Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 598191, 10 pages doi:10.1155/2008/598191
Research Article Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces
1 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, South Korea 2 Department of Mathematics and the RINS, Gyeongsang National University, Chinju 660-701, South Korea 3 Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
Yeol Je Cho,1 Shin Min Kang,2 and Haiyun Zhou3
Correspondence should be addressed to Haiyun Zhou, witman66@yahoo.com.cn
Received 1 March 2007; Accepted 27 November 2007
Recommended by H. Bevan Thompson
(cid:2) ∞ n(cid:3)0
H Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T : Ω → 2 a maximal monotone operator with T −10 /(cid:3) ∅. Let PΩ be the metric projection of H onto Ω. Suppose that, for any given xn ∈ H, βn > 0, and en ∈ H, there exists xn ∈ Ω satisfying the following set-valued mapping equation: xn (cid:4) en ∈ xn (cid:4) βnT (cid:5)xn(cid:6) for all n ≥ 0, where {βn} ⊂ (cid:5)0, (cid:4)∞(cid:6) with βn → (cid:4) ∞ as n → ∞ and {en} is regarded as an error sequence such that (cid:7)en(cid:7)2 < (cid:4)∞. Let {αn} ⊂ (cid:5)0, 1(cid:7) be a (cid:2) ∞ αn (cid:3) ∞. For any fixed u ∈ Ω, define a sequence real sequence such that αn → 0 as n → ∞ and n(cid:3)0 {xn} iteratively as xn(cid:4)1 (cid:3) αnu (cid:4) (cid:5)1 − αn(cid:6)PΩ(cid:5)xn − en(cid:6) for all n ≥ 0. Then {xn} converges strongly to a point z ∈ T −10 as n → ∞, where z (cid:3) lim t→∞ Jtu.
Copyright q 2008 Yeol Je Cho 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 H be a real Hilbert space with the inner product (cid:8)·, ·(cid:9) and norm (cid:7) · (cid:7). A set T ⊂ H × H is called a monotone operator on H if T has the following property:
∀(cid:5)x, y(cid:6), (cid:5)x(cid:10), y(cid:10)(cid:6) ∈ T. (cid:8)x − x(cid:10), y − y(cid:10)(cid:9) ≥ 0, (cid:5)1.1(cid:6)
A monotone operator T on H is said to be maximal monotone if it is not properly contained in any other monotone operator on H. Equivalently, a monotone operator T is maximal monotone if R(cid:5)I (cid:4) tT (cid:6) (cid:3) H for all t > 0. For a maximal monotone operator T , we can define the resolvent
2 Journal of Inequalities and Applications
of T by
Jt (cid:3) (cid:5)I (cid:4) tT (cid:6)−1, (cid:5)1.2(cid:6) ∀t > 0.
It is well known that Jt : H → D(cid:5)T (cid:6) is nonexpansive. Also we can define the Yosida approxima- tion Tt by
(cid:4) (cid:3) , I − Jt (cid:5)1.3(cid:6) ∀t > 0. Tt (cid:3) 1 t
0
We know that Ttx ∈ T Jtx for all x ∈ H, (cid:7)Ttx(cid:7) ≤ |T x| for all x ∈ D(cid:5)T (cid:6), where |T x| (cid:3) inf{(cid:7)y(cid:7) : y ∈ T x}, and T −10 (cid:3) F(cid:5)Jt(cid:6) for all t > 0. Throughout this paper, we assume that Ω is a nonempty closed convex subset of a real Hilbert space H and T : Ω → 2H It is well known that, for any u ∈ H, there exists uniquely y is a maximal monotone operator with T −10 /(cid:3) ∅. ∈ Ω such that
0
0 in (cid:5)1.4(cid:6), we call PΩ the metric projection of
(cid:5) (cid:5) (cid:6)(cid:5) (cid:5)u − y (cid:5) (cid:5)u − y (cid:5)1.4(cid:6) (cid:5) (cid:5) (cid:3) inf (cid:7) . : y ∈ Ω
When a mapping PΩ : H → Ω is defined by PΩu (cid:3) y H onto Ω. The metric projection PΩ of H onto Ω has the following basic properties:
0 strongly imply that P x
0
0
(cid:3) y . (cid:5)i(cid:6) (cid:8)PΩx(cid:10) − x, x(cid:10) − PΩx(cid:10)(cid:9) ≥ 0 for all x ∈ Ω and x(cid:10) ∈ H, (cid:5)ii(cid:6) (cid:7)PΩx − PΩy(cid:7)2 ≤ (cid:8)x − y, PΩx − PΩy(cid:9) for all x, y ∈ H, (cid:5)iii(cid:6) (cid:7)PΩx − PΩy(cid:7) ≤ (cid:7)x − y(cid:7) for all x, y ∈ H, 0 weakly and P xn → y (cid:5)iv(cid:6) xn → x
Finding zeroes of maximal monotone operators is the central and important topic in nonlinear functional analysis. A classical method to solve the following set-valued equation:
(cid:5)1.5(cid:6) 0 ∈ T z,
H is a maximal monotone operator, is the proximal point algorithm which, ∈ H, updates xn(cid:4)1 iteratively conforming to the following recursion:
0
where T : Ω → 2 starting with any point x
, (cid:5)1.6(cid:6) ∀n ≥ 0, xn ∈ xn(cid:4)1 (cid:4) βnT xn(cid:4)1
where {βn} ⊂ (cid:8)β, ∞(cid:6), β > 0, is a sequence of real numbers. However, as pointed out in (cid:8)1(cid:7), the ideal form of the algorithm is often impractical since, in many cases, solving the problem (cid:5)1.6(cid:6) exactly is either impossible or as difficult as the original problem (cid:5)1.5(cid:6). Therefore, one of the most interesting and important problems in the theory of maximal monotone operators is to find an efficient iterative algorithm to compute approximately zeroes of T . In 1976, Rockafellar (cid:8)2(cid:7) gave an inexact variant of the method
, ∀n ≥ 0, (cid:5)1.7(cid:6) xn (cid:4) en(cid:4)1 ∈ xn(cid:4)1 (cid:4) βnT xn(cid:4)1
(cid:2)∞ n(cid:3)0
where {en} is regarded as an error sequence. This method is called an inexact proximal point algo- (cid:7)en(cid:7) < (cid:4)∞, then the sequence {xn} defined by (cid:5)1.7(cid:6) converges rithm. It was shown that if weakly to a zero of T . G ¨uler (cid:8)3(cid:7) constructed an example showing that Rockafellar’s proximal point algorithm (cid:5)1.7(cid:6) does not converge strongly, in general. This gives rise to the following question.
Yeol Je Cho et al. 3
Question 1. How to modify Rockafellar’s algorithm so that strong convergence is guaranteed?
Xu (cid:8)4(cid:7) gave one solution to Question 1. However, this requires that the error sequence {en} is summable, which is too strong. This gives rise to the following question.
Question 2. Is it possible to establish some strong convergence theorems under the weaker assumption on the error sequence {en} given in (cid:5)1.7(cid:6)?
It is our purpose in this paper to give an affirmative answer to Question 2 under a weaker assumption on the error sequence {en} in Hilbert spaces. For this purpose, we collect some lemmas that will be used in the proof of the main results in the next section. The first lemma is standard and it can be found in some textbooks on functional analysis.
Lemma 1.1. For all x, y ∈ H and λ ∈ (cid:8)0, 1(cid:7),
(cid:5) (cid:5)x − y (cid:5) (cid:5)2. (cid:5) (cid:5)λx (cid:4) (cid:5)1 − λ(cid:6)y (cid:5) (cid:5)2 (cid:3) λ(cid:7)x(cid:7)2 (cid:4) (cid:5)1 − λ(cid:6)(cid:7)y(cid:7)2 − λ(cid:5)1 − λ(cid:6) (cid:5)1.8(cid:6)
Lemma 1.2 (cid:5)see (cid:8)5, Lemma 1(cid:7)(cid:6). For all u ∈ H, limt→∞Jtu exists and it is the point of T −10 nearest to u.
Lemma 1.3 (cid:5)see (cid:8)1, Lemma 2(cid:7)(cid:6). For any given xn ∈ H, βn > 0, and en ∈ H, there exists xn ∈ Ω conforming to the following set-valued mapping equation (in short, SVME):
xn (cid:4) en ∈ xn (cid:4) βnT xn, ∀n ≥ 0. (cid:5)1.9(cid:6)
Furthermore, for any p ∈ T −10, one has
(cid:9) (cid:8) (cid:9) (cid:8) ≥ xn − xn, xn − xn (cid:4) en (cid:5)1.10(cid:6) xn − p, xn − xn (cid:4) en (cid:5) (cid:5)2 ≤ (cid:5) (cid:5)2 − (cid:5) (cid:5)2 (cid:4) , (cid:5) (cid:5)2. (cid:5) (cid:5)xn − en − p (cid:5) (cid:5)xn − p (cid:5) (cid:5)xn − xn (cid:5) (cid:5)en
(cid:3) Lemma 1.4 (cid:5)see (cid:8)6, Lemma 1.1(cid:7)(cid:6). Let {an}, {bn}, and {cn} be three real sequences satisfying (cid:4) ≤ an (cid:4) bn (cid:4) cn, 1 − tn ∀n ≥ 0, (cid:5)1.11(cid:6) an(cid:4)1
tn (cid:3) ∞, bn (cid:3) ◦(cid:5)tn(cid:6), and where {tn} ⊂ (cid:8)0, 1(cid:7), cn < ∞. Then an → 0 as n → ∞. (cid:2)∞ n(cid:3)0 (cid:2)∞ n(cid:3)0
2. The main results
Now we give our main results in this paper.
Theorem 2.1. Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T : Ω → 2H a maximal monotone operator with T −10 /(cid:3) ∅. Let PΩ be the metric projection of H onto Ω. Suppose that, for any given xn ∈ H, βn > 0, and en ∈ H, there exists xn ∈ Ω conforming to the SVME (cid:5)1.9(cid:6), where (cid:2)∞ {βn} ⊂ (cid:5)0, (cid:4)∞(cid:6) with βn → (cid:4)∞ as n → ∞ and (cid:7)en(cid:7)2 < (cid:4)∞. Let {αn} be a real sequence in (cid:8)0, 1(cid:7) n(cid:3)0 such that
αn (cid:3) ∞. (cid:5)i(cid:6) αn → 0 as n → ∞, (cid:5)ii(cid:6) (cid:2)∞ n(cid:3)0
4 Journal of Inequalities and Applications
For any fixed u ∈ Ω, define the sequence {xn} iteratively as follows: (cid:3) (cid:3) (cid:4) , (cid:3) αnu (cid:4) (cid:4) PΩ xn − en 1 − αn ∀n ≥ 0. (cid:5)2.1(cid:6) xn(cid:4)1
Then {xn} converges strongly to a fixed point z of T, where z (cid:3) limt→∞Jtu.
0
n−1(cid:10)
Proof Claim 1. {xn} is bounded. Fix p ∈ T −10 and set M (cid:3) max{(cid:7)u − p(cid:7)2, (cid:7)x − p(cid:7)2}. First, we prove that
j(cid:3)0
(cid:5) (cid:5)2 ≤ M (cid:4) (cid:5) (cid:5)2, (cid:5) (cid:5)xn − p (cid:5) (cid:5)ej ∀n ≥ 0. (cid:5)2.2(cid:6)
When n (cid:3) 0, (cid:5)2.2(cid:6) is true. Now, assume that (cid:5)2.2(cid:6) holds for some n ≥ 0. We will prove that (cid:5)2.2(cid:6) holds for n (cid:4) 1. By using the iterative scheme (cid:5)2.1(cid:6) and Lemmas 1.1 and 1.3, we have
(cid:5) (cid:5)2 − p (cid:5) (cid:5)xn(cid:4)1 (cid:3) (cid:4) (cid:3) (cid:3) (cid:3) (cid:4)(cid:5) (cid:5)2 − p (cid:3) αn(cid:7)u − p(cid:7)2 (cid:4) (cid:4)(cid:5) (cid:5)PΩ xn − en (cid:3) (cid:5) (cid:5)2 − αn (cid:3) (cid:5) (cid:5)2 (cid:5)2.3(cid:6) ≤ αnM (cid:4) 1 − αn (cid:4)(cid:5) (cid:5)xn − p (cid:4)(cid:5) (cid:5)u − PΩ (cid:5) (cid:5) (cid:5)en (cid:5)2 (cid:4) 1 − αn
j(cid:3)0
j(cid:3)0
(cid:3) (cid:4) 1 − αn xn − en (cid:5) (cid:4)(cid:5) (cid:5)2 ≤ αnM (cid:4) (cid:5)xn − en − p n(cid:10) n(cid:10) (cid:5) (cid:5)2 (cid:3) M (cid:4) 1 − αn (cid:5) (cid:5)2. M (cid:4) ≤ αnM (cid:4) (cid:5) (cid:5)ej (cid:5) (cid:5)ej 1 − αn
∞(cid:10)
n−1(cid:10)
By induction, we assert that
j(cid:3)0
j(cid:3)0
(cid:5) (cid:5)2 ≤ M (cid:4) (cid:5) (cid:5)2 < M (cid:4) (cid:5) (cid:5)2 < (cid:4)∞, (cid:5) (cid:5)xn − p (cid:5) (cid:5)ej (cid:5) (cid:5)ej ∀n ≥ 0. (cid:5)2.4(cid:6)
xn}.
This implies that {xn} is bounded and so is {Jβn Claim 2. limn→∞(cid:8)u − z, xn(cid:4)1 xn (cid:3) βnTβn Noting that T is maximal monotone, u − Jtu (cid:3) tTtu, Ttu ∈ T Jtu, xn − Jβn − z(cid:9) ≤ 0, where z (cid:3) limt→∞Jtu, which is guaranteed by Lemma 1.2. xn, Tβn (cid:9) xn ∈ T Jβn (cid:8) xn, and βn → (cid:4)∞ (cid:5)n → ∞(cid:6), we have (cid:9) xn − Jtu u − Jtu, Jβn (cid:9) (cid:9) xn xn Tβn xn, Jtu − Jβn xn xn, Jtu − Jβn (cid:8) Ttu, Jtu − Jβn (cid:3) −t (cid:8) Ttu − Tβn (cid:3) −t (cid:8) (cid:8) − t (cid:9) (cid:5)2.5(cid:6) ≤ − xn xn − Jβn xn, Jtu − Jβn t βn
−→ 0 (cid:5)n −→ ∞(cid:6), ∀t > 0
and hence
(cid:9) xn − Jtu u − Jtu, Jβn ≤ 0. (cid:5)2.6(cid:6) (cid:8) limn→∞
(cid:5)xn (cid:4) en(cid:6) − Jβn Note that (cid:7)Jβn xn(cid:7) ≤ (cid:7)en(cid:7) → 0 as n → ∞, and so it follows from (cid:5)2.6(cid:6) that
(cid:8) (cid:3) (cid:4) xn (cid:4) en (cid:9) − Jtu u − Jtu, Jβn ≤ 0. (cid:5)2.7(cid:6) limn→∞
5 Yeol Je Cho et al.
Note that (cid:7)PΩ(cid:5)xn − en(cid:6) − Jβn (cid:5)xn (cid:4) en(cid:6)(cid:7) ≤ (cid:7)en(cid:7) → 0 as n → ∞ and so it follows from (cid:5)2.7(cid:6) that
(cid:4) u − Jtu, PΩ (cid:3) xn − en (cid:9) − Jtu (cid:5)2.8(cid:6) ≤ 0. (cid:8) limn→∞
Since αn → 0 as n → ∞, from (cid:5)2.1(cid:6) we have
(cid:3) (cid:4) − PΩ xn − en (cid:5)2.9(cid:6) −→ 0 (cid:5)n −→ ∞(cid:6). xn(cid:4)1
(cid:9) − Jtu (cid:5)2.10(cid:6) ≤ 0, ∀t > 0, u − Jtu, xn(cid:4)1 It follows from (cid:5)2.8(cid:6) and (cid:5)2.9(cid:6) that (cid:8) limn→∞
and so, from z (cid:3) limt→∞Jtu and (cid:5)2.10(cid:6), we have
(cid:9) − z (cid:5)2.11(cid:6) ≤ 0. u − z, xn(cid:4)1 (cid:8) limn→∞
Claim 3. xn → z as n → ∞. Observe that
(cid:4) (cid:3) (cid:4)(cid:3) (cid:4) (cid:4) − z − z (cid:3) (cid:3) xn − en PΩ − αn(cid:5)u − z(cid:6) (cid:5)2.12(cid:6) 1 − αn (cid:3) xn(cid:4)1
and so
2
(cid:9) (cid:3) (cid:4) (cid:3) (cid:4) (cid:8) (cid:5) (cid:5)2 ≥ − z − z , (cid:5) (cid:5)PΩ xn − en − PΩz (cid:5) (cid:5)2 − 2αn (cid:5)2.13(cid:6) 1 − αn (cid:5) (cid:5)xn(cid:4)1 u − z, xn(cid:4)1
which implies that
(cid:3) (cid:9) (cid:8) (cid:5) (cid:5)2 ≤ − z − z . (cid:4)(cid:5) (cid:5)xn − en − z (cid:5)2.14(cid:6) 1 − αn (cid:5) (cid:5)2 (cid:4) 2αn (cid:5) (cid:5)xn(cid:4)1 u − z, xn(cid:4)1
It follows from Lemma 1.3 and (cid:5)2.14(cid:6) that
(cid:3) (cid:3) (cid:9) (cid:8) (cid:5) (cid:5)2 ≤ − z − z (cid:4)(cid:5) (cid:5)xn − xn 1 − αn (cid:5) (cid:5)xn(cid:4)1 u − z, xn(cid:4)1 (cid:5)2.15(cid:6) (cid:3) (cid:5) (cid:5)2 (cid:4) (cid:9) 1 − αn (cid:8) ≤ (cid:5) (cid:5)2 (cid:4) 2αn (cid:5) (cid:5)2. − z (cid:4) (cid:4)(cid:5) (cid:5)xn − z (cid:4)(cid:5) (cid:5)xn − z (cid:5) (cid:5)en (cid:5) (cid:5)en 1 − αn (cid:5) (cid:5)2 − (cid:5) (cid:5)2 (cid:4) 2αn u − z, xn(cid:4)1
− z(cid:9), 0}. Then σn → 0 as n → ∞. Indeed, by the definition of σn, we Set σn (cid:3) max{(cid:8)u − z, xn(cid:4)1 see that σn ≥ 0 for all n ≥ 0. On the other hand, by (cid:5)2.11(cid:6), we know that for arbitrary (cid:6) > 0, there exists some fixed positive integer N such that (cid:8)u − z, xn(cid:4)1 − z(cid:9) ≤ (cid:6) for all n ≥ N. This implies that 0 ≤ σn ≤ (cid:6) for all n ≥ N, and the desired conclusion follows. Set an (cid:3) (cid:7)xn − z(cid:7)2, bn (cid:3) 2αnσn, and cn (cid:3) (cid:7)en(cid:7)2. Then (cid:5)2.15(cid:6) reduces to
(cid:4) ≤ an (cid:4) bn (cid:4) cn, ∀n ≥ 0, (cid:5)2.16(cid:6) an(cid:4)1
αn (cid:3) ∞, bn (cid:3) ◦(cid:5)αn(cid:6), and cn < (cid:4)∞. Thus it follows from Lemma 1.4 that an → 0 (cid:2)∞ n(cid:3)0 (cid:3) 1 − αn (cid:2)∞ n(cid:3)0 where as n → 0, that is, xn → z ∈ T −10 as n → ∞. This completes the proof.
H
6 Journal of Inequalities and Applications
r>0 then for any given xn ∈ Ω and βn > 0, we may find xn ∈ Ω and en ∈ H satisfying the SVME (cid:5)1.9(cid:6). Furthermore, Lemma 1.2 also holds for u ∈ Ω, and hence Theorem 2.1 still holds true for monotone operators which satisfy the range condition.
Remark 2.2. The maximal monotonicity of T is only used to guarantee the existence of solutions to the SVME (cid:5)1.9(cid:6) for any given xn ∈ H and βn > 0. If we assume that T : Ω → 2 is monotone (cid:5)need not be maximal(cid:6) and T satisfies the range condition (cid:11) D(cid:5)T (cid:6) (cid:3) Ω ⊂ R(cid:5)I (cid:4) rT (cid:6), (cid:5)2.17(cid:6)
Following the proof lines of Theorem 2.1, we can prove the following corollary.
Corollary 2.3. Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and S : Ω → Ω a continuous and pseudocontractive mapping with a fixed point in Ω. Suppose that, for any given xn ∈ Ω, βn > 0, and en ∈ H, there exists xn ∈ Ω such that (cid:3) (cid:4) xn (cid:4) en (cid:3) 1 (cid:4) βn ∀n ≥ 0, (cid:5)2.18(cid:6)
xn − βnSxn, (cid:2)∞ n(cid:3)0 (cid:7)en(cid:7)2 < (cid:4)∞. Let {αn} ⊂ (cid:5)0, 1(cid:7) be a real αn (cid:3) ∞. For any fixed u ∈ Ω, define the sequence {xn} (cid:2)∞ n(cid:3)0 where βn → ∞ (cid:5)n → ∞(cid:6) and {en} satisfies the condition sequence such that αn → 0 as n → ∞ and iteratively as follows: (cid:3) (cid:3) (cid:4) , (cid:3) αnu (cid:4) (cid:4) PΩ xn − en 1 − αn ∀n ≥ 0. (cid:5)2.19(cid:6) xn(cid:4)1
H is continuous and monotone and satisfies the range
Then {xn} converges strongly to a fixed point z of S, where z (cid:3) limt→∞Jtu, and Jt (cid:3) (cid:5)I (cid:4) t(cid:5)I − S(cid:6)(cid:6)−1 for all t > 0.
r>0
Proof. Let T (cid:3) I − S. Then T : Ω → 2 condition (cid:11) D(cid:5)T (cid:6) (cid:3) Ω ⊂ R(cid:5)I (cid:4) rT (cid:6). (cid:5)2.20(cid:6)
Now we only need to verify the last assertion. For any y ∈ Ω and r > 0, define an operator G : Ω → Ω by
Gx (cid:3) Sx (cid:4) 1 y. (cid:5)2.21(cid:6) r 1 (cid:4) r 1 (cid:4) r
Then G : Ω → Ω is continuous and strongly pseudocontractive. By Kamimura et al. (cid:8)7, Corol- lary 1(cid:7), G has a unique fixed point x in Ω, that is, x (cid:3) (cid:5)r/(cid:5)1 (cid:4) r(cid:6)(cid:6)Sx (cid:4) (cid:5)1/(cid:5)1 (cid:4) r(cid:6)(cid:6)y, which implies that y ∈ R(cid:5)I (cid:4) rT (cid:6) for all r > 0. In particular, for any given xn ∈ Ω and βn > 0, there exist xn ∈ Ω and en ∈ H such that
xn (cid:4) en (cid:3) xn (cid:4) βnT xn, ∀n ≥ 0, (cid:5)2.22(cid:6)
which means that (cid:3) (cid:4) xn (cid:4) en (cid:3) xn − βnS xn, 1 (cid:4) βn ∀n ≥ 0, (cid:5)2.23(cid:6)
and the relation (cid:5)2.18(cid:6) follows. The reminder of proof is the same as in the corresponding part of Theorem 2.1. This completes the proof.
Yeol Je Cho et al. 7
Remark 2.4. In Corollary 2.3, we do not know wether the continuity assumption on S can be dropped or not.
Remark 2.5. In Theorem 2.1, if the operator T is defined on the whole space H, then the metric projection mapping PΩ is not needed.
Remark 2.6. Our convergence results are different from those results obtained by Kamimura et al. (cid:8)7(cid:7).
Theorem 2.7. Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and T : Ω → 2H a maximal monotone operator with T −10 /(cid:3) ∅. Suppose that, for any given xn ∈ H, βn > 0, and en ∈ H, there exists xn ∈ Ω conforming to the following relation:
xn (cid:4) en ∈ xn (cid:4) βnT xn, ∀n ≥ 0, (cid:5)2.24(cid:6)
(cid:7)en(cid:7)2 < (cid:4)∞. Let {αn} be a sequence in (cid:8)0, 1(cid:7) with limn→∞αn < 1 and (cid:2)∞ n(cid:3)0 where limn→∞βn > 0 and define the sequence {xn} iteratively as follows:
0 xn(cid:4)1
∈ Ω x (cid:3) (cid:3) (cid:5)2.25(cid:6) (cid:4) , (cid:3) αnxn (cid:4) (cid:4) PΩ xn − en 1 − αn ∀n ≥ 0.
Then {xn} converges weakly to a point p ∈ T −10.
Proof Claim 1. {xn} is bounded. Since T −10 /(cid:3) ∅, we can take some w ∈ T −10. By using (cid:5)2.25(cid:6) and Lemmas 1.1 and 1.3, we obtain
(cid:5) (cid:5)2
(cid:3) (cid:4) (cid:3) (cid:4)(cid:5) (cid:5)2 (cid:5) (cid:5)xn(cid:4)1 (cid:3) αn (cid:3) xn − en (cid:5) (cid:5)2 − αn (cid:4)(cid:5) (cid:5)xn − PΩ xn − en 1 − αn (cid:3) 1 − αn (cid:3) − w (cid:5) (cid:5)2 ≤ αn 1 − αn (cid:5)2.26(cid:6) (cid:3) (cid:3) (cid:5) (cid:5)2 (cid:5) (cid:5)2 (cid:4) (cid:4)(cid:5) (cid:5)xn − xn (cid:5) (cid:5)en (cid:5) (cid:5)2 (cid:4) (cid:5) (cid:5)2 (cid:4) (cid:5) (cid:5)2 (cid:4) (cid:3) 1 − αn (cid:5) (cid:5)2 (cid:3) (cid:4)(cid:5) (cid:5)PΩ (cid:4)(cid:5) (cid:5)xn − en − w (cid:4)(cid:5) (cid:5)xn − w (cid:5) (cid:5)2 (cid:4) (cid:5) (cid:5)2 − (cid:5) (cid:5)en 1 − αn (cid:4)(cid:5) (cid:5)xn − xn
− w (cid:5) (cid:5)xn − w (cid:5) (cid:5)xn − w (cid:5) (cid:5)xn − w (cid:5) (cid:5)2 − (cid:5) (cid:5)2 (cid:4) ≤ ≤ αn (cid:5) (cid:5)xn − w (cid:5) (cid:5)xn − w 1 − αn (cid:5) (cid:5) (cid:5)2 (cid:5)en
(cid:7)en(cid:7)2 < (cid:4)∞ implies that limn→∞(cid:7)xn − w(cid:7)2 exists. Therefore, (cid:2)∞ n(cid:3)0
and so (cid:5)2.26(cid:6) together with {xn} is bounded. Claim 2. xn − Jβn xn → 0 as n → ∞.
It follows from (cid:5)2.26(cid:6) that (cid:3) (cid:5) (cid:5)2 (cid:5) (cid:5)2 ≤ (cid:5) (cid:5)2 − (cid:5) (cid:5)2 (cid:4) − w (cid:4)(cid:5) (cid:5)xn − xn (cid:5) (cid:5)xn − w (cid:5) (cid:5)en (cid:5)2.27(cid:6) 1 − αn (cid:5) (cid:5)xn(cid:4)1
and so (cid:5)2.26(cid:6) together with limn→∞αn < 1 implies that
(cid:5)n −→ ∞(cid:6). xn − xn −→ 0 (cid:5)2.28(cid:6)
8 Journal of Inequalities and Applications
Since xn (cid:3) Jβn (cid:5)xn (cid:4) en(cid:6) and Jβn is nonexpansive, we have
(cid:5) (cid:5) ≤ (cid:5) (cid:5) (cid:4) (cid:5) (cid:5) ≤ (cid:5) (cid:5) (cid:4) (cid:5) (cid:5)xn − xn xn (cid:5) (cid:5)xn − xn xn (cid:5) (cid:5)en (cid:5) (cid:5)xn − Jβn (cid:5) (cid:5)xn − Jβn (cid:5)2.29(cid:6) (cid:5) (cid:5) −→ 0
xn → 0 as n → ∞. as n → ∞ and consequently, xn − Jβn Claim 3. {xn} converges weakly to a point p ∈ T −10 as n → ∞. Set yn (cid:3) Jβn xn and let p ∈ H be a weak subsequential limit of {xn} such that {xnj } } converges weakly to p as converges weakly to a point p as j → ∞. Thus it follows that {ynj j → ∞. Observe that
1
(cid:3) (cid:4) (cid:7) (cid:5) (cid:5) (cid:5) (cid:5) (cid:3) (cid:5) (cid:5) (cid:3) (cid:5) (cid:5) (cid:3) (cid:3) (cid:5) (cid:5)yn − J yn yn yn xn (cid:5) (cid:5)Tβn (cid:5) (cid:5) ≤ inf (cid:6) (cid:7)z(cid:7) : z ∈ T yn (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5) (cid:5). (cid:5)2.30(cid:6) I − J 1 (cid:5) (cid:5)T 1 xn − yn βn
By assumption lim n→∞βn > 0, we have
1
1 is nonexpansive, by Browder’s demiclosedness principle, we assert that p ∈ F(cid:5)J
1
yn − J yn −→ 0 (cid:5)n −→ ∞(cid:6). (cid:5)2.31(cid:6)
Since J (cid:6) (cid:3) T −10. Now Opial’s condition of H guarantees that {xn} converges weakly to p ∈ T −1(cid:5)0(cid:6) as n → ∞. This completes the proof.
From Theorem 2.7 and the same proof of Corollary 2.3, we have the following corollary.
Corollary 2.8. Let H be a real Hilbert space, Ω a nonempty closed convex subset of H, and U : Ω → Ω a continuous and pseudocontractive mapping with a fixed point. Set T (cid:3) I − U. Suppose that, for any given xn ∈ Ω, βn > 0, and en ∈ H, there exists xn ∈ Ω such that
(cid:3) (cid:4) xn (cid:4) en (cid:3) xn − βnUxn, 1 (cid:4) βn ∀n ≥ 0. (cid:5)2.32(cid:6)
Define the sequence {xn} iteratively as follows:
(cid:3) (cid:4) (cid:5)2.33(cid:6) , (cid:3) αnxn (cid:4) ∈ Ω, (cid:3) (cid:4) xn − en PΩ x 0 1 − αn ∀n ≥ 0, xn(cid:4)1
(cid:7)en(cid:7)2 < (cid:4)∞. Then {xn} converges weakly to a fixed point p of U. where {αn} ⊂ (cid:8)0, 1(cid:7) with limn→∞ αn < 1, {βn} ⊂ (cid:5)0, (cid:4)∞(cid:6) with lim n→∞ βn > 0, and {en} ⊂ H with (cid:2)∞ n(cid:3)0
3. Applications
We can apply Theorems 2.1 and 2.7 to find a minimizer of a convex function f. Let H be a real Hilbert space and f : H → (cid:5)−∞, ∞(cid:7) a proper convex lower semicontinuous function. Then the subdifferential ∂f of f is defined as follows:
(cid:8) (cid:9) ∂f(cid:5)z(cid:6) (cid:3) y − z, v∗ , y ∈ H ∀ z ∈ H. (cid:7) , (cid:6) v∗ ∈ H : f(cid:5)y(cid:6) ≥ f(cid:5)z(cid:6) (cid:4) (cid:5)3.1(cid:6)
Yeol Je Cho et al. 9
Theorem 3.1. Let H be a real Hilbert space and f : H → (cid:5)−∞, ∞(cid:7) a proper convex lower semicon- tinuous function. Suppose that, for any xn ∈ H, βn > 0, and en ∈ H, there exists xn conforming to (cid:4) , xn (cid:4) en ∈ xn (cid:4) βn∂f (cid:3) xn ∀n ≥ 0, (cid:5)3.2(cid:6)
(cid:2)∞ n(cid:3)0 (cid:7)en(cid:7)2 < (cid:4)∞. Let {αn} be a αn (cid:3) ∞. For any fixed u ∈ H, let {xn} be the (cid:2)∞ n(cid:3)0 where {βn} is a sequence in (cid:5)0, ∞(cid:6) with βn → ∞ (cid:5)n → ∞(cid:6) and sequence in (cid:8)0, 1(cid:7) such that αn → 0 (cid:5)n → ∞(cid:6) and sequence generated by
0
z∈H
u, x ∈ H, (cid:12) (cid:5) (cid:5)2 (cid:13) , (cid:5) (cid:5)z − xn − en xn (cid:3) arg min (cid:5)3.3(cid:6)
(cid:3) f(cid:5)z(cid:6) (cid:4) 1 2βn (cid:4)(cid:3) (cid:4) , (cid:3) αn u (cid:4) xn − en 1 − αn ∀n ≥ 0. xn(cid:4)1
If ∂f −10 /(cid:3) ∅, then {xn} converges strongly to the minimizer of f nearest to u.
z∈H
Proof. Since f : H → (cid:5)−∞, ∞(cid:7) is a proper convex lower semicontinuous function, by (cid:8)2(cid:7), the subdifferential ∂f of f is a maximal monotone operator. Noting that (cid:13) (cid:12) (cid:5) (cid:5)2 (cid:5) (cid:5)z − xn − en (cid:5)3.4(cid:6) xn (cid:3) arg min f(cid:5)z(cid:6) (cid:4) 1 2βn
is equivalent to
(cid:4) (cid:3) (cid:4) , (cid:3) xn xn − xn − en 0 ∈ ∂f (cid:5)3.5(cid:6) (cid:4) 1 βn
we have (cid:4) , xn (cid:4) en ∈ xn (cid:4) βn∂f (cid:3) xn ∀n ≥ 0. (cid:5)3.6(cid:6)
Therefore, using Theorem 2.1, we have the desired conclusion. This completes the proof.
Theorem 3.2. Let H be a real Hilbert space and f : H → (cid:5)−∞, ∞(cid:7) a proper convex lower semicon- tinuous function. Suppose that, for any given xn ∈ H, βn > 0, and en ∈ H, there exists xn ∈ H such that (cid:4) , xn (cid:4) en ∈ xn (cid:4) βn∂f (cid:3) xn ∀n ≥ 0, (cid:5)3.7(cid:6)
(cid:7)en(cid:7)2 < (cid:4)∞. Let {αn} be a sequence (cid:2)∞ n(cid:3)0 where {βn} is a sequence in (cid:5)0, ∞(cid:6) with limn→∞βn > 0 and in (cid:8)0, 1(cid:7) with limn→∞αn < 1 and let {xn} be the sequence generated by
0
z∈H
x ∈ H, (cid:12) (cid:13) (cid:5) (cid:5)2 , (cid:5) (cid:5)z − xn − en xn (cid:3) arg min (cid:5)3.8(cid:6)
(cid:3) f(cid:5)z(cid:6) (cid:4) 1 2βn (cid:4)(cid:3) (cid:4) , (cid:3) αn xn (cid:4) xn − en 1 − αn ∀n ≥ 0. xn(cid:4)1
If ∂f −10 /(cid:3) ∅, then {xn} converges weakly to the minimizer of f nearest to u.
10 Journal of Inequalities and Applications
Proof. As shown in the proof lines of Theorem 3.1, ∂f : H → H is a maximal monotone opera- tor, and so the conclusion of Theorem 3.2 follows from Theorem 2.7.
Acknowledgment
The authors are grateful to the anonymous referee for his helpful comments which improved the presentation of this paper.
References
