THROUGHOUT this paper, we let C be a nonempty

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1 Strong Convergence Theorems of Multivalued Nonexpansive Mappings and Maximal Monotone Operators in Banach Spaces Kriengsak Wattanawitoon, Uamporn Witthayarat and Poom Kumam Abstract In this paper, we prove a strong convergence theorem for fixed points of sequence for multivalued nonexpansive mappings and a zero of maximal monotone operator in Banach spaces by using the hybrid projection method. Our results modify and improve the recent results in the literatures. Index Terms Fixed Point, multivalued nonexpansive mapping, maximal monotone operator. I. INTRODUCTION THROUGHOUT this paper, we let C be a nonempty closed convex subset of a real Banach space E. A mapping t : C C is said to be nonexpansive if tx ty x y for all x, y C. We denote by F (t) the set of fixed points of t, that is F (t) = {x C : x = tx}. A mapping t is said to be an asymptotic fixed point of t (see [11]) if C contains a sequence {x n } which converges weakly to p such that lim x n tx n = 0. The set of asymptotic fixed points of t will be denoted by F (t). A mapping t from C into itself is said to be relatively nonexpansive [9], [12], [16] if F (t) = F (t) and ϕ(p, tx) ϕ(p, x) for all x C and p F (t). The asymptotic behavior of a relatively nonexpansive mapping was studied in [2], [3]. Let N(C) and CB(C) denote the family of nonempty subsets and nonempty closed bounded subsets of C, respectively. Let H : CB(C) CB(C) R + be the Hausdorff distance on CB(C), that is H(A < B) = max{sup dist(a, A), sup dist(b, B)}, a A b B for every A, B CB(C), where dist(a, B) = inf{ a b : b B} is the distance from the point a to the subset B of C. A multi-valued mapping T : E CB(C) is said to be nonexpansive if H(T x, T y) x y, for all x, y C. An element p C is called a fixed point of T : C CB(C), if p T p. The set of fixed point T is denoted by F (T ). This work was supported by the Office of Higher Education Commission, Thailand under the Higher Education Research Promotion. K. Wattanawitoon gratefully acknowledges the support provided by the King Mongkut s University of Technology Thonburi (KMUTT) during the first author stays at the King Mongkut s University of Technology Thonburi (KMUTT) as a post- doctoral researcher (KMUTT-Post-doctoral Fellowship). K. Wattanawitoon, U. Witthayarat and P. Kumam are with the Department of Mathematics, Faculty of Science, King Mongkut s University of Technology Thonburi (KMUTT), Bang Mod, Thrung Kru, Bangkok 10140, Thailand s: kriengsak.wat@rmutl.ac.th(k.wattanawitoon), u.witthayarat@hotmail.com (U. Witthayarat) and poom.kum@kmutt.ac.th (P. Kumam) A point p C is said to be an asymptotic fixed point of T : C CB(C), if there exists a sequence {x n } C such that x n x E and d(x n T x n ) 0. Denote the set of all asymptotic fixed points of T by F (T ). T is said to be relatively nonexpansive, if F (T ), F (T ) = F (T ) and ϕ(p, z) ϕ(p, x), x C, p F (T ), z T z. A mapping T is said to be closed, if for any sequence {x n } C with x n x C and d(y, T x n ) 0 then d(y, T x) 0. T is said to be quasi-ϕ-nonexpansive if F (T ) and ϕ(p, z n ) ϕ(p, x), x C, p F (T ), z T n (x). T is said to be quasi- ϕ-asymptotically nonexpansive if F (T ) and there exists a real sequence k n [1, + ), k n 1 such that ϕ(x, z n ) k n ϕ(p, x), x C, p F (T ), z n T n x. (1) A mapping T is said to be total quasi-ϕ-asymptotically nonexpansive if F (T ) and there exist nonnegative real sequences {ν n } and {µ n } with ν n, µ n 0 as n and a strictly increasing continuous function ζ : R + R + with ζ(0) = 0 such that ϕ(x, z n ) k n ϕ(p, x) + ν n ζ[ϕ(p, x)] + µ n, x C, p F (T ), z n T n x. (2) A mapping T is said to be uniformly L-Lipschitz continuous, if there exists a constant L > 0 such that x n y n L x y, where x, y C, x n T n x, y n T n y. Let E be a real Banach space with dual E. Denote by, the duality product. The normalized duality mapping J from E to 2 E is defined by Jx = {f E : x, f = x 2 = f 2 }, for all x E. The function ϕ : E E R is defined by ϕ(x, y) = x 2 2 x, Jy + y 2, for all x, y E. (3) A mapping T is said to be hemi-relatively nonexpansive (see [12]) if F (T ) and ϕ(p, T x) ϕ(p, x), for all x C and p F (T ). A point p in C is said to be an asymptotic fixed point of T [2] if C contains a sequence {x n } which converges weakly to p such that the strong lim (x n T x n ) = 0. The set of asymptotic fixed points of T will be denoted by ˆF (T ). A hemi-relatively nonexpansive mapping T from C into itself is called relatively nonexpansive if ˆF (T ) = F (T ), see [8], [10], [14]) for more details.

2 Matsushita and Takahashi [8] introduced the iteration in a Banach space E: x 0 = x C, chosen arbitrarily, y n = J 1 (α n Jx n + (1 α n )JT x n ), C n = {z C : ϕ(z, y n ) ϕ(z, x n )}, (4) Q n = {z C : x n z, Jx Jx n 0}, x n+1 = Π Cn Q n x, n = 0, 1, 2,..., and proved that the sequence {x n } converges strongly to Π F (T ) x. Qin and Su [10] showed that the sequence {x n }, which is generated by a relatively nonexpansive mappings T in a Banach space E, as following x 0 C, chosen arbitrarily, y n = J 1 (α n Jx n + (1 α n )JT z n ), z n = J 1 (β n Jx n + (1 β n )JT x n ), C n = {v C : ϕ(v, y n ) α n ϕ(v, x n )+ (5) (1 α n )ϕ(v, z n )}, Q n = {v C : Jx 0 Jx n, x n v 0}, x n+1 = Π Cn Q n x 0, converges strongly to Π F (T ) x 0. Moreover, they also showed that the the sequence {x n }, which is generated by x 0 C, chosen arbitrarily, y n = J 1 (α n Jx 0 + (1 α n )JT x n ), C n = {v C : ϕ(v, y n ) α n ϕ(v, x 0 ) + (1 α n )ϕ(v, x n ), Q n = {v C : Jx 0 Jx n, x n v 0}, x n+1 = Π Cn Q n x 0, (6) converges strongly to Π F (T ) x 0. In 2012, Chang et al. [4] modified the Halpern-type iteration algorithm for total quasi-ϕ-asymptotically nonexpansive mapping to have the strong convergence under a limit condition in Banach space. Recently, Tang and Chang [15] introduce the concept of total quasi-ϕ-asymptotically nonexpansive multi-value mapping in Banach space, let {x n } be a sequence generated by x 0 C, is arbitrary, C 0 = C, y n = J 1 (α n Jx n + (1 α n )JT z n ), z n = J 1 (β n Jx n + (1 β n )JT w n ), C n+1 = {v C n : ϕ(v, y n ) ϕ(v, x n ) + ξ n }, x n+1 = Π Cn+1 x 0, n 0, where w n T n x n, ξ n = ν n sup p F ζ(ϕ(p, x n )) + µ n and showed that the sequence {x n } converges strongly to Π F x 0. In this paper, motivated by Tang and Chang [15], we prove strong convergence theorems for fixed points of sequence for multicalued nonexpansive mapping and a zero of maximal monotone operator in Banach space by using the hybrid projection methods. Our results extend and improve the recent results by Tang and Chang [15] and many others. II. PRELIMINARIES In this section, we will recall some basic concepts and useful well known results. A Banach space E is said to be strictly convex if x + y 2 (7) < 1, (8) for all x, y E with x = y = 1 and x y. It is said to be uniformly convex if for any two sequences {x n } and {y n } in E such that x n = y n = 1 and lim x n + y n = 2, (9) lim x n y n = 0 holds. Let U = {x E : x = 1} be the unit sphere of E. Then the Banach space E is said to be smooth if lim t 0 x + ty x, (10) t exists for each x, y U. It is said to be uniformly smooth if the limit is attained uniformly for x, y E. In this case, the norm of E is said to be Gâteaux differentiable. The space E is said to have uniformly Gâteaux differentiable if for each y U, the limit (10) is attained uniformly for y U. The norm of E is said to be uniformly Fréchet differentiable (and E is said to be uniformly smooth) if the limit (10) is attained uniformly for x, y U.. In our work, the concept duality mapping is very important. Here, we list some known facts, related to the duality mapping J, as following: (a) E (E, resp) is uniformly convex if and only if E (E, resp.) is uniformly smooth. (b) J(x) for each x E. (c) If E is reflexive, then J is a mapping of E onto E. (d) If E is strictly convex, then J(x) J(y) for all x y. (e) If E is smooth, then J is single valued. (f) If E has a Fréchet differentiable norm, then J is norm to norm continuous. (g) If E is uniformly smooth, then J is uniformly norm to norm continuous on each bounded subset of E. (h) If E is a Hilbert space, then J is the identity operator. For more information, the readers may consult [5], [13]. If C is a nonempty closed convex subset of real a Hilbert space H and P C : H C is the metric projection, then P C is nonexpansive. Alber [1] has recently introduced a generalized projection operator Π C in a Banach space E which is an analogue representation of the metric projection in Hilbert spaces. The generalized projection Π C : E C is a map that assigns to an arbitrary point x E the minimum point of the functional ϕ(y, x), that is, Π C x = x, where x is the solution to the minimization problem ϕ(x, x) = min ϕ(y, x). y C Notice that the existence and uniqueness of the operator Π C is followed from the properties of the functional ϕ(y, x) and strict monotonicity of the mapping J, and moreover, in the Hilbert spaces setting we have Π C = P C. It is obvious from the definition of the function ϕ that ( y x ) 2 ϕ(y, x) ( y + x ) 2, for all x, y E. (11) Remark II.1. If E is a strictly convex and a smooth Banach space, then for all x, y E, ϕ(y, x) = 0 if and only if x = y, see Matsushita and Takahashi [8]. To obtain our results, following lemmas is very important.

3 Lemma II.2. (Kamimura and Takahashi [6]). Let E be a uniformly convex and smooth real Banach space and let {x n }, {y n } be two sequences of E. If ϕ(x n, y n ) 0 and either {x n } or {y n } is bounded, then x n y n 0. Lemma II.3. (Kamimura and Takahashi [6]). Let E be a uniformly convex and smooth Banach space and let r > 0. Then there exists a continuous, strictly increasing and convex function g : [0, 2r] [0, ) such that g(0) = 0 and g( x y ) ϕ(x, y), for all x, y B r = {z E : z r}. Let E be a strictly convex, smooth and reflexive Banach space, let J be the duality mapping from E into E. Then J 1 is also single-valued, one-to-one, and surjective, and it is the duality mapping from E into E. Define a function V : E E R as follows (see [7]): V (x, x ) = x 2 2 x, x + x 2 (12) for all x Ex E and x E. Then, it is obvious that V (x, x ) = ϕ(x, J 1 (x )) and V (x, J(y)) = ϕ(x, y). Lemma II.4. (Kohsaka and Takahashi [7]). Let E ba a strictly convex, smooth and reflexive Banach space, and let V be as in (12). Then V (x, x ) + 2 J 1 (x x), y V (x, x + y ) (13) for all x E and x, y E. Lemma II.5. (Alber [1]). Let E be a reflexive, strict convex and smooth real Banach space, let C be a nonempty closed convex subset of E and let x E. Then ϕ(y, Π C x) + ϕ(π C x, x) ϕ(y, x), y C. (14) A set-value mapping A : E E with domain D(A) = {x E : A(x)} = and range R(A) = {x E : x A(x), x D(A)} is said to be monotone if x y, x y 0 for all x A(x), y A(y). We denote the set {s E : 0 Ax} by A 1 0. A is maximal monotone if its graph G(A) is not properly contained in the graph of any other monotone operator. If A is maximal monotone, then the solution set A 1 is closed and convex. Let E be a reflexive, strictly convex and smooth Banach space, it is known that A is a maximal monotone if and only if R(J + ra) = E for all r > 0. Define the resolvent of A by J r x = x r. In other words, J r = (J + ra) 1 J for all r > 0. J r is a single-valued mapping from E to D(A). Also, A 1 (0) = F (J r ) for all r > 0, where F (J r ) is the set os all fixed points of J r. Define, for r > 0, the Yosida approximation of A by A r = (J JJ r )/r. We know that A r x A(J r x) for all R > 0 and x E. Lemma II.6. (Kohsaka and Takahashi [7]). Let E ba a strictly convex, smooth and reflexive Banach space, let A E E be a maximal monotone operator with A 1, let r > 0 and let J r = (J + rt ) 1 J. Then ϕ(x, J r y) + ϕ(j r y, y) ϕ(x, y) (15) for all x A 1 (0) and y E. III. MAIN RESULT Theorem III.1. Let E be a real uniformly smooth and uniformly convex Banach space with Kadec-Klee property and let C be a nonempty closed convex subset of E. Let T : C CB(C) be a closed and total quasi-ϕ-asymptotically nonexpansive multivalued mapping with nonnegative real sequence {ν n }, {µ n } and a strictly increasing continuous function ζ : R + R + such that µ 1 = 0, ν n 0, µ n 0 as n and ζ(0) = 0, let t : C C be a relatively nonexpansive mapping, let A : E E be a maximal monotone operator satisfying D(A) C. Let J r = (J + ra) 1 J for r > 0 such that F := A 1 (0) F (T ) F (t) and let a sequence {x n } in C by the following algorithm: x 0 C, chosen arbitrarity and C 0 = C, w n = J 1 (β n Jx n + (1 β n )J(J rn z n )), z n T n x n, y n = J 1 (α n Jx n + (1 α n )Jtw n ), C n+1 = {z C n : ϕ(z, y n ) α n ϕ(z, x n ) +(1 α n )phi(z, w n ) ϕ(z, x n ) + ξ n }, x n+1 = Π Cn+1 x 0, (16) for n N {0}, where J is the single-valued duality mapping on E and ξ n = ν n sup u F ζ(ϕ(u, x n )) + µ n. The coefficient sequence {α n }, {β n } [0, 1] satisfying (i) 0 < β 1 β n β 2 < 1, (ii) 0 α n α < 1, (iii) lim inf r n > 0. Then {x n } converges strongly to Π F x 0, where Π F is the generalized projection from C onto F. Proof. We first show that C n+1 is closed and convex for each n 0. Obviously, from the definition of C n+1, we see that C n+1 is closed for each n 0. Now we show that C n+1 is convex for any n 0. Since ϕ(z, y n ) ϕ(z, x n ) + ξ n 2 v, Jx n Jy n + y n 2 x n 2 ξ n 0, (17) this implies that C n+1 is a convex set. Next, we show that {x n } is bounded and {ϕ(x n, x 0 )} is convergent sequence. Put u n = J rn z n, n 0, let p F := A 1 (0) F (T ) F (t). By (16), we obtain ϕ(p, y n ) = ϕ(p, J 1 (α n Jx n + (1 α n )Jtw n )) p 2 2α n p, Jx n 2(1 α n ) p, Jtw n +α n x n 2 + (1 α n ) tw n 2 α n (1 α n )g Jx n Jtw n 2 = α n ϕ(p, x n ) + (1 α n )ϕ(p, tw n ) α n (1 α n )g Jx n Jtw n 2 α n ϕ(p, x n ) + (1 α n )ϕ(p, w n ) α n (1 α n )g Jx n Jtw n 2 (18) and ϕ(p, w n ) = ϕ(p, J 1 (β n Jx n + (1 β n )Ju n )) p 2 2β n p, Jx n 2(1 β n ) p, Ju n +β n x n 2 + (1 β n ) u n 2 = β n ϕ(p, x n ) + (1 β n )ϕ(p, u n ) β n (1 β n ) g Jx n Ju n 2

4 β n ϕ(p, x n ) + (1 β n )[ϕ(p, z n ) ϕ(u n, z n )] β n ϕ(p, x n ) + (1 β n )[ϕ(p, T n x n ) ϕ(u n, z n )] β n (1 β n )g Jx n Ju n 2 β n ϕ(p, x n ) + (1 β n )[ϕ(p, x n ) +ν n ζ(ϕ(p, x n )) + µ n ϕ(u n, z n )] = ϕ(p, x n ) + ν n sup ζ(ϕ(u, x n )) + µ n u F (1 β n )ϕ(u n, z n ) = ϕ(p, x n ) + ξ n (1 β n )ϕ(u n, z n ) = ϕ(p, x n ) + ξ n, (19) where ξ n = ν n sup u F ζ(ϕ(u, x n )) + µ n. By the assumptions of {ν n }, {µ n }, we obtain ξ n = ν n sup ζ(ϕ(u, x n )) + µ n 0, as n. (20) u F Substituting (19) into (18), we have ϕ(p, y n ) α n ϕ(p, x n ) + (1 α n )[ϕ(p, x n ) + ξ n ] ϕ(p, x n ) + ξ n (1 α n )β n (1 β n )g Jx n Ju n 2 ϕ(p, x n ) + ξ n, p F. (21) This means that, p C n+1 for all n 0. As consequently, the sequence {x n } is well defined. Moreover, since x n = Π Cn x 0 and x n+1 C n+1 C n, we get ϕ(x n, x 0 ) ϕ(x n+1, x 0 ), for all n 0. Therefore, {ϕ(x n, x 0 )} is nondecreasing. By definition of x n and Lemma II.6, we have ϕ(x n, x 0 ) = ϕ(π Cn x 0, x 0 ) ϕ(p, x 0 ) ϕ(p, Π Cn x 0 ) ϕ(p, x 0 ), for all p n=0 F (T n) C n. Thus, {ϕ(x n, x 0 )} is a bounded sequence. Moreover, by (11), we know that {x n } is bounded. So, lim ϕ(x n, x 0 ) exists. Again, by Lemma II.6, we have ϕ(x n+1, x n ) = ϕ(x n+1, Π Cn x 0 ) ϕ(x n+1, x 0 ) ϕ(π Cn x 0, x 0 ) = ϕ(x n+1, x 0 ) ϕ(x n, x 0 ), for all n 0. Thus, ϕ(x n+1, x n ) 0 as n. Since {x n } is bounded and x is reflexive, there exists a subsequence {x ni } {x n } such that x ni q C. From C n is closed and convex and C n+1 C n, this implies that C n is weakly closed and q C n, for each n 0. In view of x ni = Π Cni x 0, we have ϕ(x ni, x 0 ) ϕ(q, x 0 ), n i 0. Since the norm is weakly lower semi-continuous, we have lim inf nt ϕ(x ni, x 0 ) = lim inf ni x ni 2 2 x ni, Jx 0 + x 0 2 q 2 2 q, Jx 0 + x 0 2 = ϕ(q, x 0 ). So, ϕ(q, x 0 ) lim inf ϕ(x n i, x 0 ) lim sup ϕ(x ni, x 0 ) ϕ(q, x 0 ). n i n i this implies that lim ni ϕ(x ni, x 0 ) = ϕ(q, x 0 ). By Kadec- Klee property of E, we have lim x n i = q, as n i. n i Since the sequence {ϕ(x n, x 0 )} is convergent and lim ni ϕ(x ni, x 0 ) = ϕ(q, x 0 ) which implies that lim ϕ(x n, x 0 ) = ϕ(q, x 0 ). If there exists some subsequence {x nj } {x n } such that x nj q, then from Lemma II.6 ϕ(q, q ) = lim ni,n j ϕ(x ni, x nj ) = lim ni,n j ϕ(x ni, Π Cnj x 0 ) lim ni,n j [ϕ(x ni, x 0 ) ϕ(π Cnj x 0, x 0 )] = lim ni,n j [ϕ(x ni, x 0 ) ϕ(x nj x 0, x 0 )] = ϕ(q, x 0 ) ϕ(q, x 0 ) = 0. This implies that q = q and so By definition of Π Cn x 0, we have lim x n = q. (22) ϕ(x n+1, x n ) = ϕ(x n+1, Π Cn x 0 ) ϕ(x n+1, x 0 ) ϕ(π Cn x 0, x 0 ) = ϕ(x n+1, x 0 ) ϕ(x n, x 0 ). (23) From lim ϕ(x n, x 0 ) exists, we have It follows from Lemma II.2, we get lim ϕ(x n+1, x n ) = 0. (24) lim x n+1 x n = 0. (25) By definition of C n and x n+1 = Π Cn+1 x 0 C n+1, we have ϕ(x n+1, y n ) ϕ(x n+1, x n ) + ξ n. (26) It follows from (24) and ξ n 0 as n that Again from Lemma II.2, we have lim ϕ(x n+1, y n ) = 0. (27) lim x n+1 y n = 0. (28) By using the triangle inequality, we get y n x n y n x n+1 + x n+1 x n, (29) again by (25) and (28), we also have lim y n x n = 0. (30)

5 Since J is uniformly norm-to-norm continuous, we obtain lim Jy n Jx n = 0. (31) From (21), for u F (T ) and u n = J rn z n, z n T n x n, we have ϕ(p, y n ) ϕ(p, x n )+ξ n (1 α n )β n (1 β n )g Jx n Ju n 2, and hence (1 α n )β n (1 β n )g Jx n Ju n 2 ϕ(p, x n ) ϕ(p, y n ) + ξ n. (32) On the other hand, we note that ϕ(p, x n ) ϕ(p, y n ) = x n 2 y n 2 2 p, Jx n Jy n x n y n ( x n + y n ) +2 p Jx n Jy n. (33) It follows from x n y n 0 and Jx n Jy n 0, that ϕ(p, x n ) ϕ(p, y n ) 0, as n. (34) Since condition (i), (20) and (34), it follows from (32) g Jx n Ju n 0, as n. (35) It follows from the property of g that and so Jx n Ju n 0, as n, (36) lim Jx n Ju n = lim Jx n JJ rn z n = 0. (37) Since J 1 is uniformly norm-to-norm continuous, we have lim x n u n = lim x n J rn z n = 0. (38) From (18) and (19), we obtain that and hence ϕ(p, y n ) ϕ(p, x n ) + (1 α n )ξ n (1 α n )(1 β n )ϕ(u n, z n ), (39) (1 α n )(1 β n )ϕ(u n, z n ) ϕ(p, x n ) ϕ(p, y n )+ξ n. (40) By condition (i), (ii) and (20), we have From Lemma II.2, we get lim ϕ(u n, z n ) = 0. (41) lim u n z n = lim J r n z n z n = 0. (42) Since J is uniformly norm-to-norm continuous, we have lim Ju n Jz n = lim JJ r n z n Jz n = 0. (43) From definition of C n, we have α n ϕ(z, x n ) + (1 α n )ϕ(z, w n ) ϕ(z, x n ) + ξ n ϕ(z, w n ) ϕ(z, x n ) + ξ n. Since x n+1 = Π Cn+1 x 0 C n+1, we have ϕ(x n+1, w n ) ϕ(x n+1, x n ) + ξ n. (44) It follows from (24) and ξ n 0 as n that From Lemma II.2, we have lim ϕ(x n+1, w n ) = 0. (45) lim x n+1 w n = 0. (46) By using the triangle inequality, we get w n x n w n x n+1 + x n+1 x n, (47) again by (25) and (46), we also have lim w n x n = 0. (48) Since J is uniformly norm-to-norm continuous, we obtain From (18) and (19), that and hence lim Jw n Jx n = 0. (49) ϕ(p, y n ) ϕ(p, x n ) + ξ n α n (1 α n )g Jx n Jtw n, (50) α n (1 α n )g Jx n Jtw n ϕ(p, x n ) ϕ(p, y n )+ξ n. (51) By condition (ii), (20) and (34), we obtain that g Jx n Jtw n 0, as n. (52) It follows from the property of g that Jx n Jtw n 0, as n. (53) Since J 1 is uniformly norm-to-norm continuous, we have x n tw n 0, as n. (54) By using the triangle inequality, again w n tw n w n x n + x n tw n. (55) By (48) and (54), we have w n tw n 0. (56) Since {x n } is bounded and x ni q C. It follows from (48), we have w ni q as i and t is relatively nonexpansive. We have that q F (t) = F (t). Next, we show that q A 1 (0). Indeed, since lim inf r n > 0, it follows from (42) that lim B 1 r n z n = lim z n u n = 0. (57) r n If (z, z ) A, then it holds from the monotonicity of B that z z ni, z B rni z ni 0, for all i N. Letting i, we get z q, z 0. Then, the maximality of A implies q A 1. By (38) and (42), we have From (22), we get lim x n z n = 0, (58) z n q as n. (59) From u n T n x n and let {s n } be a sequence generated by

6 s 2 T z 1 T 2 x 1 ; s 3 T z 2 T 3 x 2 ; s 4 T z 3 T 4 x 3 ;. s n T z n 1 T n x n 1 ; s n+1 T z n T n+1 x n ;. (60) By the assumption that T is uniformly L-Lipschitz continuous and any z n T n x n and a n+1 T z n T n+1 x n, we have s n+1 z n s n+1 z n+1 + z n+1 x n+1 + x n+1 x n + x n z n L x n x n+1 + z n+1 x n+1 + x n+1 x n + x n z n (L + 1) x n x n+1 + z n+1 x n+1 + x n z n. From (22), (25) and (59) that lim s n+1 z n = 0 and lim s n+1 = q. (61) In view of closedness of T, it yields that q T q. Therefore q F (T ). Finally, we show that x n q = Π F x 0. Let p = Π F x 0. Since p F C n and x n = Π Cn x 0, we have This implies that ϕ(x n, x 0 ) ϕ(p, x 0 ), n 0. ϕ(q, x 0 ) = lim ϕ(x n, x 0 ) ϕ(p, x 0 ). (62) In view of definition of Π F x 0, we have q = p. Therefore, x n q = Π F x 0. This completes the proof. Corollary III.2. Let E be a real uniformly smooth and uniformly convex Banach space with Kadec-Klee property and let C be a nonempty closed convex subset of E. Let T : C CB(C) be a closed and total quasi-ϕ-asymptotically nonexpansive multivalued mapping with nonnegative real sequence {ν n }, {µ n } and a strictly increasing continuous function ζ : R + R + such that µ 1 = 0, ν n 0, µ n 0 as n and ζ(0) = 0 such that F := F (T ) and let a sequence {x n } in C by the following algorithm: x 0 C, chosen arbitrarity and C 0 = C, w n = J 1 (β n Jx n + (1 β n )Jz n ), z n T n x n y n = J 1 (α n Jx n + (1 α n )Jw n ), C n+1 = {z C n : ϕ(z, y n ) α n ϕ(z, x n ) +(1 α n )phi(z, w n ) ϕ(z, x n ) + ξ n }, x n+1 = Π Cn+1 x 0, (63) for n N {0}, where J is the single-valued duality mapping on E and ξ n = ν n sup u F (T ) ζ(ϕ(u, x n )) + µ n. The coefficient sequence {α n }, {β n } [0, 1] satisfying (i) 0 < β 1 β n β 2 < 1, (ii) 0 α n α < 1, Then {x n } converges strongly to Π F (T ) x 0, where Π F (T ) is the generalized projection from C onto F (T ). IV. CONCLUSION We established the strong convergence theorem for fixed points of sequence for multivalued nonexpansive mappings and a zero of maximal monotone operator in Banach spaces. The results of this paper extended and improved the corresponding results given by Tang and Chang [15] and some authors in the literature. ACKNOWLEDGMENTS The authors would like to thank the Office of the Higher Education Commission for the supports under the Higher Education Research Promotion project. First author gratefully acknowledges support provided by the King Mongkut s University of Technology Thonburi (KMUTT) during the first author s stay at the King Mongkut s University of Technology Thonburi (KMUTT) as a post-doctoral researcher (KMUTT-Post-doctoral Fellowship). REFERENCES [1] Ya. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A.G. 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