Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces

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1 Available online at J. Nonlinear Sci. Appl. 9 (6), Research Article Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces Yaqiang Liu a, Shin Min Kang b,, Youli Yu c, Lijun Zhu d a School of Management, Tianjin Polytechnic University, Tianjin 3387, China. b Department of Mathematics and the RINS, Gyeongsang National University, Jinju 588, Korea. c School of Mathematics and Information Engineering, Taizhou University, Linhai 37, China. d School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 75, China. Communicated by R. Saadati Abstract In this paper, we introduce two general algorithms (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup T (s)} s in Hilbert spaces. We prove that both approaches converge strongly to a common element x of the set of the equilibrium points and the set of common fixed points of T (s)} s. Such common element x is the unique solution of some variational inequality, which is the optimality condition for some minimization problem. As special cases of the above two algorithms, we obtain two schemes which both converge strongly to the minimum norm element of the set of the equilibrium points and the set of common fixed points of T (s)} s. The results obtained in the present paper improve and extend the corresponding results by Cianciaruso et al. [F. Cianciaruso, G. Marino, L. Muglia, J. Optim. Theory. Appl., 46 (), 49 59] and many others. c 6 All rights reserved. Keywords: Equilibrium problem, variational inequality, fixed point, nonexpansive semigroup, algorithm, minimum norm. MSC: 47J5, 47J5, 47H9.. Introduction Let H be a real Hilbert space with inner product, and norm, respectively. Let C be a nonempty Corresponding author addresses: yani3579@6.com (Yaqiang Liu), smkang@gnu.ac.kr (Shin Min Kang), yuyouli@tzc.edu.cn (Youli Yu), zhulijun995@sohu.com (Lijun Zhu) Received 6--7

2 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), closed convex subset of H. Recall that a mapping f : C H be a ρ-contraction; that is, there exists a constant ρ [, ) such that f(x) f(y) ρ x y for all x, y C. A mapping T : C C is said to be nonexpansive if T x T y x y for all x, y C. Denote the set of fixed points of T by F ix(t ). Let A be a strongly positive bounded linear operator on H, i.e., there exists a constant γ > such that Ax, x γ x for all x H. Iterative methods for nonexpansive mappings are widely used to solve convex minimization problems. A typical problem is to minimize a function over the set of fixed points of a nonexpansive mapping T, min x F ix(t ) In [], Xu proved that the sequence x n } defined by Ax, x x, b. (.) x n+ = α n b + ( α n A)T x n, n, strongly converges to the unique solution of (.) under certain conditions. Recently, Marino and Xu [] introduced the viscosity approximation method x n+ = α n γf(x n ) + ( α n A)T x n, n, and proved that the sequence x n } converges strongly to the unique solution of the variational inequality (A γf)z, x z, x F ix(t ), which is the optimality condition for the minimization problem min x F ix(t ) Ax, x h(x), where h is a potential function for γf (i.e., h = γf on H). Recall also that a mapping B : C H is called α-inverse-strongly monotone if there exists a positive real number α such that Bx By, x y α Bx By, x, y C. It is clear that any α-inverse-strongly monotone mapping is monotone (that is, Bx By, x y is non-negative) and α-lipschitz continuous. Let B : C H be a nonlinear mapping and F : C C R be a bifunction. We concerned equilibrium problem is to find z C such that F (z, y) + Bz, y z, y C. (.) The solution set of (.) is denoted by Ω. If B =, then (.) reduces to the following equilibrium problem of finding z C such that F (z, y), y C. (.3) The equilibrium problem and the variational inequality problem have been investigated by many authors. To see related works, we refer the reader to [ 5, 7 5, 7, 9 8] and the references therein. The problem (.) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others. For solving equilibrium problem (.), Moudafi [] introduced an iterative algorithm and proved a weak convergence theorem. Further, Takahashi and Takahashi [9] introduced another iterative algorithm for finding an element of Ω F ix(s) and they obtained a strong convergence result. Recently, Plubtieng and Punpaeng [5] introduced the following iterative method to find an equilibrium point of F, which is also a fixed point of a nonexpansive mapping T : H H, F (u n, y) + r n x u n, u n x n, y H, x n+ = α n f(x n ) + (I α n A)T u n, n. (.4)

3 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), They proved that, with suitable conditions, both the sequences x n } and u n } defined by (.4) are strongly convergent to the unique solution z F ix(t ) Ω of the variational inequality (A γf)z, x z for all x F ix(t ) Ω, which is the optimality condition for the minimization problem min x F ix(t ) Ω Ax, x h(x), where h is a potential function for γf. In this paper, we focus on nonexpansive semigroup T (s)} s. Recall that a family S := T (s)} s of mappings of C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions: (S) T ()x = x for all x C; (S) T (s + t) = T (s)t (t) for all s, t ; (S3) T (s)x T (s)y x y for all x, y C and s ; (S4) for all x H, s T (s)x is continuous. We denote by F ix(t (s)) the set of fixed points of T (s) and by F ix(s) the set of all common fixed points of S, i.e., F ix(s) = s F ix(t (s)). It is known that F ix(s) is closed and convex. Very recently, Cianciaruso et al. [5] introduced the following implicit and explicit schemes for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup in Hilbert spaces: G(u t, y) + Implicit algorithm: r t y u t, u t x t, y H, x t = tγf(x t ) + (I ta) (.5), and Explicit algorithm: F (u n, y) + r n y u n, u n x n, y H, x n+ = α n γf(x n ) + (I α n A) λn T (s)u n ds, n. They proved that the above both approaches (.5) and (.6) have strong convergence. (Note that the integral mentioned in the present paper is the usual integral, for example, we can compute t T (s)xds as lim n n i= t n T ( t n i)x). The following interesting problem arises: can one construct some more general algorithms which unify the above algorithms? On the other hand, we also notice that it is quite often to seek a particular solution of a given nonlinear problem, in particular, the minimum-norm solution. For instance, given a closed convex subset C of a Hilbert space H and a bounded linear operator R : H H, where H is another Hilbert space. The C-constrained pseudoinverse of R, R C, is then defined as the minimum-norm solution of the constrained minimization problem R C (b) := arg min Rx b, x C which is equivalent to the fixed point problem x = P C (x λr (Rx b)), where P C is the metric projection from H onto C, R is the adjoint of R, λ > is a constant, and b H is such that P R(C) (b) R(C). It is therefore another interesting problem to invent some algorithms that can generate schemes which converge strongly to the minimum-norm solution of a given problem. In this paper, we introduce two general algorithms (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup T (s)} s in Hilbert spaces. We prove that both approaches converge strongly to a common element x of the set of the equilibrium points and the set of common fixed points of T (s)} s. Such common element x is the unique solution of some variational inequality, which is the optimality condition for some minimization (.6)

4 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), problem. As special cases of the above two algorithms, we obtain two schemes which both converge strongly to the minimum norm element of the set of the equilibrium points and the set of common fixed points of T (s)} s. The results contained in the present paper improve and extend the corresponding results by Cianciaruso et al. [5] and many others.. Preliminaries Let C be a nonempty closed convex subset of a real Hilbert space H. Throughout this paper, we assume that a bifunction F : C C R satisfies the following conditions: (H) F (x, x) = for all x C; (H) F is monotone, i.e., F (x, y) + F (y, x) for all x, y C; (H3) for each x, y, z C, lim t F (tz + ( t)x, y) F (x, y); (H4) for each x C, y F (x, y) is convex and lower semicontinuous. The metric (or nearest point) projection from H onto C is the mapping P C : H C which assigns to each point x C the unique point P C x C satisfying the property and x P C x = inf x y =: d(x, C). y C It is well known that P C is a nonexpansive mapping and satisfies x y, P C x P C y P C x P C y, x, y H. Moreover, P C is characterized by the following properties: x P C x, y P C x, (.) x y x P C x + y P C x for all x H and y C. We need the following lemmas to prove our main results. Lemma. ([8]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C C R be a bifunction which satisfies conditions (H) (H4). Let r > and x C. Then, there exists z C such that F (z, y) + y z, z x, y C. r Further, if S r (x) = z C : F (z, y) + r y z, z x, y C}, then the following hold: (i) S r is single-valued and S r is firmly nonexpansive, i.e., for any x, y H, S r x S r y S r x S r y, x y ; (ii) S F (as the set of all z C holding (.3)) is closed and convex and S F = F ix(s r ). Lemma. ([3]). Let C be a nonempty closed convex subset of a real Hilbert space H. Let the mapping B : C H be α-inverse strongly monotone and r > be a constant. Then, we have (I rb)x (I rb)y x y + r(r α) Bx By, x, y C. In particular, if r α, then I rb is nonexpansive.

5 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), Lemma.3 ([8]). Let C be a nonempty bounded closed convex subset of a Hilbert space H and let T (s)} s be a nonexpansive semigroup on C. Then, for every h, lim sup t t T (s)xds T (h) t T (s)xds t t =. x C Lemma.4 ([9]). Let C be a closed convex subset of a real Hilbert space H and let S : C C be a nonexpansive mapping. Then, the mapping I S is demiclosed. That is, if x n } is a sequence in C such that x n x weakly and (I S)x n y strongly, then (I S)x = y. Lemma.5 ([]). Assume a n } is a sequence of nonnegative real numbers such that a n+ ( γ n )a n + δ n γ n, where γ n } is a sequence in (, ) and δ n } is a sequence such that () n= γ n = ; () lim sup n δ n or n= δ nγ n <. Then lim n a n =. 3. Main results In this section we will show our main results. Theorem 3.. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S = T (s)} s be a nonexpansive semigroup on C. Let f : C H be a ρ-contraction (possibly non-self) with ρ [, ). Let A be a strongly positive linear bounded self-adjoint operator on H with coefficient γ >. Let B : C H be an α-inverse strongly monotone mapping. Let r t } <t< be a continuous net of positive real numbers such that r t [a, b] (, α). Let } <t< be a continuous net of positive real numbers such that lim t = +. Let γ and β be two real numbers such that < γ < γ/ρ and β [, ). Suppose that the function F : C C R satisfies (H)-(H4) and F ix(s) Ω. Let the nets x t } and u t } be defined by the following implicit scheme: F (u t, y) + Bx t, y u t + r t y u t, u t x t, y C, x t = P C [tγf(x t ) + βx t + (( β)i ta) ]. Then the nets x t } and u t } defined by (3.) strongly converge to x F ix(s) Ω as t and x is the unique solution of the following variational inequality: (3.) x F ix(s) Ω, (γf A)x, x x, x F ix(s) Ω. (3.) In particular, if we take f = and A = I, then the nets x t } and u t } defined by (3.) reduces to F (u t, y) + Bx t, y u t + r t y u t, u t x t, y C, x t = P C [βx t + ( β t) ]. In this case, the nets x t } and u t } defined by (3.3) converge in norm to the minimum norm element x of F ix(s) Ω, namely, the point x is the unique solution to the minimization problem: x = arg min x. x F ix(s) Ω (3.3)

6 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), Proof. First, we note that the nets x t } and u t } defined by (3.) are well-defined. As a matter of fact, from Lemma., we have u t = S rt (x t r t Bx t ). Now we define a mapping Gx := P C [ tγf(x) + βx + (( β)i ta) ] T (s)s rt (x r t Bx)ds. Since S rt and (I r t B) are nonexpansive, we have Gx Gy tγ(f(x) f(y)) + β(x y) + (( β)i ta) T (s)[s rt (x r t Bx) S rt (y r t By)]ds tγ f(x) f(y) + β x y + (( β)i ta) T (s)[s rt (x r t Bx) S rt (y r t By)]ds tγρ x y + β x y + ( β t γ) x y = ( ( γ γρ)t) x y. This implies that the mapping G is a contraction and so it has a unique fixed point. Hence, the nets x t } and u t } defined by (3.) are well-defined. Let p F ix(s) Ω. It is clear that p = S rt (p r t Bp) for all t (, ). From Lemma., we have u t p = S rt (x t r t Bx t ) S rt (p r t Bp) x t r t Bx t (p r t Bp) x t p + r t (r t α) Bx t Bp x t p. (3.4) So, we have u t p x t p. Then, we obtain [ x t p = P C tγf(x t ) + βx t + (( β)i ta) ] p ( λt t(γf(x t) Ap) + β(x t p) + (( β)i ta) p) Hence t γf(x t ) Ap + β x t p + ( β γt) T (s)u t T (s)p ds tγ f(x t ) f(p) + t γf(p) Ap + β x t p + ( β γt) u t p tγρ x t p + t γf(p) Ap + β x t p + ( β γt) x t p. x t p γf(p) Ap, γ γρ which implies that the net x t } is bounded and so is the net u t }. Set R := γ γρ γf(p) Ap. It is clear that x t} B(p, R) and u t } B(p, R). Notice that p u t p x t p R.

7 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), Moreover, we observe that if x B(p, R) then T (s)x p = T (s)x T (s)p x p R, i.e., B(p, R) is T (s)-invariant for all s. Set y t = tγf(x t ) + βx t + (( β)i ta). It follows that x t = P C [y t ]. By using the property of the metric projection (.), we have This implies that Hence, x t p = x t y t, x t p + y t p, x t p y t p, x t p = tγf(x t ) Ap, x t p + β x t p + (( β)i ta) (T (s)u t p)ds, x t p t γf(x t ) Ap x t p + β x t p + ( β t γ) u t p x t p. x t p t β γf(x t) Ap + u t p. ( ) x t p u t p t + t ( β) γf(x t) Ap + β γf(x t) Ap u t p u t p + tm x t p + r t (r t α) Bx t Bp + tm, (3.5) (3.6) where It follows that t M := sup <t< ( β) γf(x t) Ap + } β γf(x t) Ap u t p, r t x t u t. Since lim t r t = r (, α), we derive From Lemmas.,. and (3.), we obtain u t p = S rt (x t r t Bx t ) S rt (p r t Bp) which implies that r t (α r t ) Bx t Bp tm. lim t Bx t Bp =. (3.7) (x t r t Bx t ) (p r t Bp), u t p = ( (xt r t Bx t ) (p r t Bp) + u t p (x t p) r t (Bx t Bp) (u t p) ) ( xt p + u t p (x t u t ) r t (Bx t Bp) ) = ( xt p + u t p x t u t + r t x t u t, Bx t Bp r t Bx t Bp ), u t p x t p x t u t + r t x t u t, Bx t Bp r t Bx t Bp x t p x t u t + r t x t u t Bx t Bp x t p x t u t + M Bx t Bp. (3.8)

8 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), By (3.6) and (3.8), we have It follows that This together with (3.7) implies that From (3.), we deduce Note that Since we obtain x t p x t p x t u t + ( Bx t Bp + t)m. T (τ)x t x t = P C [T (τ)x t ] P C [y t ] x t u t ( Bx t Bp + t)m. lim x t u t =. t T (τ)x t y t T (τ)x t T (τ) + T (τ) + λt y t x t + T (τ) + λt y t. y t t γf(x t ) A + β x t. x t tγf(x t ) + βx t (βi + ta) t γf(x t) A + β x t, x t t β γf(x t) A. Therefore, T (τ)x t x t t β γf(x t) A + T (τ). From Lemma.3, we deduce for all τ < From (3.5), we have x t p y t p, x t p lim t T (τ)x t x t =. (3.9) = tγf(x t ) Ap, x t p + β x t p ( λt ) + (( β)i ta) p, x t p β x t p + ( β γt) u t p x t p + tγf(x t ) f(p), x t p + tγf(p) Ap, x t p [ ( γ γρ)t] x t p + tγf(p) Ap, x t p.

9 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), Therefore, x t p γ γρ γf(p) Ap, x t p, p F ix(s) Ω. From this inequality, we have immediately that ω w (x t ) = ω s (x t ), where ω w (x t ) and ω s (x t ) denote the set of weak and strong cluster points of x t }, respectively. Let t n } (, ) be a sequence such that t n as n. Put x n := x tn, u n := u tn, r n := r tn and := n. Since x n } is bounded, without loss of generality, we may assume that x n } converges weakly to a point x C. Also y n x weakly. Noticing (3.9) we can use Lemma.4 to get x F ix(s). Now we show x Ω. Since u n = S rn (x n r n Bx n ), for any y C we have From the monotonicity of F, we have F (u n, y) + r n y u n, u n (x n r n Bx n ). r n y u n, u n (x n r n Bx n ) F (y, u n ), y C. Hence, y u ni, u n i x ni r ni + Bx ni F (y, u ni ), y C. (3.) Put z t = ty + ( t)x for all t (, ] and y C. Then, we have z t C. So, from (3.) we have z t u ni, Bz t z t u ni, Bz t z t u ni, u n i x ni + Bx ni + F (z t, u ni ) r ni = z t u ni, Bz t Bu ni + z t u ni, Bu ni Bx ni z t u ni, u n i x ni + F (z t, u ni ). r ni (3.) Note that Bu ni Bx ni α u n i x ni. Further, from monotonicity of B, we have z t u ni, Bz t Bu ni. Letting i in (3.), we have From (H), (H4) and (3.), we also have and hence Letting t in (3.3), we have, for each y C, This implies that x Ω. We can rewrite (3.) as z t x, Bz t F (z t, x ). (3.) = F (z t, z t ) tf (z t, y) + ( t)f (z t, x ) tf (z t, y) + ( t)z t x, Bz t = tf (z t, y) + ( t)ty x, Bz t F (z t, y) + ( t)bz t, y x. (3.3) F (x, y) + y x, Bx. (A γf)x t = t (( β)i ta) [ x t ] + t (x t y t ).

10 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), Therefore, (A γf)x t, x t p = β t + A [ ] (I T (s)s rt (I r t B))x t (I T (s)s rt (I r t B))p, x t p ds [x t T (s)u t ]ds, x t p + t x t y t, x t p. Noting that I T (s)s rt (I r t B) is monotone and x t y t, x t p, so (A γf)x t, x t p A [x t T (s)u t ]ds, x t p Taking the limit through t := t ni, we have Ax t A, x t p = A x t x t p t β A γf(x t) A x t p. (A γf)x, x p = lim i (A γf)x ni, x ni p. Since the solution of the variational inequality (3.) is unique, hence ω w (x t ) = ω s (x t ) is singleton. Therefore, x t x. In particular, if we take f = and A = I, then it follows that x t x = P F ix(s) Ω (), which implies that x is the minimum norm fixed point of T. As a matter of fact, by (3.), we deduce that is, x, x x, x F ix(s) Ω, x x, x x x, x F ix(s) Ω. Therefore, the point x is the unique solution to the minimization problem This completes the proof. x = arg min x. x F ix(s) Ω Next we introduce an explicit algorithm to find a solution of minimization problem (.). This scheme is obtained by discretizing the implicit scheme (3.). We will show the strong convergence of this algorithm. Theorem 3.. Let C be a nonempty closed convex subset of a real Hilbert space H. Let S = T (s)} s be a nonexpansive semigroup on C. Let f : C H be a ρ-contraction (possibly non-self) with ρ [, ). Let A be a strongly positive linear bounded self-adjoint operator on H with coefficient γ >. Let B : C H be an α-inverse strongly monotone mapping. Let γ and β be two real numbers such that < γ < γ/ρ and β [, ). Suppose that the function F : C C R satisfies (H)-(H4) and F ix(s) Ω. Let x n } and u n } be defined by the following explicit algorithm: F (u n, y) + Bx n, y u n + r n y u n, u n x n, y C, x n+ = P C [α n γf(x n ) + βx n + (( β)i α n A) λn T (s)u n ds], n, (3.4) where α n } is real number sequence in [, ] and }, r n } are two sequences of positive real numbers. Suppose that the following conditions are satisfied:

11 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), (i) lim n α n =, n= α n = and n= α n α n < ; (ii) lim n = and lim n α n = ; (iii) r n [a, b] (, α) and n= r n+ r n <. Then the sequences x n } and u n } defined by (3.4) strongly converge to x F ix(s) Ω and x is the unique solution of the variational inequality (3.). In particular, if we take f = and A = I, then the sequences x n } and u n } defined by (3.4) reduces to F (u n, y) + Bx n, y u n + r n y u n, u n x n, y C, x n+ = P C [βx n + ( α n β) λn T (s)u n ds], n. (3.5) In this case, the sequences x n } and u n } defined by (3.5) converge in norm to the minimum norm element x of F ix(s) Ω. Proof. Take p F ix(s) Ω, we have x n+ p α n γf(x n ) Ap + β x n p + ( β γα n ) T (s)u n T (s)p ds α n γρ x n p + α n γf(p) Ap + β x n p + ( β α n γ) u n p. λn (3.6) From Lemma., we have u n p = S rn (x n r n Bx n ) S rn (p r n Bp) x n r n Bx n (p r n Bp) x n p + r n (r n α) Bx n Bp x n p. (3.7) So, we have u n p x n p. (3.8) By (3.6) and (3.8), we derive Using induction, it follows that x n+ p [ ( γ ργ)α n ] x n p + α n γf(p) Ap. x n p max x p, Set y n = λn T (s)u n ds for all n. From (3.4), we get } γf(p) Ap. γ γρ x n+ x n α n γf(x n ) + βx n + (( β)i α n A)y n α n γf(x n ) βx n (( β)i α n A)y n = γα n (f(x n ) f(x n )) + γ(α n α n )f(x n ) + β(x n x n ) + (( β)i α n A)(y n y n ) + (α n α n )Ay n γα n f(x n ) f(x n ) + α n α n ( γf(x n ) + Ay n ) + β x n x n + ( β α n γ) y n y n, and y n y n = λn ( [T (s)u n T (s)u n ]ds + ) λn T (s)u n ds

12 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), and + λn T (s)u n ds = λn ( [T (s)u n T (s)u n ]ds + ) λn [T (s)u n T (s)p]ds + λn [T (s)u n T (s)p]ds λn T (s)u n T (s)u n ds + λn [T (s)u n T (s)p]ds + λn T (s)u n T (s)p ds u n u n + u n p. Next, we estimate u n+ u n. From (3.4), we have F (u n, y) + Bx n, y u n + r n y u n, u n x n, y C, (3.9) F (u n+, y) + Bx n+, y u n+ + r n+ y u n+, u n+ x n+, y C. (3.) Putting y = u n+ in (3.9) and y = u n in (3.), we have and F (u n, u n+ ) + Bx n, u n+ u n + r n u n+ u n, u n x n, F (u n+, u n ) + Bx n+, u n u n+ + r n+ u n u n+, u n+ x n+. From the monotonicity of F, we have Then, F (u n, u n+ ) + F (u n+, u n ). Bx n Bx n+, u n+ u n + u n+ u n, u n x n u n+ x n+, r n r n+ and hence u n+ u n, u n u n+ + u n+ x n r n (u n+ x n+ ) + r n Bx n Bx n+, u n+ u n. r n+ It follows that u n+ u n u n+ u n, r n+ r n (u n+ x n+ ) + x n+ x n r n+ + r n Bx n Bx n+, u n+ u n = (I r n B)x n+ (I r n B)x n, u n+ u n + u n+ u n, r n+ r n (u n+ x n+ ) r n+ (I r n B)x n+ (I r n B)x n u n+ u n + r n+ r n r n+ u n+ u n u n+ x n+,

13 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), that is, Therefore, u n+ u n (I r n B)x n+ (I r n B)x n + x n+ x n + r n+ r n r n+ u n+ x n+ x n+ x n + r n+ r n a u n+ x n+. r n+ r n r n+ u n+ x n+ y n y n x n x n + r n r n a u n x n + u n p, and hence x n+ x n γα n ρ x n x n + α n α n ( γf(x n ) + Ay n ) ( + β x n x n + ( β α n γ) x n x n + r n r n a u n x n + λ ) n u n p [ ( γ γρ)α n ] x n x n + M α n α n + r n r n a + λ } n, (3.) where M > is a constant such that From Lemma.5 and (3.), we derive sup γf(x n ) + Ay n, r n u n x n, u n p } M. n lim x n+ x n =. n It follows that Note that that is, lim u n+ u n = lim y n+ y n =. n n x n y n x n+ x n + x n+ y n x n+ x n + α n γ f(x n ) + β x n y n + α n Ay n, x n y n β ( x n+ x n + α n γ f(x n ) + α n Ay n ) (as n ). Set v n = α n γf(x n ) + βx n + (( β)i α n A) λn T (s)u n ds. It follows that x n+ = P C [v n ] for all n. By using the property of the metric projection (.) and (3.4), we have x n+ p = x n+ p, x n+ p = x n+ v n, x n+ p + v n p, x n+ p v n p, x n+ p = α n γf(x n ) Ap, x n+ p + βx n p, x n+ p

14 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), which implies that Hence, + (( β)i α n A)(y n p), x n+ p α n γf(x n ) Ap x n+ p + β x n p x n+ p + ( β γα n ) y n p x n+ p (3.) α n γf(x n ) Ap x n+ p + β x n p x n+ p + ( β) y n p x n+ p α n γf(x n ) Ap x n+ p + β ( x n p + x n+ p ) + β ( y n p + x n+ p ), x n+ p α n γf(x n ) Ap x n+ p + β x n p + ( β) y n p = α n γf(x n ) Ap x n+ p + β x n p + ( β) λn [T (s)u n T (s)p]ds It follows that α n γf(x n ) Ap x n+ p + β x n p + ( β) u n p (3.3) α n γf(x n ) Ap x n+ p + β x n p + ( β)( x n p + r n (r n α) Bx n Bp ). r n (α r n )( β) Bx n Bp α n γf(x n ) Ap x n+ p From Lemmas. and., we obtain + x n p x n+ p α n γf(x n ) Ap x n+ p + x n x n+ ( x n p + x n+ p ). lim Bx n Bp =. n u n p = S rn (x n r n Bx n ) S rn (p r n Bp) (x n r n Bx n ) (p r n Bp), u n p = ( (xn r n Bx n ) (p r n Bp) + u n p (x n p) r n (Bx n Bp) (u n p) ) ( xn p + u n p (x n u n ) r n (Bx n Bp) ) = ( xn p + u n p x n u n + r n x n u n, Bx n Bp rn Bx n Bp ), which implies that u n p x n p x n u n + r n x n u n, Bx n Bp rn Bx n Bp x n p x n u n + r n x n u n Bx n Bp x n p x n u n + M Bx n Bp. (3.4) From (3.3) and (3.4), we have x n+ p α n γf(x n ) Ap x n+ p + β x n p + ( β)( x n p x n u n + M Bx n Bp ).

15 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), It follows that ( β) x n u n α n γf(x n ) Ap x n+ p + x n p x n+ p + M Bx n Bp α n γf(x n ) Ap x n+ p + x n x n+ ( x n p + x n+ p ) + M Bx n Bp. Therefore, lim x n u n =. n Note that x n } is a bounded sequence. Let x be a weak limit of x n }. Putting x = P F ix(s) Ω (I A + γf). Then there exists R such that B(x, R) contains x n }. Moreover, B(x, R) is T (s)-invariant for every s ; therefore, without loss of generality, we can assume that T (s)} s is a nonexpansive semigroup on B(x, R). We notice that, from Theorem 3., x ω w (y n ) = ω s (y n ). Then, from Lemma.3, it follows that, for every h, lim n y n T (h)y n = and from the demiclosedness principle, we have x F ix(s). By the same argument as that of Theorem 3., we can deduce that x Ω. Hence, x F ix(s) Ω. Therefore, lim supγf(x ) Ax, x n+ x = lim n n γf(x ) Ax, x x. Finally, we prove that x n x. From (3.), we have that is, x n+ x α n γf(x n ) f(x ), x n+ x + α n γf(x ) Ax, x n+ x + β x n x x n+ x + ( β γα n ) y n x x n+ x α n ργ x n x x n+ x + α n γf(x ) Ax, x n+ x + β x n x x n+ x + ( β γα n ) x n x x n+ x ( γ γρ)α n ( x n x + x n+ x ) + α n γf(x ) Ax, x n+ x, x n+ x [ ( γ γρ)α n ] x n x + α n γf(x ) Ax, x n+ x. Hence, all conditions of Lemma.5 are satisfied. Therefore, we immediately deduce that x n x. In particular, if we take f = and A = I, then it is clear that x = P F ix(s) Ω () is the unique solution to the minimization problem x = arg min x F ix(s) Ω x. This completes the proof. Remark 3.3. We observe that our algorithms presented in this paper include some algorithms in the literature as special cases: () If we take B = and β = and let S = T (s)} s be a nonexpansive semigroup on a real Hilbert space H, then our algorithms (3.) and (3.4) reduce to the algorithms (.5) and (.6) which were considered by Cianciaruso et al. [5]. () If we take B =, β =, γ = and let S = T be a nonexpansive mapping on a real Hilbert space H, then our algorithm (3.4) reduces to the algorithm (.4) which was considered by Plubtieng and Punpaeng [5]. (3) If we take A = I, B =, β =, γ =, F =, and let S = T (s)} s be a nonexpansive semigroup on a real Hilbert space H, then our algorithm (3.) reduces to the following algorithm x t = tf(x t ) + ( t) T (s)x t ds, which was considered by Plubtieng and Punpaeng [6].

16 Y. Liu, S. M. Kang, Y. Yu, L. Zhu, J. Nonlinear Sci. Appl. 9 (6), (4) If we take A = I, B =, f = u, β =, γ =, F =, and let S = T (s)} s be a nonexpansive semigroup on a real Hilbert space H, then our algorithm (3.4) reduces to the following algorithm x n+ = α n u + ( α n ) λn T (s)x n ds, n, which was considered by Shimizu and Takahashi [8]. (5) If we take B =, β =, F = and let S = T be a nonexpansive mapping on a real Hilbert space H, then our algorithms (3.) and (3.4) reduce to the following algorithms x t = tγf(x t ) + (I ta)t x t, and x n+ = α n γf(x n ) + ( α n A)T x n, n, which were considered by Marino and Xu []. Remark 3.4. From Remark 3.3, it is clear that our results contain the corresponding results in [6,, 5, 6, 8] as special cases. Acknowledgments The authors would like to thank the editor and the referees for useful comments and suggestions References [] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (994), [] L. C. Ceng, S. Al-Homidan, Q. H. Ansari, J. C. Yao, An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Comput. Appl. Math., 3 (9), [3] Y. Censor, A. N. Iusem, S. A. Zenios, An interior point method with Bregman functions for the variational inequality problem with paramonotone operators, Math. Programming, 8 (998), [4] S.-S. Chang, H. W. Joseph Lee, C. K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal., 7 (9), [5] F. Cianciaruso, G. Marino, L. Muglia, Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces, J. Optim. Theory. Appl., 46 (), ,,, 3.3 [6] V. Colao, G. L. Acedo, G. Marino, An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, Nonlinear Anal., 7 (9), [7] V. Colao, G. Marino, H.-K. Xu, An iterative method for finding common solutions of equilibrium and fixed point problems, J. Math. Anal. Appl., 344 (8), [8] P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (5), [9] K. Goebel, W. A. Kirk, Topics in metric fixed point theory, Cambridge University Press, Cambridge, (99)..4 [] J. S. Jung, Strong convergence of composite iterative methods for equilibrium problems and fixed point problems, Appl. Math. Comput., 3 (9), [] G. Marino, H.-K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 38 (6), 43 5., 3.3, 3.4 [] A. Moudafi, Krasnoselski-Mann iteration for hierarchical fixed-point problems, Inverse Problems, 3 (7), [3] N. Nadezhkina, W. Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 8 (6), 9.. [4] J.-W. Peng, J.-C. Yao, A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems, Taiwanese J. Math., (8), [5] S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput., 97 (8), , 3.3, 3.4 [6] S. Plubtieng, R. Punpaeng, Fixed point solutions of variational inequalities for nonexpansive semigroups in Hilbert spaces, Math. Comput. Modelling, 48 (8), , 3.4

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