QUADRATIC OPTIMIZATION OF FIXED POINTS FOR A FAMILY OF NONEXPANSIVE MAPPINGS IN HILBERT SPACE B. E. RHO ADES Received 10 September 2003 Given a finite family of nonexpansive self-mappings of a Hilbert space, a particular qua- dratic functional, and a strongly positive selfadjoint bounded linear operator, Yamada et al. defined an iteration scheme which converges to the unique minimizer of the qua- dratic functional over the common fixed point set of the mappings. In order to obtain their result, they needed to assume that the maps satisfy a commutative type condition. In this paper, we establish their conclusion without the assumption of any type of com- mutativity. Finding an optimal point in the intersection F of the fixed point sets of a family of nonexpansive maps is one that occurs frequently in various areas of mathematical sci- ences and engineering. For example, the well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpan- sive maps. (See, e.g., [3, 4].) The problem of finding an optimal point that minimizes a given cost function Θ : Ᏼ → R over F is of wide interdisciplinary interest and practical importance. (See, e.g., [2, 4, 5, 7, 14].) A simple algorithmic solution to the problem of minimizing a quadratic function over F is of extreme value in many applications includ- ing the set-theoretic signal estimation. (See, e.g., [5, 6, 10, 14].) The best approximation problem of finding the projection P F (a) (in the norm induced by the inner product of Ᏼ)fromanygivenpointa in Ᏼ is the simplest case of our problem. Some papers dealing with this best approximation problem are [2, 9, 11]. Let Ᏼ be a Hilbert space, C a closed convex subset of Ᏼ,andT i ,wherei = 1,2, ,N, a finite family of nonexpansive self-maps of C,withF :=∩ n i=1 Fix(T i ) =∅. Define a qua- dratic function Θ : Ᏼ → R by Θ(u):= 1 2 Au,u−b, u∀u ∈ Ᏼ,(1) where b ∈ Ᏼ and A is a selfadjoint strongly positive operator. We will also assume that B := I − A satisfies B < 1, although this is not restrictive, since µA is strongly positive Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:2 (2004) 135–147 2000 Mathematics Subject Classification: 47H10 URL: http://dx.doi.org/10.1155/S1687182004309046 136 Quadratic optimization with I − µA < 1foranyµ ∈ (0,2/A), and minimizing Θ(u):=(1/2)µAu,u−µb, u over F is equivalent to the original minimization problem. Ya m ada e t a l. [13] show that there exists a unique minimizer u ∗ of Θ over C if and only if u ∗ satisfies Au ∗ − b,u − u ∗ ≥ 0 ∀u ∈ C. (2) In their solution of this problem, Yamada et al. [13] add the restriction that the T i satisfy Fix T N ···T 1 = Fix T 1 T N ···T 3 T 2 = Fix T N−1 T N−2 ···T 1 T N . (3) There are many nonexpansive maps, with a common fixed point set, that do not satisfy (3). For example, if X = [0,1] and T 1 and T 2 are defined by T 1 x = x/2+1/4andT 2 x = 3x/4, then Fix(T 1 ,T 2 ) ={2/5},whereasFix(T 2 ,T 1 ) ={3/10}. In our solution, we are able to remove restriction (3). We will take advantage of the modified Wittmann iteration scheme developed by Atsushiba and Takahashi [1]. Let α n1 ,α n2 , ,α nN ∈ (0,1], n = 1,2, Given the mappings T 1 ,T 2 , ,T N ,onecan define, for each n,newmappingsU 1 , ,U N by U n1 = α n1 T 1 + 1 − α n1 I, U n2 = α n2 T 2 U n1 + 1 − α n2 I, . . . U n,N−1 = α n,N−1 T N−1 U n,N−2 + 1 − α n,N−1 I, W n := U nN = α nN T N U n,N−1 + 1 − α nN I. (4) From [1, Lemmas 3.1 and 3.2], if the T i are n onexpansive, so are the U ni , and both sets of functions have the same fixed point set. The iteration scheme we will use is the following. Let b ∈ C and choose any u 0 ∈ C. Define {u n } by u n+1 = λ n b + I − λ n A W n u n ,(5) where the W n are the self-maps of C generated by (4). Theorem 1. Let T i : Ᏼ → Ᏼ (i = 1, , N) be nonexpansive with nonempty common fixed point set F =∅. Assume that {λ n } and {α ni } satisfy (i) 0 ≤ λ n ≤ 1, (ii) lim λ n = 0, (iii) n≥1 λ n =∞, B. E. Rhoades 137 (iv) n≥1 |λ n − λ n−1 | < ∞, (v) n≥1 |α ni − α n−1,i | < ∞ for each i = 1,2, ,N. Then, for any point u 0 ∈ Ᏼ,thesequence{u n } generated by (5) converges strongly to the unique minimizer u ∗ of the function Θ of (1)overF. Proof. From [15], u ∗ exists and is unique. We will first assume that u 0 ∈ C u ∗ := x ∈ Ᏼ | x − u ∗ ≤ b − Au ∗ 1 −B ,(6) where A and B are as previously defined. For any x ∈ Ᏼ and 0 ≤ λ ≤ 1, define T λ (x) = λb +(I − λA)W(x). (7) Then, for any y ∈ Ᏼ, since W is nonexpansive, T λ (x) − T λ (y) = (I − λA) W(x) − W(y) ≤ 1 − λ 1 −B x − y. (8) Also, since u ∗ ∈ F, T λ u ∗ − u ∗ = λ b − Au ∗ . (9) Thus, T λ (x) − u ∗ ≤ T λ (x) − T λ u ∗ + T λ u ∗ − u ∗ ≤ 1 − λ 1 −B x − u ∗ + λ 1 −B b − Au ∗ 1 −B ≤ x − Au ∗ 1 −B . (10) If, in (7), we make the substitution λ = λ n , T λ (x) = u n+1 ,andW(x) = W n u n ,thenit follows from (9)and(10)thatu n and W n u n belong to C u ∗ for each n.Thus,{u n } and {W n u n } are bounded. Since B < 1, {BW n u n } is also bounded. Let K denote the diameter of C u ∗ . We may w rite (5)intheform u n+1 = λ n b + I − λ n (I − B) W n u n = λ n b + I − λ n W n u n + λ n BW n u n . (11) We will first show that lim u n+1 − u n = 0. (12) 138 Quadratic optimization Using (11), since each W n is nonexpansive and B < 1, u n+1 − u n = λ n b + 1 − λ n W n u n + λ n BW n u n − λ n−1 b − 1 − λ n−1 W n−1 u n−1 − λ n−1 BW n−1 u n−1 ≤ λ n − λ n−1 b + 1 − λ n W n u n − W n−1 u n + λ n − λ n−1 W n−1 u n−1 + 1 − λ n W n−1 u n − W n−1 u n−1 + λ n B W n u n − W n−1 u n + λ n B W n−1 u n − W n−1 u n−1 + λ n − λ n−1 BW n−1 u n−1 ≤ 3 λ n − λ n−1 K + 1 − λ n + λ n B × W n u n − W n−1 u n + 1 − λ n + λ n B W n−1 u n − W n−1 u n−1 . (13) From (4), since T N and U n−1,N−1 are nonexpansive, W n u n − W n−1 u n = α nN T N U n,N−1 u n + 1 − α nN u n − α n−1,N T N U n−1,N−1 u n − 1 − α n−1,N u n ≤ α nN − α n−1,N u n + α nN T N U n,N−1 u n − α n−1,N T N U n−1,N−1 u n ≤ α nN − α n−1,N u n + α nN T N U n,N−1 u n − T N U n−1,N−1 u n + α nN − α n−1,N T N U n−1,N−1 u n ≤ α nN − α n−1,N u n + α nN U n,N−1 u n − U n−1,N−1 u n + α nN − α n−1,N K ≤ 2K α nN − α n−1,N + α nN U n,N−1 u n − U n−1,N−1 u n . (14) Again, from (4), U n,N−1 u n − U n−1,N−1 u n = α n,N−1 T N−1 U n,N−2 u n + 1 − α n,N−1 u n − α n−1,N−1 T N−1 U n−1,N−2 u n − 1 − α n−1,N−1 u n ≤ α n,N−1 − α n−1,N−1 u n + α n,N−1 T N−1 U n,N−2 u n − α n−1,N−1 T N−1 U n−1,N−2 u n ≤ α n,N−1 − α n−1,N−1 u n + α n,N−1 T N−1 U n,N−2 u n − T N−1 U n−1,N−2 u n + α n,N−1 − α n−1,N−1 K ≤ 2K α n,N−1 − α n−1,N−1 + α n,N−1 U n,N−2 u n − U n−1,N−2 u n ≤ 2K α n,N−1 − α n−1,N−1 + U n,N−2 u n − U n−1,N−2 u n . (15) B. E. Rhoades 139 Therefore, U n,N−1 u n − U n−1,N−1 u n ≤ 2K α n,N−1 − α n−1,N−1 +2K α n,N−2 − α n−1,N−2 + U n,N−3 u n − U n−1,N−3 u n . . . ≤ 2K N−1 i=2 α ni − α n−1,i + U n1 u n − U n−1,1 u n = α n1 T 1 u n + 1 − α n1 u n − α n−1,1 T 1 u n − 1 − α n−1,1 u n +2K N−1 i=2 α ni − α n−1,i ≤ α n1 − α n−1,1 u n + α n1 T 1 u n − α n−1,1 T 1 u n +2K N−1 i=2 α ni − α n−1,i ≤ 2K N−1 i=1 α ni − α n−1,i . (16) Substituting (16)into(14), W n u n − W n−1 u n ≤ 2K α nN − α n−1,N +2α nN K N−1 i=1 α ni − α n−1,i ≤ 2K N i=1 α ni − α n−1,i . (17) Using (17)in(13), u n+1 − u n ≤ 1 − λ n 1 −B u n − u n−1 +3K λ n − λ n−1 +2 1 − λ n 1 −B K N i=1 α ni − α n−1,i . (18) Thus, since 0 < 1− λ n (1 −B) < 1foralln, u n+m+1 − u n+m ≤ n+m i=m 1 − λ i 1 −B u i+1 − u i +3K n+m i=m λ i − λ i−1 +2K n+m i=m N j=1 α ij − α i−1, j . (19) 140 Quadratic optimization From (iii), since the product diverges to zero, limsup n u n+1 − u n = limsup n u n+m+1 − u m+n ≤ 2K ∞ i=m λ i − λ i−1 +2K ∞ i=m N j=1 α ij − α i−1, j . (20) Therefore, taking the limsup m of both sides and using (iv) and (v), limsup n u n+1 − u n = 0, (21) and (12)issatisfied. Now, for any nonexpansive self-map T of C u ∗ ,defineG t : C u ∗ → C u ∗ by G t (x) = tb +(1− t)TG t (x)+tBTG t (x) (22) for each t ∈ (0,1]. Using an argument similar to the proof of [8, Theorem 12.2, page 45], we will now show that if T has a fixed point, then, for each x in C u ∗ , the strong limit t→0 G t (x) exists and is a fixed point of T. Define y(t) = G t (x)andletw be a fixed point of T: y(t) − w = t(b − w)+(1− t) Ty(t) − w + tBTy(t). (23) Since T is nonexpansive, y(t) − w ≤ tb − w +(1− t) Ty(t) − w + tB Ty(t) ≤ tb − w +(1− t) y(t) − w + tB Ty(t) , t y(t) − w ≤ tb − w +tB Ty(t) − w + tBw, (24) or y(t) − w ≤b − w + B y(t) − w + Bw, (25) which, since B < 1, yields y(t) − w ≤ 1 1 −B b − w + Bw , (26) and y(t) remains bounded as t → 0. Also, BTy(t) < Ty(t) ≤ Ty(t) − Tw + w≤ y(t) − w + w, (27) and both BTy(t)andTy(t) remain bounded as t → 0. Hence, y(t) − Ty(t) = t b − Ty(t)+BT y(t) −→ 0ast −→ 0. (28) B. E. Rhoades 141 Define y n = y(t n )andlett n → 0. Let µ n be a Banach limit and f : C u ∗ → R + defined by f (z) = µ n y n − z 2 . (29) Since f is continuous and convex, f (z) →∞as z→∞.SinceᏴ is reflexive, f attains it infimum over C u ∗ . Let M be the set of minimizers of f over C u ∗ .Ifu ∈ C u ∗ ,then f (Tu) = µ n y n − Tu 2 = µ n Ty n − Tu 2 ≤ µ n y n − u 2 = f (u). (30) Therefore, M is invariant under T. Since it is also bounded, closed, and convex, it must contain a fixed point of T. Denote this fixed point by v.Then, y n − Ty n , y n − v = y n − v, y n − v + v − Ty n , y n − v = y n − v 2 + Tv− Ty n , y n − v . (31) But Tv− Ty n , y n − v ≤ Tv− Ty n y n − v ≤ y n − v 2 , (32) so that y n − Ty n , y n − v ≥ 0. (33) Since y n = t n b + 1 − t n Ty n + t n BTy n , y n − b = 1 − t n Ty n − b + t n BTy n = 1 − t n Ty n − y n + 1 − t n y n − b + t n BTy n , (34) thus, t y n − b = 1 − t n Ty n − y n + t n BTy n (35) or y n − b − Bv = 1 − t n t n Ty n − y n + BTy n − Bv. (36) Therefore, from (33), y n − b − Bv, y n − v = 1 − t n t n Ty n − y n , y n − v + BTy n − Bv, y n − v ≤ BTy n − Bv, y n − v . (37) 142 Quadratic optimization For any z ∈ C u ∗ , y n − v 2 = y n − 1 − t n v − t n z + t n (z − v) − t n b − t n Bv + t n (b + Bv) 2 ≥ y n − 1 − t n v − t n b − t n Bv 2 +2t n z − v + b+ Bv, y n − 1 − t n v − t n z − t n b − t n Bv . (38) Let > 0begiven.SinceᏴ is uniformly smooth, there exists a t 0 > 0suchthat,forall t n ≤ t 0 , z − v + b+ Bv, y n − v − y n − 1 − t n v − t n z − t n b − t n Bv < . (39) Thus, from (38), z − v + b+ Bv, y n − v < + z − v + b+ Bv, y n − 1 − t n v − t n z − t n b − t n B < + 1 2t y n − v 2 − y n − 1 − t n v − t n b − t n Bv 2 . (40) Since the Gateaux derivative exists in Ᏼ,weobtain µ n z − v + b+ Bv, y n − v ≤ 0. (41) Setting z = θ in (41) and adding (37)and(41)yields µ n y n − v, y n − v ≤ µ n BTy n − Bv, y n − v (42) or µ n y n − v 2 ≤ µ n BTy n − Bv y n − v ≤ µ n B Ty n − Tv y n − v ≤ Bµ n y n − v 2 . (43) Therefore, µ n y n − v 2 = 0. Thus, there is a subsequence of {y n } converging strongly to v. Suppose that lim k→∞ y(t n k ) = v 1 and lim k→∞ y(t m k ) = v 2 .From(37), we have v 1 − b − Bv 2 ,v 1 − v 2 ≤ BTv 1 − Bv 2 ,v 1 − v 2 , v 2 − b − Bv 1 ,v 2 − v 1 ≤ BTv 2 − Bv 1 ,v 2 − v 1 . (44) Adding these inequalitites, we obtain v 1 − BTv 1 + BTv 2 − v 2 ,v 1 − v 2 ≤ 0 (45) or v 1 − v 2 ,v 1 − v 2 ≤ BTv 1 − BTv 2 ,v 1 − v 2 ; (46) B. E. Rhoades 143 that is, v 1 − v 2 2 ≤ BTv 2 − BTv 1 v 1 − v 2 ≤B Tv 2 − Tv 1 v 1 − v 2 ≤B v 2 − v 1 2 , (47) which, since B < 1, implies that v 1 = v 2 , and thus lim y n = v. Now, setting z = θ in (41), we obtain µ n b − (I − B)v, y n − v ≤ 0 (48) or µ n b − Av, y n − v ≤ 0, (49) which, from (2), implies that v = u ∗ . Let u nk denote the unique element of C u ∗ such that u nk = 1 k b + 1 − 1 k W n u nk + 1 k BW n u nk . (50) From what we have just proved, lim k u nk → u ∗ . Using (11), u n+1 − W n+1 u n+1,k = λ n b + 1 − λ n + λ n B W n u n − W n+1 u n+1,k ≤ λ n b − W n+1 u n+1,k + 1 − λ n W n u n − W n+1 u n+1,k + λ n B W n u n − W n+1 u n+1,k + λ n BW n+1 u n+1,k < 3Kλ n + 1 − λ n + λ n B W n u n − W n u nk + W n u nk − W n u n+1,k + W n u n+1,k − W n+1 u n+1,k ≤ 3Kλ n + 1 − λ n + λ n B u n − u nk + u nk − u n+1,k + W n u n+1,k − W n+1 u n+1,k . (51) As in (17), W n u n+1,k − W n+1 u n+1,k ≤ 2K N i=1 α n+1,i − α ni . (52) From the definition of u nk , u nk = 1 k b + 1 − 1 k W n u nk + 1 k BW n u nk , u n+1,k = 1 k b + 1 − 1 k W n+1 u n+1,k + 1 k BW n+1 u n+1,k , u n+1,k − u nk = 1 − 1 k W n+1 u n+1,k − W n u nk + 1 k B W n+1 u n+1,k − W n u nk . (53) 144 Quadratic optimization Therefore, since W n+1 is nonexpansive, u n+1,k − u nk ≤ 1 − 1 k + 1 k B W n+1 u n+1,k − W n u nk ≤ 1 − 1 k + 1 k B W n+1 u n+1,k − W n+1 u nk + W n+1 u nk − W n u nk ≤ 1 − 1 k + 1 k B u n+1,k − u nk + W n+1 u nk − W n u nk . (54) Thus, using (17), 1 −B k u n+1,k − u nk ≤ k − 1+B k 2K N i=1 α n+1,i − α n,i (55) or u n+1,k − u nk ≤ k − 1+B 1 −B 2K N i=1 α n+1,i − α n,i . (56) Substituting (56)and(52)into(51)yields u n+1 − W n+1 u n+1,k ≤ 3Kλ n + 1 − λ n + λ n B u n − u nk + k 1 −B 2K N i=1 α n+1,i − α n,i . (57) Thus, using (iii) and (v), we have µ n u n − W n u nk 2 = µ n u n+1 − W n+1 u n+1,k 2 ≤ µ n u n − u nk 2 . (58) From (53), u nk − u n = 1 k b − u n + 1 − 1 k W n u nk − u n + 1 k BW n u nk . (59) Hence, 1 − 1 k u n − W n u nk = u n − u nk − 1 k u n − b + 1 k BW n u nk , (60) 1 − 1 k 2 u n − W n u nk 2 ≥ u n − u nk 2 − 2 k u n − b − BW n u nk ,u n − u nk = 1 − 2 k u n − u nk 2 − 2 k u nk − b − BW n u nk ,u n − u nk . 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Wn sn un − sn By induction, un − sn ≤ u0 − s0 n 1 − λk 1 − B (76) k=1 Therefore, using (iii), lim un − sn = 0 and un − u∗ ≤ un − sn + sn − u∗ so that limun = u∗ B E Rhoades 147 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] S Atsushiba and W Takahashi, Strong convergence theorems for a finite family of nonexpansive mappings and applications, Indian J Math 41 (1999), no... (12), lim b − u∗ ,un+1 − u∗ − b − u∗ ,un − u∗ = 0 (65) We need the following result from [12] If A is a real number and {a1 ,a2 , } ∈ ∞ such that µn {an } ≤ a for all Banach limits µn and limsupn (an+1 − an ) ≤ 0, then limsupn an ≤ a Consequently, limsup b − u∗ ,un − u∗ ≤ 0 (66) Wn un − u∗ = Wn un − Wn u∗ ≤ un − u ∗ (67) n Since u∗ ∈ F, From (11), un+1 − u∗ = λn b − u∗ + 1 − λn Wn un − u∗ + λn BWn... Therefore, 1 − λn + λn B 2 Wn un − u∗ 2 ≥ un+1 − u∗ 2 − 2λn b − u∗ + Bu∗ ,un+1 − u∗ , (70) 146 Quadratic optimization which implies that un+1 − u∗ 2 2 ≤ 1 − λn + λn B Wn un − u∗ 2 + 2λn b − u∗ ,un+1 − u∗ + 2λn Bu∗ ,un+1 − u∗ (71) From (ii) and the boundedness of Cu∗ , there exists a positive integer N such that, for all n ≥ N, λn b − u∗ ,un+1 − u∗ ≤ , 4 λn Bu∗ ,un+1 − u∗ ≤ 4 (72) Therefore, for n... − u ∗ 2 (73) 2 1 − λi + λi B Using (iii), limsup un − u∗ n 2 = limsup un+m − u∗ n 2 ≤ 0 (74) Thus, {un } converges strongly to u∗ Now let u0 ∈ Ᏼ Let {sn } be another sequence generated by (11) for some s0 ∈ Cu∗ Then, by what we have just proved, lim sn = u∗ Since Wn is nonexpansive for each n, un+1 − sn+1 = λn b + 1 − λn A Wn un − λn b − 1 − λn A Wn sn ≤ 1 − λn A Wn un − Wn sn ≤ 1 − λn + λn B ≤ . QUADRATIC OPTIMIZATION OF FIXED POINTS FOR A FAMILY OF NONEXPANSIVE MAPPINGS IN HILBERT SPACE B. E. RHO ADES Received 10 September 2003 Given a finite family of nonexpansive self -mappings of a. paper, we establish their conclusion without the assumption of any type of com- mutativity. Finding an optimal point in the intersection F of the fixed point sets of a family of nonexpansive maps. u ∗ . B. E. Rhoades 147 References [1] S. Atsushiba and W. Takahashi, Strong convergence theorems for a finite family of nonexpansive mappings and applications, Indian J. Math. 41 (1999), no.