A Simple proof for Imnang's algorithms

TL;DR

This paper presents a straightforward proof of convergence for Imnang's recent iterative algorithm involving relaxed (u, v)-cocoercive mappings, simplifying the understanding of its theoretical foundation.

Contribution

The paper offers a simplified convergence proof for Imnang's algorithm, enhancing theoretical clarity and potential applicability in variational inequality problems.

Findings

Proof confirms convergence of Imnang's algorithm

Simplifies understanding of iterative method

Strengthens theoretical basis for variational inequalities

Abstract

In this paper, a simple proof of the convergence of the recent iterative algorithm by relaxed $(u, v)$-cocoercive mappings due to S. Imnang [S. Imnang, Viscosity iterative method for a new general system of variational inequalities in Banach spaces. J. Inequal. Appl., 249:18 pp., 2013.] is presented.

Equations

$$J(x)=\{f\in E^{\ast }:⟨x,f⟩=\Vert x{\Vert }^2=\Vert f{\Vert }^2\}$$

$$⟨Bx-By,j(x-y)⟩\ge (-u)\Vert Bx-By{\Vert }^2+v\Vert x-y{\Vert }^2,$$

$$\Vert Bx-By\Vert \ge \alpha \Vert x-y\Vert .$$

$$G(x)=$$

$$\displaystyle-\lambda_{1}A_{1}Q_{C}\big{(}Q_{C}(I-\lambda_{3}A_{3})x-\lambda_{2}A_{2}Q_{C}(I-\lambda_{3}A_{3})x\big{)}\big{]}$$

$$\Vert (I-\lambda A)x-(I-\lambda A)y{\Vert }^2\le \Vert x-y{\Vert }^2+2(\lambda c{L}_A^2-\lambda d+K^2{\lambda }^2{L}_A^2)\Vert x-y{\Vert }^2,$$

$$⟨Ax-Ay,j(x-y)⟩$$

$$\ge (-m{ϵ}^2)\Vert x-y{\Vert }^2+v\Vert x-y{\Vert }^2$$

$$=(v-m{ϵ}^2)\Vert x-y{\Vert }^2\ge 0,$$

$$\Vert Ax-Ay\Vert \ge (v-m{ϵ}^2)\Vert x-y\Vert ,$$

$$\{\begin{array}{ll}{z}_n=Q_C(x_n-{\lambda }_3{A}_3{x}_n),& \\ y_n=Q_C(z_n-{\lambda }_2{A}_2{z}_n),& \\ x_{n+1}=a_{n}f(x_n)+b_n{x}_n+(1-a_n-b_n)S{Q}_C(y_n-{\lambda }_1{A}_1{y}_n,& \end{array}$$

$$⟨q-f(q),J(q-p)⟩\le 0,\forall f\in {\Pi }_C,p\in \Omega .$$

$$\Vert G(x)-G(y)\Vert =$$

$$\displaystyle-\lambda_{1}A_{1}Q_{C}\big{(}Q_{C}(I-\lambda_{3}A_{3})x-\lambda_{2}A_{2}Q_{C}(I-\lambda_{3}A_{3})x\big{)}\big{]}$$

$$\displaystyle-Q_{C}\big{[}Q_{C}\big{(}Q_{C}(I-\lambda_{3}A_{3})y-\lambda_{2}A_{2}Q_{C}(I-\lambda_{3}A_{3})y\big{)}$$

$$\displaystyle-\lambda_{1}A_{1}Q_{C}\big{(}Q_{C}(I-\lambda_{3}A_{3})y-\lambda_{2}A_{2}Q_{C}(I-\lambda_{3}A_{3})y\big{)}\big{]}\big{\|}$$

$$\le$$

$$\displaystyle-\lambda_{1}A_{1}Q_{C}\big{(}Q_{C}(I-\lambda_{3}A_{3})x-\lambda_{2}A_{2}Q_{C}(I-\lambda_{3}A_{3})x\big{)}$$

$$\displaystyle-\big{[}Q_{C}\big{(}Q_{C}(I-\lambda_{3}A_{3})y-\lambda_{2}A_{2}Q_{C}(I-\lambda_{3}A_{3})y\big{)}$$

$$\displaystyle-\lambda_{1}A_{1}Q_{C}\big{(}Q_{C}(I-\lambda_{3}A_{3})y-\lambda_{2}A_{2}Q_{C}(I-\lambda_{3}A_{3})y\big{)}\big{]}\big{\|}$$

$$=$$

$$-(I-{\lambda }_1{A}_1)Q_C(I-{\lambda }_2{A}_2)Q_C(I-{\lambda }_3{A}_3)y\Vert$$

$$\le$$

$$\Vert A_{i}x-A_{i}y\Vert \ge (d_i-c_i{L}_i^2)\Vert x-y\Vert ,$$

$$\underset{n}{\lim }\Vert A_3{x}_n-A_{3}p\Vert =0,$$

$$\underset{n}{\lim }\Vert x_n-p\Vert =0.$$

$$⟨q-f(q),J(q-p)⟩\le 0,\forall f\in {\Pi }_C,p\in \Omega .$$

$$⟨q-f(q),J(q-p)⟩\le 0,\forall f\in {\Pi }_C,p\in \Omega ,$$

Full text

​​​​​​​​ ​​*∗*Corresponding author

2010 Mathematics Subject Classification: 47H09; 47H10.

Keywords: Relaxed $(u,v)$-cocoercive mapping; Strong convergence; $\alpha$-expansive mapping.

E-mail address: sori.e@lu.ac.ir(E. Soori), donal.oregan@nuigalway.ie(D. O*′*Regan), agarwal@tamuk.edu(R. P. Agarwal).

1. Introduction and Preliminaries

In this paper, a simple proof for the convergence of an iterative algorithm is presented which 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

[TABLE]

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, 4, 5]).

For a map $T$ from $E$ into itself, we denote by $\rm{Fix}(T):=\{x\in E:x=Tx\}$, 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 [2], if there exist two constants $u,v>0$ such that

[TABLE]

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 $\alpha$ such that

[TABLE]

for all $x,y\in C$, then $B$ is called $\alpha$-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 Li-Lipschitzian mapping for $i=1,2,3$. Let $G:C\rightarrow C$ be a mapping defined by

[TABLE]

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

[TABLE]

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

S. 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 $\epsilon$-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 $\epsilon$-Lipschitz continuous, for each $x,y\in C$ and $j(x-y)\in J(x-y)$, we have that

[TABLE]

and hence

[TABLE]

so $A$ is $(v-m\epsilon^{2})$-expansive. ∎

The following theorem is due to S. 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 which 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:*

[TABLE]

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)

$0<\displaystyle\lim inf_{n\rightarrow\infty}b_{n}\leq\displaystyle\lim sup_{n\rightarrow\infty}b_{n}<1.$**

Then $\left\{x_{n}\right\}$ converges strongly to $q\in\Omega$, which solves the following variational inequality:

[TABLE]

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 $\alpha_{i}$-contraction, for each $i=1,2,3$. Also, note that $Q_{C}$ is nonexpansive and $I-\lambda_{i}A_{i}$ is $\alpha_{i}$-contraction, for each $i=1,2,3$. Hence, using the proof of [2, Lemma 2.11], we conclude that

[TABLE]

and since $0<\alpha_{1}\alpha_{2}\alpha_{3}<1$ then $G$ is an $\alpha$-contraction with $\alpha=\alpha_{1}\alpha_{2}\alpha_{3}$, hence from Banach’s contraction principle $F(G)$ is a singleton set and hence, $\Omega$ 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,

[TABLE]

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

[TABLE]

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

[TABLE]

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 aim of Theorem 3.1 in  [2] are $x_{n}\rightarrow p$ and

[TABLE]

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, 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.,

[TABLE]

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 S. Imnang is presented.

4. Conclusion

In this paper, a refinement of the proof of the results due to S. Imnang is given.

5. Abbreviations

Not applicable

6. Declarations

6.1. Availability of data and material

Please contact the authors for data requests.

6.2. Competing interests

The authors declare that they have no competing interests.

6.3. Funding

Not applicable

6.4. Authors’ contributions

All authors contributed equally to the manuscript, read and approved the final manuscript.

6.5. Acknowledgements

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

7. 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.