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A simple proof for Imnang’s algorithms
Journal of Inequalities and Applications volume 2022, Article number: 165 (2022)
Abstract
In this paper, a simple proof of the convergence of the recent iterative algorithm by relaxed \((u, v)\)cocoercive mappings due to Imnang (J. Inequal. Appl. 2013:249, 2013) is presented.
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 \(\langle .,.\rangle \) denotes the pairing between E and \(E^{*}\). The normalized duality mapping \(J: E \rightarrow E^{*}\) is defined by
for each \(x \in E\). Let \(U = \{x \in E : \x\ = 1\}\). A Banach space E is called smooth if for all \(x \in U\), there exists a unique functional \(j_{x} \in E^{*}\) such that \(\langle x, j_{x}\rangle = \x\\) and \(\j_{x}\ = 1\) (see [1]).
Recall that a mapping \(f : C \rightarrow C\) is a contraction on C, if there exists a constant \(\alpha \in (0,1)\) such that \(\f (x)  f (y)\ \leq \alpha \x  y\\), \(\forall x, y \in C\). We use \(\Pi _{C}\) to denote the collection of all contractions on C, i.e., \(\Pi _{C} = \{f  f : C\rightarrow C \text{ is a contraction} \}\).
For a map T from E into itself, we denote by \(\mathrm{Fix}(T ) := \{x \in E : x = Tx\}\), the fixed point set of T.
Recall the following wellknown concepts:

(1)
Suppose that C is a nonempty closed convex subset of a real Banach space E. A mapping \(B: C \rightarrow E\) is called relaxed \((u, v)\)cocoercive [2], if there exist two constants \(u, v > 0\) such that
$$ \bigl\langle Bx  By , j(x  y)\bigr\rangle \geq (u) \Vert Bx  By \Vert ^{2}+v \Vert x  y \Vert ^{2}, $$for all \(x, y \in C\) and \(j(x  y) \in J(x  y)\).

(2)
Suppose that C is a nonempty closed convex subset of a real Banach space E and B is a selfmapping on C. If there exists a positive integer α such that
$$ \Vert Bx By \Vert \geq \alpha \Vert x  y \Vert $$for all \(x, y \in C\), then B is called αexpansive.
Lemma 1.1
([2])
Let C be a nonempty closed convex subset of a real 2uniformly smooth Banach space X with the 2uniformly smooth constant K. Let \(Q_{C}\) be the sunny nonexpansive retraction from X onto C and let \(A_{i} : C \rightarrow X\) be a relaxed \((c_{i}, d_{i})\)cocoercive and \(L_{i}\)Lipschitzian mapping for \(i = 1, 2, 3\). Let \(G : C \rightarrow C\) be a mapping defined by
If \(\lambda _{i} \leq \frac{d_{i}c_{i} L_{i}^{2}}{K^{2}L_{i}^{2}}\) for all \(i = 1, 2, 3\), then \(G : C \rightarrow C\) 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 2uniformly smooth, and the mapping \(A:C\rightarrow X\) is relaxed \((c,d)\)cocoercive and \(L_{A}\)Lipschitzian. Then,
where \(\lambda >0\). In particular, when \(d > c L^{2}_{A}\) and \(\lambda \leq \frac{dc L^{2}_{A}}{K^{2}L^{2}_{A}} \), note \(I\lambda A\) 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. Suppose that \(A: C \rightarrow E\) is a relaxed \((m, v)\)cocoercive mapping and ϵLipschitz continuous with \(vm \epsilon ^{2}>0\). Then, A is a \((vm \epsilon ^{2})\)expansive mapping.
Proof
Since A is \((m, v)\)cocoercive and ϵLipschitz continuous, for each \(x,y \in C\) and \(j(x  y) \in J(x  y)\), we have that
and hence
therefore, A is \((vm \epsilon ^{2})\)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 2uniformly smooth with the 2uniformly smooth constant K, C is a nonempty closed convex subset of X, and \(Q_{C}\) is a sunny nonexpansive retraction from X onto C. Assume that \(A_{i}: C \rightarrow X\) is relaxed \((c_{i}, d_{i})\)cocoercive and \(L_{i}\)Lipschitzian with \(0 < \lambda _{i} < \frac{d_{i}c_{i}L_{i}^{2}}{K^{2}L_{i}^{2}}\) for each \(i = 1, 2, 3\). Suppose that f is a contraction mapping with the constant \(\alpha \in (0, 1)\) and \(S: C\rightarrow C\), a nonexpansive mapping such that \(\Omega = F(S)\cap F(G)\neq \emptyset \), where G is defined as in Lemma 1.1. Suppose that \(x_{1} \in C\) and \(\{x_{n}\}\), \(\{y_{n}\}\) and \(\{z_{n}\}\) are the following sequences:
where \(\{a_{n}\}\) and \(\{b_{n}\}\) are two sequences in \((0, 1)\) such that

(C1)
\(\lim_{n \rightarrow \infty} a_{n} = 0\) and \(\sum_{n=1}^{\infty}a_{n} =\infty \);

(C2)
\(0< \lim \inf_{n\rightarrow \infty} b_{n} \leq \lim \sup_{ n\rightarrow \infty} b_{n} < 1\).
Then, \(\{x_{n} \}\) converges strongly to \(q\in \Omega \), which solves the following variational inequality:
A Simple Proof
Let \(i=1, 2,3\). Consider Theorem 2.2 and the \(L_{i}\)Lipschitz continuous and relaxed \((c_{i},d_{i} )\)cocoercive mapping \(A_{i}\) in Theorem 2.2. From the condition that \(0 < \lambda _{i} < \frac{d_{i}c_{i}L_{i}^{2}}{K^{2}L_{i}^{2}}\), we have that \(0<1+2(\lambda _{i} c_{i} L^{2}_{i}\lambda _{i} d_{i}+ K^{2}\lambda ^{2}_{i}L^{2}_{i})<1\). Note that from Lemma 1.2, we have that \(I\lambda _{i} A_{i}\) is nonexpansive when \(0<1+2(\lambda _{i} c_{i} L^{2}_{i}\lambda _{i} d_{i}+ K^{2}\lambda ^{2}_{i}L^{2}_{i})\). Then, applying the coefficients \(\alpha _{i}=1+2(\lambda _{i} c_{i} L^{2}_{i}\lambda _{i} d_{i}+ K^{2} \lambda ^{2}_{i}L^{2}_{i})\) in Lemma 1.2 we have that \(I\lambda _{i} A_{i}\) is an \(\alpha _{i}\)contraction, for each \(i=1,2,3\). Also, note that \(Q_{C}\) is nonexpansive and \(I\lambda _{i} A_{i}\) is an \(\alpha _{i}\)contraction, for each \(i=1,2,3\). Hence, using the proof of [2, Lemma 2.11], we conclude that
and since \(0< \alpha _{1}\alpha _{2}\alpha _{3} <1\) then G is an αcontraction with \(\alpha =\alpha _{1}\alpha _{2}\alpha _{3} \), hence from Banach’s contraction principle \(F(G)\) is a singleton set and hence, Ω is a singleton set, i.e., there exists an element \(p \in X\) such that \(\Omega =\{p\} \). Since \((d_{i}c_{i}L_{i}^{2})>0\), from Lemma 2.1, \(A_{i}\) is \((d_{i}c_{i} L_{i}^{2})\)expansive, i.e.,
in Theorem 2.2. The authors in [2, p.11] proved (see (3.12) in [2, p.11]) that
for \(x^{*}=p\). Now, put \(x=x_{n}\) and \(y=p\) in (1), and from (1) and (2), we have
Hence, \(x_{n} \rightarrow p\). As a result, one of the main claims of Theorem 2.2 is established (note \(\Omega =\{p\} \)).
Note that the main aims of Theorem 3.1 in [2] are \(x_{n} \rightarrow p\) and
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 \(x_{n} \rightarrow p\), i.e., the first aim of Theorem 3.1 is concluded. The second aim of the theorem, i.e.,
is clear, because \(p=q\) (\(\Omega =\{p\} \)) 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.
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References
Agarwal, R.P., O’Regan, D., Sahu, D.R.: Fixed point theory for Lipschitziantype mappings with applications. In: Topological Fixed Point Theory and Its Applications, vol. 6. Springer, New York (2009)
Imnang, S.: Viscosity iterative method for a new general system of variational inequalities in Banach spaces. J. Inequal. Appl. 2013, 249 (2013)
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. Appl. 62, 2567–2579 (2011)
Acknowledgements
The first author is grateful to the University of Lorestan for its support.
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Soori, E., O’Regan, D. & Agarwal, R.P. A simple proof for Imnang’s algorithms. J Inequal Appl 2022, 165 (2022). https://doi.org/10.1186/s1366002202904y
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DOI: https://doi.org/10.1186/s1366002202904y
MSC
 47H09
 47H10
Keywords
 Relaxed \((u, v)\)cocoercive mapping
 Strong convergence
 αexpansive mapping