A Proximal Point Method Involving Two Resolvent Operators

and Applied Analysis 3 Since weak convergence is not good for an effective algorithm, our purpose in this paper is to modify algorithm 1.3 in such a way that strong convergence is guaranteed. More precisely, for any two maximal monotone operators A and B, we define an iterative process in the following way: For x0, u ∈ H given, a sequence xn is generated using the rule x2n 1 αnu δnx2n γnJ βnx2n en for n 0, 1, . . . , 1.6 x2n J μn ( λnu 1 − λn x2n−1 e′ n ) for n 1, 2, . . . , 1.7 where αn, δn, γn, λn ∈ 0, 1 with αn δn γn 1 and βn, μn ∈ 0,∞ . We will also show that algorithm 1.6 , 1.7 contains several algorithms such as the prox-Tikhonov method, the Halpern-type proximal point algorithm, and the regularized proximal method as special cases. That is, with our algorithm, we are able to put several algorithms under one frame work. Therefore, our main results improve, generalize, and unify many related results announced recently in the literature. 2. Preliminary Results In the sequel, H will be a real Hilbert space with inner product 〈·, ·〉 and induced norm ‖ · ‖. We recall that a map T : H → H is called nonexpansive if for every x, y ∈ H we have ‖Tx − Ty‖ ≤ ‖x − y‖. We say that a map T is firmly nonexpansive if for every x, y ∈ H, we have ∥Tx − Ty∥2 ≤ ∥x − y∥2 − ∥ I − T x − I − T y∥2. 2.1 It is clear that firmly nonexpansive mappings are also nonexpansive. The converse need not be true. The excellent book by Goebel and Reich 9 is recommended to the reader who is interested in studying properties of firmly nonexpansive mappings. An operator A : D A ⊂ H → 2 is said to be monotone if 〈 x − x′, y − y′ ≥ 0, ∀x, y, x′, y′ ∈ G A , 2.2 whereG A { x, y ∈ H×H : x ∈ D A , y ∈ Ax} is the graph ofA. In other words, an operator is monotone if its graph is a monotone subset of the product spaceH ×H. An operatorA is called maximal monotone if in addition to being monotone, its graph is not properly contained in the graph of any other monotone operator. Note that ifA is maximal monotone, then so is its inverseA−1. For a maximal monotone operatorA, the resolvent ofA, defined by J β : I βA −1, is well defined on the whole spaceH, is single-valued, and is firmly nonexpansive for every β > 0. It is known that the Yosida approximation of A, an operator defined by Aβ : β−1 I − J β where I is the identity operator is maximal monotone for every β > 0. For the properties of maximal monotone operators discussed above, we refer the reader to 10 . Notations. given a sequence xn , we will use xn → x to mean that xn converges strongly to x whereas xn ⇀ x will mean that xn converges weakly to x. The weak ω-limit set of a sequence xn will be denoted by ωw xn . That is, ωw xn { x ∈ H : xnk ⇀ x for some subsequence xnk of xn } . 2.3 4 Abstract and Applied Analysis The following lemmas will be useful in proving our main results. The first lemma is a basic property of norms in Hilbert spaces. Lemma 2.1. For all x, y ∈ H, one has ∥ ∥x y ∥ ∥2 ≤ ∥y∥2 2x, x y. 2.4 The next lemma is well known, it can be found for example in 10, page 20 . Lemma 2.2. Any maximal monotone operator A : D A ⊂ H → 2 satisfies the demiclosedness principle. In other words, given any two sequences xn and yn satisfying xn → x and yn ⇀ y with xn, yn ∈ G A , then x, y ∈ G A . Lemma 2.3 Xu 11 . For any x ∈ H and μ ≥ β > 0, ∥∥x − J β x ∥∥ ≤ 2 ∥∥x − J μ x ∥∥, 2.5 where A : D A ⊂ H → 2 is a maximal monotone operator. We end this section with the following key lemmas. Lemma 2.4 Boikanyo and Moroşanu 12 . Let sn be a sequence of nonnegative real numbers satisfying sn 1 ≤ 1 − αn 1 − λn sn αnbn λncn dn, n ≥ 0, 2.6 where αn , λn , bn , cn , and dn satisfy the conditions: i αn, λn ∈ 0, 1 , with ∏∞ n 0 1−αn 0, ii lim supn→∞bn ≤ 0, iii lim supn→∞cn ≤ 0, and iv dn ≥ 0 for all n ≥ 0with ∑∞ n 0 dn < ∞. Then limn→∞sn 0. Remark 2.5. Note that if limn→∞αn 0, then ∏∞ n 0 1 − αn 0 if and only if ∑∞ n 0 αn ∞. Lemma 2.6 Maingé 13 . Let sn be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence snj of sn such that snj < snj 1 for all j ≥ 0. Define an integer sequence τ n n≥n0 as τ n max{n0 ≤ k ≤ n : sk < sk 1}. 2.7 Then τ n → ∞ as n → ∞ and for all n ≥ n0, max { sτ n , sn } ≤ sτ n 1. 2.8 Abstract and Applied Analysis 5 3. Main Results Wefirst begin by giving a strong convergence result associatedwith the exact iterative processand Applied Analysis 5 3. Main Results Wefirst begin by giving a strong convergence result associatedwith the exact iterative process v2n 1 αnu δnv2n γnJ βnv2n for n 0, 1, . . . , 3.1 v2n J μn λnu 1 − λn v2n−1 for n 1, 2, . . . , 3.2 where αn, δn, γn, λn ∈ 0, 1 with αn δn γn 1, βn, μn ∈ 0,∞ and v0, u ∈ H are given. The proof of the following theorem makes use of some ideas of the papers 12–15 . Theorem 3.1. LetA : D A ⊂ H → 2 and B : D B ⊂ H → 2 be maximal monotone operators with A−1 0 ∩ B−1 0 : F / ∅. For arbitrary but fixed vectors v0, u ∈ H, let vn be the sequence generated by 3.1 , 3.2 , where αn, δn, γn, λn ∈ 0, 1 with αn δn γn 1, and βn, μn ∈ 0,∞ . Assume that i limn→∞ αn 0, γn ≥ γ for some γ > 0 and limn→∞ λn 0, ii either ∑∞ n 0 αn ∞ or ∑∞ n 0 λn ∞, and iii βn ≥ β and μn ≥ μ for some β, μ > 0. Then vn converges strongly to the point of F nearest to u. Proof. Let p ∈ F. Then from 3.2 and the fact that the resolvent operator of a maximal monotone operator B is nonexpansive, we have ∥v2n − p ∥ ≤ ∥λn ( u − p 1 − λn ( v2n−1 − p )∥ ≤ λn ∥u − p∥ 1 − λn ∥v2n−1 − p ∥. 3.3 Again using the fact that the resolvent operator J βn : H → H is nonexpansive, we have from 3.1 ∥v2n 1 − p ∥ ≤ αn ∥u − p∥ δn ∥v2n − p ∥ γn ∥∥∥JA βnv2n − p ∥∥∥ ≤ αn ∥u − p∥ 1 − αn ∥v2n − p ∥ ≤ αn 1 − αn λn ∥u − p∥ 1 − αn 1 − λn ∥v2n−1 − p ∥ 1 − 1 − αn 1 − λn ∥u − p∥ 1 − αn 1 − λn ∥v2n−1 − p ∥, 3.4 where the last inequality follows from 3.3 . Using a simple induction argument, we get ∥v2n 1 − p ∥ ≤ [ 1 − n ∏ k 1 1 − αk 1 − λk ] ∥u − p∥ ∥v1 − p ∥ n ∏ k 1 1 − αk 1 − λk . 3.5 This shows that the subsequence v2n 1 of vn is bounded. In view of 3.3 , the subsequence v2n is also bounded. Hence the sequence vn must be bounded. Now from the firmly nonexpansive property of J βn : H → H, we have for any p ∈ F ∥∥JA βnv2n − p ∥∥ 2 ≤ ∥v2n − p ∥∥2 − ∥∥∥v2n − J βnv2n ∥∥∥ 2 , 3.6 6 Abstract and Applied Analysis which in turn gives 2 〈 v2n − p, J βnv2n − p 〉 ∥ ∥v2n − p ∥ ∥2 ∥ ∥ ∥J βnv2n − p ∥ ∥ ∥ 2 − ∥ ∥ ∥v2n − J βnv2n ∥ ∥ ∥ 2 ≤ 2 (∥ ∥v2n − p ∥ ∥2 − ∥ ∥ ∥v2n − J βnv2n ∥ ∥ ∥ 2 ) . 3.7 Again by using the firmly nonexpansive property of the resolvent J βn : H → H, we see that ∥ ∥ ∥δn ( v2n − p ) γn ( J βnv2n − p )∥ ∥ 2 δ2 n ∥v2n − p ∥∥2 γ2 n ∥∥∥JA βnv2n − p ∥∥∥ 2 2γnδn 〈 v2n − p, J βnv2n − p 〉 ≤ 1 − αn 2 ∥v2n − p ∥2 − γn ( γn 2δn ∥∥∥v2n − J βnv2n ∥∥ 2 . 3.8 Now from 3.1 and Lemma 2.1, we have ∥v2n 1 − p ∥2 ≤ ∥∥∥δn ( v2n − p ) γn ( J βnv2n − p ∥∥∥ 2 2αn 〈 u − p, v2n 1 − p 〉 ≤ 1 − αn ∥v2n − p ∥2 − ε ∥∥∥v2n − J βnv2n ∥∥ 2 2αn 〈 u − p, v2n 1 − p 〉 , 3.9 where ε > 0 is such that γn γn 2δn ≥ ε. On the other hand, we observe that 3.2 is equivalent to v2n − p μnBv2n λn ( u − p 1 − λn ( v2n−1 − p ) . 3.10 Multiplying this inclusion scalarly by 2 v2n − p and using the monotonicity of B, we obtain 2 ∥v2n − p ∥2 ≤ 2λn 〈 u − p, v2n − p 〉 2 1 − λn 〈 v2n−1 − p, v2n − p 〉 1 − λn [∥v2n−1 − p ∥2 ∥v2n − p ∥2 − ‖v2n − v2n−1‖ ] 2λn 〈 u − p, v2n − p 〉 , 3.11