+1 x (1 a n)( x n x 2 + x n+1 x 2 ), which implies that x n+1 x 2 (1 a n ) x n x 2 +2a n v x, x n+1 x. It follows from Lemma 2.2 and (3.19) that {x n } converges strongly to x. This completes the proof. If A = B in Theorem 3.2, then we obtain the following result. Corollary 3.3. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C C to R satisfying (A1)-(A5) and ϕ : C R {+ } be a proper lower semicontinuous and convex function. Let A : C H be a L A -Lipschitzian and relaxed (c, d)-cocoercive mapping. Let {T i } N i=1 be a finite family of nonexpansive self-mappings of C such that Ω= N i=1 F (T i) Φ MEP(F, ϕ) Ø. For all j {1, 2,...,N}, let α (n) j =(α n,j 1,αn,j 2,αn,j 3 ) be such that αn,j 1,αn,j 2,αn,j 3 [0, 1], α n,j 1 +αn,j 2 +αn,j 3 =1, {α n,j 1 } N 1 j=1 [η 1,θ 1 ] with 0 < η 1 θ 1 < 1, {α n,n 1 } [η N, 1] with 0 < η N 1 and {α n,j 2 } N j=1, {α n,j 3 } N j=1 [0,θ 2 ] with 0 θ 2 < 1. Let S n be the S-mappings generated by T 1,T 2,...,T N and α (n) 1,α(n) 2,...,α(n) N. Assume that either (B1) or (B2) holds and that v is an arbitrary point in C. Let x 1 C and {x n }, {u n }, {y n } be the sequences generated by F (u n,y)+ϕ(y) ϕ(u n )+ 1 r n y u n,u n x n 0, y C, y n = P C (u n μau n ), x n+1 = a n v + b n x n +(1 a n b n )S n P C (y n λay n ), n 1, where 0 <λ,μ< 2(d cl2 A ). If the sequences {r L 2 n }, {a n }, {b n } and {α n,j 1 } N j=1 are A as in Theorem 3.2, then {x n } converges strongly to x = P Ω v and (x, y) is a solution of problem (1.3), where y = P C (x μax). If N =1, T 1 = S, ϕ = 0 and α n,1 2, α n,1 3 =0 n N in Theorem 3.2, then we obtain the following result.

19 Iterative method for a finite family Corollary 3.4. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F be a function from C C to R satisfying (A1)-(A5). Let A : C H be a L A -Lipschitzian and relaxed (c, d)-cocoercive mapping and B : C H be a L B -Lipschitzian and relaxed (c,d )-cocoercive mapping. Let S be a nonexpansive self-mappings of C such that Ω=F (S) GV I(C, A, B) EP(F ) Ø. Assume that v is an arbitrary point in C. Letx 1 C and {x n }, {u n }, {y n } be the sequences generated by F (u n,y)+ 1 r n y u n,u n x n 0, y C, y n = P C (u n μbu n ), x n+1 = a n v + b n x n +(1 a n b n )SP C (y n λay n ), n 1. If λ, μ and the sequences {r n }, {a n }, {b n } are as in Theorem 3.2, then {x n } converges strongly to x P Ω v and (x, y) is a solution of problem (1.2), where y = P C (x μbx). 4 Applications In this section, we will apply Theorem 3.2 to obtain two strong convergence theorems in a real Hilbert space. Theorem 4.1. Let H be a real Hilbert space. Let F be a function from H H to R satisfying (A1)-(A5) and ϕ : H R {+ } be a proper lower semicontinuous and convex function. Let A : H H be a L A -Lipschitzian and relaxed (c, d)-cocoercive mapping. Let {T i } N i=1 be a finite family of nonexpansive selfmappings of H such that Ω= N i=1 F (T i) A 1 0 MEP(F, ϕ) Ø. For all j {1, 2,...,N}, let α (n) j =(α n,j 1,α n,j 2,α n,j 3 ) be such that α n,j 1,α n,j 2,α n,j 3 [0, 1], α n,j 1 + α n,j 2 + α n,j 3 = 1, {α n,j 1 } N 1 j=1 [η 1,θ 1 ] with 0 < η 1 θ 1 < 1, {α n,n 1 } [η N, 1] with 0 < η N 1 and {α n,j 2 } N j=1, {αn,j 3 } N j=1 [0,θ 2] with 0 θ 2 < 1. Let S n be the S-mappings generated by T 1,T 2,...,T N and α (n) 1,α (n) 2,...,α (n). Assume that either (B1) or (B2) holds and that v is an N arbitrary point in H. Letx 1 H and {x n }, {u n }, {y n } be the sequences defined by F (u n,y)+ϕ(y) ϕ(u n )+ 1 r n y u n,u n x n 0, y H, y n = u n λau n, x n+1 = a n v + b n x n +(1 a n b n )S n (y n λay n ), n 1, where 0 <λ< 2(d cl2 A ). Suppose that the following conditions hold: L 2 A (C1) lim n a n =0and n=1 a n = ; (C2) 0 < lim inf n b n lim sup n b n < 1;

20 122 S. Imnang (C3) lim inf n r n > 0 and lim n r n+1 r n =0; (C4) lim n α n+1,i 1 α n,i 1 =0for all i {1, 2,...,N} and lim n α n+1,j 3 α n,j 3 =0for all j {2, 3,...,N}. Then {x n } converges strongly to x = P Ω v. Proof. Put λ = μ, C = H, B = A and P H = I, we have A 1 0=VI(H, A). In this case, there holds the following: problem (1.2) problem (1.3) VI(A, H). Indeed, it is sufficient to show that problem (1.3) VI(A, H). Suppose that there is (x,y ) H H such that { λay + x y,x x 0, x H, λax + y x,x y 0, x H. Then we have { x = P H (y λay ), y = P H (x λax ), this is { x = y λay, y = x λax. (4.1) We claim that x = y. Otherwise, from (4.1) it follows that Ax 0,Ay 0 and Ax + Ay = 0. Again from (4.1), we obtain x y 2 = (y x ) λ(ay Ax ) 2 = y x 2 + λ 2 Ay Ax 2 2λ y x,ay Ax y x 2 + λ 2 Ay Ax 2 2λ[ c Ay Ax 2 + d y x 2 ] = y x 2 + λ 2 Ay Ax 2 +2λc Ay Ax 2 2λd y x 2 y x 2 + λ 2 L 2 A y x 2 +2λcL 2 A y x 2 2λd y x 2 =(1+2λcL 2 A 2λd + λ2 L 2 A ) y x 2 y x 2, which hence leads to a contradiction. Therefore x = y. Thus, problem (1.3) VI(A, H). Thus, by Theorem 3.2 we obtain the desired result.

21 Iterative method for a finite family Recall that the resolvent of the maximal monotone mapping B : H 2 H is defined by Jr B =(I + rb) 1 for all r>0, it is known that F (Jr B)=B 1 0 and is nonexpansive. J B r Theorem 4.2. Let H be a real Hilbert space. Let F be a function from H H to R satisfying (A1)-(A5) and ϕ : H R {+ } be a proper lower semicontinuous and convex function. Let A : H H be a L A -Lipschitzian and relaxed (c, d)-cocoercive mapping and let B 1,B 2,...,B N : H 2 H be maximal monotone mappings such that Ω= N i=1 B 1 i 0 A 1 0 MEP(F, ϕ) Ø. For all i {1, 2,...,N} and r i > 0, let J B i r i be the resolvents of B i. For all j {1, 2,...,N}, let α (n) j =(α n,j 1,αn,j [0, 1], α n,j 1 + α n,j 2 + α n,j 3 = 1, {α n,j 1 } N 1 2,αn,j 3 ) be such that αn,j 1,αn,j 2,αn,j 3 j=1 [η 1,θ 1 ] with 0 < η 1 θ 1 < 1, {α n,n 1 } [η N, 1] with 0 < η N 1 and {α n,j 2 }N j=1, {αn,j 3 }N j=1 [0,θ 2] with 0 θ 2 < 1. Let S n be the S-mappings generated by J B 1 r 1,J B 2 r 2,...,J B N r N and α (n) 1,α(n) 2,...,α(n) N. Assume that either (B1) or (B2) holds and that v is an arbitrary point in H. Letx 1 H and {x n }, {u n }, {y n } be the sequences defined by F (u n,y)+ϕ(y) ϕ(u n )+ 1 r n y u n,u n x n 0, y H, y n = u n λau n, x n+1 = a n v + b n x n +(1 a n b n )S n (y n λay n ), n 1, where 0 <λ< 2(d cl2 A ). Suppose that the following conditions hold: L 2 A (C1) lim n a n =0and n=1 a n = ; (C2) 0 < lim inf n b n lim sup n b n < 1; (C3) lim inf n r n > 0 and lim n r n+1 r n =0; (C4) lim n α n+1,i 1 α n,i 1 =0for all i {1, 2,...,N} and lim n α n+1,j 3 α n,j 3 =0for all j {2, 3,...,N}. Then {x n } converges strongly to x = P Ω v. Proof. For all i =1, 2,...,N and r i > 0, we have F (J B i r i P H = I and T i = J B i r i desired result. 0. Putting for all i =1, 2,...,N, by Theorem 4.1, we obtain the )=B 1 i Acknowledgement. The author would like to thank Professor Dr. Suthep Suantai and the reviewer for careful reading, valuable comment and suggestions on this paper. This work was supported by Thaksin University in the year The author also would like to thank the Commission on Higher Education, the Thailand Research Fund, the Centre of Excellence in Mathematics for their financial support.

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