PHƯƠNG PHÁP CHIẾU THU HẸP GIẢI BÀI TOÁN ĐIỂM BẤT ĐỘNG CHUNG TÁCH TRONG KHÔNG GIAN HILBERT

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A SHRINKING PROJECTION METHOD FOR SOLVING THE SPLIT COMMON FIXED POINT PROBLEM IN HILBERT SPACES.. Mai Thi Ngoc Ha University of Agriculture and Forestry – TNU.[r] (1) ISSN: 1859-2171 e-ISSN: 2615-9562 TNU Journal of Science and Technology 203(10): 31 - 35 http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn 31 A SHRINKING PROJECTION METHOD FOR SOLVING THE SPLIT COMMON FIXED POINT PROBLEM IN HILBERT SPACES Mai Thi Ngoc Ha University of Agriculture and Forestry – TNU ABSTRACT We study the split common fixed point problem in two Hilbert spaes Let H1 and H2 be two real Hilbert spaces Let S1 : H1 → H1, and S2 : H2 → H2, be two nonexpansive mappings on H1 and H2, respectively Consider the following problem: find an element x† ∈ H1 such that x† ∈ Ω := Fix(S1) ∩ T−1( Fix(S2)) ≠ ∅, where T : H1 → H2 is a given bounded linear operator from H1 to H2 Using the shrinking projection method, we propose a new algorithm for solving this problem and establish a strong convergence theorem for that algorithm Key words: Hilbert space, metric projection, monotone operator, nonexpansive mapping, split common fixed point problem Received: 12/6/2019; Revised: 22/8/2019; Published: 30/8/2019 PHƯƠNG PHÁP CHIẾU THU HẸP GIẢI BÀI TOÁN ĐIỂM BẤT ĐỘNG CHUNG TÁCH TRONG KHÔNG GIAN HILBERT Mai Thị Ngọc Hà Trường Đại học Nôg Lâm – ĐH Thái Nguyên TĨM TẮT Trong báo này, chúng tơi nghiên cứu toán điểm bất động chung tách không gian Hilbert Cho H1 H2 hai không gian Hilbert thực Cho S1: H1 H1, S2: H2 H2, hai ánh xạ không giãn khơng gian H1 H2 tương ứng Bài tốn đặt là: tìm phần tử x† ∈ H1 cho: x† ∈ Ω := Fix(S1) ∩ T−1( Fix(S2)) ≠ ∅, Khi T : H1 H2 ánh xạ tuyến tính bị chặn cho trước từ H1 vào H2 Sử dụng phương pháp chiếu thu hẹp, chúng tơi đề xuất thuật tốn (Thuật toán 3.1) để giải toán thiết lập định lý hội thụ mạnh cho thuật toán (Định lý 3.3) Từ khóa: Khơng gian Hilbert, phép chiếu metric, tốn tử đơn điệu, ánh xạ khơng giãn, toán điểm bất động chung tách Ngày nhận bài: 12/6/2019;Ngày hoàn thiện: 22/8/2019; Ngày đăng: 30/8/2019 Email: maithingocha@tuaf.edu.vn (2)1 Introduction Let K and Q be nonempty, closed and convex subsets of two real Hilbert spaces H1 and H2, respectively Let T : H1 −→ H2 be a bounded linear operator and let T∗ : H2 −→ H1 be its adjoint The split convex feasibility problem (SCFP) is formulated as follows: Find an element x∗ ∈ K such that T x∗ ∈ Q (1.1) The SCFP was first introduced by Y Cen-sor and T Elfving [1] for modeling certain inverse problems It plays an important role in medical image reconstruction and in sig-nal processing (see [2, 3]) Several iterative algorithms for solving (1.1) were presented and analyzed in [2–14], and in references therein It is known that the SCFP is a special case of the split common fixed point problem (SCFPP), which is formulated as follows Let S1 : H1 −→ H1 and S2 : H2 −→ H2 be two nonexpansive mappings and let T : H1 −→ H2 be a bounded linear operator such that Ω = Fix(S1) ∩ T−1(Fix(S2)) 6= ∅ The SCFPP is to find an element x∗ ∈ Ω. In this paper, by combining the prox-imal point algorithm with the shrinking projection method, we introduce and an-alyze a new iterative method for solving the SCFPP in Hilbert spaces Using these methods, we also remove the assumptions imposed on the norm kT k (see Section below) 2 Preliminaries Let C be a nonempty, closed and convex subset of a real Hilbert space H It is well known that for each x ∈ H, there is unique point PH C x∈ C such that (2.1) kx − PH C xk = inf u∈Ckx − uk The mapping PH C : H −→ C defined by (2.1) is called the metric projection of H onto C Moreover, we have (see, for exam-ple, Section in [15]) (2.2) hx − PCHx, y− P H Cxi ≤ ∀x ∈ H, y ∈ C Recall that a mapping T : C −→ C is said to be nonexpansive if kT x − T yk ≤ kx − yk for all x, y ∈ C We denote the set of fixed points of T by Fix(T ), that is, Fix(T ) :=x ∈ C : T x = x The following lemma is used in the se-quel in the proofs of the main result of this paper From (2.2), we have the following Lemma Lemma 2.1 Let H be a real Hilbert space and let C be a nonempty, closed and convex subset of H Then for all x∈ H and y ∈ C, we have kx − PCHxk + ky − PCHxk ≤ kx − yk2 3 Main results Let H1and H2be two real Hilbert spaces Let S1 : H1 −→ H1, and S2 : H2 −→ H2, be two nonexpansive mappings on H1 and H2, respectively Consider the following problem: find an element x†∈ H 1 such that (3.1) x†∈ Ω := Fix(S1) ∩ T−1(Fix(S2)) 6= ∅, where T : H1 −→ H2 is a given bounded linear operator from H1 to H2 Using the shrinking projection method, we introduce in this section a new algorithm for solving Problem (3.1) Algorithm 3.1 For any initial guess x0= x∈ H1, C0 = D0 = H1, define the sequence {xn} by yn= S1(xn), zn= S2(T yn), (3)Dn+1=z ∈ Dn : kzn− T zk ≤ kT yn− T zk, xn+1= PCHn+11 ∩Dn+1x0, n≥ The following theorem yields the strong convergence of the sequence generated by Algorithm 3.1 Theorem 3.1 The sequence {xn} gener-ated by Algorithm 3.1 converges strongly to PH1 Ω x0 Proof We divide the proof of this theorem into four steps Step The sequence {xn} is well defined First, we claim that Cnand Dnare closed and convex subsets of H1 for all n ≥ To see this, we rewrite, for each integer n ≥ 0, the subsets Cn+1and Dn+1in the following forms: Cn+1 = Cn∩ n z ∈ H1 : hxn − yn, zi ≤ 1 2(kxnk 2− ky nk2) o , Dn+1 = Dn∩ n z ∈ H1 : hT yn− zn, T zi ≤ 1 2(kT ynk 2− kz nk2) o , = Dn ∩ n z ∈ H1 : hT∗(T yn − zn), zi ≤ 1 2(kT ynk 2− kz nk2) o , respectively Now, using induction and the fact that C0 = D0 = H1, we see that Cn and Dn are indeed closed and convex subsets of H1 for all n ≥ 0, as claimed Next, we show that Ω ⊂ Cn∩ Dn for all n≥ It is clear that Ω ⊂ C0∩ D0 = H1 Suppose that Ω ⊂ Cn∩ Dn for some n ≥ Taking any point p ∈ Ω, we have S1(p) = p and S2(T p) = T p Therefore, the nonex-pansivity of S1 and S2 implies that kyn− pk = kS1(xn) − S1(p)k ≤ kxn− pk kzn− T pk = kS2(T yn) − S2(T p)k ≤ kT yn− T pk Hence the definitions of Cn+1, Dn+1 and the fact that Ω ⊂ Cn ∩ Dn imply that Ω ⊂ Cn+1∩ Dn+1 Hence, by induction, we obtain that Ω ⊂ Cn∩ Dn for all n ≥ and hence that Cn∩ Dn is a nonempty, closed and convex subset of H1 for each integers n≥ This implies that the sequence {xn} is indeed well defined, as asserted Step kxn+1− xnk → as n → ∞ We first show that the sequence {xn} is bounded Indeed, let x† = P Ωx0 It follows from the fact that Ω ⊂ Cn∩ Dn, x† ∈ Cn∩ Dn for all n ≥ Thus, using xn= PCn∩Dnx0, we obtain that kx0− xnk ≤ kx0− x†k for all n ≥ (3.2) Hence the sequence {xn} is bounded Next, using xn+1 = PCn+1∩Dn+1x0 ∈ Cn∩ Dn, xn = PCn∩Dnx0 and Lemma 2.1, we obtain that kxn− x0k2 ≤ kxn+1− x0k2− kxn+1− xnk2 ≤ kxn+1− x0k2 This implies that the sequencekxn− x0k is increasing The boundedness of {xn} now implies that the limit of {kxn− x0k} exists and is finite Next, we show that sequence {xn} con-verges strongly to some point p ∈ H1 In-deed, for all m ≥ n, we have Cm ∩ Dm ⊂ Cn∩ Dn Thus, xm ∈ Cn∩ Dn By Lemma 2.1, we have kxm− xnk2≤ kxm− x0k 2 − kxn− x0k 2 → as m, n→ ∞ So, {xn} is Cauchy sequence Hence there exists the limit limn→∞xn = q Thus we have kxn+1− xnk ≤ kxn+1− qk + kxn− qk → 0, which implies that kxn+1 − xnk → as n→ ∞, as claimed Step kxn−ynk → and kzn−T ynk → as n → ∞ From xn+1 = P H1 Cn∩Dnx0 ∈ Cn and the definition of Cn, we have (4)So, from limn→∞kxn+1− xnk = 0, we ob-tain that kxn+1− ynk → (3.3) Since kxn− ynk ≤ kxn+1− ynk + kxn+1− xnk, it follows that kxn− ynk → (3.4) From xn+1 = PCn∩Dnx0 ∈ Dn and the definition of Dn we have kzn− T xn+1k ≤ kT yn− T xn+1k ≤ kT kkxn+1− ynk It now follows from (3.3) that kzn− T xn+1k → (3.5) So, using (3.3) and the estimate kzn− T ynk ≤ kzn− T xn+1k + kT xn+1− T ynk ≤ kzn− T xn+1k + kT kkxn+1− ynk, we obtain kzn− T ynk → (3.6) Step xn→ x†= PΩx0 as n → ∞ Since xn → q and T is bounded linear operator, T xn → T q It follows from (3.4), (3.6), the continuity S1 and S2 that q ∈ Ω Letting n → ∞ in (3.2), we get that kx0− pk ≤ kx0− x†k and the uniqueness of x†yields the equality p= x†. This completes the proof  The following result which concerns find-ing a fixed point of a nonexpansive mappfind-ing in a real Hilbert space Corollary 3.2 Let H be a real Hilbert space and let S : H −→ H be a nonex-pansive mapping such that Ω = Fix(S) 6= ∅ Then the sequence {xn} is generated by x0= x ∈ H, C0= H1 and yn= S(xn), Cn+1 =z ∈ Cn: kyn− zk ≤ kxn− zk, xn+1= PCn+1x0, n≥ 0, converges strongly to x†= PH1 Ω x0 Proof We obtain this result by applying Theorem 3.1 with H1 = H2 = H, S = S1, S2 = IH and T = IH, the identity op-erator on H  We now have the following result for solv-ing the SCFP in Hilbert sapces Let H1and H2be two real Hilbert spaces, and let K and Q, be two closed and convex subsets of H1 and H2, respectively Letting T : H1 −→ H2 be a bounded linear opera-tor such that Ω = K ∩ T−1(Q) 6= ∅, we now consider the following problem: Find an element x†∈ Ω (3.7) Using Theorem 3.1, we obtain the follow-ing result concernfollow-ing Problem (3.7) Theorem 3.3 The sequence {xn} gener-ated by x0∈ H1, C0= D0 = H1 and yn = PKH1xn, zn= PQH2T yn, Cn+1=z ∈ Cn: kyn− zk ≤ kxn− zk, Dn+1=z ∈ Dn: kzn− T zk ≤ kT yn− T zk, xn+1= PCHn1+1∩Dn+1x0, n≥ 0, converges strongly to x†= PH1 Ω x0 References [1] Y Censor, T Elfving, A multi projection al-gorithm using Bregman projections in a prod-uct space, Numer Algorithms, 8, pp 221–239 (1994) [2] C Byrne, Iterative oblique projection onto con-vex sets and the split feasibility problem, In-verse Problems, 18, pp 441–453, 2002 [3] C Byrne, A unified treatment of some itera-tive algorithms in signal processing and image reconstruction, Inverse Problems, 18, pp 103– 120 (2004) (5)[5] Y Censor, T Elfving, N Kopf, T Bortfeld, The multiple-sets split feasibility problem and its application, Inverse Problems, 21, pp 2071– 2084 (2005) [6] Y Censor, A Gibali, S Reich, Algorithms for the split variational inequality problems, Nu-mer Algorithms, 59, pp 301–323 (2012) [7] V Dadashi, Shrinking projection algorithms for the split common null point problem, Bull Aust Math Soc., 99, pp 299–306 (2017) [8] S Takahashi, W Takahashi, The split common null point problem and the shrinking projection method in Banach spaces, Optimization, 65, pp 281–287 (2016) [9] W Takahashi, The split feasibility problem and the shrinking projection method in Banach spaces, J Nonlinear Convex Anal., 16, pp 1449–1459 (2015) [10] W Takahashi, The split common null point problem in Banach spaces, Arch Math., 104, pp 357–365 (2015) [11] F Wang, H.-K Xu, Cyclic algorithms for split feasibility problems in Hilbert spaces, Nonlinear Anal., 74, pp 4105–4111 (2011) [12] H.-K Xu, A variable Krasnosel’skii-Mann al-gorithm and the multiple-set split feasibility problem, Inverse Problems, 22, pp 2021–2034 (2006) [13] H.-K Xu, Iterative methods for the split fea-sibility problem in infinite dimensional Hilbert spaces, Inverse Problems, 26, 105018 (2010) [14] Q Yang, The relaxed CQ algorithm for solving the split feasibility problem, Inverse Problems, 20, pp 1261–1266 (2004)
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