EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6837 ISSN 1307-5543 – ejpam.com Published by New York Business Global Further Study on R-Sets Operator in Acyclic Fashion Salihah Thabet Alwadani1,* 1 Mathematics, Yanbu Industrial College, The Royal Commission for Jubail and Yanbu, Yanbu, Saudi Arabia Abstract. We provide a distinct and detailed proof of the weak convergence of the acyclic Dou- glas–Rachford iteration to a point whose nearest-point projections onto each of the N convex sets coincide. Our analysis shows that the cyclic Douglas–Rachford operator is asymptotically regular, that its fixed-point set coincides with the intersection of the individual fixed-point sets when this intersection is nonempty, and that the iteration converges weakly to such a point. Special cases highlight when the method coincides with alternating projections and when it diverges from von Neumann’s scheme. 2020 Mathematics Subject Classifications: 47H09, 47H05, 47A06, 90C25 Key Words and Phrases: Nonexpansive mapping, acyclic Douglas–Rachford method, cyclic Dou- glas–Rachford operator, fixed point theory, convex analysis, projection algorithms, alternating projections, weak convergence, asymptotic regularity 1. Introduction The Douglas Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. It is also used to solve the convex feasibility problem. That is, given convex subsets C1, C2, . . . , Cm and C = ∩Ci ̸= ∅, Fid x ∈ C (1) For more information about feasibility problems, we refer the reader to [1] which pro- vides a thorough treatment of feasibility problems, especially in Hilbert spaces, which are common in signal processing, image recovery, and optimization. It dicusses projection methods, such as Douglas -Rachford algorithm and alternating projections, which are standard techniques used to solve feasibility problems involving convex sets. See also [2–4] for more details, where [2] presents projection methods and their use in solving large-scale feasibility problems, particulary in applications such as image reconstruction and medical imaging. [3] provides a unified treatment of algorithms for feasibility and ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6837 Email addresses: salihah.s.alwadani@gmail.com (S. Th. Alwadani) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 2 of 15 inverse problems where [4] focuses on feasibility problems in signal processing . It ex- plores how projection-based algrithms can be used to recover signal that satisfy multiple constraints represented as convex sets. Throughout this paper, we shall assume that X = H is a real Hilbert space with the product ⟨·, ·⟩ and induced norm ∥ · ∥ (2) In this paper, we provide a different, detailed proof to the weak convergence of Acyclic Douglas-Rachford iteration schema to a point whose nearest point projections onto each of the N sets coincide using the assumption in (2) and convex analysis. This paper is distributed as follows: Section 2 presents standard material and basic facts and collects some useful properties from convex analysis and algebra, which are useful in our later proofs. We designate all of the known results as facts with explicit references. In Section 3, we visit the Cyclic Douglas–Rachford iteration scheme that is defined in [5] and show that even with the case N = 2 the the Cyclic Douglas–Rachford iteration is dif- ferent from the Douglas–Rachford iteration see Proposition 2, Example 1, and Example 2. The main results are in Section 4, where can be summarized as follows: • We show that the Cyclic Douglas–Rachford operator T[C1C2 ...CN ] is asymptotically regular, see Section 4. • Section 4 shows that the fixed point sets of the Cyclic Douglas–Rachford opera- tor are equal to the intersection of the individual fixed point sets of the individual operators under the assumption that the intersection is not empty. • The cyclic Douglas–Rachford iteration converges weakly to a point in the fixed point sets of the Cyclic Douglas–Rachford operator, see Theorem 1 for more details. • Proposition 4 illustrates that if the initial point belongs to the first set, then the Cyclic Douglas-Rachford method coincides with the alternating projection method. Additionally, if the Cyclic Douglas–Rachford schema defined on to two closed affine subspaces C2 and C2 is equal to the averaged of TC1,C2 and TC2,C1 , see Lemma 1 for more details. • Example 3 indicates that if x0 /∈ C1, then the cyclic Douglas–Rachford iteration need not coincide with von Neumann’s alternating projection method. 2. Background Recall X = H is a real Hilbert space with the product ⟨·, ·⟩ and induced norm ∥ · ∥. The identity operator on H is denoted by Id. Let C ⊂ H is closed and convex set, the projector onto the set C is the mapping PC : H → C defined as, PC := argmin c∈C ∥x − c∥ = { z ∈ C : ∥x − z∥ = inf c∈C ∥x − c∥}, for all x ∈ H (3) S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 3 of 15 The reflector with respect to the set C is a set valued mapping PC : H → H defined as, RC := PC +(PC − Id) = 2 PC − Id, for all x ∈ H (4) Let T : H → H be an operator. Then a fixed point of T is a point x ∈ H that map a point to itself. That is, Tx = x. The set of fixed points of the operator T is denoted by Fix T, i.e., Fix T := {x ∈ H : T(x) = x} ̸= ∅, for all x ∈ H (5) Definition 1. [1, Definition 4.1] Let C be a nonempty, closed and convex subset of H. Let T : C → H then T is; (i) nonexpansive on C if it is Lipschitz continuous with constant 1, i.e., (∀x ∈ C)(∀y ∈ C) ∥Tx − Ty∥ ≤ ∥x − y∥; (6) (ii) firmly nonexpansive if (∀x ∈ C)(∀y ∈ C) ∥Tx − Ty∥2 + ∥ ( Id−T ) x − ( Id−T ) y∥ ≤ ∥x − y∥2; (7) (iii) quasinonexpansive if T is Fejér montone with respect to Fix T, i.e., (∀x ∈ C)(∀y ∈ Fix T) ∥Tx − y∥ ≤ ∥x − y∥; (8) (iv) strictly quasinonexpansive if (∀x /∈ Fix T)(∀y ∈ Fix T) ∥Tx − y∥ < ∥x − y∥; (9) (v) α- avaraged for α ∈ (0, 1), if there exisits anonexpansive operator N : C → H such that T = (1 − α) Id+αN (10) Definition 2. [6, Definition 4.8-1] A sequence (xn)n∈N in a normed space is said to be convergent (strongly convergent or convergent in the norm) if there is an x∗ ∈ H such that lim n→∞ ∥xn − x∗∥ = 0 This is written lim n→∞ xn = x∗, or simply as xn → x∗. Definition 3. [6, Definition 4.8-2] A sequence (xn)n∈N in a normed space is said to be weakly convergent if there is an x∗ ∈ H such that for every bounded linear functional f on H, lim n→∞ f (xn) = f (x∗). This is written xn ⇀ x∗. S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 4 of 15 Definition 4. Let T : H → H. We recall that T is asymptotically regular if Tnx− Tn+1x → 0, in norm, for all x ∈ H. Definition 5. [7, Fact 3.52 ]Let C1, C2, . . . , Cn be closed and convex subset of H with n⋂ i=1 Ci ̸= ∅. The Douglas-Rachford operator associated with the ordered tuple ( C1, C2, . . . , Cn ) is TC1,C2,...,Cn := 1 2 ( Id+RCn RCn−1 . . . RC2 RC1 ) . For n = 2, let C1 and C2 closed and convex subset of H with C1 ∩ C2 ̸= ∅. The Douglas-Rachford operator associated with the ordered pair ( C1, C2 ) is: TC1,C2 := 1 2 ( Id+RC2 RC1 ) , (11) and the generated sequence ( xn ) n∈N is( ∀n ∈ N ) xn+1 = TC1,C2 xn where x0 ∈ H, also called the (DRA) sequence. For more information about Douglas Rachford algorithm you can see [8] where the original theoretical foundation of the Dougals- Rachford algo- rithm for monotone operator splitting. A comprehensive analysis linking the Dougals- Rachford method to the proximal point algorithm is provided in [9]. In 2004, Bauschke, Combettes, and Luke analyze the application of Douglas-Rachford to convex feasibil- ity and best approximation problems. see [10] In 2005 Combettes and Wajs introduce proximal splitting methods that are closely related to and extend the Douglas-Rachford algorithm, see [11]. 6 years later Combettes and Pesquet applies Douglas-Rachford and related algorithms to signal processing and inverse problems, see [12]. In 2017, Bauschke and Combettes comes up with a textbook-level comprehensive treatment of the Douglas- Rachford method and its role in convex feasibility and optimization, see [1]. Proposition 1. Let C1 and C2 be a nonempty closed convex subsets of H. Then PC1 is firmly nonexpansive, RC1 is nonexpansive, N = RC2 RC1 is nonexpansive and TC1,C2 := 1 2 ( Id+RC2 RC1 ) is firmly nonexpansive. Proof. See [1, Lemma 222]. ■ Definition 6. [1, Definition 5.1] Let C be a nonempty subset of H and let (xn)n∈N be a sequence in H. Then (xn)n∈N is Fejér monotone with respect to C if( ∀x ∈ C ) ( ∀n ∈ N ) ∥xn+1 − x∥ ≤ ∥xn − x∥. [1, Proposition 5.7 ] Let (xn)n ∈ N be a sequence in H and let C be a nonempty closed convex subset of H. Suppose that (xn)n ∈ N is Fejér monotone with respect to C. Then the shadow sequence (P xn)n ∈ N converges strongly to a point in C. [1, Corollary 5.8 ] Let (xn)n ∈ N be a sequence in H, let C be a nonempty closed convex subset of H, and let x ∈ C. Suppose that (xn)n ∈ N is Fejér monotone with respect to C and that xn ⇀ x. Then PC xn → x. S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 5 of 15 3. The Cyclic Douglas- Rachford method In order to solve the feasibility problem (1), where Ci are closed and convex subsets of H with nonempty intersection, we employ the Cyclic Douglas–Rachford iteration scheme that generates a sequence (xn)n∈N by (∀n ∈ N) xn+1 = T[C1C2···CN]xn (12) and where T[C1C2...CN ] := TCN ,C1 TCN−1,CN . . . TC2,C3 TC1,C2 . (13) Proposition 2. Assume N = 2, let C1 and C2 closed and convex subset of H with C1 ∩ C2 ̸= ∅. Recall (12), (13), and (11). Then T[C1C2] ̸= TC1,C2 . Proof. Let N = 2, C1 and C2 closed and convex subset of H with C1 ∩ C2 ̸= ∅. Using (12), (13), and (11) gives T[C1C2] := TC2,C1 TC1,C2 (14) = ( Id+RC1 RC2 2 )( Id+RC2 RC1 2 ) (15) Observe that T[C1C2] ̸= TC1,C2 . ■ Example 1. Suppose that X = R2, C1 = R × {0} and C2 = {x ∈ R2 | ∥x − 3∥ ≤ 3}. Then 2 ∩ i=1 Ci ̸= ∅, and for starting point x0 ∈ ]−∞, 1[ × {1}, the DRA sequence (xn)n∈N with respect to (C1, C2) satisfies (∀n ∈ {2, 3, · · · }) xn = (0, n) and PC1 xn = (0, 0) ∈ 2 ∩ i=1 Ci. The CDRA sequence (xn)n∈N with respect to (C1, C2) will converge to (0, 0). See Fig. 1 for an illustration, created with GeoGebra [13]. Example 2. Suppose that X = R2, C1 = R · (1, 1), C2 = {0} × R and C3 = R · (1,−1). Then the 3-set Douglas-Rachford sequence (xn)n∈N with respect to (C1, C2, C3) will fail to converge to a point (0, 0) ∈ 3 ∩ i=1 Ci. The CDRA sequence (xn)n∈N with respect to (C1, C2, C3) will converge to x∗ = (0, 0) ∈ 3 ∩ i=1 Ci. See Fig. 2 for an illustration, created with GeoGebra [13]. The notation employed in this paper is standard and closely aligned with that in [14], [7], [15], and [16]. S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 6 of 15 Figure 1: An illustration for Example 1 with the starting point x0 = (−4, 1). In the left, the DRA sequence (xn)n∈N converges to x = (0, 2.4) /∈ 2 ∩ i=1 Ci. However, the shadew sequence PC1 (xn) converges to (0, 0) = 2 ∩ i=1 Ci. In the right, the CDRA sequece (xn)n∈N converges to (0, 0) = 2 ∩ i=1 Ci. 4. Main Results Let Ti : H → H be firmly nonexpansive, for each i. Recall from (13) that T[C1C2 ...CN ]: = TCN ,C1 TCN−1,CN . . . TC2,C3 TC1,C2 with Fix T[C1C2...CN ] ̸= ∅. Then T[C1C2 ...CN ] is asymptotically regular. Proof. From (13) and Proposition 1 we have TCi ,Ci+1 is firmly nonexpansive for all i. We also have, ∅ ̸= Fix T[C1C2...CN ]. Let y ∈ Fix T[C1C2 ...CN ] then; ∥Txn − Ty∥2 † ≤ ∥TCN−1,CN · · · TC2,C3 TC1,C2 xn − TCN−1,CN · · · TC2,C3 TC1,C2 y∥2 − ∥(Id−TCN ,C1)xn − (Id−TCN ,C1)y∥ 2 †By using the definition of T[C1C2···CN ] and the fact that (∀i) TCi ,Ci+1 is firmly nonexpansive. S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 7 of 15 Figure 2: An illustration for Example 2 with the starting point x0 = (0, 2). The graph in the left side describes the 3- sets DRA iterations which faill to converge to (0, 0). However, the graph in the right side describes the 3- sets CDRA iteration which converges to (0, 0). ≤ ∥TCN−2,CN−1 · · · TC2,C3 TC1,C2 xn − TCN−2,CN−1 · · · TC2,C3 TC1,C2 y∥2 − ∥(Id−TCN−1,CN )TCN ,C1 xn − (Id−TCN−1,CN )TCN ,C1 y∥2 − ∥(Id−TCN ,C1)xn − (Id−TCN ,C1)y∥ 2 ≤ ... ≤ ∥TC1,C2 xn − TC1,C2 y∥2 − ∥(Id−TC1,C2)TC2,C3 · · · TCN ,C1 xn − (Id−TC1,C2)TC2,C3 · · · TCN ,C1 y∥2 − · · · − ∥(Id−TCN−1,CN )TCN ,C1 xn − (Id−TCN−1,CN )TCN ,C1 y∥2 − ∥(Id−TCN ,C1)xn − (Id−TCN ,C1)y∥ 2 ‡ ≤ ∥xn − y∥2 − ∥(Id−TCN ,C1)xn − (Id−TCN ,C1)y∥ 2 − ∥(Id−TCN−1,CN )TCN ,C1 xn − (Id−TCN−1,CN )TCN ,C1 y∥2 − · · · − ∥(Id−TC2,C3)TC3,C4 · · · TCN ,C1 xn − (Id−TC2,C3)TC3,C4 · · · TCN ,C1 y∥2 − ∥(Id−TC1,C2)TC2,C3 · · · TCN ,C1 xn − (Id−TC1,C2)TC2,C3 · · · TCN ,C1 y∥2 (16) Therefore, (xn)n∈N is Fejér montone with respect to FixT[C1C2...CN ] and, (Id−TCN ,C1)xn − (Id−TCN ,C1)y → 0 (17) (Id−TCN−1,CN )TCN ,C1 xn − (Id−TCN−1,CN )TCN ,C1 y → 0 (18) ... (19) (Id−TC1,C2)TC2,C3 · · · TCN ,C1 xn − (Id−TC1,C2)TC2,C3 . . . TCN ,C1 y → 0 (20) Adding (17) - - (20), we obtain xn − TCN ,C1 TCN−1,CN . . . TC1,C2 xn → 0 ■ ‡Because TC1,C2 is firmly nonexpansive which means it is nonexpansive. S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 8 of 15 Let TCi ,Ci+1 :H→H be firmly nonexpansive for each i and recall from (13) that T[C1C2 ...CN ] = TCN ,C1 . . . TC2,C3 TC1,C2 . If N+1 ∩ i=1 Fix TCi ,Ci+1 ̸=∅, then Fix T[C1C2···CN ] = N+1 ∩ i=1 Fix TCi ,Ci+1 . Proof. Since TCi ,Ci+1 is firmly nonexpansive for each i, then TCi ,Ci+1 is α- avaraged with α = 1 2 for each i. Moreover, ∅ ̸= N ∩ i=1 Ci ⊆ N+1 ∩ i=1 Fix TCi ,Ci+1 The inclusion N+1 ∩ i=1 Fix TCi ,Ci+1 ⊆ Fix T[C1C2 ...CN ] is obvious. Now we show that the con- verse inclusion also holds. When N = 1 let y ∈ Fix TC1,C2 and let x ∈ Fix T[C1C2] := Fix TC2,C1 TC1,C2 . Then; ◦ If x ∈ Fix TC1,C2 ⇒ TC1,C2 x = x. Therefore, TC2,C1 x = TC2,C1 TC1,C2 x = x ∈ Fix TC1,C2 . Therefore, under this case we have Fix T[C1C2] ⊆ Fix TC1,C2 . ◦ If TC1,C2 x ∈ Fix TC2,C1 ⇒ TC1,C2 x = TC2,C1 TC1,C2 x = x ∈ Fix TC1,C2 ⇒ Fix T[C1C2] ⊆ Fix TC1,C2 . ◦ Let x /∈ Fix TC2,C1 and TC2,C1 x /∈ Fix TC1,C2 . Since TC1,C2 and TC2,C1 are firmly nonex- pansive and by [1, Corollary 2.15] they are strictly quasinonexpansive. Therefore, ∥x − y∥ = ∥TC2,C1 TC1,C2 x − y∥ < ∥TC1,C2 x − y∥ < ∥x − y∥ which is not true. There- fore, Fix TC2,C1 TC1,C2 = Fix TC1,C2 . Hypothesis induction assumption: for n ≥ 2 the result holds up to N operators. We have Fix T[C1C2···CN−1] = N ∩ i=1 Fix TCi ,Ci+1 . Then show the results hold for N + 1 operators. Let S1 = TCN−1,CN . . . TC2,C3 TC1,C2 and let S2 = TCN ,C1 because S2 is quasinonexpansive with Fix S2 = Fix TCN ,C1 and by the induction hypothesis we have, Fix S1 = N ∩ i=1 Fix TCi ,Ci+1 . Therefore, by the fact that S1S2 = TCN ,C1 TCN−1,CN · · · TC2,C3 TC1,C2 is strictly quasinonexpansive then , Fix TCN ,C1 TCN−1,CN . . . TC2,C3 TC1,C2 = Fix S1S2 = Fix S1 ∩ S2 = N+1 ∩ i=1 Fix TCi ,Ci+1 ■ Theorem 1. Let TCi ,Ci+1 :H→H be firmly nonexpansive for each i, with N+1 ∩ i=1 Fix TCi ,Ci+1 ̸=∅. Then, for any x0 ∈ H, the sequence Tn [C1C2 ...CN ] x0 ⇀ x ∈ N+1 ∩ i=1 Fix TCi ,Ci+1 . S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 9 of 15 Proof. First, show that every weak cluster point x of (xn)∈N lies in Fix T[C1C2...CN ]. By Section 4 (xn)n∈N is Fejér montone with respect to Fix T[C1C2...CN ], it is bounded. Let x be a weak sequential cluster point of (xn)n∈N. Then, there exists a subsequence xnk of xn such that xnk ⇀ x.∥∥∥x − T[C1...CN ]x ∥∥∥2 = ∥∥∥xnk − T[C1...CN ]x ∥∥∥2 − ∥xnk − x∥2 − 2 〈 xnk − x, x − T[C1 ...CN ]x 〉 = ∥∥∥xnk − T[C1...CN ]x + T[C1 ...CN ]xnk − T[C1...CN ]xnk ∥∥∥2 − ∥xnk − x∥2 − 2 〈 xnk − x, x − T[C1 ...CN ]x 〉 = ∥∥∥xnk − T[C1...CN ]xnk ∥∥∥2 + 2 〈 xnk − T[C1 ...CN ]xnk , T[C1...CN ]xnk − T[C1···CN ]x 〉 + ∥∥∥T[C1...CN ]xnk − T[C1 ...CN ]x ∥∥∥2 − ∥xnk − x∥2 − 2 〈 xnk − x, x − T[C1...CN ]x 〉 § ≤ ∥∥∥xnk − T[C1...CN ]xnk ∥∥∥2 + 2 〈 xnk − T[C1 ...CN ]xnk , T[C1...CN ]xnk − T[C1 ...CN ]x 〉 + ∥xnk − x∥2 − ∥xnk − x∥2 − 2 〈 xnk − x, x − T[C1 ...CN ]x 〉 = ∥∥∥xnk − T[C1...CN ]xnk ∥∥∥2 + 2 〈 xnk − T[C1 ...CN ]xnk , T[C1...CN ]xnk − T[C1 ...CN ]x 〉 − 2 〈 xnk − x, x − T[C1...CN ]x 〉 Note that; ∥∥∥〈xnk − T[C1...CN ]xnk , T[C1...CN ]xnk − T[C1 ...CN ]x 〉∥∥∥2 ¶ ≤ ∥∥∥xnk − T[C1 ...CN ]xnk ∥∥∥2 ∥∥∥T[C1 ...CN ]xnk − T[C1...CN ]x ∥∥∥2 || ≤ ∥∥∥xnk − T[C1 ...CN ]xnk ∥∥∥2 ∥xnk − x∥2 = ∥∥∥xnk − T[C1 ...CN ]xnk ∥∥∥2 ( ∥xnk∥2 − 2 ⟨xnk , x⟩ + ∥x∥2 ) ≤ ∥∥∥xnk − T[C1 ...CN ]xnk ∥∥∥2 ( ∥xnk∥2 − 2∥xnk∥∥x∥ + ∥x∥2 ) Therefore, sup∥xnk∥ = M < ∞. Also, xnk⇀x. Then ∥x∥ = lim | ⟨xnk , x⟩ | ≤ lim ∥xnk∥<∞. §Follows from the nonexpansiveness of T[C1 ...CN ]. ¶Follows from Cauchy-Schwarz inequality | ⟨x, y⟩ |≤ ∥x∥∥y∥. ∥Follows from the nonexpansiveness of T[C1 ...CN ]. S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 10 of 15 Therefore, ( ∥xnk∥2 − 2∥xnk∥∥x∥ + ∥x∥2 ) < ∞. Hence, taking the limit as n → ∞, we have ∥xnk − T[C1...CN ]xnk∥2 ( ∥xnk∥2 − 2∥xnk∥∥x∥+ ∥x∥2 ) → 0 Moreover, xnk − T[C1...CN ]xnk → 0 because T[C1...CN is asympotically regular by Section 4. Also, by Definition 3, 〈 xnk − x, x − T[C1 ...CN ]x 〉 = 〈 xnk , x − T[C1 ...CN ]x 〉 − 〈 x, x − T[C1...CN ]x 〉 = 0. Therefore, x − T[C1 ...CN ]x = 0 Next, show that (xn)n∈N cannot have two distinct weak sequential cluster points in Fix T[C1 ...CN ]. Let x and y be weak sequential cluster points of (xn)n∈N ∈ Fix T[C1...CN ], say xnk ⇀ x and xnl ⇀ y Then, by monotonicity the sequences (∥xn − x∥2)n∈N and (∥xn − y∥2)n∈N converge. Since ∥x − y∥2 = ∥xn − y∥2 − ∥xn − x∥2 − 2 ⟨xn − x, x − y⟩ ⇒ ∥x − y∥2 + 2 ⟨xn, x − y⟩ = ∥xn − y∥2 − ∥xn − x∥2 + 2 ⟨xn, x − y⟩ − 2 ⟨xn − x, x − y⟩ = ∥xn − y∥2 − ∥xn − x∥2 + 2 ⟨x, x − y⟩ 2 ⟨xn, x − y⟩ = ∥xn − y∥2 − ∥xn − x∥2 + 2 ⟨x, x − y⟩ − ∥x − y∥2 = ∥xn − y∥2 − ∥xn − x∥2 + 2 ⟨x, x − y⟩ − ⟨x − y, x − y⟩ = ∥xn − y∥2 − ∥xn − x∥2 + ∥x∥2 − ∥y∥2 (∀n ∈ N) 2 ⟨xn, x − y⟩ = ∥xn − y∥2 − ∥xn − x∥2 + ∥x∥2 − ∥y∥2 (21) (⟨xn, x − y⟩)n∈N converges as well. Let ⟨xn, x − y⟩ → m. Taking the limit along (xnk) and (xnl ) repectively, we have m = ⟨x, x − y⟩ = ⟨y, x − y⟩ Therefore, ∥x − y∥2 = 0. Hence, Tn [C1C2 ...CN ] x0 ⇀ x ∈ Fix T[C1...CN ] and from Section 4 Tn [C1C2 ...CN ] x0 ⇀ x ∈ N+1 ∩ i=1 Fix TCi ,Ci+1 . Finally, by using Section 2 and Section 2, we get that shadow sequence (PFix T[C1...CN ] )n∈N converges strongly to a point in x ∈ Fix T[C1...CN ] = N+1 ∩ i=1 Fix TCi ,Ci+1 . ■ S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 11 of 15 Proposition 3. Let C1, C2 · · ·CN ⊆ H be closed and convex set with non empty intersec- tion. Recall from Definition 5 that TC1,C2,...,Cn := 1 2 ( Id+RCn RCn−1 . . . RC2 RC1 ) . If x ∈ Ci, then TCi ,Ci+1 x = PCi+1 x. (22) Proof. Let x ∈ Ci, then TCi ,Ci+1 x = 2−1(x + RCi+1 RCi x ) ** = 2−1(x + RCi+1 x ) = 2−1(x + 2 PCi+1 x − x ) by (4) = PCi+1 x, as required. ■ Lemma 1. Let C1 and C2 be two closed affine subspaces. Recall from (13) that T[C1C2...CN ] := TCN ,C1 TCN−1,CN . . . TC2,C3 TC1,C2 . Then T[C1C2] = 2−1(TC1,C2 + TC2,C1 ) . Proof. Using (13) with N = 2 and (11) give T[C1C2] = TC2,C1 TC1,C2 = 2−1( Id+RC1 RC2 ) TC1,C2 = 2−1(TC1,C2 + RC1 RC2 TC1,C2 ) = 2−1 ( TC1,C2 + RC1 RC2 ( 2−1( Id+RC2 RC1 ))) = 2−1 ( TC1,C2 + RC1 ( 2−1(RC2 + RC2 RC2 RC1 ))) †† = 2−1 ( TC1,C2 + 2−1(RC1 RC2 + RC1 RC1 )) = 2−1 ( TC1,C2 + 2−1(RC1 RC2 + Id )) = 2−1(TC1,C2 + TC2,C1 ) . ■ Proposition 4. When x0 ∈ C1, the cyclic Douglas- Rachford method coincides with alter- nating projection method. ∗∗By assumption that x ∈ Ci ††By using the fact that C2 is closed an affine therefore R2 C2 = Id. Similarly for C1. S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 12 of 15 Proof. Let x0 ∈ C1, then by (13) we have, T[C1C2...CN−1CN ]x0 = TCN ,C1 TCN−1,CN . . . TC1,C2 x0 Note: TC1,C2 x0 = 2−1 ( x0 + RC2(RC1 x0) ) = 2−1 ( x0 + RC2 x0 ) = 2−1 ( x0 + 2PC2 x0 − x0 ) = PC2 x0 ∈ C2 TC2,C3(PC2 x0) = 1 2 ( PC2 x0 + RC3(RC2(PC2 x0)) ) = 1 2 ( PC2 x0 + RC3(PC2 x0) ) = 1 2 ( PC2 x0 + 2PC3(PC2 x0)− PC2 x0 ) = PC3 PC2 x0 ∈ C3 Keep doing that we have TCN ,C1 TCN−1,CN . . . TC2,C3 PC2 x0 (22) = PC1 PN . . . PC3 P2 x0 ∈ C1 ■ The next example indicates that if x0 /∈ C1, then the cyclic Douglas–Rachford iteration need not coincide with von Neumann’s alternating projection method. Example 3. Let C1 = {x ∈ H | ⟨a, x⟩ ≤ 0}, and C2 = {x ∈ H | ⟨a, x⟩ = 0}, where a ∈ H and ∥a∥ = 1. If x0 /∈ C1 ∪ C2, then 〈 a, T[C1C2]x 〉 ̸= 0. Proof. The projection to C1 and C2, see [1, Example 28.15 and Example 28.16], are PC1 x = { x − ⟨a, x⟩ a if ⟨a, x⟩ > 0 x if ⟨a, x⟩ ≤ 0 and , PC2 x = x − ⟨a, x⟩ a TC1,C2 = 2−1(x + RC2 RC1 x ) . (23) RC1 x = { x − 2 ⟨a, x⟩ a if ⟨a, x⟩ > 0 x if ⟨a, x⟩ ≤ 0 S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 13 of 15 and, RC2(RC1 x) = { x if ⟨a, x⟩ > 0 x − 2 ⟨a, x⟩ a if ⟨a, x⟩ ≤ 0 Then plugging this result in (23) we get that TC1,C2 x = { x if ⟨a, x⟩ > 0 x − ⟨a, x⟩ a if ⟨a, x⟩ ≤ 0 Similarly, TC2,C1 x = { x if ⟨a, x⟩ > 0 x − ⟨a, x⟩ a if ⟨a, x⟩ ≤ 0 Then, by Lemma 1 we have, 2 〈 a, T[C1C2]x 〉 = 2 〈 a, 2−1(TC1,C2 + TC2,C1)x 〉 = ⟨a, TC1,C2 x⟩+ ⟨a, TC2,C1 x⟩ (24) When ⟨a, x⟩ > 0; 〈 a, T[C1C2] 〉 (24) = ⟨a, x⟩+ ⟨a, x⟩ = 2 ⟨a, x⟩ . (25) When ⟨a, x⟩ ≤ 0 ;〈 a, T[C1C2] 〉 (24) = ⟨a, x⟩ − ⟨a, x⟩ ∥a∥2 + ⟨a, x⟩ − ⟨a, x⟩ ∥a∥2 = 0. (26) Therefore (25) and (26) show that if x0 /∈ C1 ∪ C2 then the Douglas- Rachford iterates will not lie in C1 or C2. Hence, if ⟨a, x⟩ ̸≤ 0, then 〈 a, T[C1C2] 〉 ̸= 0. ■ 4.1. A product version of the Cyclic Douglas–Rachford method Consider the Hilbert space HN = H×H× · · · × H. Define two closed and convex subsets C and D of HN , where C ∩ D ̸= ∅, by C := {(x1, x2, . . . , xn) ∈ HN | xi ∈ Ci}, and D := {(x, x, . . . , x) ∈ HN | x ∈ H}. (1) will be solved by find x ∈ (C ∩ D) ⊆ HN . The projection on C and D will be PC = (PC1 x1, PC2 x2, . . . , PCN xN ), and PD = ( 1 N N ∑ i=1 xi, 1 N N ∑ i=1 xi, . . . , 1 N N ∑ i=1 xi). See [1, Proposition 25.4(iii) and (iv)] for more details. S. Th. Alwadani / Eur. J. Pure Appl. Math, 18 (4) (2025), 6837 14 of 15 Lemma 2. The iteration for the product version of the cyclic Douglas-Rachford method will be define as T[D C]x = x − PD x + 2 PD PC TD,Cx − PC TD,Cx + PC RDx − PD PC RDx. (27) Proof. Then RDRCTDCx = (2 PD − Id)RCTDCx = 2 PD RCTDCx − RCTD,Cx = 2 PD(2 PC − Id)TDCx − RCTDCx = 4 PD PC TDCx − 2 PD TDCx − RCTDCx = 4 PD PC TDCx − 2 PD TDCx − (2 PC − Id)TDCx = 4 PD PC TDCx − 2 PD TDCx − 2 PC TDCx + TDCx (28) Using the result from (28) in (??), we have T[D C]x = TD,Cx + 2 PD PC TD,Cx − PD TD,Cx − PC TD,Cx (29) However, TDCx = 2−1(x + RCRDx ) (30) = 2−1(x + (2 PC − Id)RDx ) (31) = 2−1(x + 2 PC RDx − RDx ) (32) = x − PD x − PC RDx. (33) Moreover, −PD TDCx = −2−1 PD x − 2−1 PD(RCRDx) = −2−1 PD x − 2−1 PD ( (2 PC − Id)RDx ) = −2−1 PD x − PD PC RDx + 2−1 PD RDx = −2−1 PD x − PD PC RDx + 2−1 PD(2 PD − Id) = −2−1 PD x − PD PC RDx + PD x − 2−1 PD x = −PD PC RDx (34) By (33) and (34), the updated formula for equation (29) will be T[D C]x = x − PD x + 2 PD PC TD,Cx − PC TD,Cx + PC RDx − PD PC RDx. ■ 5. Clarification There is no conflict of interest and there is no data were used to support this study. 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Introduction Background The Cyclic Douglas- Rachford method Main Results A product version of the Cyclic Douglas–Rachford method Clarification