THE complementarity problem is one of the important

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1 The common solutions of complementaity poblems and a zeo point of maximal monotone opeatos by using the hybid pojection method Khanittha Pomluang, Pongus Phuangphoo, and Poom Kumam, Abstact In this pape, we popose a pojection method fo solving nonlinea complementaity poblems and a zeo point of maximal monotone opeatos. Stong convegence theoems ae established fo solving the common solutions of complementaity poblems and a zeo point of maximal monotone opeatos togethe with a system of genealized equilibium, vaiational inequality and fixed point poblems in a unifomly smooth and 2-unifomly convex eal Banach space. Moeove, we also apply the esult to Hilbet spaces. Keywods Complementaity poblems; a zeo point; invesestongly monotone mapping; vaiational inequality poblem; equilibium poblem; fixed point poblem. I. INTRODUCTION AND PRELIMINARIES THE complementaity poblem is one of the impotant poblems and deep connections with nonlinea analysis and by its inteesting applications in aeas example such as: Optimization Theoy, Engineeing, Stuctual Mechanics, Elasticity Theoy, Economics, Equilibium Theoy, Nonlinea Dynamics, Stochastic Optimal Contol. Some natual connections between the complementaity poblem and some special fixed point theoems ae used to pove seveal existance theoems. Recently, In 2015, Phuangphoo and Kumam [10] intoduced a new iteative sequence which is constucted by using the hybid pojection method fo solving the common solution fo a system of genealized equilibium poblems of invese stongly monotone mappings and a system of bifunctions satisfying cetain the conditions, the common solution fo the families of quasi -φ- asymptoticallonexpansive and unifomly Lipschitz continuous and the common solution fo a vaiational inequality poblem. Stong convegence theoems ae poved on appoximating a common solution of a system of genealized equilibium poblems, fixed point poblems fo two countable families and a vaiational inequality poblem in a unifomly smooth and 2-unifomly convex eal Banach space. Let E be a Banach space with nom, C be a nonempty closed and convex subset of E and let E denote by the dual Manuscipt eceived Decembe 25, This eseach was suppoted by the Theoetical and Computational Science (TaCS Cente, Faculty of Science, King Mongkut s Univesity of Technology Thonbui (KMUTT. Depatment of Mathematics & Theoetical and Computational Science (TaCS Cente, Faculty of Science, King Mongkut s Univesity of Technology Thonbui (KMUTT, 126 Pacha-Uthit Road, Bang Mod, Thung Khu, Bangkok 10140, Thailand. s: k.pomluang@gmail.com (K. Pomluang and poom.kum@kmutt.ac.th (P. Kumam Depatment of Mathematics, Faculty of Education, Bansomdejchaopaya Rajabhat Univesity (BSRU 1061 Issaaphap Rd., Hianuchi, Thonbui, Bangkok 10600, Thailand. s: p.phuangphoo@gmail.com $ The coesponding autho: poom.kum@kmutt.ac.th (P. Kumam of E. One of the majo poblems in the theoy of monotone opeatos is as follows: find a point z E such that 0 Az, (I.1 whee A is an opeato fom E into E. A point z E is called a zeo point of opeato A. The set of zeoes of the opeato A is denoted by A 1 0. This poblem contains numeous poblems in optimization, economics, physics and seveal aeas of engineeing. The poblem of finding a zeo point fo monotone opeatos play an impotant ole in moden optimization and nonlinea analysis. This is because it can be elated to many kinds of impotant poblems, such as convex optimization poblem, image pocessing, equilibium poblem and vaiational inequality poblems. In ode to appoximate the solution to this poblems, many authos have studies the convegence of such poblems in seveal setting. It is well known that the metic pojection opeato plays an impotant ole in nonlinea functional analysis, optimization theoy, vaiational inequality, and complementaity poblems, etc. In 2012, Saewan and Kumam [14] intoduced a modified hybid block pojection algoithm fo finding a common element of the set of the solution of the complementaity poblems, the set of solutions of the system of equilibium poblems and the set of common fixed points of an infinite family in a 2-unifomly convex and unifomly smooth Banach space. In this pape by the peviously mentioned above esults, we will apply the esults of Phuangphoo and Kumam [10], this wok binging in the esults to applied in othe poblems as well complementaity poblems and we shall use the esult to study the stong convegence theoem of zeo point of maximal monotone opeatos togethe with a system of genealized equilibium, vaiational inequality and fixed point poblems in a unifomly smooth and 2-unifomly convex eal Banach space. Moeove, we also apply the esult to obtain in Hilbet spaces. Thoughout this pape, we assume that R and J ae denoted by the set of eal numbes and the set of {1, 2, 3,..., M}, espectively, whee M is any given positive intege. Let {F k } k J : C C R be a bifunction, and {B k } k J : C E be a monotone mapping. The system of genealized equilibium poblems, is to find x C such that F k (x, y + y x, B k x 0, k J, y C. (I.2 ISSN:

2 The set of solutions of (I.2 is denoted by SGEP (F k, B k, that is SGEP (F k, B k = {x C : F k (x, y + y x, B k x 0, y C}, k J. If J is a singleton, then poblem (I.2 educes to the genealized equilibium poblems, is to find x C such that F (x, y + y x, Bx 0, y C. (I.3 The set of solutions of (I.3 is denoted by GEP (F, B, that is GEP (F, B = {x C : F (x, y+ y x, Bx 0, } (I.4 y C. If B 0 the poblem (I.3 educes into the equilibium poblem fo F, denoted by EP (F, is to find x C such that F (x, y 0, y C. (I.5 If F 0 the poblem (I.3 educes into vaiational inequality of Bowde type, denoted by V I(C, B, is to find x C such that y x, Bx 0, y C. (I.6 Let {x n } be a sequence in E. We denote by x n x and x n x that the stong convegence and weak convegence of {x n }, espectively. The nomalized duality mapping J fom E to 2 E is defined by Jx = {f E : x, f = x 2 = f 2 }, x E.By the Hahn-Banach theoem, Jx fo each x E. A Banach space E is said to be stictly convex if x+y 2 < 1 fo all x, y E with x = y = 1 and x y. It is said to be unifomly convex if lim n x n = 0 fo any two sequences {x n } and { } in E such that x n 1, 1 and lim n x n + 2 = 1. (I.7 Let U E = {x E : x = 1} be the unit sphee of E. Then, the Banach space E is said to be smooth if lim t 0 x + ty x t (I.8 exists fo each x, y U E. It is said to be unifomly smooth if the limit (I.8 is attained unifomly fo all x, y U E. The function ρ E : R + R + is { said to be the modulus of smoothness of E if ρ E (t = sup x+y + x y 2 1 : x = } 1, y = t. The space E is said to be smooth if ρ E (t > 0, t > 0 and is said to be unifomly smooth if and only if lim ρ E(t t 0 + t = 0. The modulus of convexity of { E is the function δ E : [0, 2] [0, 1] defined by δ E (ɛ = inf 1 x+y 2 : x 1, y } 1 ; x y ɛ. A Banach space E is said to be unifomly convex if and only if δ E (ɛ > 0 fo all ɛ (0, 2]. Let a eal numbe p > 1. Then, E is said to be p - unifomly convex if thee exists a constant c > 0 such that δ E (ɛ cɛ p, fo all ɛ [0, 2]. Obseve that evey p-unifomly convex space is unifomly convex. It is wellknown fo example that { L p (l p o Wm p p-unifomly convex, if p 2; is 2-unifomly convex, if 1 < p 2. One should note that no a Banach space is p-unifomly convex fo 1 < p < 2. It is known that a Hilbet space is unifomly smooth and 2-unifomly convex. Now let E be a smooth and stictly convex eflexive Banach space. As Albe (see [1] and Kamimua and Takahashi (see [7] did, the Lyapunov functional φ : E E R + is defined by φ(x, y = x 2 2 x, Jy + y 2, x, y E. It follows fom Kohsaka and Takahashi (see [8] that φ(x, y = 0 if and only if x = y, and that ( x y 2 φ(x, y ( x + y 2. (I.9 If E is a Hilbet space, then φ(x, y = x y 2, fo all x, y E. Futhe suppose that C is nonempty closed convex subset of E. The genealized pojection (Albe [1] Π C : E C is a map that assigns to an abitay point x E the minimum point of the functional φ(x, y, that is, Π C x = x, whee x is the solution to the following minimization poblem φ( x, x = inf φ(y, x, (I.10 y C existence and uniqueness of the opeato Π C follows fom the popeties of the functional φ(x, y and stict monotonicity of the mapping J. So, the genealized pojection Π C : E C is defined by fo each x E, Π C (x = ag min φ(x, y. y C As well know that if C is a nonempty closed convex subset of a Hilbet space H and P C : H C is the metic pojection of H onto C, then P C is nonexpansive. Remak I.1. If E is a eflexive, stictly convex and smooth Banach space, then fo x, y E, φ(x, y = 0 if and only if x = y. It is sufficient to show that if φ(x, y = 0 then x = y. Fom Lyapunov functional, we have x = y. This implies that x, Jy = x 2 = Jy 2. Fom the definition of J, one has Jx = Jy. Theefoe, we have x = y; see [5] fo moe details. The following basic popeties can be found in Cioanescu [5]. If E is a stictly convex, then J is stictly monotone. If E is unifomly smooth, then J is unifomlom-tonom continuous on each bounded subset of E. If E is eflexive smooth and stictly convex, then the nomalized duality mapping J is single valued, one-toone and onto. If E be a eflexive stictly convex and smooth Banach space and J is the duality mapping fom E into E, then J 1 is also single-value, bijective and is also the duality mapping fom E into E and thus JJ 1 = I E and J 1 J = I E. If E is unifomly smooth, then E is smooth and eflexive. ISSN:

3 E is unifomly smooth if and only if E is unifomly convex. If E is a eflexive and stictly convex Banach space, then J 1 is nom-weak -continuous. Recall that, a mapping S : C C is said to be nonexpansive if Sx Sy x y, x, y C. A mapping A : C E is said to be δ-invese-stongly monotone, if thee exists a constant δ > 0 such that x y, Ax Ay δ Ax Ay 2, x, y C. A mapping S : C C is said to be closed if fo each {x n } C, x n x and Sx n y imply Sx = y. Remak I.2. Let T is a nonexpansive of C into itself and I is the identity mapping of a eal Banach space E. Then, a mapping A = I T is 1 2-invese-stongly monotone mapping. Remak I.3. Let a mapping A k : R R by A k x = kx, x R and k {1, 2, 3,..., n}. Then, a mapping A k : R R is a finite family of 1 k -invese-stongly monotone. A mapping S : C C is said to be quasi -φ- nonexpansive (elatively quasi-nonexpansive if F ix(s, and φ(u, Sx φ(u, x, x C, u F ix(s. A mapping S : C C is said to be quasi -φasymptotically nonexpansive (asymptotically elatively nonexpansive if F ix(s, and thee exists a sequence {k n } [1, with k n 1 such that φ(u, S n x k n φ(u, x, x C, u F ix(s, n 1. It is easy to see that if A : C E is δ-invese-stongly monotone, then A is 1 δ -Lipschitz continuous. Example I.4. Let Π c be the genealized pojection fom a smooth, stictly convex and eflexive Banach space E onto a nonempty closed and convex subset C of E. Then, we get Π c is closed and quasi -φ- asymptoticallonexpansive mapping fom E onto C with F ix(π c = C. Example I.5. Let C := [ 1 π, 1 π ] and define T : C C by { x T x = 2 sin( 1 x, if x 0; x, if x = 0. Then, T is quasi -φ- asymptoticallonexpansive mapping. The class of quasi -φ- asymptoticallonexpansive mappings contains popely the class of elativelonexpansive mappings (see Matsushita and Takahashi [9] as a subclass. Let E be a smooth, stictly convex and eflexive Banach space, C be a nonempty closed convex subset of E, T : C C be a mapping and F ix(t be the set of fixed points of T. A point p C is said to be an asymptotic fixed point of T if thee exists a sequence {x n } C such that x n p and x n T x n 0. We denoted the set of all asymptotic fixed points of T by F ix(t. A point p C is said to be a stong asymptotic fixed point of T, if thee exists a sequence {x n } C such that x n p and x n T x n 0. We denoted the set of all stong asymptotic fixed points of T by F ix(t. A mapping T : C C is said to be elativelonexpansive [9], [11], if F ix(t, F ix(t = F ix(t and φ(p, T x φ(p, x, x C, p F ix(t. A mapping T : C C is said to be weak elatively nonexpansive [15], if F ix(t, F ix(t = F ix(t and φ(p, T x φ(p, x, x C, p F ix(t. Remak I.6. If E is a eal Hilbet space H, then φ(x, y = x y 2 and Π C = P C (the metic pojection of H onto C. Remak I.7. Diect fom the definition of a mapping, so it is easy to see that Each elativelonexpansive mapping is closed. Evey quasi -φ- nonexpansive mapping is quasi -φasymptotically nonexpansive mapping with {k n = 1}, but the convese is not tue. Each weak elativelonexpansive mapping is a quasi -φ- nonexpansive mapping (because it does not equie the condition F ix(t = F ix(t, but the convese is not tue. Evey elativelonexpansive mapping is a weak elatively nonexpansive mappings, but the convese is not tue. Evey countable family of weak elativelonexpansive mappings is a countable family of of unifomly closed and quasi -φ- nonexpansive mappings, and so it is a countable family of unifomly closed and quasi -φasymptotically nonexpansive mappings. Definition I.8. Let {S i } i=1 : C C be a sequence of mappings. {S i } i=1 is said to be a family of unifomly quasi -φasymptotically nonexpansive mappings, if i=1 F ix(s i and thee exists a sequence {k n } [1, with k n 1 such that fo each i 1, φ(u, Si nx k nφ(u, x, u i=1 F ix(s i, x C, n 1. Definition I.9. A mapping S : C C is said to be unifomly L-Lipschitz continuous, if thee exists a constant L > 0 such that S n x S n y L x y, x, y C, n 1. Lemma I.10. (see Albe [1]. Let C be a nonempty closed and convex subset of a smooth and stictly convex eflexive Banach space E, and let x E. Then, φ(x, Π C (y + φ(π C (y, y φ(x, y, x C, y E. Lemma I.11. (see Kamimua and Takahashi [7]. Let C be a nonempty closed and convex subset of a smooth and stictly convex eflexive Banach space E, and let x E and u C. Then, u = Π C (x u y, Jx Ju 0, y C. We make use of the function V : E E R defined by V (x, x = x 2 2 x, x + x 2, x E, x E. Obseve that V (x, x = φ(x, J 1 x fo all x E and x E. The following lemma is well-known. Lemma I.12. (see Albe [1] Let E be a smooth and stictly convex eflexive Banach space with E as its dual, then V (x, x + 2 J 1 x x, y V (x, x + y, fo all x E and x, y E. ISSN:

4 Lemma I.13. (see Kamimua and Takahashi [7]. Let E be a unifomly convex and smooth eal Banach space and let {x n } and { } be two sequences of E. If φ(x n, 0 and eithe {x n } o { } is bounded, then x n 0. Fo solving the genealized equilibium poblem, let us assume that the mapping B : C E is δ-invese-stongly monotone mapping and the bifunction F : C C R satisfies the following conditions: (A1 F (x, x = 0, fo all x C; (A2 F is monotone, i.e., F (x, y + F (y, x 0, x, y C; (A3 lim sup F (x + t(z x, y F (x, y, x, y, z C; t 0 (A4 fo any y C, the function y F (x, y is convex and lowe semicontinuous. Lemma I.14. Let E be a unifomly smooth and stictly convex Banach space with the Kadec-Klee popety, {x n } and { } be two sequences of E, and p E. If x n p and φ(x n, 0, then p. Lemma I.15. (see Blum and Oettli [2]. Let C be a nonempty closed and convex subset of a smooth and stictly convex eflexive Banach space E, and let F : C C R be a bifunction satisfying the following conditions (A1 - (A4. Let > 0 be any given numbe and x E be any point. Then, thee exists a z C such that F (z, y + 1 y z, Jz Jx 0, y C. (I.11 Lemma I.16. (see Chang et al. [4] Let C be a nonempty closed and convex subset of a smooth and stictly convex eflexive Banach space E, and let B : C E be a δ- invese-stongly monotone mapping and F : C C R be a bifunction satisfying the following conditions (A1-(A4. Let > 0 be any given numbe and x E be any point. Then, thee exists a point z C such that F (z, y + y z, Bz + 1 y z, Jz Jx 0, y C. (I.12 Lemma I.17. (see Chang et al. [4] Let C be a nonempty closed and convex subset of a smooth and stictly convex eflexive Banach space E, and let B : C E be a δ- invese-stongly monotone mapping and F : C C R be a bifunction satisfying the following conditions (A1-(A4. Let > 0 and x E, and we define a mapping T F : E C as follows: fo any x C, T F x = {z C : F (z, y + y z, Bz + 1 y z, Jz Jx 0, } y C. Then, the following conclusions hold: (1 T F is single-valued; (2 T F is a fimlonexpansive type mapping, i.e., T F x T F y, JT F x JT F y T F x T F y, Jx Jy, x, y E (3 F ix(t F = F ix(t F = EP ; (4 EP is a closed and convex set of C; (5 φ(p, T F x + φ(t F x, x φ(p, x, p F ix(t F ; (6 fo each n 1, n > d > 0 and u n C with lim n u n = lim n T n u n = u, we have F (u, y + y u, Bu 0, y C. Lemma I.18. (see Cioanescu [6] Let C be a nonempty closed and convex subset of a eal unifomly smooth and stictly convex Banach space E with the Kadec-Klee popety, S : C C be a closed and quasi -φ- asymptotically nonexpansive mapping with a sequence {k n } [1, and k n 1. Then, F ix(s is closed and convex in C. Lemma I.19. (see Chang et al. [3] Let E be a unifomly convex Banach space, > 0 be a positive numbe and B (0 be a closed ball of E. Then, fo any given sequence {x n } n=1 B (0 and fo any given { n } n=1 (0, 1 with n=1 n = 1, thee exists a continuous, stictly inceasing and convex function g : [0, 2 [0, with g(0 = 0 such that fo any positive integes i, j with i < j, 2 n x n n x n 2 i j g( x i x j. n=1 n=1 Lemma I.20. (see Xu [17] Let E be a 2-unifomly convex eal Banach space, then fo all x, y E, we have x y 2 Jx Jy, c2 whee J is the nomalized duality mapping of E and 0 < c 1, and 1 c is called the 2-unifomly convex constant of E. We note that evey unifomly convex Banach space has the Kadec-Klee popety. Fo moe details on Kadec-Klee popety, the eade is efeed to [6], [16]. Example I.21. Let E be a unifomly smooth and stictly convex Banach space and A E E be a maximal monotone mapping such that its zeo set A 1 (0 is nonempty. Then, we get J = (J + A 1 J is closed and quasi -φasymptotically nonexpansive mapping fom E onto D(A and F ix(j = A 1 (0. A monotone A is said to be maximal if its gaph G(A = {(x, y : y Ax} is not popely contained in the gaph of any othe monotone opeato. If A is maximal monotone, then the solution set A 1 0 is closed and convex. Let E be a eflexive, stictly convex and smooth Banach space, it is known that A is a maximal monotone if and only if R(J + A = E fo all > 0. The esolvent of monotone opeato A is defined by J = (J + A 1 J, > 0. A well-known method fo solving zeoes of maximal monotone opeato is poximal point algoithm. Let A be a maximal monotone opeato in a Hilbet space H. The poximal point algoithm geneates, fo stating x 1 = x H, a sequence {x n } in H by x n+1 = J n x n, n 1, (I.13 whee { n } (0, and J n = (I + n A 1. Rockafella [13] poved that the sequence {x n } defined by (I.13 conveges weakly to an element of A 1 0. Let E be a smooth stictly convex and eflexive Banach space, C be a nonempty closed convex subset of E and A E E be a monotone opeato satisfying the following: D(A C J 1 ( >0 R(J + A. ISSN:

5 Then the esolvent J : C D(A of A is defined by J x = {z D(A : Jx Jz + Az, x C}. J is a single-valued mapping fom E to D(A. Also, A 1 (0 = F (J fo all > 0, whee F (J is the set of all fixed points of J. Fo any > 0, the Yosida appoximation A : C E of A is define by A x = Jx JJx fo all x C. We know that A x A(J x fo all > 0 and x E. Lemma I.22. (see Kohsaka and Takahashi [8] Let E be a smooth stictly convex and eflexive Banach space, C be a nonempty closed convex subset of E and A E E be a monotone opeato satisfying D(A C J 1 ( >0 R(J + A. Fo any > 0, let J and A be the esolvent and the Yosida appoximation of A, espectively. Then the following hold: (1 φ(p, J x + φ(j x, x φ(p, x fo all x C and p A 1 0; (2 (J x, A x A fo all x C; (3 F (J = A 1 0. Lemma I.23. (see Rockafella [12] Let E be a eflexive stictly convex and smooth Banach space. Then an opeato A E E is maximal monotone if and only if R(J+A = E fo all > 0. Let A be an invese-stongly monotone mapping of C into E which is said to be hemicontinuous it fo all x, y C, the mapping F : [0, 1] E, defined by F (t = A(tx+(1 ty is continuous with espect to the weak topology of E. We define N C (v the nomal cone fo C at a point v C, that is, N C (v = {x E : v y, x 0, y C}. Lemma I.24. (see Rockafella [12] Let C be a nonempty, closed and convex subset of a Banach space E and A is monotone, hemicontinuous opeato of C into E. Let U E E be an opeato defined as follows: { Av + NC (v, v C; Uv =, v / C. Then, U is maximal monotone and U 1 (0 = V I(C, A. II. THE SYSTEM OF GENERALIZED EQUILIBRIUM PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES In this section, we efe a stong convegence theoem by [10] which solves the poblem of finding a common solution of the system of genealized equilibium poblems and fixed point poblems in Banach spaces. Fo example the following the condition is satisfied. Example II.1. Let A : R R by given by { 0, if x 0; A := 4x, if x > 0. 1 Then, A is 4-invese-stongly monotone mapping with V I(R, A = A 1 (0 = (, 0]. We give an example fo nonlinea mappings to illustate the next esults. Example II.2. Let S : R R be given by S := (I +B 1, fo > 0, whee x + 1, if x (, 1]; Bx := 0, if x ( 1, 0], 2x, if x (0,. Then, we get that J := (I+B 1 = S is unifomly L- Lipschitz continuous and quasi -φ- asymptoticallonexpansive with {k n } = 1 fo each n 1 and F ix(j = B 1 (0 = F ix(s = [ 1, 0]. Now, we emak that, let C be a subset of a eal Banach space E and A : C E be an invese stongly monotone mapping satisfying Ax Ax Ap, fo all x C and p V I(C, A, then V I(C, A = A 1 (0 = {p C : Ap = 0}. Theoem II.3. Let C be a nonempty, closed and convex subset of a unifomly smooth and 2-unifomly convex eal Banach space E. Suppose that (B1 Let B k : C E fo each k = 1, 2, 3,..., M be a finite family of δ k -invese-stongly monotone mappings, and let F k : C C R be a bifunction which satisfies conditions (A1-(A4. (B2 Let {T i } i=1 and {S j} j=1 : C C be countable families of unifomly closed and ω i,µ j -Lipschitz continuous and quasi -φ- asymptoticallonexpansive mappings with sequences {k n }, {l n } [1, and k n 1, l n 1, espectively. (B3 Let A n : C E fo each n = 1, 2, 3,..., N be a let γ = min{γ ( n : n = 1, 2, 3,..., N}. ( (B4 Ω := i=1 F ix(t i j=1 F ix(s j ( M SGEP (F ( N k, B k V I(C, A n k=1 n=1 is a nonempty and bounded in C. x 0 C chosen abitay, C 0 = C z n = Π C J 1 (Jx n n A n x n = J 1 (β (1 i=1 β(2 n,i JT n i x n + j=1 β(3 n,j JSn j z n M 1,n...T F2 2,n T F1 1,n C n+1 = {v C n : φ(v, u n φ(v, x n + θ n } x n+1 = Π Cn+1 (x 0, n 0, (II.1 k,n : E C, k = 1, 2, 3,..., M, is a mapping defined by (I.17 with F = F k and = k,n and it is the solutions to the following system of genealized equilibium poblem: F k (z, y + y z, B k z + 1 y z, Jz Jx 0, y C, k = 1, 2, 3,..., M. k,n [d,, fo some d > 0, θ n = sup p Ω (max{k n, l n } 1φ(p, x n, A n A n (mod N, A n x A n x A n p, fo all x C and p Ω. Let { n } be a sequence in [0,1] such that 0 < n < c2 γ 2, whee 1 c is the 2-unifomly convex constant of E. Let {β (1 n,0 }, {β(2 n,i }, {β(3 n,j } be sequences in [0,1] satisfying the following conditions: ISSN:

6 lim inf n β (1 n,0 β(3 n,j > 0, i, j 1, i j. Then, the sequence {x n } conveges stongly to p = Π Ω (x 0. Poof: See the complete this poof in [10]. III. THE COMPLEMENTARITY PROBLEMS Complementaity poblems ae used to model seveal poblems of economics, physics, optimization theoy and engineeing. Definition III.1. Let K be a nonempty, closed and convex cone in E. We define the pola K of K as follows: K = {y E : x, y 0, x K}. Let a mapping A : K E is an opeato, then an element u K is called a solution of complementaity poblems if [16] Au K, and u, Au = 0. The set of solutions of the complementaity poblem is denoted by CP (K, A. Seveal poblems aising in diffeent fields such as mathematical pogamming, mechanics, game theoy ae to find the solutions of complementaity poblem and the vaiational type poblem. Lemma III.2. Let X be a nonempty, closed convex cone in a locally convex topological vecto space E and let X be the pola of X. Let T be a mapping of X into E. Then, the set of solutions of the complementaity poblem equals the set of solutions of the vaiational inequality. That is {x X : T x X and x, T x = 0} = {x X : u x, T x 0, u X}. Poof: Let x X be a solution of complementaity poblem. Then, fo any u X, we have u x, T x = u, T x x, T x = u, T x 0. Theefoe, x is a solution of vaiational inequality poblem. Convesely, let x X be a solution of vaiational inequality poblem. Then, we have u x, T x 0, u X. In the paticula, if u = 0, we have x, T x 0. If u = x, with = 1, we have x x, T x = ( 1 x, T x 0. Hence, x, T x 0. Theefoe, x, T x = 0. Next, we show that T x X. If not, thee exists u 0 X such that x, T x = 0. On the othe hand, u 0 x, T x 0, So that, we have 0 > u 0, T x x, T x = 0. This is a contadiction. This implies that x is a solution of complementaity poblem. Theoem III.3. Let K be a nonempty, closed and convex subset of a unifomly smooth and 2-unifomly convex eal Banach space E. Suppose that (C1 Let B k : K E fo each k = 1, 2, 3,..., M be a finite family of δ k -invese-stongly monotone mappings, and let F k : K K R be a bifunction which satisfies conditions (A1-(A4. (C2 Let {T i } i=1 and {S j} j=1 : K K be countable families of unifomly closed and ω i,µ j -Lipschitz continuous and quasi -φ- asymptoticallonexpansive mappings with sequences {k n }, {l n } [1, and k n 1, l n 1, espectively. (C3 Let A n : K E fo each n = 1, 2, 3,..., N be a let γ = min{γ ( n : n = 1, 2, 3,..., N}. ( (C4 Ω := i=1 F ix(t i j=1 F ix(s j ( M SGEP (F ( N k, B k CP (K, A n k=1 n=1 is a nonempty and bounded in C. x 0 K chosen abitay, K 0 = K z n = Π K J 1 (Jx n n A n x n = J 1 (β (1 i=1 β(2 n,i JT i nx n + j=1 β(3 n,j JSn j z n M 1,n...T F2 2,n T F1 1,n K n+1 = {v K n : φ(v, u n φ(v, x n + θ n } x n+1 = Π Kn+1 (x 0, n 0, (III.1 k,n : E K, k = 1, 2, 3,..., M, is a mapping defined by (I.17 with F = F k and = k,n and it is the solutions to the following system of genealized equilibium poblem: F k (z, y + y z, B k z + 1 y z, Jz Jx 0, y C, k = 1, 2, 3,..., M. k,n [d,, fo some d > 0, θ n = sup(max{k n, l n } 1φ(p, x n, p Ω A n A n (mod N, A n x A n x A n p, fo all x C and p Ω. Let { n } be a sequence in [0,1] such that 0 < n < c2 γ 2, whee 1 c is the 2-unifomly convex constant of E. Let {β (1 n,0 }, {β(2 n,i }, {β(3 n,j } be sequences in [0,1] satisfying the following conditions: lim inf n β (1 n,0 β(3 n,j > 0, i, j 1, i j. Then, the sequence {x n } conveges stongly to p = Π Ω (x 0. Poof: As in the poof of Lemma III.2, we have V I(C, A n = CP (K, A n, So, we obtain the above the esults. ISSN:

7 IV. A ZERO POINT OF MAXIMAL MONOTONE OPERATORS Definition IV.1. Let E be a unifomly smooth and stictly convex Banach space and A be a maximal monotone opeato fom E to E. Fo each > 0, we can define a single valued mapping J : E D(A by J = (J + A 1 J and such a mapping J is called the esolvent of A. It is easy to show that F ix(j = A 1 (0, fo all > 0. It is well known that if E is a smooth stictly convex and eflexive Banach space and A is a continuous monotone opeato with A 1 0, then J is weak elativelonexpansive mappings. We know that F (J is closed and convex. One of the most inteesting and impotant poblems in the theoy of maximal monotone opeatos is to find an efficient iteative algoithm to compute appoximately zeoes of maximal monotone opeatos. The poximal point algoithm of Rockafella [13] is ecognized as a poweful and successful algoithm fo finding a solution of maximal monotone opeatos. Theoem IV.2. Let C be a nonempty, closed and convex subset of a unifomly smooth and 2-unifomly convex eal Banach space E. Suppose that (D1 Let B k : C E fo each k = 1, 2, 3,..., M be a finite family of δ k -invese-stongly monotone mappings, and let F k : C C R be a bifunction which satisfies conditions (A1-(A4. (D2 Let {T i } i=1 and {S j} j=1 : C C be countable families of unifomly closed and weak elativelonexpansive mappings. (D3 Let A n : C E fo each n = 1, 2, 3,..., N be a let γ = min{γ ( n : n = 1, 2, 3,..., N}. ( (D4 Ω := i=1 F ix(t i j=1 F ix(s j ( M SGEP (F ( N k, B k V I(C, A n k=1 n=1 is a nonempty and bounded in C. x 0 C chosen abitay, C 0 = C z n = Π C J 1 (Jx n n A n x n = J 1 (β (1 i=1 β(2 n,i JT ix n + j=1 β(3 n,j JS jz n M 1,n...T F2 2,n T F1 1,n C n+1 = {v C n : φ(v, u n φ(v, x n } x n+1 = Π Cn+1 (x 0, n 0, (IV.1 k,n : E C, k = 1, 2, 3,..., M, is a mapping defined by (I.17 with F = F k and = k,n and it is the solutions to the following system of genealized equilibium poblem: F k (z, y + y z, B k z + 1 y z, Jz Jx 0, y C, k = 1, 2, 3,..., M. k,n [d,, fo some d > 0, A n A n (mod N, A n x A n x A n p, fo all x C and p Ω. Let { n } be a sequence in [0,1] such that is the 2-unifomly convex constant of E. Let {β (1 n,0 }, {β(2 n,i }, {β(3 n,j } be sequences in [0,1] satisfying the following conditions: 0 < n < c2 γ 2, whee 1 c lim inf n β (1 n,0 β(3 n,j > 0, i, j 1, i j. Then, the sequence {x n } conveges stongly to p = Π Ω (x 0. Poof: Since {T i } i=1 and {S j} j=1 ae countable families of unifomly closed and weak elativelonexpansive mappings, By Remak I.7, it is countable families of unifomly closed and quasi -φ- nonexpansive mappings, and it is countable families of unifomly closed and quasi -φasymptotically nonexpansive mappings. Theefoe, it can be obtained fom Theoem II.3 immediately. If T i = T, S j = S, F k = F, B k = B and A n = A whee i, j N, k = 1, 2, 3,..., M and n = 1, 2, 3,..., N in Theoem II.3, then the Theoem II.3 is educed to the following coollay. Theoem IV.3. Let C be a nonempty, closed and convex subset of a unifomly smooth and 2-unifomly convex eal Banach space E. Suppose that (E1 Let B k : C E fo each k = 1, 2, 3,..., M be a finite family of δ k -invese-stongly monotone mappings, and let F k : C C R be a bifunction which satisfies conditions (A1-(A4. (E2 Let A and B be two maximal monotone opeatos fom E to E and let J A and J B be the esolvent of A and B, espectively, whee > 0. (E3 Let A n : C E fo each n = 1, 2, 3,..., N be a let γ = min{γ n : n = 1, 2, 3,..., N}. (E4 Ω := A 1 (0 B 1 (0 ( M SGEP (F k, B k k=1 ( N n=1 n V I(C, A is a nonempty and bounded in C. x 0 C chosen abitay, C 0 = C z n = Π C J 1 (Jx n n A n x n = J 1 (β (1 i=1 β(2 n,i JJA i x n + j=1 β(3 n,j JJB j z n M 1,n...T F2 2,n T F1 1,n C n+1 = {v C n : φ(v, u n φ(v, x n + θ n } x n+1 = Π Cn+1 (x 0, n 0, (IV.2 k,n : E C, k = 1, 2, 3,..., M, is a mapping defined by (I.17 with F = F k and = k,n and it is the solutions to the following system of genealized equilibium poblem: F k (z, y + y z, B k z + 1 y z, Jz Jx 0, y C, k = 1, 2, 3,..., M. k,n [d,, fo some d > 0, θ n = sup p Ω (max{k n, l n } 1φ(p, x n, A n A n (mod N, A n x A n x A n p, fo all x C and p Ω. Let { n } be a sequence in [0,1] such that is the 2-unifomly convex constant of E. Let {β (1 n,0 }, {β(2 n,i }, {β(3 n,j } be sequences in [0,1] satisfying the following conditions: 0 < n < c2 γ 2, whee 1 c lim inf n β (1 n,0 β(3 n,j > 0, i, j 1, i j. Then, the sequence {x n } conveges stongly to p = Π Ω (x 0. ISSN:

8 Poof: It is well-known that fo each i 1, J A i is a elativelonexpansive mapping (see, fo example [9], [11]. Theefoe, fo each p F ix(j A i and w E, we have φ(p, J A i w φ(p, w. Again, by the same method we can pove that the set of stong asymptotic fixed points, so that F ix({j A i } i=1 = F ix(j A i = A 1 (0. i=1 This implies that {J A i } i=1 is a countable family of weak elativelonexpansive mapping with the common fixed point set i=1 F ix(j A i = A 1 (0. By the simila way, we can pove that {J B j } j=1 is a countable family of weak elativelonexpansive mapping with the common fixed point set j=1 F ix(j B j = B 1 (0. Hence, the conclusion of the Theoem IV.3 can be obtained fom Theoem IV.2 immediately. A. Application to Hilbet spaces Theoem IV.4. Let C be a nonempty, closed and convex subset of a Hilbet space H. Suppose that (F1 Let B k : C H fo each k = 1, 2, 3,..., M be a finite family of δ k -invese-stongly monotone mappings, and let F k : C C R be a bifunction which satisfies conditions (A1-(A4. (F2 Let {T i } i=1 and {S j} j=1 : C C be countable families of unifomly closed and ω i,µ j -Lipschitz continuous and quasi -φ- asymptoticallonexpansive mappings with sequences {k n }, {l n } [1, and k n 1, l n 1, espectively. (F3 Let A n : C H fo each n = 1, 2, 3,..., N be a let γ = min{γ ( n : n = 1, 2, 3,..., N}. ( (F4 Ω := i=1 F ix(t i j=1 F ix(s j ( M SGEP (F ( N k, B k V I(C, A n k=1 n=1 is a nonempty and bounded in C. x 0 C chosen abitay, C 0 = C z n = P C (x n n A n x n = β (1 i=1 β(2 n,i T i nx n + j=1 β(3 n,j Sn j z n M 1,n...T F2 2,n T F1 1,n C n+1 = {v C n : v u n v x n + θ n } x n+1 = P Cn+1 (x 0, n 0, (IV.3 k,n : H C, k = 1, 2, 3,..., M, is a mapping defined by (I.17 with F = F k and = k,n and it is the solutions to the following system of genealized equilibium poblem: F k (z, y + y z, B k z + 1 y z, Jz Jx 0, y C, k = 1, 2, 3,..., M. k,n [d,, fo some d > 0, θ n = sup p Ω (max{k n, l n } 1 p x n, A n A n (mod N, A n x A n x A n p, fo all x C and p Ω. Let { n } be a sequence in [a, b] fo some a, b such that 0 < a < b < n 2. Let {β(1 n,0 }, {β(2 n,i }, {β(3 n,j } be sequences in [0,1] satisfying the following conditions: lim inf n β (1 n,0 β(3 n,j > 0, i, j 1, i j. Then, the sequence {x n } conveges stongly to p = P Ω (x 0. Poof: If E = H, a Hilbet space, then E is 2-unifomly convex (we can choose c = 1 and unifomly smooth Banach space. Moeove, J = I, identity mapping on H and Π C = P C, pojection mapping fom H to C. Thus, the conclusion of the Theoem IV.4 can be obtained fom Theoem II.3 immediately. V. ACKNOWLEDGEMENTS The second autho would like to thank the Bansomdejchaopaya Rajabhat Univesity (BSRU fo suppot. Moeove, the thid autho was suppoted by the Theoetical and Computational Science (TaCS Cente. REFERENCES [1] Y. I. Albe, Metic and genealized pojection opeatos in Banach space : popeties and applications. In : Katosato, AG(ed. Theoy and Applications of Nonlinea opeatos of Accetive and Monotone Type, Lectue Notes in Pue and Applied Mathematics, Macel Dekke, Newyok,vol.178, pp , [2] E. Blum and W. Oettli, Fom optimization and vaiational inequalities to equilibium poblems, The Mathematics Student, vol.63, pp , [3] S. S. Chang, J. K. Kim and X. R. 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9 [17] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinea Analysis, vol.16, pp , [18] H. Y. Zhou and G. L. Gao, Convegence theoems of a modified hybid algoithm fo a family of quasi -φ- asymptoticallonexpansive mappings. Jounal of Applied Mathematics and Computing, vol. 32, pp , Khanittha Pomluang eceived maste s degee of education in majo Mathematics Education(M.E fom the Chiang Mai Univesity, Chiang Mai, Thailand, in 2004 and bachelo s degee of education in majo Science-Mathematics Education fom the Buapha Univesity, Chonbui, Thailand, in She stated he caee as a lectue at the depatment of leaning management, faculty of education, Buapha Univesity (BUU, Chonbui, Thailand in She is cuently a PhD. student of faculty of science, majoing in applied mathematics at King Mongkuts Univesity of Technology Thonbui (KMUTT, Bangkok, Thailand. The eseach topic is egulaization method and poximal point algoithm fo invese poblems and vaiational inequalities of nonlinea opeatos. Pongus Phuangphoo D.Pongus Phuangphoo eceived his Bachelo of Education in Mathematics, Fist Class Hono fom Bansomdejchaopaya Rajabhat Univesity, Bangkok, Thailand in 1998, his Maste of Education in Mathematics fom Sinakhainwiot Univesity, Bangkok, Thailand, in 2002, his Docto of Philosophy in Applied Mathematics fom King Mongkuts Univesity of Technology Thonbui, Bangkok, Thailand, in 2012, unde the supevision of Associate Pofesso Poom Kumam, (Ph.D.. He stated his caee as a lectue at the Depatment of Mathematics, Faculty of Eduacation, Bansomdejchopaya Rajabhat Univesity (BSRU, Bangkok, Thailand, in Next, in 2014, he was an adjunct Assistant Pofesso in Depatment of Mathematics, Faculty of Eduacation, Bansomdejchopaya Rajabhat Univesity (BSRU, and he eceived the cetificate about the outstanding eseache of Faculty of Education in academic yea and Gyeongsang National Univesity, Republic of Koea. In 2012, he took the same ole, a Visiting Pofesso, at the Univesity of Albeta, AB, Canada. Moeove, he also a Visiting Pofesso at Kyushu Institute of Technology (KIT, Tobata Campus, Kitakyushu, Japan, a Visiting Pofesso at, The Univesity of Newcastle, Austalia, 1-15 Febuay 2013 and a Visiting Pofesso at Univesity of Jean, Andalucia, Spain, 1-20 July 2014 and July D. Poom is the Associate Edito-in-Chief of the Thai Jounal of Mathematics, Assistance Editos fo The Jounal of Nonlinea Analysis and Optimization: Theoy & Applications, Edito fo The Jounal of Nonlinea Science and Applications (SCIE, The jounal BULLETIN of the Malaysian Mathematical Sciences Society (SCIE and The editoial boad of Fixed Point Theoy and Applications (July 2012-Apil 2015 (SCIE, Ceative Mathematics and Infomatics, Spingeplus (SCIE and Guest Editos of Abstact and Applied Analysis. He won two of the most impotant awads fo mathematicians. The fist one is the TRF-CHE-Scopus Young Reseache Awad in 2010 which is the awad given by the copoation fom thee oganizations: Thailand Reseach Fund (TRF, the Commission of Highe Education (CHE, and Elsevie Publishe (Scopus. The second awad was in 2012 when he eceived the TWAS Pize fo Young Scientist in Thailand which is given by the Academy of Sciences fo the Developing Wold TWAS (UNESCO togethe with the National Reseach Council of Thailand. In 2014, the thid awad is the Fellowship Awad Fo Outstanding Contibution to Mathematics fom Intenational Academy of Physical Science, Allahabad, India. In 2015 D. Poom Kumam has been awaded Thailand Fontie Autho Awad 2015, Awad fo outstanding eseache who has published woks and has often been used as a efeence o evaluation citeia of the database Web of Science. His eseach inteest focuses on fixed-point theoy with elated with optimisation poblems in both pue science and applied science. D. Poom has ove 300 scientific papes and pojects eithe pesented o published. Poom Kumam eceived his B.S. degee in mathematics education fom the Buapha Univesity, Chonbui, Thailand, in 2000, his M.S. degee (mathematics fom the Chiang Mai Univesity, Chiang Mai, Thailand, in 2002, unde the supevision of Pofesso Sompong Dhompongsa, and his Ph.D. degee (mathematics fom the Naesuan Univesity, Phitsanulok, Thailand, in 2007, unde the supevision of Pofesso Somyot Plubtieng. He stated his caee as a lectue at the Depatment of Mathematics, King Mongkuts Univesity of Technology Thonbui (KMUTT, Bangkok, Thailand, in In 2008, he eceived a gant fom Fanco- Thai Coopeation fo shot-tem eseach at Laboatoie de Mathmatiques, Univesit de Betagne Occidentale, Fance. Poom Kumam is an Adjunct Associate Pofesso at Depatment of Mathematics, Faculty of Science, King Mongkut s Univesity of Technology Thonbui (KMUTT, and a Head of Theoetical and Computational Science (TaCS Cente of KMUTT. In 2014 he is cuently the Associate Dean fo Reseach fo the Faculty of Science, KMUTT. Futhemoe, in 2011, Kumam had an oppotunity to take a ole as a Visiting Pofesso at Kyungnam Univesity ISSN: