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).
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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.
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Received: 29 September 2022 Accepted: 19 December 2022
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