**Authors**
Donal O’Regan
Ebrahim Soori
Ravi P. Agarwal

**License**
CC-BY-4.0

Soori et al. Journal of Inequalities and Applications (2022) 2022:165 https://doi.org/10.1186/s13660-022-02904-y RESEARCH Open Access A simple proof for Imnang’s algorithms Ebrahim Soori1* , Donal O’Regan2 and Ravi P. Agarwal3 * Correspondence: sori.e@lu.ac.ir 1 Department of Mathematics, Abstract Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran In this paper, a simple proof of the convergence of the recent iterative algorithm by Full list of author information is relaxed (u, v)-cocoercive mappings due to Imnang (J. Inequal. Appl. 2013:249, 2013) is available at the end of the article presented. MSC: 47H09; 47H10 Keywords: Relaxed (u, v)-cocoercive mapping; Strong convergence; α -expansive mapping 1 Introduction and preliminaries In this paper, a simple proof for the convergence of an iterative algorithm is presented that improves and reﬁnes the original proof. Suppose that C is a nonempty closed convex subset of a real normed linear space E and E∗ is its dual space. Suppose that ., . denotes the pairing between E and E∗ . The normalized duality mapping J : E → E∗ is deﬁned by J(x) = f ∈ E∗ : x, f = x2 = f 2 for each x ∈ E. Let U = {x ∈ E : x = 1}. A Banach space E is called smooth if for all x ∈ U, there exists a unique functional jx ∈ E∗ such that x, jx = x and jx = 1 (see [1]). Recall that a mapping f : C → C is a contraction on C, if there exists a constant α ∈ (0, 1) such that f (x) – f (y) ≤ αx – y, ∀x, y ∈ C. We use C to denote the collection of all contractions on C, i.e., C = {f |f : C → C is a contraction}. For a map T from E into itself, we denote by Fix(T) := {x ∈ E : x = Tx}, the ﬁxed point set of T. Recall the following well-known concepts: (1) Suppose that C is a nonempty closed convex subset of a real Banach space E. A mapping B : C → E is called relaxed (u, v)-cocoercive [2], if there exist two constants u, v > 0 such that Bx – By, j(x – y) ≥ (–u)Bx – By2 + vx – y2 , for all x, y ∈ C and j(x – y) ∈ J(x – y). © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Soori et al. Journal of Inequalities and Applications (2022) 2022:165 Page 2 of 5 (2) Suppose that C is a nonempty closed convex subset of a real Banach space E and B is a self-mapping on C. If there exists a positive integer α such that Bx – By ≥ αx – y for all x, y ∈ C, then B is called α-expansive. Lemma 1.1 ([2]) Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X with the 2-uniformly smooth constant K . Let QC be the sunny nonexpan- sive retraction from X onto C and let Ai : C → X be a relaxed (ci , di )-cocoercive and Li - Lipschitzian mapping for i = 1, 2, 3. Let G : C → C be a mapping deﬁned by G(x) = QC QC QC (x – λ3 A3 x) – λ2 A2 QC (x – λ3 A3 x) – λ1 A1 QC QC (I – λ3 A3 )x – λ2 A2 QC (I – λ3 A3 )x . di –ci L2i If λi ≤ K 2 L2i for all i = 1, 2, 3, then G : C → C is nonexpansive. Lemma 1.2 ([3, Lemma 2.8]) Suppose that C is a nonempty closed convex subset of a real Banach space X that is 2-uniformly smooth, and the mapping A : C → X is relaxed (c, d)- cocoercive and LA -Lipschitzian. Then, 2 (I – λA)x – (I – λA)y ≤ x – y2 + 2 λcL2A – λd + K 2 λ2 L2A x – y2 , d–cL2A where λ > 0. In particular, when d > cL2A and λ ≤ K 2 L2A , note I – λA is nonexpansive. In this paper, using relaxed (u, v)-cocoercive mappings, a new proof for the iterative algorithm [2] is presented. 2 A simple proof for the theorem Imnang [2] considered an iterative algorithm for ﬁnding a common element of the set of ﬁxed points of nonexpansive mappings and the set of solutions of a variational inequality. Our argument will rely on the following lemma. Lemma 2.1 Suppose that C is a nonempty closed convex subset of a Banach space E. Sup- pose that A : C → E is a relaxed (m, v)-cocoercive mapping and -Lipschitz continuous with v – m 2 > 0. Then, A is a (v – m 2 )-expansive mapping. Proof Since A is (m, v)-cocoercive and -Lipschitz continuous, for each x, y ∈ C and j(x – y) ∈ J(x – y), we have that Ax – Ay, j(x – y) ≥ (–m)Ax – Ay2 + vx – y2 ≥ –m 2 x – y2 + vx – y2 = v – m 2 x – y2 ≥ 0, Soori et al. Journal of Inequalities and Applications (2022) 2022:165 Page 3 of 5 and hence Ax – Ay ≥ v – m 2 x – y, therefore, A is (v – m 2 )-expansive. The following theorem is due to Imnang [2] that solves the viscosity iterative problem for a new general system of variational inequalities in Banach spaces: Theorem 2.2 (i.e., Theorem 3.1, from [2, §3, p.7]) Suppose that X is a Banach space that is uniformly convex and 2-uniformly smooth with the 2-uniformly smooth constant K , C is a nonempty closed convex subset of X, and QC is a sunny nonexpansive retraction from X onto C. Assume that Ai : C → X is relaxed (ci , di )-cocoercive and Li -Lipschitzian with di –ci L2i 0 < λi < K 2 L2i for each i = 1, 2, 3. Suppose that f is a contraction mapping with the constant α ∈ (0, 1) and S : C → C, a nonexpansive mapping such that = F(S) ∩ F(G) = ∅, where G is deﬁned as in Lemma 1.1. Suppose that x1 ∈ C and {xn }, {yn } and {zn } are the following sequences: ⎧ ⎪ ⎪ ⎨zn = QC (xn – λ3 A3 xn ), yn = QC (zn – λ2 A2 zn ), ⎪ ⎪ ⎩ xn+1 = an f (xn ) + bn xn + (1 – an – bn )SQC (yn – λ1 A1 yn , where {an } and {bn } are two sequences in (0, 1) such that (C1) limn→∞ an = 0 and ∞ n=1 an = ∞; (C2) 0 < lim infn→∞ bn ≤ lim supn→∞ bn < 1. Then, {xn } converges strongly to q ∈ , which solves the following variational inequality: q – f (q), J(q – p) ≤ 0, ∀f ∈ C , p ∈ . A Simple Proof Let i = 1, 2, 3. Consider Theorem 2.2 and the Li -Lipschitz continuous and relaxed (ci , di )-cocoercive mapping Ai in Theorem 2.2. From the condition that 0 < λi < di –ci L2i K 2 L2i , we have that 0 < 1 + 2(λi ci L2i – λi di + K 2 λ2i L2i ) < 1. Note that from Lemma 1.2, we have that I – λi Ai is nonexpansive when 0 < 1 + 2(λi ci L2i – λi di + K 2 λ2i L2i ). Then, applying the coeﬃcients αi = 1 + 2(λi ci L2i – λi di + K 2 λ2i L2i ) in Lemma 1.2 we have that I – λi Ai is an αi -contraction, for each i = 1, 2, 3. Also, note that QC is nonexpansive and I – λi Ai is an αi -contraction, for each i = 1, 2, 3. Hence, using the proof of [2, Lemma 2.11], we conclude that G(x) – G(y) = QC QC QC (I – λ3 A3 )x – λ2 A2 QC (I – λ3 A3 )x – λ1 A1 QC QC (I – λ3 A3 )x – λ2 A2 QC (I – λ3 A3 )x – QC QC QC (I – λ3 A3 )y – λ2 A2 QC (I – λ3 A3 )y – λ1 A1 QC QC (I – λ3 A3 )y – λ2 A2 QC (I – λ3 A3 )y ≤ QC QC (I – λ3 A3 )x – λ2 A2 QC (I – λ3 A3 )x – λ1 A1 QC QC (I – λ3 A3 )x – λ2 A2 QC (I – λ3 A3 )x Soori et al. Journal of Inequalities and Applications (2022) 2022:165 Page 4 of 5 – QC QC (I – λ3 A3 )y – λ2 A2 QC (I – λ3 A3 )y – λ1 A1 QC QC (I – λ3 A3 )y – λ2 A2 QC (I – λ3 A3 )y = (I – λ1 A1 )QC (I – λ2 A2 )QC (I – λ3 A3 )x – (I – λ1 A1 )QC (I – λ2 A2 )QC (I – λ3 A3 )y ≤ α1 α2 α3 x – y, and since 0 < α1 α2 α3 < 1 then G is an α-contraction with α = α1 α2 α3 , hence from Banach’s contraction principle F(G) is a singleton set and hence, is a singleton set, i.e., there exists an element p ∈ X such that = {p}. Since (di – ci L2i ) > 0, from Lemma 2.1, Ai is (di – ci L2i )-expansive, i.e., Ai x – Ai y ≥ di – ci L2i x – y, (1) in Theorem 2.2. The authors in [2, p.11] proved (see (3.12) in [2, p.11]) that lim A3 xn – A3 p = 0, (2) n for x∗ = p. Now, put x = xn and y = p in (1), and from (1) and (2), we have lim xn – p = 0. n Hence, xn → p. As a result, one of the main claims of Theorem 2.2 is established (note = {p}). Note that the main aims of Theorem 3.1 in [2] are xn → p and q – f (q), J(q – p) ≤ 0, ∀f ∈ C , p ∈ . Next, we show that the main aim of Theorem 3.1 in [2] can be concluded from the relations (3.12) in [2, page 11] and the proof in Theorem 2.2 can be simpliﬁed even further using the above. Note that the part of the proof between the relations (3.12) in [2, page 11] to the end of the proof of Theorem 3.1 can be removed from the proof. Indeed, since immediately from (3.12) in [2], we conclude that xn → p, i.e., the ﬁrst aim of Theorem 3.1 is concluded. The second aim of the theorem, i.e., q – f (q), J(q – p) ≤ 0, ∀f ∈ C , p ∈ , is clear, because p = q ( = {p}) and J(0) = {0}. Consequently, the relations between (3.12) in [2, page 11] to the end of the proof of Theorem 3.1 in [2, page 11] can be removed. 3 Discussion In this paper, a simple proof for the convergence of an algorithm by relaxed (u, v)- cocoercive mappings due to Imnang is presented. 4 Conclusion In this paper, a reﬁnement of the proof of the results due to Imnang is given. Soori et al. Journal of Inequalities and Applications (2022) 2022:165 Page 5 of 5 Acknowledgements The ﬁrst author is grateful to the University of Lorestan for its support. Funding Not applicable. Abbreviations Not applicable. Availability of data and materials Please contact the authors for data requests. Declarations Competing interests The authors declare no competing interests. Author contributions All authors reviewed the manuscript. Author details 1 Department of Mathematics, Lorestan University, P.O. Box 465, Khoramabad, Lorestan, Iran. 2 School of Mathematical and Statistical Sciences, University of Galway, Galway, Ireland. 3 Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., MSC 172 Kingsville, Texas, USA. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations. Received: 29 September 2022 Accepted: 19 December 2022 References 1. Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed point theory for Lipschitzian-type mappings with applications. In: Topological Fixed Point Theory and Its Applications, vol. 6. Springer, New York (2009) 2. Imnang, S.: Viscosity iterative method for a new general system of variational inequalities in Banach spaces. J. Inequal. Appl. 2013, 249 (2013) 3. Cai, G., Bu, S.: Strong convergence theorems based on a new modiﬁed extragradient method for variational inequality problems and ﬁxed point problems in Banach spaces. Comput. Math. 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