RESEARC H Open Access Weak and strong convergence theorems of implicit iteration process on Banach spaces Lai-Jiu Lin 1* , Chih-Sheng Chuang 1 and Zenn-Tsun Yu 2 * Correspondence: maljlin@cc.ncue. edu.tw 1 Department of Mathematics, National Changhua University of Education, Changhua 50058, Taiwan Full list of author information is available at the end of the article Abstract In this article, we first consider weak convergence theorems of implicit iterative processes for two nonexpansive mappings and a mapping which satisfies condition (C). Next, we consider strong convergence theorem of an implicit-shrinking iterative process for two nonexpansive mappings and a relative nonexpansive mapping on Banach spaces. Note that the conditions of strong convergence theorem are different from the strong convergence theorems for the implicit iterative processes in the literatures. Finally, we discuss a strong convergence theorem concerning two nonexpansive mappings and the resolvent of a maximal monotone operator in a Banach space. 1 Introduction Let E be a Banach spa ce, and let C be a nonempty closed convex subset of E.Amap- ping T: C ® Eisnone xpansive if ||Tx - Ty|| ≤ ||x - y|| for every x, y Î C. Let F(T): = {x Î C: x = Tx} denote the set of fixed points of T.Besides,amappingT: C ® E is quasinonexpansive if F( T) = ∅ and ||Tx - y|| ≤ ||x - y|| for all x Î C and y Î F(T). In 2008, Suzuki [1 ] introduced the following generalized nonexpansive mapping on Banach spaces. A mapping T: C ® E is said to satisfy condition (C) if for all x, y Î C, 1 2 ||x − Tx|| ≤ ||x − y|| ⇒ ||Tx − Ty|| ≤ ||x − y||. In fact, eve ry nonexpansive mapping satisfies condition (C), but the converse may be false [1, Example 1]. Besides, i f T: C ® E satisfies condition (C)and F( T) = ∅ , then T is a quasinonexpansive mapping. However, the converse may be false [1, Example 2]. Construction of approximating fixed points of nonlinear mappings is an important subject in the theory of nonlinear mappings and its applications in a number of applied areas. Let C be a nonempty closed convex subset of a real Hilbert space H, and let T: C ® C be a mapping. In 1953, Mann [2] gave an iteration process: x n+1 = α n x n +(1− α n )Tx n , n ≥ 0, (1:1) where x 0 is taken in C arbitrarily, and {a n } is a sequence in [0,1]. In 2001, Soltuz [3] introduced the following Mann-type implicit process for a nonex- pansive mapping T: C ® C: x n = α n x n−1 +(1− α n )Tx n , n ∈ N, (1:2) Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 © 2011 L in et al; licen see Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. where x 0 is taken in C arbitrarily, and {t n } is a sequence in [0,1]. In 2001, Xu and Ori [4] have introduced an implicit iteration process for a finite family of nonexpansive mappings. Let T 1 , T 2 , ,T N be N self-mappings of C and sup- pose that F := ∩ N i=1 F( T i ) = ∅ , the set of common fixed points of T i , i = 1, 2, , N. Let I: = {1, 2, , N}. Xu and Ori [4] gave an implicit iteration process for a finite family of nonexpansive mappings: x n = t n x n−1 +(1− t n )T n x n , n ∈ N, (1:3) where x 0 is taken in C arbitrarily, {t n } is a sequence in [0,1], and T k = T k mod N . (Here the mod N functio n takes values in I.) And they proved the weak convergence of p ro- cess (1.3) to a common fixed point in the setting of a Hilbert space. In 2010, K han et al. [5] presented an implicit iterative process for two nonexpansive mappings in Banach spaces. Let E be a Banach space, and let C beanonemptyclosed convex subset of E,andletT, S: C ® C be two nonexpansive mappings. Khan et al. [5] considered the following implicit iterative process: x n = α n x n−1 + β n Sx n + γ n Tx n , n ∈ N, (1:4) where {a n }, {b n }, and {g n } are sequences in [0,1] with a n + b n + g n =1. Motivated by the above works in [5], we want to consider the following implicit iterative process. Let E be a Banach space, C be a nonempty closed convex subset of E, and let T 1 , T 2 : C ® C be two nonexpansive mappings, and let S: C ® C be a mapping which satisfy condition (C). We first consider the weak convergen ce theorems for the following implicit iterative process:  x 0 ∈ C chosen arbitrary, x n = a n x n−1 + b n Sx n−1 + c n T 1 x n + d n T 2 x n , (1:5) where {a n }, {b n }, {c n }, and {d n } are sequences in [0,1] with a n + b n + c n + d n =1. Next, we also consider weak convergence theorems for another implicit iterative pro- cess: ⎧ ⎨ ⎩ x 0 ∈ C chosen arbitrary, y n = a n x n−1 + b n T 1 y n + c n T 2 y n , x n = d n y n +(1− d n )Sy n , (1:6) where {a n }, {b n }, {c n }, and {d n } are sequences in [0,1] with a n + b n + c n =1. In fact, for the above implicit iterative processes, most researchers always considered weak convergence theorems, and fe w researchers considered strong co nvergence theo- rem under suitable conditions. For example, one can see [ 5-7]. However, some condi- tions are not natural. For this reason, we consider the following shrinking-implicit iterative processes and study the strong convergence theorem. Let {x n } be defined by ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ x 0 ∈ C chosen arbitrary and C 0 = D 0 = C, y n = a n x n−1 + b n T 1 y n + c n T 2 y n , z n = J −1 (d n Jy n +(1− d n )JSy n ), C n = {z ∈ C n−1 : φ(z , z n ) ≤ φ(z, y n )}, D n = {z ∈ D n−1 : ||y n − z|| ≤ ||x n−1 − z||}, x n =  C n ∩D n x 0 , (1:7) Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 2 of 20 where {a n }, {b n }, {c n }, and {d n } are sequences in (0, 1) with a n + b n + c n =1. In this article, we first co nsider weak convergence theorems of implicit iterative pro- cesses for two nonexpansive mappings and a mapping which satisfy condition (C). And we generalize Khan et al.’s result [5] as special case. Next, we consider strong conver- gence theorem of an implicit-shrinking iterative process for two non-expansive map- pings and a relative nonexpansive mapping on Banach spaces. Note that the conditions of strong convergence theorem are different from the strong convergence theorems for the implicit iterative processes in the literatures. Finally, we discuss a strong conver- gence theorem concerning two nonexpansive mappings and the resolvent of a maximal monotone operator in a Banach space. 2 Preliminaries Throughout this article, let N and ℝ be the sets of all positive integers and real num- bers, respectively. Let E be a Banach space and let E* b e the dual space of E. For a sequence {x n }ofE and a point x Î E, the weak convergence of {x n }tox and the strong convergence of {x n }tox are denoted by x n ⇀ x and x n ® x, respectively. A Banach space E is said to satisfy Opial’s condition if {x n } is a sequence in E with x n ⇀ x, then lim sup n→∞ ||x n − x|| < lim sup n→∞ ||x n − y||, ∀y ∈ E, y = x. Let E be a Banach space. Then, the duality mapping J : E  E ∗ is defined by Jx :  x ∗ ∈ E ∗ :  x, x ∗  = ||x || 2 = ||x ∗ || 2  , ∀x ∈ E. Let S(E) be the unit sphere centered at the origin of E. Then, the space E is said to be smooth if the limit lim t→0 ||x + ty|| − ||x|| t exists for all x, y Î S(E). It is also said to be uniformly smooth if th e limit exists uni- formly in x, y Î S(E). A Banach space E is said to be strictly convex if    x + y 2    < 1 whenever x, y Î S(E) and x ≠ y. It is said to be uniformly convex if for each ε Î (0, 2], there exists δ >0suchthat    x + y 2    < 1 − δ whenev er x, y Î S(E)and||x - y|| ≥ ε. Furthermore, we know that [8] (i) if E in smooth, then J is single-valued; (ii) if E is reflexive, then J is onto; (iii) if E is strictly convex, then J is one-to-one; (iv) if E is strictly convex, then J is strictly monotone; (v) if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E. A Banach space E is said to have Kadec-Klee property if a sequence {x n }ofE satisfy- ing that x n ⇀ x and ||x n || ® ||x||, then x n ® x. It is known that if E uniformly convex, then E has the Kadec-Klee property [8]. Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 3 of 20 Let E be a smooth, strictly convex and reflexive Banach space and let C be a none- mpty closed convex subset of E. Throughout this article, define the function j: C × C ® ℝ by φ(x, y):=||x|| 2 − 2  x, Jy  + ||y|| 2 , ∀x, y ∈ E. Observe that, in a Hilbert space H, j(x, y)=||x - y|| 2 for all x , y Î H.Furthermore, for each x, y, z, w Î E, we know that: (1) (||x|| - ||y||) 2 ≤ j(x, y) ≤ (||x|| + ||y||) 2 ; (2) j(x, y) ≥ 0; (3) j(x, y)=j( x, z)+j(z, y)+2〈x - z, Jz - Jy〉; (4) 2〈x - y, Jz - Jw〉 = j(x, w)+j(y, z)-j(x, z)-j(y, w); (5) if E is additionally assumed to be strictly convex, then φ(x, y) = 0 if and only if x = y; (6) j(x, J -1 (lJy +(1-l)Jz)) ≤ lj (x, y) + (1 - l)j(x, z). Lemma 2.1. [9] Let E be a uniformly convex Banach space and let r > 0. Then, there exists a strictly increasing, co ntinuous, and convex function g:[0,2r] ® [0, ∞)such that g(0) = 0 and ||ax + by + cz + dw || 2 ≤ a ||x || 2 + b||y|| 2 + c||z|| 2 + d||w|| 2 − abg(||x − y||) for all x, y, z, w Î B r and a, b, c, d Î [0,1] with a + b + c + d = 1, where B r :={z Î E:||z|| ≤ r}. Lemma 2.2.[10]LetE beauniformlyconvexBanachspaceandletr >0.Then, there exists a strictly increasing, continuous, and convex function g:[0,2r] ® [0, ∞) such that g(0) = 0 and φ(x, J −1 (λJy +(1− λ)Jz)) ≤ λφ(x, y)+(1− λ)φ(x , z) − λ(1 − λ)g(||Jy − Jz||) for all x, y, z Î B r and lÎ[0,1], where B r :={z Î E:||z|| ≤ r}. Lemma 2.3.[11]LetE be a uniformly convex Banach space, let {a n }beasequence of real numbers such that 0 <b ≤ a n ≤ c <1foralln Î N,andlet{x n }and{y n }be sequences of E such that lim sup n®∞ ||x n || ≤ a,limsup n®∞ ||y n || ≤ a,andlim n®∞ || a n x n +(1-a n )y n || = a for some a ≥ 0. Then, lim n®∞ ||x n - y n || = 0. Lemma 2.4. [12] Let E be a smooth and uniformly convex Banach space, and let {x n } and {y n } be sequences in E such that either {x n }or{y n } is bounded. If lim n®∞ j(x n , y n ) = 0, then lim n®∞ ||x n - y n || = 0. Remark 2 .1. [13] Let E be a uniformly convex and uniformly smooth Banach space. If {x n } and { y n } are bounded sequences in E, then lim n→∞ φ(x n , y n )=0⇔ lim n→∞ ||x n − y n || =0⇔ lim n→∞ ||Jx n − Jy n || =0. Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E. For an arbitrary point x of E, the set Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 4 of 20  z ∈ C : φ(z, x) = min y∈C φ(y, x)  is always nonempty and a singleton [14]. Let us de fine the mapping Π C from E onto C by Π C x = z, that is, φ( C x, x) = min y∈C φ(y, x) for every x Î E. Such Π C is called the generalized projection from E onto C [14]. Lemma 2.5. [14,15] Let C be a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach space E, and let (x, z) Î E × C. Then: (i) z = Π C x if and only if 〈y - z, Jx - Jz〉 ≤ 0 for all y Î C; (ii) j(z, Π C x)+j(Π C x, x) ≤ j(z, x). Lemma 2.6. [16] Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and T: C ® C is a nonexpansive m apping. Let {x n }beasequence in C with x n ⇀ x Î C and lim n®∞ ||x n - Tx n || = 0. Then, Tx = x. Lemma 2.7.[1]LetC be a nonempty subset of a Banach space E with the Opial property. Assume that T: C ® E satisfies condition (C). Let {x n }beasequenceinC with x n ⇀ x Î C and lim n®∞ ||x n - Tx n || = 0. Then, Tx = x. Lemma 2.8.[1]LetT be a mapping on a c losed subset C of a Banach space E. Assume that T satisfies condition (C). Then, F(T) is a closed set. Moreover, if E is strictly convex and C is convex, then F(T) is also convex. Lemma 2.9. [17] Let C be a nonempty closed convex subset of a strictly convex Banach space E,andT: C ® C be a nonexpansive mapping. Then, F(T) is a closed convex subset of C. 3 Weak convergence theorems Lemma 3.1.LetE be a un iformly convex Banach space, C be a nonem pty closed con- vex subset of E,andletT 1 , T 2 : C ® C be two nonexpansive mappings, and let S: C ® C be a mapping with condition (C). Let {a n }, {b n }, {c n }, and {d n }besequenceswith 0<a ≤ a n , b n , c n , d n ≤ b <1anda n + b n + c n + d n =1.Supposethat  := F(S) ∩ F(T 1 ) ∩ F( T 2 ) = ∅ . Define a sequence {x n }by  x 0 ∈ C chosen arbitrary, x n = a n x n−1 + b n Sx n−1 + c n T 1 x n + d n T 2 x n . Then, we have: (i) lim n→∞ ||x n − p|| exists for each p Î Ω. (ii) lim n→∞ ||x n − Sx n || = lim n→∞ ||x n − T 1 x n || = lim n→∞ ||x n − T 2 x n || =0 . Proof. First, we show t hat {x n } is well-defined. Now, let f(x): = a 1 x 0 +b 1 Sx 0 +c 1 T 1 x +d 1 T 2 x. Then, ||f (x)− f(y)|| ≤ c 1 ||T 1 x −T 1 y|| +d 1 ||T 2 x −T 2 y|| ≤ (c 1 + d 1 )||x −y || ≤ (1− 2a)||x−y||. Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 5 of 20 By Banach contraction principle, the existence of x 1 is established. Similarly, the exis- tence of {x n } is well-defined. (i) For each p Î Ω and n Î N, we have: ||x n − p|| ≤ a n ||x n−1 − p|| + b n ||Sx n−1 − p|| + c n ||T 1 x n − p|| + d n ||T 2 x n − p|| ≤ a n ||x n−1 − p|| + b n ||x n−1 − p|| +(c n + d n )||x n − p||. This implies that (1 -c n - d n )||x n - p|| ≤ (a n +b n )||x n-1 -p||. Hence, ||x n -p||≤ ||x n-1 - p||, lim n ®∞ ||x n -p|| exists, and {x n } is a bounded sequence. (ii) Take any p Î Ω and l et p be fixed. Suppose that lim n→∞ ||x n − p|| = d . Clearly, lim sup n→∞ ||T 2 x n − p|| ≤ d , and we have: lim n→∞ ||x n − p|| = lim n→∞ ||a n x n−1 + b n Sx n−1 + c n T 1 x n + d n T 2 x n − p|| = lim n→∞     (1 − d n )  a n 1 − d n (x n−1 − p)+ b n 1 − d n (Sx n−1 − p)+ c n 1 − d n (T 1 x n − p)  + d n (T 2 x n − p)     . Besides, lim sup n→∞     a n 1 − d n (x n−1 − p)+ b n 1 − d n (Sx n−1 − p)+ c n 1 − d n (T 1 x n − p)     ≤ lim sup n→∞ a n 1 − d n ||x n−1 − p|| + b n 1 − d n ||Sx n−1 − p|| + c n 1 − d n ||T 1 x n − p|| ≤ lim sup n→∞ a n 1 − d n ||x n−1 − p|| + b n 1 − d n ||Sx n−1 − p|| + c n 1 − d n ||T 1 x n − p|| ≤ lim sup n→∞ a n + b n 1 − d n ||x n−1 − p|| + c n 1 − d n ||x n − p|| ≤ lim sup n→∞ a n + b n + c n 1 − d n ||x n−1 − p|| = d. By Lemma 2.3, lim n→∞     a n 1 − d n (x n−1 − p)+ b n 1 − d n (Sx n−1 − p)+ c n 1 − d n (T 1 x n − p) − (T 2 x n − p)     =0. This implies that lim n®∞ ||x n - T 2 x n || = 0. Similarly, lim n®∞ ||x n - T 1 x n || = 0. Since {x n } is bounded, there exists r > 0 such that 2 sup{||x n -p||:n Î N}≤ r. By Lemma 2.1, there exists a strictly increasing, continuous, and convex function g: [0, 2r] ® [0, ∞) such that g(0) = 0 and ||x n − p|| 2 ≤ a n ||x n−1 − p|| 2 + b n ||Sx n−1 − p|| 2 + c n ||T 1 x n − p|| 2 + d n ||T 2 x n − p|| 2 −a n b n g(||x n−1 − Sx n−1 ||) ≤ (a n + b n )||x n−1 − p|| 2 +(c n + d n )||x n − p|| 2 − a n b n g(||x n−1 − Sx n−1 ||). Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 6 of 20 This implies that a n b n g(||x n−1 − Sx n−1 ||) ≤ (a n + b 2 )(||x n−1 − p|| 2 −||x n − p|| 2 ). By the properties of g and lim n®∞ ||x n - p|| = d, we get lim n®∞ ||x n - Sx n || = 0. Theorem 3.1.LetE be a uniformly conve x Banach space with Opial’s condition, C be a nonempty closed convex subset of E, and let T 1 , T 2 : C ® C be two nonexpansive mappings, and let S: C ® C be a mapping with condition (C). Let {a n }, {b n }, {c n }, and {d n } be sequences with 0 <a ≤ a n , b n , c n , d n ≤ b < 1 and a n +b n +c n +d n = 1. Suppose that  := F(S) ∩ F(T 1 ) ∩ F( T 2 ) = ∅ . Define a sequence {x n }by  x 0 ∈ C chosen arbitrary, x n = a n x n−1 + b n Sx n−1 + c n T 1 x n + d n T 2 x n . Then, x n ⇀ z for some z Î Ω. Proof. By Lemma 3.1, {x n } is a bounded sequence. Then, there exists a subsequence {x n k } of {x n }andz Î C such that x n k  z . By Lemmas 2.6, 2.7, and 3.1, we know that z Î Ω. Since E has Opial’s condition, it is easy to see that x n ⇀ z. Hence, the proof is completed. Remark 3.1. The conclus ion of Theorem 3.1 is still true if S: C ® C is a quasi-non- expansive mapping, and I - S is demiclosed at zero, that is, x n ⇀ x and (I-S)x n ⇀ 0 implies that (I - S)x =0. In Theorem 3.1, if S = I, then we get the following result. Hence, Theorem 3.1 gen- eralizes Theorem 4 in [ 5]. Corol lary 3 .1. [5] Let E be a uniforml y convex Banach space with Opial’s condition, C be a nonempty closed convex subset of E, and let T 1 , T 2 : C ® C be two nonexpan- sive mappings. Let { a n }, {b n }, and {c n } be sequences with 0 <a ≤ a n , b n , c n ≤ b < 1 and a n +b n +c n = 1. Suppose that  := F(T 1 ) ∩ F ( T 2 ) = ∅ . Define a sequence {x n }by  x 0 ∈ C chosen arbitrary, x n = a n x n−1 + b n T 1 x n + c n T 2 x n . Then, x n ⇀ z for some z Î Ω. Besides, it is easy to get the following result from Theorem 3.1. Corollary 3.2 .LetE be a uniformly convex Banach space with Opial’s condition, C be a nonempty closed convex subset of E,andletS: C ® C be a mapping with condi- tion (C). Let {a n }beasequencewith0<a ≤a n ≤b < 1. Suppose that F( S) = 0 .Definea sequence {x n }by  x 0 ∈ C chosen arbitrary, x n = a n x n−1 +(1− a n )Sx n−1 . Then, x n ⇀ z for some z Î F(S). Proof.LetT 1 = T 2 = I,whereI is the identity mapping. For each n Î N,weknow that x n = a n 2 x n−1 + 1 − a n 2 Sx n−1 + 1 4 T 1 x n + 1 4 T 2 x n . By Theorem 3.1, it is easy to get the conclusion. Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 7 of 20 Theorem 3.2.LetE be a uniformly conve x Banach space with Opial’s condition, C be a nonempty closed convex subset of E, and let T 1 , T 2 : C ® C be two nonexpansive mappings, and let S: C ® C be a mapping with condition (C). Let {a n }, {b n }, {c n }, and {d n } be sequences with 0 <a ≤ a n , b n , c n , d n ≤ b < 1 and a n +b n +c n = 1. Suppose that  := F(S) ∩ F(T 1 ) ∩ F( T 2 ) = ∅ . Define a sequence {x n } by ⎧ ⎨ ⎩ x 0 ∈ C chosen arbitrary, y n = a n x n−1 + b n T 1 y n + c n T 2 y n , x n = d n y n +(1− d n )Sy n . Then, x n ⇀ z for some z Î Ω. Proof. Following the same argument as in Lemma 3.1, we k now that {y n }iswell- defined. Take any w Î Ω and let w be fixed. Then, for each n Î N, we have ||y n − w|| = ||a n x n−1 + b n T 1 y n + c n T 2 y n − w|| ≤ a n ||x n−1 − w|| + b n ||T 1 y n − w|| + c n ||T 2 y n − w|| ≤ a n ||x n−1 − w|| +(b n + c n )||y n − w||. This implies that ||y n - w|| ≤ ||x n-1 - w|| for each n Î N. Besides, we also have ||x n − w|| = ||d n y n +(1− d n )Sy n − w|| ≤ d n ||y n − w|| +(1− d n )||Sy n − w|| ≤||y n − w||. Hence, ||x n - w|| ≤ ||y n - w|| ≤ ||x n-1 - w|| for each n Î N. So, lim n®∞ ||x n - w|| and lim n®∞ ||y n - w|| exist, and {x n }, {y n } are bounded sequences. Suppose that lim n®∞ ||x n -w|| = lim n®∞ ||y n -w|| = d. Clearly, lim sup n®∞ ||T 2 y n -w|| ≤ d, and we have lim n→∞ ||y n − w|| = lim n→∞ ||a n x n−1 + b n T 1 y n + c n T 2 y n − w|| = lim n→∞     (1 − c n )  a n 1 − c n (x n−1 − w)+ b n 1 − c n (T 1 y n − w)  + c n (T 2 y n − w)     . Besides, lim sup n→∞     a n 1 − c n (x n−1 − w)+ b n 1 − c n (T 1 y n − w)     ≤ lim sup n→∞ a n 1 − c n ||x n−1 − w|| + b n 1 − c n ||T 1 y n − w|| ≤ lim sup n→∞ a n 1 − c n ||x n−1 − w|| + b n 1 − c n ||y n − w|| ≤ lim sup n→∞ ||x n−1 − w|| = d. By Lemma 2.3, lim n→∞     a n 1 − c n (x n−1 − w)+ b n 1 − c n (T 1 y n − w) − (T 2 y n − w)     =0. This implies that lim n®∞ ||y n - T 2 y n || = 0. Similarly, lim n®∞ ||y n - T 1 y n || = 0. Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 8 of 20 Since {x n } and { y n } are bounded sequences, there exists r > 0 such that 2sup{||x n ||, ||y n ||, ||x n − w||, ||y n − w|| : n ∈ N}≤r. By Lemma 2.1, there exists a strictly increasing, continuous, and convex function g: [0, 2r] ® [0, ∞) such that g(0) = 0 and ||d n y n +(1−d n )Sy n −w|| 2 ≤ d n ||y n −w|| 2 +(1−d n )||Sy n −w|| 2 −d n (1−d n )g(||y n −Sy n ||). This implies that d n (1 − d n )g(||y n − Sy n ||) ≤||y n − w|| 2 −||x n − w|| 2 . Since lim n®∞ ||x n - w|| = lim n®∞ ||y n - w|| = d, and the properties of g,weget lim n®∞ ||y n - Sy n || = 0. Besides, ||x n − y n || = ||d n y n +(1− d n )Sy n − y n || =(1− d n )||y n − Sy n ||. Hence, lim n®∞ ||x n -y n || = 0. Final ly, fo llowing the same argument as in the proof of Theorem 3.1, we know that x n ⇀ z for some z Î Ω. Next, we give the following examples for Theorems 3.1 and 3.2. Example 3.1.LetE = ℝ, C:=[0,3],T 1 x = T 2 x = x,andletS: C ® C be the same as in [1]: Sx :=  0ifx =3, 1ifx =3. For each n, let a n = b n = c n = d n = 1 4 . Let x 0 = 1. Then, for the sequence {x n }, in The- orem 3.1, we know that x n = 1 2 n for all n Î N,andx n ® 0, and 0 is a common fixed point of S, T 1 , and T 2 . Example 3.2.LetE, C, T 1 ,T 2 , S be the same as in E xample 3.1. For each n,let a n = b n = c n = 1 3 ,and d n = 1 2 .Letx 0 =1.Then,forthesequence{x n } in Theorem 3.1, we know that x n = 1 2 n for all n Î N,andx n ® 0, and 0 is a common fixed point of S, T 1 , and T 2 . Example 3.3.LetE, C,{a n }, {b n }, {c n }, {d n }, and let S: C ® C be the same as in Example 3.1. Let T 1 x = T 2 x = 0 for each x Î C. Then, for the sequence {x n }inTheo- rem 3.1, we know that x n = 1 4 n for all n Î N. Example 3.4.LetE, C,{a n }, {b n }, {c n }, {d n }, and let S: C ® C be the same as in Example 3.2. Let T 1 x = T 2 x = 0 for each x Î C. Then, for the sequence {x n }inTheo- rem 3.2, we know that x n = 1 6 n for all nÎ N. Remark 3.2. (i) For the rate of convergence, by Examples 3.3 and 3.4, we know that the iteration process in Theorem 3.2 may be faster than the iteration process in Theorem 3.1. Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 9 of 20 But, the times of iteration process for Theorem 3.2 is much than ones in Theorem 3.1. (ii) The conclusion of Theorem 3.2 is still true if S: C ® C is a quasi-nonexpansive mapping, and I-Sis demiclosed at zero, that is, x n ⇀ x and (I-S)x n ® 0 implies that (I - S)x =0. (iii) Corollaries 3.1 and 3.2 are special cases of Theorem 3.2. Definition 3.1. [18] Let C be a nonempty subset of a Banach space E. A mapping T: C ® E satisfy condition (E) if there exists μ ≥ 1 such that for all x, y Î C, ||x − Ty|| ≤ μ||x − Tx|| + ||x − y||. By Lemma 7 in [1], we know that if T satisfies condition (C), then T satisfi es condi- tion (E). But, the converse may be false [18, Example 1]. Furthermore, we also observe the following result. Lemma 3.2. [18] Let C be a nonempty subset of a Banac h space E. Let T: C ® E be a mapping. Assume that: (i) lim n→∞ ||x n − Tx n || =0 and x n ⇀ x; (ii) T satisfies condition (E); (iii) E has Opial condition. Then, Tx = x. By Lemma 3.2, if S satisfies condition (E), then the conclusions of Theorems 3.1 and 3.2 are still true. Hence, we can use the following condition to replace condition (C)in Theorems 3.1 and 3.2 by Proposition 19 in [19]. Definition 3.2. [19] Let T be a mapping on a subset C of a Banach space E. Then, T is said to satisfy (SKC)-condition if 1 2 ||x − Tx|| ≤ ||x − y|| ⇒ ||Tx − Ty|| ≤ N( x , y), where N( x , y):=max{||x − y ||, 1 2 (||x − Tx|| + ||Ty − y||), 1 2 (||Tx − y|| + ||x − Ty||)} for all x, y Î C. 4 Strong convergence theorems (I) Let C be a nonem pty closed convex subset of a Banach space E. A point p in C is said to be an asymptotic fixed point of a mapping T: C ® C if C contains a sequence {x n } which converges weakly to p such th at lim n®∞ ,||x n - Tx n || = 0. The set of asymptotic fixed points of T will be denoted by ˆ F( T) . A mapping T: C ® C is called relatively nonexpansive [20] if F( T) = 0, ˆ F(T)=F(T) , and j(p,Tx) ≤ j(p,x) for all x Î C and p Î F(T). Note that every identity mapping is a relatively nonexpansive mapping. Lemma 4.1.[21]LetE be a strictly convex and smooth Banach space, let C be a closed convex subset of E,andletT: C ® C be a relatively nonexpansive mapping. Then, F(T) is a closed and convex subset of C. The following property is motivated by the property (Q 4 ) in [22]. Lin et al. Fixed Point Theory and Applications 2011, 2011:96 http://www.fixedpointtheoryandapplications.com/content/2011/1/96 Page 10 of 20 [...]... also said to be maximal monotone if A is monotone and there is no monotone operator from E into E* whose graph properly contains the graph of A It is known that if A ⊆ E × E* is maximal monotone, then A -10 is closed and convex Lemma 6.1 [30] Let E be a reflexive, strictly convex, and smooth Banach space and let A ⊆ E × E* be a monotone operator Then, A is maximal monotone if and only if R (J + rA) = E*... M: Convergence of an implicit algorithm for two families of nonexpansive mappings Comput Math Appl 59, 3084–3091 (2010) doi:10.1016/j.camwa.2010.02.029 6 Chidume, CE, Shahzad, N: Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings Nonlinear Anal 65, 1149–1156 (2005) 7 Fukhar-ud-din, H, Khan, SH: Convergence of iterates with errors of asympotically quasi-nonexpansive... doi:10.1186/1687-1812-2011-96 Cite this article as: Lin et al.: Weak and strong convergence theorems of implicit iteration process on Banach spaces Fixed Point Theory and Applications 2011 2011:96 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous peer review 7 Immediate publication on acceptance 7 Open access: articles freely available online 7 High visibility within the field... Nilsrakoo, W, Saejung, S: Strong convergence theorems by Halpern-Mann iterations for relatively nonexpansive mappings in Banach spaces Appl Math Comput 217, 6577–6586 (2011) doi:10.1016/j.amc.2011.01.040 14 Alber, YaI: Metric and generalized projection operators in Banach spaces: properties and applications Theory and Applications of Nonlinear Operators of Accretive and Monotone Type pp 15–50.Dekker,... generalization of Theorem 4.1 in [24] But, it is a special case of Theorem 3.1 in [25] 5 Strong convergence theorems (II) In this section, we need the following important lemmas Lemma 5.1 [26] Let E be a reflexive Banach space and f: E ® ℝ ∪ {+∞} be a convex and lower semicontinuous function Let C be a nonempty bounded and closed convex subset of E Then, the function f attains its minimum on C That is,... uniformly convex and uniformly smooth Banach space with Opial’s condition, C be a nonempty closed convex subset of E, and let T1, T2 : C ® C be two nonexpansive mappings, and let S: C ® C be a mapping with condition (C) Let {an}, {bn}, and {cn} be sequences in (a, b) for some 0 . consider weak convergence theorems of implicit iterative processes for two nonexpansive mappings and a mapping which satisfies condition (C). Next, we consider strong convergence theorem of an implicit- shrinking. iterative process for two nonexpansive mappings and a relative nonexpansive mapping on Banach spaces. Note that the conditions of strong convergence theorem are different from the strong convergence theorems. Lin et al.: Weak and strong convergence theorems of implicit iteration process on Banach spaces. Fixed Point Theory and Applications 2011 2011:96. Submit your manuscript to a journal and benefi