Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 484050, 13 pages doi:10.1155/2008/484050 Research Article Composite Implicit General Iterative Process for a Nonexpansive Semigroup in Hilbert Space Lihua Li, 1 Suhong Li, 1 and Yongfu Su 2 1 Department of Mathematic and Physics, Hebei Normal University of Science and Technology Qinhuangdao, Hebei 066004, China 2 Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Correspondence should be addressed to Lihua Li, lilihua103@eyou.com Received 19 March 2008; Accepted 14 August 2008 Recommended by H ´ el ` ene Frankowska Let C be nonempty closed convex subset of real Hilbert space H. Consider C a nonexpansive semigroup I  {Ts : s ≥ 0} with a common fixed point, a contraction f with coefficient 0 <α<1, and a strongly positive linear bounded operator A with coefficient γ>0. Let 0 <γ<γ/α. It is proved that the sequence {x n } generated iteratively by x n I − α n A1/t n   t n 0 Tsy n ds  α n γfx n ,y n I − β n Ax n  β n γfx n  converges strongly to a common fixed point x ∗ ∈ FI which solves the variational inequality γf − Ax ∗ ,z− x ∗ ≤0forallz ∈ FI. Copyright q 2008 Lihua Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and preliminaries Let C be a closed convex subset of a Hilbert space H, recall that T : C → C is nonexpansive if Tx − Ty≤x − y for all x, y ∈ C. Denote by FT the set of fixed points of T,thatis, FT : {x ∈ C : Tx  x}. Recall that a family I  {Ts | 0 ≤ s<∞} of mappings from C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions: i T0x  x for all x ∈ C; ii Ts  tTsTt for all s, t ≥ 0; iii Tsx − Tsy≤x − y for all x, y ∈ C and s ≥ 0; iv for all x ∈ C, s |→Tsx is continuous. We denote by FI the set of all common fixed points of I,thatis,FI ∩ 0≤s<∞ FTs. It is known that FI is closed and convex. 2 Fixed Point Theory and Applications Iterative methods for nonexpansive mappings have recently been applied t o solve convex minimization problems see, e.g., 1–5 and the references therein. A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H: min x∈C 1 2 Ax, x−x, b, 1.1 where C is the fixed point set of a nonexpansive mapping T on H,andb is a given point in H. Assume that A is strongly positive, that is, there is a constant γ>0 with the property Ax, x≥ γx 2 ∀x ∈ H. 1.2 It is well known that FT is closed convex cf. 6.In3see also 4,itisprovedthatthe sequence {x n } defined by the iterative method below, with the initial guess x 0 ∈ H chosen arbitrarily, x n1   I − α n A  Tx n  α n b, n ≥ 0 1.3 converges strongly to the unique solution of the minimization problem 1.1 provided that the sequence {α n } satisfies certain conditions. On the other hand, Moudafi 7 introduced the viscosity approximation method for nonexpansive mappings see 8 for further developments in both Hilbert and Banach spaces.Letf be a contraction on H. Starting with an arbitrary initial x 0 ∈ H, define a sequence {x n } recursively by x n1   1 − σ n  Tx n  σ n f  x n  ,n≥ 0, 1.4 where {σ n } is a sequence in 0, 1. It is proved 7, 8 that under certain appropriate conditions imposed on {σ n }, the sequence {x n } generated by 1.4 strongly converges to the unique solution x ∗ in C of the variational inequality  I − fx ∗ ,x− x ∗  ≥ 0,x∈ C. 1.5 Recently, Marino and Xu 9 combined the iterative method 1.3 with the viscosity approximation method 1.4 considering the following general iteration process: x n1   I − α n A  Tx n  α n γf  x n  ,n≥ 0, 1.6 and proved that if the sequence {α n } satisfies appropriate conditions, then the sequence {x n } Lihua Li et al. 3 generated by 1.6 converges strongly to the unique solution of the variational inequality  A − γfx ∗ ,x− x ∗  ≥ 0,x∈ C, 1.7 which is the optimality condition for the minimization problem min x∈C 1 2 Ax, x−hx, 1.8 where h is a potential function for γf i.e., h  xγfx,forx ∈ H. In this paper, motivated and inspired by the idea of Marino and Xu 9, we introduce the composite implicit general iteration process 1.9 as follows: x n   I − α n A  1 t n  t n 0 Tsy n ds  α n γf  x n  , y n   I − β n A  x n  β n γf  x n  , 1.9 where {α n }, {β n }⊂0, 1, and investigate the problem of approximating common fixed point of nonexpansive semigroup {Ts : s ≥ 0} which solves some variational inequality. The results presented in this paper extend and improve the main results in Marino and Xu 9, and the methods of proof given in this paper are also quite different. In what follows, we will make use of the following lemmas. Some of them are known; others are not hard to derive. Lemma 1.1 Marino and Xu 9. Assume that A is a strongly positive linear bounded operator on a Hilbert space H with coefficient γ>0 and 0 <ρ≤A −1 .ThenI − ρA≤1 − γ. Lemma 1.2 Shimizu and Takashi 10. Let C be a nonempty bounded closed convex subset of H and let I  {Ts :0≤ s<∞} be a nonexpansive semigroup on C, then for any h ≥ 0, lim t →∞ sup x∈C     1 t  t 0 Tsxds − Th  1 t  t 0 Tsxds       0. 1.10 Lemma 1.3. Let C be a nonempty bounded closed convex subset of a Hilbert space H and let I  {Tt :0≤ t<∞} be a nonexpansive semigroup on C.If{x n } is a sequence in C satisfying the following properties: i x n z; ii lim sup t →∞ lim sup n →∞ Ttx n − x n   0, where x n zdenote that {x n } converges weakly to z,thenz ∈ FI. Proof. This lemma is the continuous version of Lemma 2.3 of Tan and Xu 11. This proof given in 11 is easily extended to the continuous case. 4 Fixed Point Theory and Applications 2. Main results Lemma 2.1. Let H be a Hilbert space, C a closed convex subset of H,letI  {Ts : s ≥ 0} be a nonexpansive semigroup on C, {t n }⊂0, ∞ is a sequence, then I − 1/t n   t n 0 Tsds is monotone. Proof. In fact, for all x, y ∈ H,  x − y,  I − 1 t n  t n 0 Tsds  x −  I − 1 t n  t n 0 Tsds  y   x − y 2 −  x − y, 1 t n  t n 0 Tsxds − 1 t n  t n 0 Tsyds  ≥x − y 2 −x − y 1 t n  t n 0     Tsx − Tsy     ds ≥x − y 2 −x − y 2  0. 2.1 Theorem 2.2. Let C be nonempty closed convex subset of real Hilbert space H, suppose that f : C → C is a fixed contractive mapping with coefficient 0 <α<1, and I  {Ts : s ≥ 0} is a nonexpansive semigroup on C such that FI is nonempty, and A is a strongly positive linear b ounded operator with coefficient γ>0, {α n }, {β n }⊂0, 1, {t n }⊂0, ∞ are real sequences such that lim n →∞ α n  0,β n  ◦  α n  , lim n →∞ t n  ∞, 2.2 then for any 0 <γ< γ/α, there is a unique {x n }∈C such that x n   I − α n A  1 t n  t n 0 Tsy n ds  α n γf  x n  , y n   I − β n A  x n  β n γf  x n  , 2.3 and the iteration process {x n } converges strongly to the unique solution x ∗ ∈ FI of the variational inequality γf − Ax ∗ ,z− x ∗ ≤0 for all z ∈ FI. Proof. Our proof is divided into five steps. Since α n → 0, β n → 0asn →∞, we may assume, with no loss of generality, that α n < A −1 , β n < A −1 for all n ≥ 1. i {x n } is bounded. Firstly, we will show that the mapping T f n : C → C defined by T f n   I − α n A  1 t n  t n 0 Ts  I − β n A   β n γf  ds  α n γf 2.4 Lihua Li et al. 5 is a contraction. Indeed, from Lemma 1.1,wehaveforanyx, y ∈ C that   T f n x − T f n y   ≤   I − α n A   1 t n  t n 0   Ts  I − β n A  x  β n γfx  − Ts  I − β n A  y  β n γfy    ds  α n γ   fx − fy   ≤  1 − α n γ     I − β n A  x  β n γfx  −  I − β n A  y  β n γfy     α n γαx − y ≤  1 − α n γ    I − β n A   x − y  β n γαx − y   α n γα   x − y   ≤  1 − α n γ  1 − β n  γ − γα  x − y  α n γαx − y   1 − α n γ  1 − β n  γ − γα   α n γα  x − y   1 − α n  γ − γα  −  1 − α n γ  β n  γ − γα  x − y <  1 − α n  γ − γα  x − y < x − y. 2.5 Let x n ∈ C be the unique fixed point of T f n .Thus, x n   I − α n A  1 t n  t n 0 Tsy n ds  α n γf  x n  , y n   I − β n A  x n  β n γf  x n  2.6 is well defined. Next, we will show that {x n } is bounded. Pick any z ∈ FI to obtain   x n − z         I − α n A   1 t n  t n 0 Tsy n ds − z   α n  γf  x n  − Az      ≤   I − α n A   1 t n  t n o   Tsy n − z   ds  α n  γ   f  x n  − fz      γfz − Az    ≤  1 − α n γ    y n − z    α n  γ   f  x n  − fz      γfz − Az    , 2.7   x n − z   ≤  1 − α n γ    y n − z    α n γα   x n − z    α n   γfz − Az   . 2.8 Also   y n − z   ≤   I − β n A     x n − z    β n   γf  x n  − Az   ≤  1 − β n γ    x n − z    β n γα   x n − z    β n   γfz − Az     1 − β n  γ − γα    x n − z    β n   γfz − Az   . 2.9 6 Fixed Point Theory and Applications Substituting 2.9 into 2.8,weobtainthat   x n − z   ≤  1 − α n γ  1 − β n  γ − γα    x n − z     1 − α n γ  β n   γfz − Az    α n γα   x n − z    α n   γfz − Az     1 − α n γ  1 − β n  γ − γα   α n γα    x n − z     1 − α n γ  β n  α n    γfz − Az     1 −  γ − γα  α n   1 − α n γ  β n    x n − z     α n   1 − α n γ  β n    γfz − Az   ,  γ − γα  α n   1 − α n γ  β n    x n − z   ≤  α n   1 − α n γ  β n    γfz − Az   ,   x n − z   ≤ 1 γ − γα   γfz − Az   . 2.10 Thus {x n } is bounded. ii lim n →∞ x n − Tsx n   0. Denote that z n :1/t n   t n 0 Tsy n ds,since{x n } is bounded, z n − z≤y n − z and {Az n }, {fx n } are also bounded, From 2.6 and lim n →∞ α n  0, we have   x n − z n    α n   γf  x n  − Az n   −→ 0 n −→ ∞ . 2.11 Let K  {w ∈ C : w −z≤1/ γ − γαγfz − Az}, then K is a nonempty bounded closed convex subset of C and Ts-invariant. Since {x n }⊂K and K is bounded, there exists r>0 such that K ⊂ B r , it follows from Lemma 1.2 that lim n →∞   z n − Tsz n    0 ∀s ≥ 0. 2.12 From 2.11 and 2.12, we have lim n →∞   x n − Tsx n    0. 2.13 iii There exists a subsequence {x n k } of {x n } such that x n k x ∗ ∈ FI and x ∗ is the unique solution of the following variational inequality:  A − γfx ∗ ,x ∗ − z  ≤ 0 ∀z ∈ FI. 2.14 Firstly since   y n − x n    β n   γf  x n  − A  x n    . 2.15 Lihua Li et al. 7 From condition β n → 0 and the boundedness of {x n },weobtainthaty n − x n →0. Again by boundedness of {x n }, we know that there exists a subsequence {n k } of {n} such that x n k x ∗ . Then y n k x ∗ .FromLemma 1.3 and step ii, we have that x ∗ ∈ FI. Next we will prove that x ∗ solves the variational inequality 2.14. Since x n   I − α n A  1 t n  t n 0 Tsy n ds  α n γf  x n  , 2.16 we derive that A − γfx n  − 1 α n  I − α n A   I − 1 t n  t n 0 Tsds  y n  1 α n  I − α n A  y n −  I − α n A  x n  . 2.17 It follows that, for all z ∈ FI,  A − γf  x n ,y n − z   − 1 α n   I − α n A   I − 1 t n  t n 0 Tsds  y n ,y n − z   1 α n  I − α n A  y n −  I − α n A  x n ,y n − z   − 1 α n  I − 1 t n  t n 0 Tsds  y n −  I − 1 t n  t n 0 Tsds  z, y n − z    A  I − 1 t n  t n 0 Tsds  y n ,y n − z   1 α n  I − α n A  y n −  I − α n A  x n ,y n − z  . 2.18 Using Lemma 2.1, we have from 2.18 that  A − γfx n ,y n − z  ≤  A  I − 1 t n  t n 0 Tsds  y n ,y n − z   1 α n  I − α n A  y n − x n  ,y n − z  ≤  A  I − 1 t n  t n 0 Tsds  y n ,y n − z   1 α n β n   γf  x n  − A  x n      y n − z   . 2.19 Now replacing n in 2.19 with n k and letting k →∞,wenoticethat  I − 1 t n k  t n k 0 Tsds  y n k  0, 2.20 8 Fixed Point Theory and Applications and from condition β n  ◦α n  and boundedness of {x n }, we have 1 α n k β n k   γf  x n k  − A  x n k      y n k − z   −→ 0. 2.21 For x ∗ ∈ FI,weobtain  A − γfx ∗ ,x ∗ − z  ≤ 0. 2.22 From 9, Theorem 3.2, we know that the solution of the variational inequality 2.14 is unique. That is, x ∗ ∈ FI is a unique solution of 2.14. iv lim sup n →∞  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗  ≤ 0, 2.23 where x ∗ is obtained in step iii. To see this, there exists a subsequence {n i } of {n} such that lim sup n →∞  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗   lim i →∞  1 t n i  t n i 0 Tsy n i ds − x ∗ ,γf  x ∗  − Ax ∗  , 2.24 we may also assume that x n i z, then 1/t n i   t n i 0 Tsy n i ds  z,notefromstepii that z ∈ FI in virtue of Lemma 1.2. It follows from the variational inequality 2.14 that lim sup n →∞  1 t n  t n 0 Tsx n ds − x ∗ ,γf  x ∗  − Ax ∗    z − x ∗ ,γf  x ∗  − Ax ∗  ≤ 0. 2.25 So 2.23 holds thank to 2.14. v x n → x ∗ n →∞. Finally, we will prove x n → x ∗ . Since   y n − x ∗   2     I − β n A  x n − x ∗   β n  γf  x n  − Ax ∗    2 ≤   I − β n A     x n − x ∗   2  β n   γf  x n  − Ax ∗   2 ≤  1 − β n γ    x n − x ∗   2  β n   γf  x n  − Ax ∗   2 . 2.26 Lihua Li et al. 9 Next, we calculate   x n − x ∗   2       I − α n A   1 t n  t n 0 Tsy n ds − x ∗   α n  γf  x n  − Ax ∗      2       I − α n A   1 t n  t n 0 Tsy n ds − x ∗      2  α 2 n    γf  x n  − Ax ∗    2  2α n   I − α n A   1 t n  t n 0 Tsy n ds − x ∗  ,γf  x n  − Ax ∗  ≤  1 − α n γ  2   y n − x ∗   2  α 2 n    γf  x n  − Ax ∗    2  2α n   I − α n A   1 t n  t n 0 Tsy n ds − x ∗  ,γf  x n  − Ax ∗ . 2.27 Thus it follows from 2.26 that   x n − x ∗   2 ≤  1 − α n γ  2  1 − β n γ    x n − x ∗   2   1 − α n γ  2 β n   γf  x n  − Ax ∗   2  α 2 n   γf  x n  − Ax ∗   2  2α n  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x n  − Ax ∗  − 2α 2 n  A  1 t n  t n 0 Tsy n ds − x ∗  ,γf  x n  − Ax ∗  ≤  1 − α n γ  2  1 − β n γ    x n − x ∗   2   1 − α n γ  2 β n   γf  x n  − Ax ∗   2  α 2 n   γf  x n  − Ax ∗   2  2α n γ  1 t n  t n 0 Tsy n ds − x ∗ ,f  x n  − f  x ∗    2α n  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗  − 2α 2 n  A  1 t n  t n 0 Tsy n ds − x ∗  ,γf  x n  − Ax ∗  ≤  1 − α n γ  2  1 − β n γ    x n − x ∗   2  2α n γα   y n − x ∗     x n − x ∗     1 − α n γ  2 β n   γf  x n  − Ax ∗   2  α n  2  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗   α n    γf  x n  − Ax ∗   2  2     A  1 t n  t n 0 Tsy n ds − x ∗      ·   γf  x n  − Ax ∗    10 Fixed Point Theory and Applications ≤  1 − α n γ  2  1 − β n γ    x n − x ∗   2  2α n γα  1 − β n  γ − γα    x n − x ∗   2  2α n γαβ n   γf  x ∗  − Ax ∗     x n − x ∗     1 − α n γ  2 β n   γf  x n  − Ax ∗   2  α n  2  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗   α n    γf  x n  − Ax ∗   2  2     A  1 t n  t n 0 Tsy n ds − x ∗      ·   γf  x n  − Ax ∗      1 − α n γ  2  1 − β n γ    x n − x ∗   2  2α n γα  1 − β n γ    x n − x ∗    2α n β n α 2 γ 2   x n − x ∗   2  2α n γαβ n   γf  x ∗  − Ax ∗     x n − x ∗     1 − α n γ  2 β n   γf  x n  − Ax ∗   2  α n  2  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗   α n    γf  x n  − Ax ∗   2  2     A  1 t n  t n 0 Tsy n ds − x ∗      ·   γf  x n  − Ax ∗      1 − α n γ  2  2α n γα  1 − β n γ    x n − x ∗   2  2α n β n α 2 γ 2   x n − x ∗   2  2α n γαβ n   γf  x ∗  − Ax ∗     x n − x ∗     1 − α n γ  2 β n   γf  x n  − Ax ∗   2  α n  2  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗   α n    γf  x n  − Ax ∗   2  2     A  1 t n  t n 0 Tsy n ds − x ∗      ·   γf  x n  − Ax ∗    <  1 − α n γ  2  2α n γα    x n − x ∗   2  2α n β n α 2 γ 2   x n − x ∗   2  2α n γαβ n   γf  x ∗  − Ax ∗     x n − x ∗     1 − α n γ  2 β n   γf  x n  − Ax ∗   2  α n  2  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗   α n    γf  x n  − Ax ∗   2  2α n     A  1 t n  t n 0 Tsy n ds − x ∗      ·   γf  x n  − Ax ∗      1 − 2  γ − γα  α n    x n − x ∗   2  2α n β n α 2 γ 2   x n − x ∗   2  2α n γαβ n   γf  x ∗  − Ax ∗     x n − x ∗     1 − α n γ  2 β n   γf  x n  − Ax ∗   2  α n  2  1 t n  t n 0 Tsy n ds − x ∗ ,γf  x ∗  − Ax ∗   α n    γf  x n  − Ax ∗   2  2     A  1 t n  t n 0 Tsy n ds − x ∗      ·   γf  x n  − Ax ∗    γ 2   x n − x ∗   2  . 2.28 [...]... Cambridge, UK, 1990 7 A Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol 241, no 1, pp 46–55, 2000 8 H.-K Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol 298, no 1, pp 279–291, 2004 9 G Marino and H.-K Xu, A general iterative method fornonexpansive mappings... mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol 318, no 1, pp 43–52, 2006 10 T Shimizu and W Takahashi, “Strong convergence to common fixed points of families of nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol 211, no 1, pp 71–83, 1997 11 K.-K Tan and H.-K Xu, “The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach... Mathematics, pp 473–504, North-Holland, Amsterdam, The Netherlands, 2001 5 I Yamada, N Ogura, Y Yamashita, and K Sakaniwa, “Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space,” Numerical Functional Analysis and Optimization, vol 19, no 1-2, pp 165–190, 1998 6 K Goebel and W A Kirk, Topics in Metric Fixed Point Theory, vol 28 of Cambridge Studies in Advanced Mathematics, Cambridge... Optimization Theory and Applications, vol 116, no 3, pp 659–678, 2003 4 I Yamada, “The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings,” in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D Butnariu, Y Censor, and S Reich, Eds., vol 8 of Studies in Computational Mathematics,... “Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings,” Numerical Functional Analysis and Optimization, vol 19, no 1-2, pp 33–56, 1998 Lihua Li et al 13 2 H.-K Xu, Iterative algorithms for nonlinear operators,” Journal of the London Mathematical Society, vol 66, no 1, pp 240–256, 2002 3 H.-K Xu, “An iterative approach to quadratic optimization,” Journal... x∗ − Ax∗ 2.32 0 ◦ αn and 2.23 , we conclude that lim sup xn − x∗ n→∞ 2 ≤ 0 2.33 So xn → x∗ This completes the proof of the Theorem 2.2 It follows from the above proof that Theorem 2.2 is valid for nonexpansive mappings Thus, we have that Corollaries 2.3 and 2.4 are two special cases of Theorem 2.2 Corollary 2.3 Let T be a nonexpansive mapping from nonempty closed convex subset C of a Hilbert space... of a Hilbert space H to C, {xn } is generated by the following algorithm: xn yn I − αn A T yn I − βn A xn αn γf xn , 2.36 βn γf xn , where {αn } is a sequence in (0, 1) satisfying the following condition: limn → ∞ αn 0, then the sequence {xn } converges strongly to the unique solution x∗ ∈ F T of the variational inequality I −f x∗ , x∗ − z ≤ 0 for all z ∈ F T References 1 F Deutsch and I Yamada, “Minimizing... generated by the following algorithm: xn I − αn A T yn yn I − βn A xn αn γf xn , βn γf xn , 2.34 where {αn }, {βn } ⊂ 0, 1 are real sequences such that lim αn n→∞ 0, βn ◦ αn , 2.35 then for any 0 < γ < γ/α, the sequence {xn } above converges strongly to the unique solution x∗ ∈ F T of the variational inequality γf − A x∗ , z − x∗ ≤ 0 for all z ∈ F T Corollary 2.4 Let T be a nonexpansive mapping from... 2.29 Since {xn } is bounded, we can take a constant L1 , L2 , L3 > 0 such that L1 ≥ 2α2 γ 2 xn − x∗ 2γα γf x∗ − Ax∗ · xn − x∗ , 2 L2 ≥ γf xn − Ax∗ 2 L3 ≥ γf xn − Ax∗ 2 , 2 A 1 tn tn T s yn ds − x∗ · γf xn − Ax∗ γ 2 xn − x∗ 2 0 2.30 for all n ≥ 0 It then follows from 2.29 that 2 γ − γα xn − x∗ 2 ≤ βn L1 βn L2 αn 2 1 tn tn T s yn ds − x∗ , γf x∗ − Ax∗ αn L3 0 2.31 12 Fixed Point Theory and Applications...Lihua Li et al 11 Thus 2 γ − γα xn − x∗ ≤ 2βn α2 γ 2 xn − x∗ 2 1 − αn γ 2 tn 1 tn 2 βn αn 2γαβn γf x∗ − Ax∗ · xn − x∗ 2 γf xn − Ax∗ 2 T s yn ds − x∗ , γf x∗ − Ax∗ 0 γf xn − Ax∗ αn 2 2 A 1 tn · γf xn − Ax∗ ≤ βn 2α2 γ 2 xn − x∗ 2 βn γf xn − Ax∗ αn 2 tn 1 tn tn T s yn ds − x∗ 0 γ 2 xn − x∗ 2 2γα γf x∗ − Ax∗ · xn − x∗ 2 T s yn ds − x∗ , γf x∗ − Ax∗ 0 αn γf xn − Ax∗ 2 2 A · γf xn − Ax∗ 1 tn tn T . The Netherlands, 2001. 5 I. Yamada, N. Ogura, Y. Yamashita, and K. Sakaniwa, “Quadratic optimization of fixed points of nonexpansive mappings in Hilbert space,” Numerical Functional Analysis and Optimization,. “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004. 9 G. Marino and H K. Xu, A general iterative. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 484050, 13 pages doi:10.1155/2008/484050 Research Article Composite Implicit General Iterative Process