Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007, Article ID 32870, 8 pages doi:10.1155/2007/32870 Research Article Iterative Algorithm for Approximating Solutions of Maximal Monotone Operators in Hilbert Spaces Yonghong Yao and Rudong Chen Received 11 October 2006; Revised 8 December 2006; Accepted 11 December 2006 Recommended by Nan-Jing Huang We first introduce and analyze an algorithm of approximating solutions of maximal monotone operators in Hilbert spaces. Using this result, we consider the convex mini- mization problem of finding a minimizer of a proper lower-semicontinuous convex func- tion and the variational problem of finding a solution of a variational inequality. Copyright © 2007 Y. Yao and R. Chen. 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 Throughout this paper, we assume that H is a real Hilbert space and T : H → 2 H is a maximal monotone operator. A well-known method for solving the equation 0 ∈ Tv in a Hilbert space H is the proximal point algorithm: x 1 = x ∈ H and x n+1 = J r n x n , n = 1,2, , (1.1) where {r n }⊂(0,∞)andJ r = (I + rT) −1 for all r>0. This algorithm was first introduced by Martinet [1]. Rockafellar [2] proved that if liminf n→∞ r n > 0andT −1 0 =∅, then the sequence {x n } defined by (1.1) converges weakly to an element of T −1 0. Later, many re- searchers have studied the convergence of the sequence defined by (1.1)inaHilbertspace; see, for instance, [3–6] and the references mentioned therein. In particular, Kamimura and Takahashi [7] proved the following result. Theorem 1.1. Let T : H → 2 H be a maximal monotone operator. Let {x n } be a seque nce defined as follows: x 1 = u ∈ H and x n+1 = α n u + 1 − α n J r n x n , n = 1,2, , (1.2) 2 Fixed Point Theory and Applications where {α n }⊂[0,1] and {r n }⊂(0,∞) satisfy lim n→∞ α n = 0, ∞ n=1 α n =∞,andlim n→∞ r n = ∞ .IfT −1 0 =∅, then the sequence {x n } defined by (1.2)convergesstronglytoPu,whereP is the metric projection of H onto T −1 0. Motivated and inspired by the above result, in this paper, we suggest and analyze an it- erative algorithm which has strong convergence. Furt her, using this result, we consider the convex minimization problem of finding a minimizer of a proper lower-semicontinuous convex function and the variational problem of finding a solution of a var iational in- equality. 2. Preliminaries Recall that a mapping U : H → H is said to be nonexpansive if Ux− Uy≤x − y for all x, y ∈ H. We denote the set of all fixed points of U by F(U). A multivalued operator T : H → 2 H with domain D(T) and range R(T) is said to be monotone if for each x i ∈ D( T)andy i ∈ Tx i , i = 1,2, we have x 1 − x 2 , y 1 − y 2 ≥0. A monotone operator T is said to be maximal if its graph G(T) ={(x, y):y ∈ Tx} is not properly contained in the graph of any other monotone operator. Let I denote the identity operator on H and let T : H → 2 H be a maximal monotone operator. Then we can define, for each r>0, a nonexpansive single-valued mapping J r : H → H by J r = (I + rT) −1 . It is called the resolvent (or the proximal mapping) of T. We also define the Yosida approximation A r by A r = (I − J r )/r.WeknowthatA r x ∈ TJ r x and A r x≤inf{y : y ∈ Tx} for all x ∈ H. Before starting the main result of this paper, we include some lemmas. Lemma 2.1 (see [8]). Let {x n } and {z n } be bounded sequences in a Banach space X and let {α n } beasequencein[0,1] with 0 < liminf n→∞ α n ≤ limsup n→∞ α n < 1.Supposex n+1 = α n x n +(1− α n )z n for all integers n ≥ 0 and limsup n→∞ (z n+1 − z n −x n+1 − x n ) ≤ 0. Then, lim n→∞ z n − x n =0. Lemma 2.2 (the resolvent identity). For λ,μ>0, there holds the identity J λ x = J μ μ λ x + 1 − μ λ J λ x , x ∈ X. (2.1) Lemma 2.3 (see [9]). Let E be a real Banach space. Then for all x, y ∈ E x + y 2 ≤x 2 +2 y, j(x + y) ∀ j(x + y) ∈ J(x + y). (2.2) Lemma 2.4 (see[10]). Let {a n } be a sequence of nonnegative real numbers satisfying the property a n+1 ≤ (1 − s n )a n + s n t n , n ≥ 0,where{s n }⊂(0,1) and {t n } are such that (i) ∞ n=0 s n =∞, (ii) either limsup n→∞ t n ≤ 0 or ∞ n=0 |s n t n | < ∞. Then {a n } converges to zero. Y. Yao and R. Chen 3 3. Main result Let T : H → 2 H be a maximal monotone operator and let J r : H → H be the resolvent of T for each r>0. Then we consider the following algorithm: for fixed u ∈ H and given x 0 ∈ H arbitrarily, let the sequence {x n } is generated by y n ≈ J r n x n , x n+1 = α n u + β n x n + δ n y n , (3.1) where {α n }, {β n }, {δ n } are three real numbers in [0,1] and {r n }⊂(0,∞). Here the crite- rion for the approximate computation of y n in (3.1)willbe y n − J r n x n ≤ σ n , (3.2) where ∞ n=0 σ n < ∞. Theorem 3.1. Let T : H → 2 H be a maximal monotone operator. Assume {α n }, {β n }, {δ n }, and {r n } satisfy the following control conditions: (i) α n + β n + δ n = 1; (ii) lim n→∞ α n = 0 and ∞ n=0 α n =∞; (iii) 0 < liminf n→∞ β n ≤ limsup n→∞ β n < 1; (iv) r n ≥ > 0 for all n and r n+1 − r n → 0(n →∞). If T −1 0 =∅, then {x n } defined by (3.1) under criterion (3.2)convergesstronglytoPu,where P is the metric projection of H onto T −1 0. Proof. From T −1 0 =∅, there exists p ∈ T −1 0suchthatJ s p = p for all s>0. Then we have x n+1 − p ≤ α n u − p + β n x n − p + δ n y n − p ≤ α n u − p + β n x n − p + δ n σ n + J r n x n − p ≤ α n u − p + β n x n − p + δ n x n − p + δ n σ n = α n u − p + 1 − α n x n − p + δ n σ n . (3.3) An induction g ives that x n − p ≤ max u − p, x 0 − p + n k=0 σ k (3.4) for all n ≥ 0. This implies that {x n } is bounded. Hence {J r n x n } and {y n } are also bounded. Define a sequence {z n } by x n+1 = β n x n + 1 − β n z n , n ≥ 0. (3.5) 4 Fixed Point Theory and Applications Then we observe that z n+1 − z n = x n+2 − β n+1 x n+1 1 − β n+1 − x n+1 − β n x n 1 − β n = α n+1 1 − β n+1 − α n 1 − β n u + δ n+1 1 − β n+1 y n+1 − y n + δ n+1 1 − β n+1 − δ n 1 − β n y n . (3.6) If r n−1 ≤ r n ,fromLemma 2.2, using the resolvent identity J r n x n = J r n−1 r n−1 r n x n + 1 − r n−1 r n J r n x n , (3.7) we obtain J r n x n − J r n−1 x n−1 ≤ r n−1 r n x n − x n−1 + r n − r n−1 r n J r n x n − x n−1 ≤ x n − x n−1 + 1 r n−1 − r n J r n x n − x n−1 . (3.8) Similarly, we can prove that the last inequality holds if r n−1 ≥ r n . On the other hand, from (3.2), we have y n+1 − y n ≤ y n+1 − J r n+1 x n+1 + y n − J r n x n + J r n+1 x n+1 − J r n x n ≤ σ n+1 + σ n + J r n+1 x n+1 − J r n x n . (3.9) Thus it follows from (3.5)that z n+1 − z n − x n+1 − x n ≤ α n+1 1 − β n+1 − α n 1 − β n u + y n + δ n+1 1 − β n+1 x n+1 − x n + δ n+1 1 − β n+1 1 r n+1 − r n × J r n+1 x n+1 − x n + σ n+1 + σ n − x n+1 − x n ≤ α n+1 1 − β n+1 − α n 1 − β n u + y n + σ n+1 + σ n + δ n+1 1 − β n+1 1 r n+1 − r n × J r n+1 x n+1 − x n , (3.10) which implies that limsup n→∞ (z n+1 − z n −x n+1 − x n ) ≤ 0. Hence, by Lemma 2.1,we have lim n→∞ z n − x n =0. Consequently, it follows from (3.5)that lim n→∞ x n+1 − x n = lim n→∞ 1 − β n z n − x n = 0. (3.11) On the other hand, x n − y n ≤ x n+1 − x n + x n+1 − y n ≤ x n+1 − x n + α n u − y n + β n x n − y n , (3.12) Y. Yao and R. Chen 5 and so, by (ii), (iii), (3.11), and (3.12), we have lim n→∞ x n − y n =0. It follows that A r n x n ≤ 1 r n x n − y n + y n − J r n x n ≤ 1 x n − y n + σ n −→ 0. (3.13) We next prove that limsup n→∞ u − Pu,x n+1 − Pu ≤ 0, (3.14) where P is the metric projection of H onto T −1 0. To prove this, it is sufficient to show limsup n→∞ u − Pu,J r n x n − Pu≤0, because x n+1 − J r n x n → 0. Now there exists a subse- quence {x n i }⊂{x n } such that lim i→∞ u − Pu,J r n i x n i − Pu = limsup n→∞ u − Pu,J r n x n − Pu . (3.15) Since {J r n x n } is bounded, we may assume that {J r n i x n i } converges weakly to some v ∈ H. Then it follows that v ∈ T −1 0. Indeed, since A r n x n ∈ TJ r n x n and T is monotone, we have s − J r n i x n i ,s − A r n i x n i ≥0, where s ∈ Ts.FromA r n x n → 0, we obtain s − v,s ≥0 whenever s ∈ Ts. Hence, from the maximality of T,wehavev ∈ T −1 0. Since P is the metric projection of H onto T −1 0, we obtain limsup n→∞ u − Pu,J r n x n − Pu = lim i→∞ u − Pu,J r n i x n i − Pu = u − Pu,v − Pu≤0. (3.16) That is, ( 3.14)holds. Finally, to prove that x n → p,weapplyLemma 2.3 to get x n+1 − Pu 2 ≤ β n x n − Pu + δ n y n − Pu 2 +2α n u − Pu,x n+1 − Pu ≤ β n x n − Pu + δ n x n − Pu + δ n σ n 2 +2α n u − Pu,x n+1 − Pu = 1 − α n x n − Pu + δ n σ n 2 +2α n u − Pu,x n+1 − Pu ≤ 1 − α n x n − Pu 2 +2α n u − Pu,x n+1 − Pu + Mσ n , (3.17) where M>0 is some constant such that 2(1 − α n )δ n x n − Pu + δ 2 n σ n ≤ M.Anapplica- tion of Lemma 2.4 yields that x n − Pu→0. This completes the proof. Remark 3.2. It is clear that the algorithm (3.1)includesthealgorithm(1.2) as a special case. Our result can be considered as a complement of Kamimura and Takahashi [7]and others. 4. Applications Let f : H → (−∞,∞] be a proper lower semicontinuous convex function. Then we can define the subdifferential ∂f of f by ∂f(x) = z ∈ H : f (y) ≥ f (x)+y − x,z∀y ∈ H (4.1) 6 Fixed Point Theory and Applications for all x ∈ H. It is well known that ∂f is a maximal monotone operator of H into itself; see Minty [11]andRockafellar[12, 13]. In this section, we investigate our algorithm in the case of T = ∂f. Our discussion fol- lows Rockafellar [14,Section4].IfT = ∂f,thealgorithm(3.1) is reduced to the following algorithm: y n ≈ argmin z∈H f (z)+ 1 2r n z − x n 2 , x n+1 = α n u + β n x n + δ n y n , n ∈ N, (4.2) with the following criterion: d 0,S n y n ≤ σ n r n , (4.3) where ∞ n=0 σ n < ∞, S n (z) = ∂f(z)+(z − x n )/r n ,andd(0,A) = inf{x : x ∈ A}.About (4.3), the following lemma was proved in Rockafellar [2, Proposition 3]. Lemma 4.1. If y n is chosen according to criterion (4.3), then y n − J r n x n ≤σ n holds, where J r n = (I + r n ∂f) −1 . Theorem 4.2. Let f : H → (−∞, ∞] be a proper lower semicontinuous convex function. Assume {α n }, {β n }, {δ n },and{r n } satisfy the same conditions (i)–(iv) as in Theorem 3.1. If (∂f) −1 0 =∅, then {x n } defined by (4.2)withcriterion(4.3)convergesstronglytov ∈ H, which is the minimizer of f nearest to u. Proof. Putting g n (z) = f (z)+z − x n 2 /2r n ,weobtain ∂g n (z) = ∂f(z)+ 1 r n z − x n = S n (z) (4.4) for all z ∈ H and J r n x n = (I + r n ∂f) −1 x n = argmin z∈H g n (z). It follows from Theorem 3.1 and Lemma 4.1 that {x n } converges strongly to v ∈ H and f (v) = min z∈H f (z). This com- pletes the proof. Next we consider a variational inequality. Let C be a nonempty closed convex subset of H and let T be a single-valued operator of C into H. We denote by VI(C, T) the set of solutions of the variational inequality, that is, VI(C,T) = w ∈ X : s − w,Tw≥0, ∀s ∈ C . (4.5) A single-valued operator T is called semicontinuous if T is continuous from each line segment of C to H with the weak topology. Let F be a single-valued monotone and semi- continuous operator of C into H and let N C z be the normal cone to C at z ∈ C, that is, N C z ={w ∈ H : z − s,w≥0, ∀s ∈ C}.Letting Az = ⎧ ⎨ ⎩ Fz+N C z, z ∈ C, ∅, z ∈ H \ C, (4.6) Y. Yao and R. Chen 7 we have that A is a maximal monotone operator (see Rockafellar [14, Theorem 3]). We can also check that 0 ∈ Av if and only if v ∈ VI(C,F) and that J r x = VI(C,F r,x )forall r>0andx ∈ H,whereF r,x z = Fz +(z − x)/r for all z ∈ C. Then we have the following result. Corollary 4.3. Let F be a single-valued monotone and semicontinuous operator of C into H.Forfixedu ∈ H,letthesequence{x n } be generated by y n ≈ VI C,F r n ,x n , x n+1 = α n u + β n x n + δ n y n . (4.7) Here the criterion for the approximate computation of y n in (4.7)willbe y n − VI C,F r n ,x n ≤ σ n , (4.8) where ∞ n=0 σ n < ∞.Assume{α n }, {β n }, {δ n },and{r n } satisfy the same conditions (i)–(iv) as in Theorem 3.1.If VI(C,F) =∅, then {x n } defined by (4.7) with criterion (4.8)converges strongly to the point of VI(C,F) nearest to u. References [1] B. Martinet, “R ´ egularisation d’in ´ equations variationnelles par approximations successives,” Re- vue Franc¸aise d’Automatique et Informatique. 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Rockafellar, “Characterization of the subdifferentials of convex functions,” Pacific Journal of Mathematics, vol. 17, pp. 497–510, 1966. [13] R. T. Rockafellar, “On the maximal monotonicity of subdifferential mappings,” Pacific Journal of Mathematics, vol. 33, pp. 209–216, 1970. [14] R. T. Rockafellar, “On the maximality of sums of nonlinear monotone operators,” Transactions of the American Mathematical Society, vol. 149, no. 1, pp. 75–88, 1970. Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address: yuyanrong@tjpu.edu.cn Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Email address: chenrd@tjpu.edu.cn . first introduce and analyze an algorithm of approximating solutions of maximal monotone operators in Hilbert spaces. Using this result, we consider the convex mini- mization problem of finding a minimizer. of proximal point iterations in a Hilbert space,” Mathematical Programming, vol. 87, no. 1, pp. 189–202, 2000. [7] S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators. the convex minimization problem of finding a minimizer of a proper lower-semicontinuous convex function and the variational problem of finding a solution of a var iational in- equality. 2. Preliminaries Recall