ITERATIVE ALGORITHMS WITH ERRORS FOR ZEROS OF ACCRETIVE OPERATORS IN BANACH SPACES. Jong Soo Jung. 1. Introduction

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1 J. Appl. Math. & Computing Vol. 20(2006), No. 1-2, pp Website: ITERATIVE ALGORITHMS WITH ERRORS FOR ZEROS OF ACCRETIVE OPERATORS IN BANACH SPACES Jong Soo Jung Abstract. The iterative algorithms with errors for solutions to accretive operator inclusions are investigated in Banach spaces, including a modification of Rockafellar s proximal point algorithm. Some applications are given in Hilbert spaces. Our results improve the corresponding results in [1, 15 17, 29, 35]. AMS Mathematics Subject Classification : 47H06, 47H10, 47J25, 49M05, 90C25. Key words and phrases : Iterative algorithms with errors, resolvent, proximal point algorithm, m-accretive operator, maximal monotone operator, sunny and nonexpansive retraction. 1. Introduction Let E be a real Banach space, let A E E be an m-accretive operator and let J r be the resolvent of A of r>0. It is well known that many problems in nonlinear analysis and optimization can be formulated as the problem: find an x such that 0 Ax. This problem has been investigated by many researchers: see, for instance, Benavides et al. [1], Brézis and Lions [2], Bruck [3], Bruck and Passty [4], Ha and Jung [10], Jung and Takahashi [13, 14], Nevanlinna and Reich [22], Reich [24, 25, 27], Rockafellar [29], Takahashi and Ueda [31], Xu [35] and others. One popular method of solving 0 Ax is the proximal point algorithm. The proximal point algorithm generates, for any starting point x 0 = x E, a sequence {x n } by the rule x n+1 = J rn x n, n 0, (1) Received June 17, Revised November 26, This work was supported by Korea Research Foundation Grant (KRF C00032). c 2006 Korean Society for Compuational & Applied Mathematics and Korean SIGCAM. 369

2 370 Jong Soo Jung where {r n } is a sequence of positive real numbers. Some of them dealt with the weak convergence of the sequence {x n } generated by (1) and others proved strong convergence theorems by imposing strong assumptions on A. In particular, in 1976, Rockafellar [29] devised the proximal point algorithm which generates, starting with an arbitrary initial x 0 in a Hilbert space H, a sequence {x n } satisfying: x n+1 := J rn x n + e n, n 0, (2) where A is a maximal monotone operator in H, r n > 0 is a real number, and e n is an error vector. Rockafellar proved the weak convergence of algorithm (2) if the sequence {r n } is bounded away from zero and if the sequence of the errors satisfies the condition: n e n <. In 1991, Güler [9] gave an example showing that Rockafellar s proximal point algorithm does not converge strongly. Solovov and Svaitor [30] in 2000 proposed a modified proximal point algorithm which converges strongly to a solution of equation 0 Ax by using projection method. Motivated by iterative algorithms of Halpern s type [11] and Mann s type [20], Kamimura and Takahashi [15, 17] introduced the iterative algorithms in Hilbert spaces and Banach spaces: x n+1 := α n x 0 +(1 α n )J rn x n, n 0 (3) and x n+1 := α n x n +(1 α n )J rn x n, n 0, (4) and showed that the sequence {x n } generated by (3) converges strongly to some v A 1 0 and the sequence {x n } generated by (4) converges weakly to some v A 1 0. Moreover, in 2000, Kamimura and Takahasi [16] considered the following algorithm with errors in Banach spaces: x n+1 := α n x 0 +(1 α n )J rn x n + e n, n 0, (5) and x n+1 := α n x n +(1 α n )J rn x n + e n, n 0, (6) and studied the strong and weak convergence of the algorithms (5) and (6), respectively, by using Reich s result [27]. In, 2002, Xu [35] also considered the algorithms (5) and (6) with errors e n replaced by (1 α n )e n in Hilbert space and, by using the methods slightly different from Kamimura and Takahashi [15] and Solodov and Svaiter [30], established the strong and weak convergence of the algorithms provided that the sequences {α n }, {r n } of real numbers and the sequence {e n } of errors are chosen appropriately. Recently, Benavides et al. [1] extended the results of Kamimura and Takahashi [15] to a Banach space setting with the initial datum u which is not necessary the x 0 in the iteration scheme (3).

3 Iterative algorithms with errors for zeros 371 In this paper, first we introduce the iterative algorithm (5) with mixed errors e n = e n + e n and the initial datum u x 0 for m-accretive operators in Banach spaces. Then we show that the sequence {x n } generated by (5) converges strongly to some v A 1 0 in a reflexive Banach space with a uniformly Gâteaux differentiable norm and a weakly sequentially continuous duality mapping. Second we prove that the sequence {x n } generated by the iterative algorithm(6) with errors e n converges weakly to some v A 1 0 in a uniformly convex Banach space with a Fréchet differentiable norm. Finally we give some applications of main results in a Hilbert space. Our results improve and unify results in [1, 16, 17] in Banach spaces and results in [15, 29, 35] in framework of a Hilbert space, respectively. 2. Preliminaries and Lemmas Let E be a real Banach space with norm and let E be its dual. The value of f E at x E will be denoted by x, f. When {x n } is a sequence in E, then x n x (resp. x n x, x n x) will denote strong (resp. weak, weak ) convergence of the sequence {x n } to x. The modulus of convexity of E is defined by { } x + y δ(ε) = inf 1 : x 1, y 1, x y ε 2 for every ε with 0 ε 2. A Banach space E is said to uniformly convex if δ(ε) > 0 for every ε>0. If E is uniformly convex, then ( )) x + y ε 2 (1 r δ r for every x, y E with x r, y r and x y ε. We also know that if C is a closed convex subset of a uniformly convex Banach space E, then for each x E, there exists a unique element u = Px C with x u = inf{ x y : y C}. Such a P is called the metric projection of E onto C. The norm of E is said to be Gâteaux differentiable (and E is said to be smooth) if x + ty x lim (7) t 0 t exists for each x, y in its unit sphere U = {x E : x =1}. It is said to be Fréchet differentiable if for each x U, this limit is obtained uniformly for y U. The norm is said to be uniformly Gâteaux differentiable if for y U, the limit is attained uniformly for x U. The space E is said to have a uniformly Fréchet differentiable norm (and E is said to be uniformly smooth) if the limit in (7) is attained uniformly for (x, y) U U.

4 372 Jong Soo Jung It is relevant to the first theorem of this paper to note that while every uniformly smooth Banach space is a reflexive Banach space with a uniformly Gâteaux differentiable norm, the converse does not hold. Indeed there are reflexive spaces with a uniformly Gâteaux differentiable norm that are not even isomorphic to a uniformly smooth space. To see this consider E to be the direct sum l 2 (l pn ), the class of all those sequences x = {x n } with x n l pn and ( ) 1/2 x = x n 2 (see [5]). Now, if 1 <p n < for all n 1, where n< either lim sup p n = or lim inf p n = 1, then E is a reflexive Banach space with a uniformly Gâteaux differentiable norm, but is not uniformly smooth (see [5, 37]) We also observe that spaces with enjoy the fixed point property for nonexpansive self-mappings are not necessarily spaces with a uniformly Gâteaux differentiable norm. On the other hand, the converse of this fact appears to be unknown as well. For these facts, see also [21]. The (normalized) duality mapping J from E into the family of nonempty (by Hahn-Banach theorem) weak-star compact subsets of its dual E is defined by J(x) ={f E : x, f = x 2 = f 2 }. for each x E. It is single valued if and only if E is smooth. It is also wellknown that if E has a uniformly Gâteaux differentiable norm, J is uniformly norm to weak continuous on each bounded subsets of E. Suppose that J is single valued. Then J is said to be weakly sequentially continuous if for each {x n } E with x n x, J(x n ) J(x). We need the following lemma. For the proof, see also [12]. Lemma 1. Let E be a real Banach space and let J be the duality mapping. Then for any given x, y E, we have for all j(x + y) J(x + y). x + y 2 x 2 +2 y, j(x + y) (8) Let C be a nonempty closed convex subset of E. A mapping T : C C is nonexpansive if Tx Ty x y for all x, y C. We denote the set of all fixed points of T by F (T ) (that is, F (T )={x C : x = Tx}). A closed convex subset C of E is said to have the fixed point property for nonexpansive mappings if every nonexpansive mapping of a bounded closed convex subset D of C into itself has a fixed point in D. It is well-known (cf. [7, P. 45]) that every weakly compact convex subset of a uniformly smooth Banach space has the fixed point property for nonexpansive mappings. Let I denote the identity operator on E. An operator A E E with domain D(A) ={z E : Az } and range R(A) = {Az : z D(A)}

5 Iterative algorithms with errors for zeros 373 is said to be accretive if for each x i D(A) and y i Ax i, i =1, 2, there exists j J(x 1 x 2 ) such that y 1 y 2,j 0. If A is accretive, then we have x 1 x 2 x 1 x 2 +r(y 1 y 2 ) for all x i D(A), y i Ax i,i=1, 2 and r>0. If A is accretive, then we can define, for each r>0, a nonexpansive single valued mapping J r : R(I+rA) D(A) by the J r =(I+rA) 1. It is called the resolvent of A. We also define the Yosida approximation A r by A r = 1 r (I J r). It is well known that A r x A(J r x) for all x R(I + ra) and A r x inf{ y : y Ax} for all x D(A) R(I +ra). It is also well known that for an accretive operator A which satisfies the range condition, A 1 0=F (J r ) for all r>0. If A 1 0 0, that is, 0 R(A), then the inclusion 0 Ax is solvable. An accretive operator A is said to be m-accretive if R(I + ra) =E for all r>0. In a Hilbert space, an operator A is m-accretive if and only if A is maximal monotone. A Banach space E is said to satisfy Opial s condition ([23]) if for any sequence {x n } in E, x n ximplies lim sup x n x < lim sup x n y for all y E with y x. It is well-known that if E admits a weakly sequentially continuous duality mapping, then E satisfies Opial s condition and the duality mapping J is single-valued (see [6, 8]). Recall that a mapping T defined on a subset C of a Banach space E ( and taking values in E) is said to be demiclosed if for any sequence {u n } in C the following implication holds: u n u and lim Tu n w =0 implies u C and Tu= w. The following lemma can be found in [6, p. 108]. Lemma 2. Let E be a reflexive Banach space which satisfies Opial s condition, let C be a nonempty closed convex subset of E, and suppose T : C E is nonexpansive. Then the mapping I T is demiclosed on C, where I is the identity mapping. A mapping Q of C into C is said to be a retraction if Q 2 = Q. If a mapping Q of C into itself is a retraction, then Qz = z for every z R(Q), where R(Q) is range of Q. Let D be a subset of C and let Q be a mapping of C into D. Then Q is said to be sunny if each point on the ray {Qx + t(x Qx) : t>0} is mapped by Q back onto Qx, in other words, Q(Qx + t(x Qx)) = Qx

6 374 Jong Soo Jung for all t 0 and x C. A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction of C onto D. In a smooth Banach space E, it is known (cf. [7, p. 48]) that Q is a sunny nonexpansive retraction of C onto D if and only if the following condition holds: x Qx, J(y Qx) 0, x C y D; (9) for more details, see [7]. It is known [27, Theorem 1] (see also [7, P ] [10, Corollary 3], [31, Theorem 1]) that if E be a reflexive Banach space with a uniformly Gâteaux differentiable norm, every weakly compact convex subset of E has the fixed point property for nonexpansive mappings, and D is the fixed point set of a nonexpansive self-mapping of a closed convex subset C of E, then there exists a (unique) sunny nonexpansive retraction of C onto D. Finally, we need the following lemma, which was proved by Xue et al. [35]. Lemma 3 ([35]). Let {s n } be a sequence of non-negative real numbers satisfying s n+1 (1 α n )s n + λ n s n + β n + γ n, n 0, where {α n }, {λ n }, {β n } and {γ n } satisfying the condition: (i) {α n } [0, 1] and α n = or, equivalently, (1 α n ) := lim (1 α k )=0; (ii) β n = (α n ); (iii) λ n 0(n 0), (iv) γ n 0(n 0), Then lim s n =0. λ n < ; γ n <. 3. Iterative algorithm of Halpern s type n k=0 In the sequel, unless otherwise stated, we assume that A E E is an m-accretive operator and J r is the resolvent of A for r>0. Now we study the strong convergence of sequence {x n } generated by the following algorithm with mixed errors: for u, x 0 E, x n+1 := α n u +(1 α n )J rn x n + e n, n 0, (IA1) where {α n } [0, 1], {r n } (0, ) and the computational mixed errors {e n } E, e n = e n + e n.

7 Iterative algorithms with errors for zeros 375 Remark 1. The iterative algorithm (IA1) is inspired by the Rockafellar s proximal point algorithms [28] for maximal monotone operators in a Hilbert space, Liu s iteration process with errors [19] for strongly accretive operator, and Halpern s iteration process [11] (later developed by Lions [18], Wittmann [33] and others): x n+1 := α n u +(1 α n )Tx n, n 0, where T is a nonexpansive self-mapping of a closed bounded convex subset of a Hilbert space. Theorem 1. Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that E has a weakly sequentially continuous duality mapping J. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } E, satisfy the following conditions: n (i) lim α n =0and α n = or equivalently, (1 α n ) := lim (1 k=0 α k )=0; (ii) lim r n = ; (iii) e n = e n + e n for any sequences {e n}, {e n} in E and for n 0 with e n < and e n = (α n ). Let u, x 0 E and let {x n } be a sequence generated by (IA1). If A 1 0 and there exists a sunny nonexpansive mapping Q of E onto A 1 0, then {x n } converges strongly to Qu. Proof. Since e n = (α n ), we see that e n = α n w n, where w n E and w n 0 as n. Let y n = J rn x n. Then Eq. (IA1) can be re-written as x n+1 := α n (u + w n )+(1 α n )y n + e n, n 0. Now, we proceed with three steps. Step 1: {x n } is bounded. Indeed, let z A 1 0 and d = sup u + w n z + x 0 z, M = d + n 0 e n. Then we have x 1 z α 0 u + w 0 z +(1 α 0 ) y 0 z + e 0 α 0 u + w 0 z +(1 α 0 ) x 0 z + e 0 α 0 d +(1 α 0 )d + e 0 = d + e 0.

8 376 Jong Soo Jung By induction, we obtain x n+1 z d + n k=0 e k M, n 0. Hence, it follows from e n < that {x n} is bounded, and so is {y n }. Step 2: lim sup u Qu, J(y n Qu) 0. To prove this, we take a subsequence {y nj } of {y n } be such that lim u Qu, J(y n j Qu) = lim sup u Qu, J(y n Qu) j and y nj x for some x E. We also have A rn x n Ay n and lim y n J 1 y n = lim (I J 1)y n = lim A 1y n lim inf{ z : z Ay n} lim A r n x n = lim x n y n r n = lim x n J rn x n r n =0. It follows from Lemma 2 that x A 1 0=F (J 1 ). Since A 1 0 is sunny nonexpansive retract of E, by weakly sequentially continuity of duality mapping J and (9), we have lim sup u Qu, J(y n Qu) = lim u Qu, J(y n j Qu) j = u Qu, J( x Qu) 0. Step 3: lim x n Qu = 0. Since (x n+1 Qu) =(1 α n )(y n Qu)+α n (u+ w n Qu)+e n, by using the inequality (8) in Lemma 1, we have x n+1 Qu 2 (1 α n )(y n Qa)+α n (u + w n Qu) 2 +2 e n,j(x n+1 Qu) (1 α n ) 2 y n Qu 2 +2α n u + w n Qu, J(x n+1 Qu e n +2 e n,j(x n+1 Qu) (1 α n ) x n Qu 2 +2α n u + w n Qu, J(x n+1 Qu e n +2 e n,j(x n+1 Qu) (10)

9 Iterative algorithms with errors for zeros 377 Now, observe that 2 e n,j(x n+1 Qu) 2 e n x n+1 Qu e n (1 + x n+1 Qu 2 ) (11) and u + w n Qu, J(x n+1 Qu e n u + wn Qu = 1+ x n Qu,J ( xn+1 Qu e n 1+ x n Qu ) ( J y n Qu 1+ x n Qu ) (1 + x n Qu ) 2 + u + w n Qu, J(y n Qu) L n (1 + x n Qu ) 2 + w n y n Qu + u Qu, J(y n Qu), (12) where L n = u + wn Qu 1+ x n Qu,J ( xn+1 Qu e ) ( n J 1+ x n Qu y n Qu 1+ x n Qu ). Now we show L n 0as. Indeed, { u + w n Qu /(1 + x n Qu } is bounded. Since { u+w n Qu /(1+ x n Qu )} and { y n Qu /(1+ x n Qu )} are bounded, we have x n+1 Qu e n 1+ x n Qu y n Qu 1+ x n Qu ( u + wn Qu α n 1+ x n Qu + y ) n Qu 0 1+ x n Qu as n. Thus, by uniform norm to weak continuity of J on each bounded subsets of E, we have L n 0asn. By using (10) (12) with (1 + x n Qu ) 2 2(1 + x n Qu 2 ), we have x n+1 Qu 2 (1 α n ) x n Qu 2 +2α n u + w n Qu, J(x n+1 Qu e n +2 e n,j(x n+1 Qu) (1 α n ) x n Qu 2 +4α n L n x n Qu 2 +4α n L n +2α n w n y n Qu +2α n u Qu, J(y n Qu) + e n (1 + x n+1 Qu 2 ) (1 α n +4α n L n ) x n Qu 2 +4α n L n +2α n w n B +2α n u Qu, J(y n Qu) + e n (1 + x n+1 Qu 2 ), (13)

10 378 Jong Soo Jung where B = sup n 0 y n Qu. Choosing a positive integer N so large that 1 e n > 0 for all n N, from (13), we have x n+1 Qu 2 1 e n α n +4α n L n + e n 1 e x n Qu 2 + 4α nl n n 1 e n + 2α n w n B 1 e n + 2α n u Qu, J(y n Qu) 1 e n + e n 1 e n ( 1 1 4α nl n 1 e n α n + α n ) x n Qu 2 + e n 1 e n x n Qu 2 u Qu, J(y n Qu) 1 e n 4L n +2 w n B + 2 lim sup (14) + e n 1 e n. In (14), put s n = x n Qu 2, λ n = e n 1 e n = γ n 4L n +2 w n B + 2 lim sup u Qu, J(y n Qu) β n = α n 1 e n t n = 1 4α nl n 1 e n. Then, since α n,l n, w n 0asn, we see that β n = (α n ) with Step 2, and for some 0 <k<1, there exists N >N such that t n = 1 4α nl n 1 e n k for all n N. Then the inequality (14) reduces to s n+1 (1 kα n )s n + λ n s n + β n + γ n, where β n = (α n ) and λ n = γ n <. Applying Lemma 3, we have n=n n=n lim x n Qu = 0. This completes the proof. Using Theorem 1, we have the following results.

11 Iterative algorithms with errors for zeros 379 Corollary 1. Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings and E has a weakly sequentially continuous duality mapping J. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } H, e n = e n + e n are the same as in Theorem 1. Let u, x 0 E and let {x n } be a sequence generated by (IA1). If A 1 0, then {x n } converges strongly to Qu, where Q is a sunny nonexpansive retraction of E onto A 1 0. Proof. It follows from Reich [27, Theorem 1] that there exists a sunny nonexpansive retraction Q from E onto A 1 0 (also see [7, 10, 31]). Thus the result follows from Theorem 1. Corollary 2. Let E be a uniformly smooth Banach space with a weakly sequentially continuous duality mapping J. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } E, e n = e n + e n are the same as in Theorem 1. Let u, x 0 E and let {x n } be a sequence generated by (IA1). If A 1 0, then {x n } converges strongly to Qu, where Q is a sunny nonexpansive retraction of E onto A 1 0. Corollary 3. Let H be a Hilbert space and let A H H be a maximal monotone operator. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } H, e n = e n + e n are the same as in Theorem 1. Let u, x 0 H and let {x n } be a sequence generated by (IA1). If A 1 0, then {x n } converges strongly to Pu, where P is the metric projection of H onto A 1 0. Proof. Note that the metric projection P of H onto A 1 0 is a sunny nonexpansive retraction. Thus the result follows from Theorem 1. Let T be a nonexpansive mapping of E into itself. Then A = I T is an m-accretive operator. Then, putting A = I T in Theorem 1, we have the following result. Corollary 4. Let E be a reflexive Banach space with a uniformly Gâteaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings and E has a weakly sequentially continuous duality mapping J. Let T be a nonexpansive mapping from E into itself. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } E,

12 380 Jong Soo Jung e n = e n + e n are the same as in Theorem 1. Let u, x 0 E and let {x n } be a sequence generated by { yn = 1 1+r n x n + rn 1+r n T (y n ), x n+1 = α n u +(1 α n )y n + e n, n 0. If F (T ), then {x n } converges strongly to Qu, where Q is a sunny nonexpansive retraction of E onto F (T ). Remark 2. (1) In our algorithm (IA1), the initial datum x 0 is not necessarily the u in the iterative algorithm. It is not difficult to see that if the iterative algorithm (IA1) converges, then the iterative algorithm with the same initial y 0 = u also converges. Indeed, If {x n } and {y n } denote the respective sequences generated the algorithm (IA1) and the algorithm with the same initial y 0 = u, n 1 then x n y n x 0 y 0 (1 α k ) 0as. k=0 (2) Theorem 1 (and Corollary 1) extends Theorem 2 of [16] to a mixed error version. Moreover, our proof lines of Theorem 1 are different from those of Theorem 2 of [16], in which the Reich s result [27] was utilized with the initial datum x 0 = u. (3) Corollary 1 and Corollary 2 generalize Corollary 4 of [17] and Theorem 2.1 of [1] to a mixed error version, respectively. (4) Corollary 3 of [16] is a special case of Corollary 3 with e n =0. (5) Corollary 3 and Corollary 4 improve Theorem 1 of [15], Corollary 3 of [17], and Theorem 5.1 of [35] together with mixed errors. 4. Iterative algorithm of Mann s type In this section, we study the weak convergence of sequence {x n } generated by the following algorithm with errors: for x 0 = u E, x n+1 := α n x n +(1 α n )J rn x n + e n, n 0, (IA2) where {α n } [0, 1], {r n } (0, ) and the computational errors {e n } E. Remark 3. The iterative algorithm (IA2) is motivated by the Rockafellar s proximal point algorithms [31] for maximal monotone operators in a Hilbert space, Liu s iteration process with errors [19] for strongly accretive operator, and Mann s iteration process [20]: x n+1 := α n x n +(1 α n )f(x n ), n 0,

13 Iterative algorithms with errors for zeros 381 where f is real-valued function in R. This Mann s iteration process has extensively been studied over the last twenty years for constructions of fixed points of nonlinear mappings and of solutions of nonlinear operator equations involving monotone, accretive and pseudo-contractive operators and others. We need the following lemma, which was proved by Reich [26, Proposition]. Lemma 4 ([26]). Let C be a closed convex subset of a uniformly convex Banach space with a Fréchet differentiable norm and let {T n } be a sequence of nonexpansive self-mappings of C with a nonempty common fixed point set F. If x 1 C and x n+1 = T n x n for n 1, then lim x n,j(f 1 f 2 ) exists for all f 1, f 2 F. In particular, q 1 q 2,J(f 1 f 2 ) =0, where f 1, f 2 F and q 1, q 2 are weak limit points of {x n }. Now we prove the following theorem for weak convergence. mainly due to Benavides et al. [1] and Brézis and Lions [2]. The proof is Theorem 2. Let E be a uniformly convex Banach space with a Fréchet differentiable norm. Suppose that E has a weakly sequentially continuous duality mapping J. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } E satisfy the following conditions: (i) lim α n =0; (ii) lim r n = ; (iii) e n <. Let x 0 E and let {x n } be a sequence generated by (IA2). If A 1 0, then {x n } converges weakly to a point in A 1 0. Proof. First, following the same idea as in the proof of [1, Theorem 3.1], we prove theorem in the case e n 0. Let p be an element of A 1 0 and y n = J rn x n. Then lim x n p exists. In particular, {x n } is bounded, and so is {y n }. Indeed we have by nonexpansivity of J rn x n x n+1 p α n x n p +(1 α n ) y n p x n p, and so lim x n p exists. Let v be a weak subsequential limit of {x n } such that x nj v. Noting x n+1 y n α n x n y n 0.

14 382 Jong Soo Jung we get y nj 1 v. We also have A rn x n Ay n and lim y n J 1 y n = lim (I J 1)y n = lim A 1y n lim inf{ z : z Ay n} lim A r n x n = lim x n y n r n = lim It follows from Lemma 2 that v A 1 0=F (J 1 ). Putting x n J rn x n r n =0. n=1 T n := α n I +(1 α n )J rn, we have x n+1 = T n x n and F (T n )=A 1 0. Let v 1, v 2 be any weak subsequential limits of {x n }. Then by above fact, we have v 1, v 2 A 1 0 and v 1 v 2,J(v 1 v 2 ) =0, that is, v 1 = v 2 by Lemma 4. Therefore {x n } converges weakly to a point in A 1 0. Next we show the theorem in the case e n 0. As in [16], we follow an idea of Brézis and Lions [2, Remarque 14]. Let U n z = T n z + e n for all z E and n 1. Then the sequence {x n } generated by (IA2) satisfies x n+1 = U n x n, n 1. we define, for every m 1, the sequence {z n (m)} by z 0 (m) =x m and z n+1 (m) =T n+m z n (m), n 1. Then we know that {z n (m)} converges weakly to some z(m) A 1 0. From definition, we have z n (m +1) z n+1 (m) = T n+m T n+m 1 T m+1 x m+1 T n+m T n+m 1 T m x m x m+1 T m x m = e m for all n, m 1. This implies z(m +1) z(m) e m for all m 1. Then, from e n <, it follows {z(m)} is a Cauchy sequence and hence {z(m)} converges strongly to p A 1 0. Since x n+m+1 z n+1 (m) = U n+m U n+m 1 U m x m T n+m T n+m 1 T m x m n+m i=m e i,

15 Iterative algorithms with errors for zeros 383 we have x n+m+1 p, f = x n+m+1 z n+1 (m),f + z n+1 (m) z(m),f + z(m) p, f ( n+m ) e i + z(m) p f + z n+1 (m) z(m),f i=m for all f E and n, m 1. This implies lim sup x n p, f = lim sup x n+m+1 p, f ( ) e i + z(m) p f i=m for all f E and m 1. Hence, from e n <, we conclude that {x n } converges weakly to p A 1 0 in the case of e n 0. This completes the proof. If we take {α n } to be away from 0 and 1, we can weaken the weak sequential continuity of the duality mapping J. By using the same argument of Theorem 2 along with the proof of [1, Theorem 3.2] (or [17, Theorem 7]) for e n 0, we can obtain an error version of [1, Theorem 3.2]. So we give the following result without proof. Theorem 3. Let E be a uniformly convex Banach space which either has a Fréchet differentiable norm or satisfies Opial s condition. Assume that the sequences {α n } [0, 1] and {r n } (0, ) and {e n } E satisfy the following conditions: for some ε>0, (i) ε α n 1 ε for n 1; (ii) r n ε for n 1: (iii) e n <. Let x 0 E and let {x n } be a sequence generated by (IA2). If A 1 0, then {x n } converges weakly to a point in A 1 0. We need the following result in the proof of Corollary 5. We refer to [16, Proposition 7] for the proof (see also [17, Proposition 8] (or [1, Theorem 3.10 (i)]) for e n 0). So we omit the proof.

16 384 Jong Soo Jung Proposition 1. Let E be a uniformly convex Banach space. Assume that {α n } [0, 1], {r n } (0, ) and {e n } E satisfies e n <. Let x 0 E and let {x n } be a sequence generated by (IA2). IfA 1 0 and P is the metric projection of E onto A 1 0, then {Px n } converges strongly to a point of A 1 0. Combining Theorem 2 and Proposition 1, we obtain the following results. Corollary 5. Let E be a uniformly convex Banach space with a Fréchet differentiable norm. Suppose that E has a weakly sequentially continuous duality mapping J. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } E are the same as in Theorem 2. Let x 0 E and let {x n } be a sequence generated by (IA2). If A 1 0 and P is the metric projection of E onto A 1 0, then {x n } converges weakly to v A 1 0, where v = lim Px n. Proof. From Theorem 2, {x n } converges weakly some v A 1 0 and from Proposition 1, {Px n } converges strongly to some v A 1 0. We know from [7, Proposition 3.1] that A 1 0 is a Chebyshev set. Since P is metric projection of E onto A 1 0, from [7, p. 14] we have w Px n,j(x n Px n ) 0, for n 1 for all w A 1 0. By weak sequential continuity of J, we obtain w v,j(v v ) = lim w Px n,j(x n Px n ) 0. Putting w = v, we have v v 2 0 and hence v = v. This completes the proof. Corollary 6. Let E be a uniformly convex Banach space which satisfies Opial s condition. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } E are the same as in Theorem 2. Let x 0 E and let {x n } be a sequence generated by (IA2). If A 1 0 and P is the metric projection of E onto A 1 0 and if {x n } converges weakly to v, then v = lim Px n. Proof. We know that if v is a weak limit of {x n }, then v A 1 0. Let v be a strong limit of {Px n }. Then we see that lim x n v = lim x n Px n lim x n v. So Opial s condition must imply that v = v. This completes the proof.

17 Iterative algorithms with errors for zeros 385 Corollary 7. Let E be a uniformly convex Banach space with a Fréchet differentiable norm. Suppose that E has a weakly sequentially continuous duality mapping J. LetT be a nonexpansive mapping from E into itself. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } E are the same as in Theorem 2. Let x 0 E and let {x n } be a sequence generated by { yn = 1 1+r n x n + rn 1+r n Ty n, x n+1 = α n x n +(1 α n )y n + e n, n 0. If F (T ), then {x n } converges weakly to a point of F (T ). Remark 4. (1) Theorem 2, Theorem 3 and Corollary 5 generalize Theorem 3.1 of [1], Theorem 3 of [15], Theorem 7 and Corollary 10 of [17], Theorem 5.2 of [35] to error versions. (2) In the case that {α n } is only away from 1 in Theorem 3, Kamimura and Takahashi gave a result [16, Theorem 6]. Even though the conditions on {α n } in Theorem 3 are more restrictive than those of Theorem 6 in [16], the proof lines of [1] in the case e n 0 are simple and different from those of [16] (or [17, Theorem 7]) because of using Xu s inequality for uniform convexity [34]. (3) Corollary 6 and Corollary 7 improve Theorem 3.3 (ii) of [1], Corollary 8 of [16] and Corollary 9 of [17], respectively. 5. Applications In this section, as in [16], we give some applications of Theorem 1 and Theorem 2. Throughout this section, we assume that H is a Hilbert space. First we consider a minimization problem. Let f : H (, ] be a proper lower semicontinuous convex function. The subdifferential f of f is defined by f(z) ={w H : f(y) y z,w, for any y H} for all z H. IfA = f, then A is maximal monotone operator (see Rockaleffar [29, Theorem 4]). We also know that 0 Av if and only if v = arg min z H f(z) and that J r x = arg min z H {f(z)+ z x 2 /2r} for all r>0 and x H. As consequences of Theorem 1 and Theorem 2, we obtain the following results.

18 386 Jong Soo Jung Corollary 8. Let f : H (, ] be a proper lower semicontinuous convex function. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } H, e n = e n + e n, are the same as in Theorem 1. Let x H and let {x n} be a sequence generated by x 0 = x H, y n = arg min x H { f(z)+ 1 2r n z x n 2 }, x n+1 = α n x +(1 α n )y n + e n, n 1. If ( f) 1 0, then {x n } converges strongly to the minimizer of f nearest to x. Corollary 9 ([16, Corollary 10]). Let f : H (, ] be a proper lower semicontinuous convex function. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } H are the same as in Theorem 2. Let x H and let {x n } be a sequence generated by x 0 = x H, y n = arg min x H { f(z)+ 1 2r n z x n 2 }, x n+1 = α n x n +(1 α n )y n + e n, n 1. If ( f) 1 0 and P is the metric projection of H onto ( f) 1 0, then {x n } converges weakly to v ( f) 1 0, where v = lim Px n. Next, we consider a variational inequality. Let X be a nonempty closed convex subset of H and let T be a single valued operator of x into H. We denote by VI(X, T) the set of solutions of the variational inequality, that is, VI(X, T) ={w X : u w, Tw 0, for any u X}. A single valued operator T is called hemicontinuous if T is continuous from each line segment of X to H with weak topology. Let F be a single valued, monotone and hemicontinuous operator of X into H and let N X z be the normal cone to X at z X, that is, N X z = {w H : z u, w 0, for all u = inx. Letting { Fz+ NX z, z H, Az =, z H \ X, we have that A is a maximal monotone operator (see Rockafellar [28, Theorem 3]). We can also check that 0 Av if and only if v VI(X, F) and that J r = VI(X, F r,x ) for all r>0 and x H, where F r,x z = Fz+(z x)/r for all z X. Then we have the following results.

19 Iterative algorithms with errors for zeros 387 Corollary 10. Let X be a nonempty closed convex subset of H and let F be a single valued, monotone and hemicontinuous operator of X into H. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } H, e n = e n + e n are the same as in Theorem 1. Let x H and let {x n } be a sequence generated by x 0 = x H, y n = VI(X, F rn,x n ), x n+1 = α n x +(1 α n )y n + e n, n 1. If VI(X, F), then {x n } converges strongly to the point of VI(X, F) nearest to x. Corollary 11 ([16, Corollary 12]). Let X be a nonempty closed convex subset of H and let F be a single valued, monotone and hemicontinuous operator of X into H. Assume that the sequences {α n } [0, 1], {r n } (0, ) and {e n } H are the same as in Theorem 2. Let x H and let {x n } be a sequence generated by x 0 = x H, y n = VI(X, F rn,x n ), x n+1 = α n x n +(1 α n )y n + e n, n 1. If VI(X, F) and P is the metric projection of H onto VI(X, T), then {x n } converges weakly to the point v VI(X, F), where v = lim Px n. Remark 5. (1) Corollary 9 and Corollary 11 of [16] are special cases of Corollary 8 and Corollary 10 with e n = 0, respectively. (2) Corollary 8 is also a mixed error version of [15, Theorem 6]. References 1. T. D. Benavides, G. L. Acedo and H. K. Xu, Iterative solutions for zeros of accretive operators, Math. Nachr (2003), H. Brézis and P. L. Lions, Produits infinis de resolvants, Israel J. Math. 29 (1978), R. F. Bruck, A strongly convergent iterative solution of 0 U(x) for a maximal monotone operator u in Hilbert space, J. Math. Anal. Appl. 48 (1974), R. E. Bruck and Passty, Almost convergence of the infinite product of resolvents in Banach spaces, Nonlinear Anal. 3 (1979), M. M. Day, Reflexive Banach space not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), K. Goebel and W. A. Kirk, Topics in metric fixed point theory in Cambridge Studies in Advanced Mathematics, Vol. 28 Cambridge Univ. Press, Cambridge, UK, 1990.

20 388 Jong Soo Jung 7. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York and Basel, J. P. Gossez and E. L. Dozo, Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40(3) (1972), O. Güler, On the convergence of the proximal point algorithm for convex minimization, SIAM J. Control Optim. 29 (1991), K. S. Ha and J. S. Jung, Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990), B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967), J. S. Jung and C. Morales, The Mann process for perturbed m-accretive operators in Banach spaces Nonlinear Anal. 46 (2001), J. S. Jung and W. Takahashi, Dual convergence theorems for the infinite products of resolvents in Banach spaces, Kodai Math. J. 14 (1991), J. S. Jung and W. Takahashi, On the asymptotic behavior of the infinite products of resolvents in Banach spaces, Nonlinear Anal. 20 (1993), S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory 106 (2000), S. Kamimura and W. Takahashi, Weak and strong convergence of solutions to accretive operator inclusions and applications, Set-Valued Anal. 8 (2000), S. Kamimura and W. Takahashi, Iterative schemes for approximating solutions of accretive operators in Banach spaces Sci. Math. 3(1) (2000), P. L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Sér A-B, Paris 284 (1977), L. S. Liu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), C. H. Morales and J. S. Jung, Convergence of paths for pseudo-contractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), O. Nevanlinna and S. Reich, Strong convergence of contraction semigroups and of iterative methods for accretive operators in Banach spaces, Israel Math. J. 32 (1979), Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), S. Reich, On infinite products of resolvents, Atti. Acad. Naz. Lincei 63 (1977), S. Reich, An iterative procedure for constructing zeros of accretive sets in Banach spaces, Nonlinear Anal. 2 (1978), S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Amer. Math. Soc. 149 (1970), R. T. Rockafellar, Monotone operators and the proximal point algorithms, SIAM J. Control Optim. 14 (1976), M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program. Ser. A 87 (2000), W. Takahashi and Y. Ueda, On Reich s strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl. 104 (1984), K. K. Tan and X. H. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 59 (1992),

21 Iterative algorithms with errors for zeros H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66(3) (2002), Z. Q. Xue, H. Y. Zhou and Y. J. Cho, Iterative solutions of nonlinear equations for m- accretive operators in Banach spaces, J. Nonlinear Convex Anal. 1 (2000), V. Zizler, On some rotundity and smoothness properties of Banach spaces, Dissert. Math. 87 (1971), Jong Soo Jung received his BS from Pusan National University in 1979, MS from Seoul National University in 1981, and Ph.D at Pusan National University under the direction of Professor Ki Sik Ha in Since 1982, he has been at Dong-A University. In 1990, he was a post-doctorial research fellow at Tokyo Institute of Technology and in 1998, he was a visiting professor at the University of Alabama in Huntsville for one year under the financial support of LG Yonam Foundation. His research interests focus on Nonlinear Analysis, in particular, the fixed point theory, variational principles and inequalities, nonlinear evolution equation, approximation methods. Department of Mathematics, Dong-A University, Busan , Korea jungjs@mail.donga.ac.kr