# A simple proof for Imnang’s algorithms

## 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

\begin{aligned} J(x)=\bigl\{ f \in E^{*}: \langle x, f \rangle = \Vert x \Vert ^{2}= \Vert f \Vert ^{2} \bigr\} \end{aligned}

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 ).

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 well-known concepts:

1. (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 , 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. (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

$$\Vert Bx -By \Vert \geq \alpha \Vert x - y \Vert$$

for all $$x, y \in C$$, then B is called α-expansive.

### Lemma 1.1

()

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 $$Q_{C}$$ 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

\begin{aligned} G(x) ={} & Q_{C} \bigl[Q_{C} \bigl(Q_{C}(x-\lambda _{3}A_{3}x)-\lambda _{2}A_{2}Q_{C}(x- \lambda _{3}A_{3}x) \bigr) \\ & {}-\lambda _{1}A_{1}Q_{C} \bigl(Q_{C}(I- \lambda _{3}A_{3})x-\lambda _{2}A_{2}Q_{C}(I- \lambda _{3}A_{3})x \bigr) \bigr]. \end{aligned}

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 2-uniformly smooth, and the mapping $$A:C\rightarrow X$$ is relaxed $$(c,d)$$-cocoercive and $$L_{A}$$-Lipschitzian. Then,

$$\bigl\Vert (I-\lambda A)x-(I-\lambda A)y \bigr\Vert ^{2} \leq \Vert x-y \Vert ^{2}+2\bigl(\lambda cL^{2}_{A}- \lambda d+K^{2}\lambda ^{2}L^{2}_{A} \bigr) \Vert x-y \Vert ^{2},$$

where $$\lambda >0$$. In particular, when $$d > c L^{2}_{A}$$ and $$\lambda \leq \frac{d-c L^{2}_{A}}{K^{2}L^{2}_{A}}$$, note $$I-\lambda A$$ is nonexpansive.

In this paper, using relaxed $$(u, v)$$-cocoercive mappings, a new proof for the iterative algorithm  is presented.

## 2 A simple proof for the theorem

Imnang  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 ϵ-Lipschitz continuous with $$v-m \epsilon ^{2}>0$$. Then, A is a $$(v-m \epsilon ^{2})$$-expansive mapping.

### Proof

Since A is $$(m, v)$$-cocoercive and ϵ-Lipschitz continuous, for each $$x,y \in C$$ and $$j(x - y) \in J(x - y)$$, we have that

\begin{aligned} \bigl\langle Ax-Ay, j(x-y) \bigr\rangle & \geq (-m) \Vert Ax-Ay \Vert ^{2}+v \Vert x-y \Vert ^{2} \\ &\geq \bigl(-m \epsilon ^{2}\bigr) \Vert x-y \Vert ^{2} +v \Vert x-y \Vert ^{2} \\ &= \bigl(v-m \epsilon ^{2}\bigr) \Vert x-y \Vert ^{2} \geq 0, \end{aligned}

and hence

$$\Vert Ax-Ay \Vert \geq \bigl(v-m \epsilon ^{2}\bigr) \Vert x-y \Vert ,$$

therefore, A is $$(v-m \epsilon ^{2})$$-expansive. □

The following theorem is due to Imnang  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 $$Q_{C}$$ 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:

$$\textstyle\begin{cases} z_{n}=Q_{C}(x_{n}-\lambda _{3}A_{3}x_{n}), \\ y_{n} = Q_{C}(z_{n}-\lambda _{2}A_{2}z_{n}), \\ x_{n+1} = a_{n} f(x_{n}) + b_{n} x_{n} + (1-a_{n}- b_{n})SQ_{C}(y_{n}- \lambda _{1}A_{1}y_{n}, & \end{cases}$$

where $$\{a_{n}\}$$ and $$\{b_{n}\}$$ are two sequences in $$(0, 1)$$ such that

1. (C1)

$$\lim_{n \rightarrow \infty} a_{n} = 0$$ and $$\sum_{n=1}^{\infty}a_{n} =\infty$$;

2. (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:

$$\bigl\langle q-f(q), J(q-p)\bigr\rangle \leq 0,\quad \forall f \in \Pi _{C} , p \in \Omega .$$

### 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

\begin{aligned} \bigl\Vert G(x) - G(y) \bigr\Vert ={}& \bigl\Vert Q_{C} \bigl[Q_{C} \bigl(Q_{C}(I-\lambda _{3}A_{3})x- \lambda _{2}A_{2}Q_{C}(I-\lambda _{3}A_{3})x \bigr) \\ &{} -\lambda _{1}A_{1}Q_{C} \bigl(Q_{C}(I- \lambda _{3}A_{3})x-\lambda _{2}A_{2}Q_{C}(I- \lambda _{3}A_{3})x \bigr) \bigr] \\ &{} - Q_{C} \bigl[Q_{C} \bigl(Q_{C}(I-\lambda _{3}A_{3})y-\lambda _{2}A_{2}Q_{C}(I- \lambda _{3}A_{3})y \bigr) \\ &{} -\lambda _{1}A_{1}Q_{C} \bigl(Q_{C}(I- \lambda _{3}A_{3})y-\lambda _{2}A_{2}Q_{C}(I- \lambda _{3}A_{3})y \bigr) \bigr] \bigr\Vert \\ \leq {}& \bigl\Vert Q_{C} \bigl(Q_{C}(I-\lambda _{3}A_{3})x-\lambda _{2}A_{2}Q_{C}(I- \lambda _{3}A_{3})x \bigr) \\ & {}-\lambda _{1}A_{1}Q_{C} \bigl(Q_{C}(I- \lambda _{3}A_{3})x-\lambda _{2}A_{2}Q_{C}(I- \lambda _{3}A_{3})x \bigr) \\ &{} - \bigl[Q_{C} \bigl(Q_{C}(I-\lambda _{3}A_{3})y- \lambda _{2}A_{2}Q_{C}(I- \lambda _{3}A_{3})y \bigr) \\ &{} -\lambda _{1}A_{1}Q_{C} \bigl(Q_{C}(I- \lambda _{3}A_{3})y-\lambda _{2}A_{2}Q_{C}(I- \lambda _{3}A_{3})y \bigr) \bigr] \bigr\Vert \\ = {}& \bigl\Vert (I-\lambda _{1} A_{1})Q_{C}(I- \lambda _{2} A_{2})Q_{C}(I- \lambda _{3} A_{3})x \\ &{}- (I-\lambda _{1} A_{1})Q_{C}(I-\lambda _{2} A_{2})Q_{C}(I-\lambda _{3} A_{3})y \bigr\Vert \\ \leq{} & \alpha _{1}\alpha _{2}\alpha _{3} \Vert x-y \Vert , \end{aligned}

and since $$0< \alpha _{1}\alpha _{2}\alpha _{3} <1$$ then G is an α-contraction 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.,

$$\Vert A_{i}x -A_{i}y \Vert \geq \bigl(d_{i} -c_{i} L_{i}^{2}\bigr) \Vert x - y \Vert ,$$
(1)

in Theorem 2.2. The authors in [2, p.11] proved (see (3.12) in [2, p.11]) that

$$\lim_{n} \Vert A_{3}x_{n}-A_{3}p \Vert =0,$$
(2)

for $$x^{*}=p$$. Now, put $$x=x_{n}$$ and $$y=p$$ in (1), and from (1) and (2), we have

$$\lim_{n} \Vert x_{n}-p \Vert =0.$$

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  are $$x_{n} \rightarrow p$$ and

$$\bigl\langle q-f(q), J(q-p)\bigr\rangle \leq 0, \quad \forall f \in \Pi _{C} , p \in \Omega .$$

Next, we show that the main aim of Theorem 3.1 in  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 , 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.,

$$\bigl\langle q-f(q), J(q-p)\bigr\rangle \leq 0,\quad \forall f \in \Pi _{C} , p \in \Omega ,$$

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.

## 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. Appl. 62, 2567–2579 (2011)

## Acknowledgements

The first author is grateful to the University of Lorestan for its support.

Not applicable.

## Author information

Authors

### Contributions

All authors reviewed the manuscript.

### Corresponding author

Correspondence to Ebrahim Soori.

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### Competing interests

The authors declare no competing interests. 