CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS

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1 An. Şt. Univ. Ovidius Constanţa Vol. 18(1), 2010, CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS Yan Hao Abstract In this paper, a demiclosed principle for total asymptotically nonexpansive mappings is established. An implicit iterative method for the class of total asymptotically nonexpansive mappings is considered. Weak and strong convergence theorems are established in a real Hilbert space. As applications of main results, an equilibrium problem is considered based on implicit iterative process. 1. Introduction and Preliminaries Throughout this paper, we assume that H is a real Hilbert space, whose inner product and norm are denoted by, and. We also assume that ψ : [0, ) [0, )isstrictlyincreasingcontinuousfunctionwithψ(0) = 0. and are denoted by strong convergence and weak convergence, respectively. Let C be a nonempty closed and convex subset of H and T : C C a mapping. In this paper, we use F(T) to denote the fixed point set of the mapping T. Recall that T is said to be nonexpansive if Tx Ty x y, x,y C. (1.1) Key Words: nonexpansive mapping; asymptotically nonexpanvie mapping; total asymptotically nonexpansive mapping; fixed point; equilibrium Mathematics Subject Classification: 47H09, 47H10, 47J25. Received: June, 2009 Accepted: January,

2 164 Yan Hao T is said to be asymptotically nonexpansive if there exists a positive sequence h n [1, ) with lim h n = 1 such that T n x T n y h n x y, x,y C,n 1. (1.2) The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [14] as a generalization of the class of nonexpansive mappings. They proved that if C is a nonempty closed convex and bounded subset of a real uniformly convex Banach space and T is an asymptotically nonexpansive mapping on C, then T has a fixed point. T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds: lim sup Observe that if we define sup ( T n x T n y x y ) 0. (1.3) x,y C σ n = sup ( T n x T n y x y ) and ν n = max{0,σ n }, x,y C then ν n 0 as n. It follows that (1.3) is reduced to T n x T n y x y +ν n, x,y C,n 1. (1.4) The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck, Kuczumow and Reich [2]. It is known [16] that if C is a nonempty closed convex bounded subset of a uniformly convex Banach space E and T is asymptotically nonexpansive in the intermediate sense, then T has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings. Recently, Alber, Chidume and Zegeye [1] introduced the concept of total asymptotically nonexpansive mappings. Recall that T is said to be total asymptotically nonexpansive if T n x T n y x y +µ n ψ( x y )+ν n, x,y C, (1.5) where {µ n } and {ν n } are nonnegative real sequences such that µ n 0 and ν n 0 as n. From the definition, we see that the class of total asymptotically nonexpansive mappings include the class of asymptotically nonexpansive mappings as a special case; see also [13] for more details. Recently, weak convergence problems of implicit (or non-implicit) iterative processes to a common fixed point for a finite family of nonexpansive mappings

3 CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 165 and asymptotically nonexpansive mappings have been considered by a number of authors (see, for example, [1],[8],[9],[12],[15],[19],[21],[23],[24],[27]-[32],[34]- [36],[38]). In 2001, Xu and Ori [34] introduced the following implicit iteration process for afinite family ofnonexpansive mappings {T 1,T 2,...,T N }, with {α n }areal sequence in (0,1), and an initial point x 0 C: x 1 = α 1 x 0 +(1 α 1 )T 1 x 1, x 2 = α 2 x 1 +(1 α 2 )T 2 x 2,. x N = α N x N 1 +(1 α N )T N x N, x N+1 = α N+1 x N +(1 α N+1 )T 1 x N+1,. which can be re-written in the following compact form: x n = α n x n 1 +(1 α n )T n x n, n 1, (1.6) where T n = T n(modn) (here the mod N function takes values in {1,2,...,N}). They obtained the following results in a real Hilbert space. Theorem XO. Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and T : C C be a finite family of nonexpansive self-mappings on C such that F = N i=1 F(T i). Let {x n } be defined by (1.6). If {α n } is chosen so that α n 0, as n, then {x n } converges weakly to a common fixed point of the family of {T i } N i=1. In 2006, Chang et al. [9] improved the results of Xu and Ori [34] from the class of nonexpansive mappings to the class of asymptotically nonexpansive mappings which is defined on a nonempty closed and convex subset C of H such that C + C C. To be more precise, they considered the following implicit iterative process for a finite family of asymptotically nonexpansive mappings{t 1,T 2,...,T N },with{α n }arealsequencein(0,1),{u n }abounded

4 166 Yan Hao sequence in C, and an initial point x 0 C: x 1 = α 1 x 0 +(1 α 1 )T 1 x 1 +u 1, x 2 = α 2 x 1 +(1 α 2 )T 2 x 2 +u 2,. x N = α N x N 1 +(1 α N )T N x N +u N, x N+1 = α N+1 x N +(1 α N+1 )T 1 x N+1 +u N+1,. x 2N = α 2N x 2N 1 +(1 α 2N )T 2 Nx 2N +u 2N, x 2N+1 = α 2N+1 x 2N +(1 α 2N+1 )T 3 1x 2N+1 +u 2N+1,. Since for each n 1, it can be written as n = (k 1)N +i, where i = i(n) {1,2,...,N}, k = k(n) 1 is a positive integer and k(n) as n. Hence the above table can be rewritten in the following compact form: x n = α n x n 1 +(1 α n )T k(n) i(n) x n +u n, n 1. (1.7) They obtained weak convergence theorems of the implicit iterative scheme (1.7)forafinitefamilyofasymptoticallynonexpansivemappings{T 1,T 2,...,T N }; see [9] for more details. The purpose of this paper is to establish weak and strong convergence theorems of the implicit iteration process (1.7) for a finite family of uniformly Lipschitz total asymptotically nonexpansive mappings in a real Hilbert space. Next, we recall some well-known concepts. Recall that a space X is said to satisfy Opial s condition [18] if for each sequence {x n } in X, the condition that the sequence x n x weakly implies that lim inf x n x < liminf x n y for all y E and y x. Recall that a mapping T : C C is semicompact if any sequence {x n } in C satisfying lim x n Tx n = 0 has a convergent subsequence. Recall that the mapping T is said to be demiclosed at the origin if for each sequence {x n } in C, the condition x n x 0 weakly and Tx n 0 strongly implies Tx 0 = 0.

5 CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 167 Next, we show that the process (1.7) is well-defined if the control sequence satisfies 0 < L 1 L < α n < 1, where L = max 1 i N {L i }. Indeed, define a mapping It follows that W n x = α n x n 1 +(1 α n )T k(n) i(n) x+u n, n 1, x C. W n x W n y (1 α n )L x y, x,y C. Since (1 α n )L < 1, it follows that W n is a contractive mapping and hence has a unique fixed point x n in C. This is, the process (1.7) is well-defined. In order to prove our main results, we also need the following lemmas. Lemma 1.1 ([30]). Let {a n }, {b n } and {c n } be three nonnegative sequences satisfying the following condition: a n+1 (1+b n )a n +c n, n n 0, where n 0 is some nonnegative integer, n=0 c n < and n=0 b n <. Then lim a n exists. Lemma 1.2 ([27]). Let H be a real Hilbert space and 0 < p t n q < 1 for all n 0. Suppose that {x n } and {y n } are sequences of H such that and lim sup x n r, limsup y n r lim t nx n +(1 t n )y n = r hold for some r 0. Then lim x n y n = 0. Lemma 1.3. Let C be a nonempty closed convex and bounded subset of a real Hilbert space H and T be a L-Lipschitz continuous and total asymptotically nonexpansive mapping with the function ψ and sequences {µ n } and {ν n } such that µ n 0 and ν n 0 as n. Then I T is demiclosed at zero. Proof. Let {x n } be a sequence in C such that x n x and x n Tx n 0 as n. Next, we show that x C and x = Tx. Since C is closed and convex, we see that x C. It is sufficient to show that x = Tx. Choose 1 α (0, 1+L ) and define y α,m = (1 α)x + αt m x for arbitrary but fixed m 1. Note that x n T m x n x n Tx n + Tx n T 2 x n + + T m 1 x n T m x n ml x n Tx n.

6 168 Yan Hao It follows that Note that lim x n T m x n = 0. (1.8) x y α,m,y α,m T m y α,m = x x n,y α,m T m y α,m + x n y α,m,y α,m T m y α,m = = x x n,y α,m T m y α,m + x n y α,m,t m x n T m y α,m x n y α,m,x n y α,m + x n y α,m,x n T m x n x x n,y α,m T m y α,m + + x n y α,m ( x n y α,m +µ n ψ( x n y α,m )+ν m ) x n y α,m 2 + x n y α,m x n T m x n x x n,y α,m T m y α,m +µ m Mψ(M)+ν m M+ + x n y α,m x n T m x n, (1.9) where M = sup n 0 { x n y α,m }. Since x n x and (1.8), we arrive at x y α,m,y α,m T m y α,m µ m Mψ(M)+ν m M. (1.10) On the other hand, we have x y α,m,(x T m x ) (y α,m T m y α,m ) (1+L) x y α,m 2 Note that x T m x 2 = x T m x,x T m x = = (1+L)α 2 x T m x 2. (1.11) = 1 α x y α,m,x T m x = = 1 α x y α,m,(x T m x ) (y α,m T m y α,m ) + (1.12) + 1 α x y α,m,y α,m T m y α,m. Substituting (1.10) and (1.11) into (1.12), we arrive at x T m x 2 (1+L)α x T m x 2 + µ mmψ(m)+ν m M. α This implies that α[1 (1+L)α] x T m x 2 µ m Mψ(M)+ν m M, m 1. (1.13)

7 CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 169 Letting m in (1.13), we see that T m x x. Since T is uniformly L-Lipschitz, we obtain that x = Tx. This completes the proof. 2. Main results Theorem 2.1. Let H be a real Hilbert space and C be a nonempty closed convex and bounded subset of H such that C + C C. Let T i : C C be a uniformly L i -Lipschitz total asymptotically nonexpansive mapping with the function ψ i and sequences {µ (i) n }, {ν n (i) } for each i {1,2,...,N}. Assume that n=1 µ(i) n < and n=1 ν(i) n < for each i {1,2,...,N}. Let {u n } be a bounded sequence in C such that n=1 u n < and {α n } a sequence in [ L 1 L,a], where L = max 1 i N{L i } > 1 and a is some constant in (0,1). Assume that F = N i=1 F(T i). Let {x n } be a sequence generated by (1.7). Then the sequence {x n } converges weakly to some point x F. Proof. Define the following sequences µ n = max{µ (1) n,µ (2) n,...,µ (N) n } and ν n = max{ν n (1),ν n (2),...,ν n (N) }. It is easy to see that n=1 µ n < and n=1 ν n <. Fixing p F, we have x n p α n x n 1 p +(1 α n ) T k(n) i(n) x n p + u n where α n x n 1 p +(1 α n ) ( x n p +µ k(n) ψ i(n) ( x n p )+ν k(n) ) + un x n 1 p +µ k(n) ψ i(n) ((diam C))+ν k(n) + u n leq x n 1 p +µ k(n) ψ r ((diam C))+ν k(n) + u n, ψ r ((diam C)) = max{ψ 1 ((diam C)),ψ 2 ((diam C)),...,ψ N ((diam C))}. (2.1) In view of Lemma 1.1, we obtain that the limit of the sequence { x n p } exits. Next, we assume that where d > 0 is some constant. It follows that lim sup x n 1 p u n limsup d = lim x n p, (2.2) x n 1 p +limsup u n = d. (2.3)

8 170 Yan Hao Note that T k(n) i(n) x n p+u n T k(n) i(n) x n p + u n x n p +µ k(n) ψ i(n) ( x n p )+ν k(n) + u n x n p +µ k(n) ψ r ((diam C))+ν k(n) + u n. It follows that lim sup T k(n) i(n) x n p+u n d. (2.4) On the other hand, we have d = lim x n p = lim α n(x n 1 p u n )+(1 α n )(T k(n) i(n) x n p+u n ). Combining (2.3), (2.4) with (2.4) and from Lemma 1.2, we obtain that Note that From (2.6), we arrive at (2.5) lim x n 1 T k(n) i(n) x n = 0. (2.6) x n x n 1 (1 α n ) x n 1 T k(n) i(n) x n + u n. On the other hand, we have lim x n x n 1 = 0. (2.7) T k(n) i(n) x n x n T k(n) i(n) x n x n 1 + x n 1 x n, which combined with (2.6) gives that From (2.7), we also have lim Tk(n) i(n) x n x n = 0. (2.8) lim x n x n+l = 0, l = 1,2,...,N. (2.9) For any positive n > N, it can be rewritten as n = (k(n) 1)N + i(n), i(n) {1,2,...,N}. Note that x n 1 T n x n x n 1 T k(n) i(n) x n + T k(n) i(n) x n T n x n x n 1 T k(n) i(n) x n +L T k(n) 1 i(n) x n x n x n 1 T k(n) i(n) x n +L{ T k(n) 1 i(n) x n T k(n) 1 i(n N) x n N + + T k(n) 1 i(n N) x n N x (n N) 1 + x (n N) 1 x n }. (2.10)

9 CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 171 On the other hand, we have n N = ((k(n) 1) 1)N +i(n) = ((k(n) 1) 1)N +i(n N), i.e., It follows that and k(n N) = k(n) 1 and i(n N) = i(n). T k(n) 1 i(n) x n T k(n) 1 i(n N) x n N L x n x n N (2.11) T k(n) 1 i(n N) x n N x (n N) 1 = T k(n N) i(n N) x n N x (n N) 1. (2.12) Combining (2.6), (2.9) with (2.10), we see that On the other hand, we have that lim x n 1 T n x n = 0. (2.13) x n T n x n x n x n 1 + x n 1 T n x n. From (2.7) and (2.13), we obtain that Consequently, for any j = 1,2,...,N, we see that lim x n T n x n = 0. (2.14) x n T n+j x n x n x n+j + x n+j T n+j x n+j + T n+j x n+j T n+j x n (1+L) x n x n+j + x n+j T n+j x n+j. From (2.9) and (2.14), we arrive at lim x n T n+j x n = 0. Therefore, for i {1,2,...,N}, there exists some e {1,2,...,N} such that n+e = i(mod N). It follows that lim x n T i x n = lim x n T n+e x n = 0. (2.15) Since {x n } is bounded, we see that there exists a subsequence {x ni } {x n } such that x ni x. From Lemma 1.3, we can obtain that x F. Next we prove that {x n } converges weakly to x. Suppose the contrary. Then we see that there exists some subsequence {x nj } {x n } such that {x nj } converges

10 172 Yan Hao weakly to x C and x x. From Lemma 1.3, we also have x F. Put w = lim x n x. Since H is an Opial s space, we see that w = lim n i x n i x < lim n i x n i x = = lim n j x n j x < lim n j x n j x = = lim n i x n i x = w. This derives a contradiction. It follows that x = x. This completes the proof. Remark 2.2. Theorem 2.1 improves Theorem 1 of Chang et al. [9] from asymptotically nonexpansive mappings to total asymptotically nonexpansive mappings. Corollary 2.3. Let H be a real Hilbert space and C be a nonempty closed convex and bounded subset of H. Let T i : C C be a uniformly L i -Lipschitz total asymptotically nonexpansive mapping with the function ψ i and sequences {µ (i) n }, {ν n (i) } for each i {1,2,...,N}. Assume that n=1 µ(i) n < and n=1 ν(i) n < for each i {1,2,...,N}. Let {α n } be a sequence in [ L 1 L,a], where L = max 1 i N {L i } > 1 and a is some constant in (0,1). Assume that F = N i=1 F(T i). Let {x n } be a sequence generated by the following manner: x 0 C, x n = α n x n 1 +(1 α n )T k(n) i(n) x n, n 1. (2.16) Then the sequence {x n } converges weakly to some point x F. Next, we prove a strong convergence theorem under the condition of semicompactness. Theorem 2.4. Let H be a real Hilbert space and C a nonempty closed convex and bounded subset of H such that C + C C. Let T i : C C be a uniformly L i -Lipschitz total asymptotically nonexpansive mapping with the function ψ i and sequences {µ (i) n }, {ν n (i) } for each i {1,2,...,N}. Assume that n=1 µ(i) n < and n=1 ν(i) n < for each i {1,2,...,N}. Let {u n } be a bounded sequence in C such that n=1 u n < and {α n } a sequence in [ L 1 L,a], where L = max 1 i N{L i } > 1 and a is some constant in (0,1). Assume that F = N i=1 F(T i) and at least there exists a mapping T r which is semicompact. Let {x n } be a sequence generated by (1.7). Then the sequence {x n } converges weakly to some point x F. Proof. Without loss of generality, we may assume that T 1 is semicompact. It follows from (2.15) that lim x n T 1 x n = 0.

11 CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 173 By the semicompactness of T 1, we have there exists a subsequence {x ni } of {x n } such that x ni x C strongly. From (2.15), we have lim x n i T l x ni = x T l x = 0, n i for all l = 1,2,,N. This implies that x F. From Theorem 2.1, we know that lim x n p exists for each p F. This shows that lim x n x exists. From x ni x, we have lim x n x = 0. This completes the proof of Theorem 2.4. Corollary 2.5. Let H be a real Hilbert space and C be a nonempty closed convex bounded subset of H. Let T i : C C be a uniformly L i -Lipschitz total asymptotically nonexpansive mapping with the function ψ i and sequences {µ n (i) }, {ν n (i) } for each i {1,2,...,N}. Assume that n=1 µ(i) n < and n=1 ν(i) n < for each i {1,2,...,N}. Let {α n } be a sequence in [ L 1 L,a], where L = max 1 i N {L i } > 1 and a is some constant in (0,1). Assume that F = N i=1 F(T i) and at least there exists a mapping T r which is semicompact. Let {x n } be a sequence generated by (2.16). Then the sequence {x n } converges weakly to some point x F. 3. Applications Let A : C H be a mapping. Recall that the classical variational inequality problem is to find x C such that Ax,y x 0, y C. (3.1) It is known that x C is a solution to the variational inequality problem (3.1) if and only if x is a fixed point of the mapping P C (I ρa), where P C is the metric projection from H onto C, ρ > 0 is a constant and I is the identity mapping. This implies that the variational inequality problem (3.1) is equivalent to a fixed point problem. This alternative formula is very important form the numerical analysis point of view. Recently, many authors studied the variational inequality (3.1) by iterative methods. Let F be a bifunction of C C into R, where R is the set of real numbers. We consider the following equilibrium problem: Find x C such that F(x,y) 0, y C. (3.2)

12 174 Yan Hao In this paper, the set of such x C is denoted by EP(F), i.e., EP(F) = {x C : F(x,y) 0, y C}. Numerous problems in physics, optimization and economics reduce to find a solution of (1.1); see ([3],[5],[10],[11]). Approximating solutions of the problem (3.2) based on iterative methods was studied by many authors, see, for example, ([4]-[7],[17],[20],[22],[25],[26],[33],[37]) and the reference therein. Putting F(x,y) = Ax,y x, x,y C, we see that z FP(F) if and only if Az,y z 0, y C. That is, z is a solution to the variational inequality (3.1). Numerous problems in physics, optimization, and economics reduce to find a solution of the problem (3.2). To study the problem (3.2), we may assume that the bifunction F : C C R satisfies the following conditions: (A1) F(x,x) = 0 for all x C; (A2) F is monotone, i.e., F(x,y)+F(y,x) 0 for all x,y C; (A3) for each x,y,z C, lim t 0 F(tz +(1 t)x,y) F(x,y); (A4) for each x C, y F(x,y) is convex and weakly lower semi-continuous. Next, we consider the convergence of implicit iterative process (1.7) for the equilibrium problem (3.2). To prove the main results in this section, we need the following lemma which can be found in [3] and [4]. Lemma 3.1. Let C be a nonempty closed convex subset of H ad let F : C C R be a bifunction satisfying (A1)-(A4). Then, for any r > 0 and x H, there exists z C such that F(z,y) + 1 r y z,z x 0, y C. Further, define a mapping T r x = {z C : F(z,y)+ 1 y z,z x 0, y C} r for all r > 0 and x H. Then, the following hold: (1) T r is single-valued; (2) T r is firmly nonexpansive, i.e., for any x,y H, (3) F(T r ) = EP(F); (4) EP(F) is closed and convex. T r x T r y 2 T r x T r y,x y ;

13 CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 175 Theorem 3.2. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let F 1 be a bifunction from C C to R satisfying (A1)-(A4) such that EP(F 1 ) is nonempty. Let {x n } be a sequence generated in the following manner: x 0 H, chosen arbitrily F 1 (y n,y)+ 1 r n y y n,y n x n 0, y C, (3.3) x n = α n x n 1 +(1 α n )y n +u n, n 1. where {r n } (0, ), {α n } (0,1) and {u n } is a bounded sequence. Assume that the following conditions are satisfied (1) a α n b, where 0 < a < b < 1; (2) liminf r n > 0; (3) n=1 u n <. Then the sequence {x n } generated in (3.3) converges weakly to some point in EP(F). Proof. Putting y n = T rn x n for each n 1, we from Lemma 3.1 see that T rn is firmly nonexpansive. Whenever needed, we shall equivalently write the implicit iteration (3.3) as x 0 H, x n = α n x n 1 +(1 α n )T rn x n +u n, n 1, (3.4) Fixing p EP(F), we see that x n p α n x n 1 p +(1 α n ) T rn x n p + u n This in turn implies that α n x n 1 p +(1 α n ) x n p + u n. x n p x n 1 p + u n a. From Lemma 1.1, we obtain that lim x n p exits. Next, we assume that lim x n p = r > 0. Note that lim sup x n 1 p+u n r (3.5) and lim sup T rn x n p+u n limsup( x n p + u n ) r. (3.6)

14 176 Yan Hao On the other hand, we have lim x n p = lim α n(x n 1 p+u n )+(1 α n )(T rn x n p+u n ) = r. (3.7) Combining (3.5), (3.6) with (3.7), from Lemma 1.2, we obtain that From (3.1), we arrive at lim x n 1 T rn x n = 0. (3.8) x n x n 1 = (1 α n )(T rn x n x n 1 )+u n. From the conditions (3) and (3.8), we see that Note that lim x n 1 x n = 0. (3.9) x n T rn x n x n x n 1 + x n 1 T rn x n. In view of (3.8) and (3.9), we obtain that lim x n T rn x n = 0. (3.9) Since the sequence {x n } is a bounded, we see that there exists a subsequence {x ni } of {x n } such that x ni q. Next, we show that q EP(T). Since y n = T rn x n, we have It follows from (A2) that F(y n,y)+ 1 r n y y n,y n x n 0, y C. y y n, y n x n r n F(y,y n ) and hence y y ni, y n i x ni r ni F(y,y ni ). Since yn i xn i r ni 0, y ni q and (A4), we have F(y,q) 0 for all y C. For t with 0 < t 1 and y C, let y t = ty +(1 t)q. Since y C and q C, we have y t C and hence F(y t,q) 0. So, from (A1) and (A4), we have 0 = F(y t,y t ) tf(y t,y)+(1 t)f(y t,q) tf(y t,y).

15 CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 177 That is, F(y t,y) 0. It follows from (A3) that F(q,y) 0 for all y C and hence q EP(F). This proves that q EP(T). Finally, we show that the sequence {x n } converges weakly to q. Suppose the contrary holds. It follows that there exists some subsequence {x nj } of {x n } such that x nj q and q q. By the same method as given above, we can prove that q EP(T). Put lim x n q = d 1 and lim x n q = d 2, where d 1 and d 2 are two nonnegative numbers. In view of Opial s condition, we see that d 1 = liminf i x n i q < liminf j x n i q = liminf j x n j q < liminf j x n j q = d 1. This is a contradiction. Hence q = q. This shows that the sequence {x n } converges weakly to q. The proof is completed. Acknowledgments The author are extremely grateful to the referee for useful suggestions that improved the contents of the paper. References [1] Ya.I. Alber, C.E. Chidume, H. Zegeye, Approximating fixed point of total asymptotically nonexpansive mappings, Fixed Point Theory Appl., (2006) Art. ID [2] R.E. Bruck, T. Kuczumow, S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. Math., 65 (1993), [3] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud., 63 (1994), [4] P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), [5] L.C. Ceng, J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math., 214 (2008),

16 178 Yan Hao [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., 71 (2009), [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., 334 (2008), [8] S.S. Chang, Y.J. Cho, H.Y. Zhou, Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings, J. Korean Math. Soc., 38 (2001), [9] S.S. Chang, K.K. Tan, H.W.J. Lee, C.K. Chan, On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 313 (2006), [10] S.S. Chang, H.W.J. Lee, C.K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal., 70 (2009), [11] O. Chadli, N.C.Wong, J.C. Yao, Equilibrium problems with applications to eigenvalue problems, J. Optim. Theory Appl., 117 (2003), [12] C.E. Chidume, N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal., 62 (2005), [13] C.E. Chidume, E.U. Ofoedu, Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings, J. Math. Anal. Appl., 333 (2007), [14] K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer.Math. Soc., 35 (1972), [15] J. Gornicki, Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces, Comment. Math. Univ. Carolin., 301 (1989), [16] W.A. Kirk, Fixed point theorems for non-lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math., 17 (1974), [17] P. Kumam, A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping, Nonlinear Anal., 2 (2008),

17 CONVERGENCE THEOREMS FOR TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 179 [18] Z. Opial, Weak convergence of the sequence of successive appproximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), [19] S. Plubtieng, K. Ungchittrakool, R. Wangkeeree, Implicit iterations of two finite families for nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 28 (2007), [20] S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput., 197 (2008), [21] X. Qin, Y.J. Cho, M. Shang, Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings, Appl. Math. Comput., 210 (2009), [22] X. Qin, M. Shang, Y. Su, Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems, Math. Comput. Model., 48 (2008), [23] X. Qin, Y. Su, Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl., 329 (2007), [24] X. Qin, Y.J. Cho, J.I. Kang, S.M. Kang, Strong convergence theorems for an infinite family of nonexpansive mappings in Banach spaces, J. Comput. Appl. Math., 230 (2009), [25] X. Qin, S.Y. Cho, S.M. Kang, Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications, J. Comput. Appl. Math., 233 (2009), [26] X. Qin, Y.J. Cho, S.M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math., 225 (2009), [27] J. Schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), [28] Z.H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasinonexpansive mappings, J. Math. Anal. Appl., 286 (2003), [29] N. Shahzad, H. Zegeye, Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps, Appl. Math. Comput., 189 (2007),

18 180 Yan Hao [30] K.K. Tan, H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process, J. Math. Anal. Appl., 178 (1993), [31] S. Thianwan, S. Suantai, Weak and strong convergence of an implicity iteration process for a finite family of nonexpansive mappings, Sci. Math. Japon., 66 (2007), [32] B.S. Thakur, Weak and strong convergence of composite implicit iteration process, Appl. Math. Comput., 190 (2007), [33] S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2007), [34] H.K. Xu, M.G. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22 (2001), [35] L.P. Yang, Convergence of the new composite implicit iteration process with random errors, Nonlinear Anal., 69 (2008), [36] Y.Y. Zhou, S.S. Chang, Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Optim., 23 (2002), [37] Y. Yao, M. Aslam Noor, S. Zainab, Y.C. Liou, Mixed equilibrium problems and optimization problems, J. Math. Anal. Appl., 354 (2009), [38] L.C. Zeng, J.C. Yao, Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings, Nonlinear Anal., 64 (2006), Zhejiang Ocean University School of Mathematics, Physics and Information Science Zhoushan , China zjhaoyan@yahoo.cn