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 [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 well-known 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 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

([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 $Q_{C}$ 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 [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 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 [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 $Q_{C}$ 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

• (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:

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

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 [2] 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 [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.,

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

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.