Projection methods of iterative solutions in Hilbert spaces

Abstract

In this paper, zero points of the sum of two monotone mappings, solutions of a monotone variational inequality, and fixed points of a nonexpansive mapping are investigated based on a hybrid projection iterative algorithm. Strong convergence of the purposed iterative algorithm is obtained in the framework of real Hilbert spaces without any compact assumptions.

MSC:47H05, 47H09, 47J25, 90C33.

1 Introduction and preliminaries

Throughout this paper, we always assume that H is a real Hilbert space with an inner product $ã€ˆâ‹\dots ,â‹\dots ã€‰$ and a norm $âˆ¥â‹\dots âˆ¥$. Let C be a nonempty, closed, and convex subset of H. Let $S:Câ†’C$ be a nonlinear mapping. $F\left(S\right)$ stands for the fixed point set of S; that is, $F\left(S\right):=\left\{xâˆˆC:x=Tx\right\}$.

Recall that S is said to be nonexpansive iff

$âˆ¥Sxâˆ’Syâˆ¥â‰¤âˆ¥xâˆ’yâˆ¥,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

If C is a bounded, closed, and convex subset of H, then $F\left(S\right)$ is not empty, closed, and convex; see [1].

Let $A:Câ†’H$ be a mapping. Recall that A is said to be inverse-strongly monotone iff there exists a constant $\mathrm{Î±}>0$ such that

$ã€ˆAxâˆ’Ay,xâˆ’yã€‰â‰¥\mathrm{Î±}{âˆ¥Axâˆ’Ayâˆ¥}^{2},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

For such a case, A is also said to be Î±-inverse-strongly monotone.

A is said to be monotone iff

$ã€ˆAxâˆ’Ay,xâˆ’yã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.$

Recall that the classical variational inequality is to find an $xâˆˆC$ such that

$ã€ˆAx,yâˆ’xã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆC.$
(1.1)

In this paper, we use $VI\left(C,A\right)$ to denote the solution set of (1.1). It is known that $xâˆˆC$ is a solution of (1.1) if and only if x is a fixed point of the mapping ${Proj}_{C}\left(Iâˆ’rA\right)$, where $r>0$ is a constant, I stands for the identity mapping, and ${Proj}_{C}$ stands for the metric projection from H onto C. If A is Î±-inverse-strongly monotone and $râˆˆ\left(0,2\mathrm{Î±}\right]$, then the mapping ${Proj}_{C}\left(Iâˆ’rA\right)$ is nonexpansive; see [2] for more details. It follows that $VI\left(C,A\right)$ is closed and convex.

Monotone variational inequality theory has emerged as a fascinating branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, ecology, social, regional, pure, and applied sciences. In recent years, much attention has been given to developing efficient numerical methods for treating solution problems of monotone variational inequality. The gradient-projection method is a powerful tool for solving constrained convex optimization problems and has extensively been studied; see [3â€“5] and the references therein. It has recently been applied to solving split feasibility problems which find applications in image reconstructions and the intensity modulated radiation theory; see [6â€“9] and the references therein. However, the gradient-projection method requires the operator to be strongly monotone and Lipschitz continuous. These strong conditions rule out many applications. Extra gradient-projection method which was first introduce by Korpelevich [10] in the finite dimensional Euclidean space has been studied recently for relaxing the strong monotonicity of operators; see [11â€“13] and the references therein.

Recall that a set-valued mapping $M:Hâ‡‰H$ is said to be monotone iff, for all $x,yâˆˆH$, $fâˆˆMx$ and $gâˆˆMy$ imply $ã€ˆxâˆ’y,fâˆ’gã€‰>0$. A monotone mapping $M:Hâ‡‰H$ is maximal iff the graph $Graph\left(M\right)$ of R is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping M is maximal if and only if, for any $\left(x,f\right)âˆˆHÃ—H$, $ã€ˆxâˆ’y,fâˆ’gã€‰â‰¥0$, for all $\left(y,g\right)âˆˆGraph\left(M\right)$ implies $fâˆˆRx$.

For a maximal monotone operator M on H and $r>0$, we may define the single-valued resolvent ${J}_{r}:Hâ†’D\left(M\right)$, where $D\left(M\right)$ denotes the domain of M. It is known that ${J}_{r}$ is firmly nonexpansive, and ${M}^{âˆ’1}\left(0\right)=F\left({J}_{r}\right)$, where $F\left({J}_{r}\right):=\left\{xâˆˆD\left(M\right):x={J}_{r}x\right\}$ and ${M}^{âˆ’1}\left(0\right):\left\{xâˆˆH:0âˆˆMx\right\}$.

Recently, variational inequalities, fixed point problems, and zero point problems have been investigated by many authors based on iterative methods; see, for example, [14â€“32] and the references therein. In this paper, zero point problems of the sum of a maximal monotone operator and an inverse-strongly monotone mapping, solution problems of a monotone variational inequality, and fixed point problems of a nonexpansive mapping are investigated. A hybrid iterative algorithm is considered for analyzing the convergence of iterative sequences. Strong convergence theorems are established in the framework of real Hilbert spaces without any compact assumptions.

In order to prove our main results, we also need the following definitions and lemmas.

Lemma 1.1 Let C be a nonempty, closed, and convex subset of H. Then the following inequality holds:

${âˆ¥xâˆ’{Proj}_{C}xâˆ¥}^{2}+{âˆ¥yâˆ’{Proj}_{C}âˆ¥}^{2}â‰¤{âˆ¥xâˆ’yâˆ¥}^{2},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}xâˆˆH,yâˆˆC.$

Lemma 1.2[1]

Let C be a nonempty, closed, and convex subset of H. Let$S:Câ†’C$be a nonexpansive mapping. Then the mapping$Iâˆ’S$is demiclosed at zero, that is, if$\left\{{x}_{n}\right\}$is a sequence in C such that${x}_{n}â‡€\stackrel{Â¯}{x}$and${x}_{n}âˆ’S{x}_{n}â†’0$, then$\stackrel{Â¯}{x}âˆˆF\left(S\right)$.

Lemma 1.3 Let C be a nonempty, closed, and convex subset of H, $B:Câ†’H$be a mapping, and$M:Hâ‡‰H$be a maximal monotone operator. Then$F\left({J}_{r}\left(Iâˆ’sB\right)\right)={\left(B+M\right)}^{âˆ’1}\left(0\right)$.

Proof Notice that

$\begin{array}{r}pâˆˆF\left({J}_{r}\left(Iâˆ’sB\right)\right)\phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}p={J}_{r}\left(Iâˆ’sB\right)p\phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}pâˆ’sBpâˆˆp+sMp\\ \phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}0âˆˆ{\left(B+M\right)}^{âˆ’1}\left(0\right)\phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}pâˆˆ{\left(B+M\right)}^{âˆ’1}\left(0\right).\end{array}$

This completes the proof.â€ƒâ–¡

Lemma 1.4[33]

Let C be a nonempty, closed, and convex subset of H, $A:Câ†’H$be a Lipschitz monotone mapping, and${N}_{C}x$be the normal cone to C at$xâˆˆC$; that is, ${N}_{C}x=\left\{yâˆˆH:ã€ˆxâˆ’u,yã€‰,\mathrm{âˆ€}uâˆˆC\right\}$. Define

$Wx=\left\{\begin{array}{cc}Ax+{N}_{C}x,\hfill & xâˆˆC,\hfill \\ \mathrm{âˆ\dots },\hfill & xâˆ‰C.\hfill \end{array}$

Then W is maximal monotone and$0âˆˆWx$if and only if$xâˆˆVI\left(C,A\right)$.

2 Main results

Now, we are in a position to give our main results.

Theorem 2.1 Let C be a nonempty, closed, and convex subset of H. Let$S:Câ†’C$be a nonexpansive mapping with a nonempty fixed point set, $A:Câ†’H$be an Î±-Lipschitz continuous and monotone mapping, and$B:Câ†’H$be a Î²-inverse-strongly monotone mapping. Let$M:Hâ‡‰H$be a maximal monotone operator such that$D\left(M\right)âŠ‚C$. Assume that$\mathcal{F}:=F\left(S\right)âˆ©{\left(B+M\right)}^{âˆ’1}\left(0\right)âˆ©VI\left(C,A\right)$is not empty. Let$\left\{{x}_{n}\right\}$be a sequence generated by the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {C}_{1}=C,\hfill \\ {z}_{n}={Proj}_{C}\left({J}_{{s}_{n}}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)âˆ’{r}_{n}A{J}_{{s}_{n}}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)\right),\hfill \\ {y}_{n}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{Proj}_{C}\left({J}_{{s}_{n}}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)âˆ’{r}_{n}A{z}_{n}\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}nâ‰¥0,\hfill \end{array}$
(2.1)

where${J}_{{s}_{n}}={\left(I+{s}_{n}M\right)}^{âˆ’1}$, $\left\{{r}_{n}\right\}$is a sequence in$\left(0,\frac{1}{\mathrm{Î±}}\right)$, $\left\{{s}_{n}\right\}$is a sequence in$\left(0,2\mathrm{Î²}\right)$, and$\left\{{\mathrm{Î±}}_{n}\right\}$is a sequence in$\left(0,1\right)$. Assume that the following restrictions are satisfied:

1. (a)

$0;

2. (b)

$0;

3. (c)

$0â‰¤{\mathrm{Î±}}_{n}â‰¤e<1$,

where a, b, c, d, and e are real constants. Then the sequence$\left\{{x}_{n}\right\}$converges strongly to${Proj}_{\mathcal{F}}{x}_{1}$.

Proof First, we show that ${C}_{n}$ is closed and convex for each $nâ‰¥1$. From the assumption, we see that ${C}_{1}=C$ is closed and convex. Suppose that ${C}_{m}$ is closed and convex for some $mâ‰¥1$. We show that ${C}_{m+1}$ is closed and convex for the same m. Let ${v}_{1},{v}_{2}âˆˆ{C}_{m+1}$ and $v=t{v}_{1}+\left(1âˆ’t\right){v}_{2}$, where $tâˆˆ\left(0,1\right)$. Notice that

$âˆ¥{y}_{m}âˆ’vâˆ¥â‰¤âˆ¥{x}_{m}âˆ’vâˆ¥$

is equivalent to

${âˆ¥{y}_{m}âˆ¥}^{2}âˆ’{âˆ¥{x}_{m}âˆ¥}^{2}âˆ’2ã€ˆv,{y}_{m}âˆ’{x}_{m}ã€‰â‰¥0.$

It is clear that $vâˆˆ{C}_{m+1}$. This shows that ${C}_{n}$ is closed and convex for each $nâ‰¥1$.

Next, we show that $\mathcal{F}âŠ‚{C}_{n}$ for each $nâ‰¥1$. Put ${u}_{n}={Proj}_{C}\left({v}_{n}âˆ’{r}_{n}A{z}_{n}\right)$, where ${v}_{n}={J}_{{s}_{n}}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)$. From the assumption, we see that $\mathcal{F}âŠ‚C={C}_{1}$. Suppose that $\mathcal{F}âŠ‚{C}_{m}$ for some $mâ‰¥1$. For any $pâˆˆ\mathcal{F}âŠ‚{C}_{m}$, we see from Lemma 1.1 that

$\begin{array}{rl}{âˆ¥{u}_{m}âˆ’pâˆ¥}^{2}â‰¤& {âˆ¥{v}_{m}âˆ’{r}_{m}A{z}_{m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{m}âˆ’{r}_{m}A{z}_{m}âˆ’{u}_{m}âˆ¥}^{2}\\ =& {âˆ¥{v}_{m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{m}âˆ’{u}_{m}âˆ¥}^{2}+2{r}_{m}ã€ˆA{z}_{m},pâˆ’{u}_{m}ã€‰\\ =& {âˆ¥{v}_{m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{m}âˆ’{u}_{m}âˆ¥}^{2}+2{r}_{m}\left(ã€ˆA{z}_{m}âˆ’Ap,pâˆ’{z}_{m}ã€‰+ã€ˆAp,pâˆ’{z}_{m}ã€‰\\ +ã€ˆA{z}_{m},{z}_{m}âˆ’{u}_{m}ã€‰\right)\\ â‰¤& {âˆ¥{v}_{m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{m}âˆ’{z}_{m}+{z}_{m}âˆ’{u}_{m}âˆ¥}^{2}+2{r}_{m}ã€ˆA{z}_{m},{z}_{m}âˆ’{u}_{m}ã€‰\\ =& {âˆ¥{v}_{m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{m}âˆ’{z}_{m}âˆ¥}^{2}âˆ’{âˆ¥{z}_{m}âˆ’{u}_{m}âˆ¥}^{2}\\ +2ã€ˆ{v}_{m}âˆ’{z}_{m}âˆ’{r}_{m}A{z}_{m},{u}_{m}âˆ’{z}_{m}ã€‰.\end{array}$
(2.2)

Notice that A is Lipschitz continuous. In view of ${z}_{m}={Proj}_{C}\left({v}_{m}âˆ’{r}_{m}A{v}_{m}\right)$, we find that

$\begin{array}{r}ã€ˆ{v}_{m}âˆ’{z}_{m}âˆ’{r}_{m}A{z}_{m},{u}_{m}âˆ’{z}_{m}ã€‰\\ \phantom{\rule{1em}{0ex}}=ã€ˆ{v}_{m}âˆ’{z}_{m}âˆ’{r}_{m}A{v}_{m},{u}_{m}âˆ’{z}_{m}ã€‰+ã€ˆ{r}_{m}A{v}_{m}âˆ’{r}_{m}A{z}_{m},{u}_{m}âˆ’{z}_{m}ã€‰\\ \phantom{\rule{1em}{0ex}}â‰¤{r}_{m}\mathrm{Î±}âˆ¥{v}_{m}âˆ’{z}_{m}âˆ¥âˆ¥{u}_{m}âˆ’{z}_{m}âˆ¥.\end{array}$
(2.3)

Substituting (2.3) into (2.2), we obtain that

$\begin{array}{rl}{âˆ¥{u}_{m}âˆ’pâˆ¥}^{2}& â‰¤{âˆ¥{v}_{m}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{m}âˆ’{z}_{m}âˆ¥}^{2}âˆ’{âˆ¥{z}_{m}âˆ’{u}_{m}âˆ¥}^{2}+2{r}_{m}\mathrm{Î±}âˆ¥{v}_{m}âˆ’{z}_{m}âˆ¥âˆ¥{u}_{m}âˆ’{z}_{m}âˆ¥\\ â‰¤{âˆ¥{v}_{m}âˆ’pâˆ¥}^{2}âˆ’\left(1âˆ’{r}_{m}^{2}{\mathrm{Î±}}^{2}\right){âˆ¥{v}_{m}âˆ’{z}_{m}âˆ¥}^{2}.\end{array}$
(2.4)

This in turn implies from restriction (a) that

$\begin{array}{rl}{âˆ¥{y}_{m}âˆ’pâˆ¥}^{2}& â‰¤{\mathrm{Î±}}_{m}{âˆ¥{x}_{m}âˆ’pâˆ¥}^{2}+\left(1âˆ’{\mathrm{Î±}}_{m}\right){âˆ¥S{u}_{m}âˆ’pâˆ¥}^{2}\\ â‰¤{\mathrm{Î±}}_{m}{âˆ¥{x}_{m}âˆ’pâˆ¥}^{2}+\left(1âˆ’{\mathrm{Î±}}_{m}\right){âˆ¥{u}_{m}âˆ’pâˆ¥}^{2}\\ â‰¤{\mathrm{Î±}}_{m}{âˆ¥{x}_{m}âˆ’pâˆ¥}^{2}+\left(1âˆ’{\mathrm{Î±}}_{m}\right)\left({âˆ¥{v}_{m}âˆ’pâˆ¥}^{2}âˆ’\left(1âˆ’{r}_{m}^{2}{\mathrm{Î±}}^{2}\right){âˆ¥{v}_{m}âˆ’{z}_{m}âˆ¥}^{2}\right)\\ â‰¤{âˆ¥{x}_{m}âˆ’pâˆ¥}^{2}âˆ’\left(1âˆ’{\mathrm{Î±}}_{m}\right)\left(1âˆ’{r}_{m}^{2}{\mathrm{Î±}}^{2}\right){âˆ¥{v}_{m}âˆ’{z}_{m}âˆ¥}^{2}\\ â‰¤{âˆ¥{x}_{m}âˆ’pâˆ¥}^{2}.\end{array}$
(2.5)

This shows that $pâˆˆ{C}_{m+1}$. This proves that $\mathcal{F}âŠ‚{C}_{n}$ for each $nâ‰¥1$. Note ${x}_{n}={Proj}_{{C}_{n}}{x}_{1}$. For each $pâˆˆ\mathcal{F}âŠ‚{C}_{n}$, we have $âˆ¥{x}_{1}âˆ’{x}_{n}âˆ¥â‰¤âˆ¥{x}_{1}âˆ’pâˆ¥$. Since B is inverse-strongly monotone, we see from Lemma 1.3 that ${\left(B+M\right)}^{âˆ’1}\left(0\right)$ is closed and convex. Since A is Lipschitz continuous, we find that $VI\left(C,A\right)$ is close and convex. This proves that $\mathcal{F}$ is closed and convex. It follows that

$âˆ¥{x}_{1}âˆ’{x}_{n}âˆ¥â‰¤âˆ¥{x}_{1}âˆ’{Proj}_{\mathcal{F}}{x}_{1}âˆ¥.$
(2.6)

This implies that $\left\{{x}_{n}\right\}$ is bounded. Since ${x}_{n}={Proj}_{{C}_{n}}{x}_{1}$ and ${x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1}âˆˆ{C}_{n+1}âŠ‚{C}_{n}$, we have

$\begin{array}{rl}0& â‰¤ã€ˆ{x}_{1}âˆ’{x}_{n},{x}_{n}âˆ’{x}_{n+1}ã€‰\\ =ã€ˆ{x}_{1}âˆ’{x}_{n},{x}_{n}âˆ’{x}_{1}+{x}_{1}âˆ’{x}_{n+1}ã€‰\\ â‰¤âˆ’{âˆ¥{x}_{1}âˆ’{x}_{n}âˆ¥}^{2}+âˆ¥{x}_{1}âˆ’{x}_{n}âˆ¥âˆ¥{x}_{1}âˆ’{x}_{n+1}âˆ¥.\end{array}$

It follows that

$âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥â‰¤âˆ¥{x}_{n+1}âˆ’{x}_{1}âˆ¥.$

This proves that ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥$ exists. Notice that

$\begin{array}{r}{âˆ¥{x}_{n}âˆ’{x}_{n+1}âˆ¥}^{2}\\ \phantom{\rule{1em}{0ex}}={âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥}^{2}+2ã€ˆ{x}_{n}âˆ’{x}_{1},{x}_{1}âˆ’{x}_{n+1}ã€‰+{âˆ¥{x}_{1}âˆ’{x}_{n+1}âˆ¥}^{2}\\ \phantom{\rule{1em}{0ex}}={âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥}^{2}âˆ’2{âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥}^{2}+2ã€ˆ{x}_{n}âˆ’{x}_{1},{x}_{n}âˆ’{x}_{n+1}ã€‰+{âˆ¥{x}_{1}âˆ’{x}_{n+1}âˆ¥}^{2}\\ \phantom{\rule{1em}{0ex}}â‰¤{âˆ¥{x}_{1}âˆ’{x}_{n+1}âˆ¥}^{2}âˆ’{âˆ¥{x}_{n}âˆ’{x}_{1}âˆ¥}^{2}.\end{array}$

It follows that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’{x}_{n+1}âˆ¥=0.$
(2.7)

In view of ${x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1}âˆˆ{C}_{n+1}$, we see that

$âˆ¥{y}_{n}âˆ’{x}_{n+1}âˆ¥â‰¤âˆ¥{x}_{n}âˆ’{x}_{n+1}âˆ¥.$

This implies that

$âˆ¥{y}_{n}âˆ’{x}_{n}âˆ¥â‰¤âˆ¥{y}_{n}âˆ’{x}_{n+1}âˆ¥+âˆ¥{x}_{n}âˆ’{x}_{n+1}âˆ¥â‰¤2âˆ¥{x}_{n}âˆ’{x}_{n+1}âˆ¥.$

From (2.7), we find that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥=0.$
(2.8)

Since B is Î²-inverse-strongly monotone, we see from restriction (b) that

$\begin{array}{rl}{âˆ¥\left(Iâˆ’{s}_{n}B\right)xâˆ’\left(Iâˆ’{s}_{n}B\right)yâˆ¥}^{2}& ={âˆ¥xâˆ’yâˆ¥}^{2}âˆ’2{s}_{n}ã€ˆxâˆ’y,Bxâˆ’Byã€‰+{s}_{n}^{2}{âˆ¥Bxâˆ’Byâˆ¥}^{2}\\ â‰¤{âˆ¥xâˆ’yâˆ¥}^{2}âˆ’{s}_{n}\left(2\mathrm{Î²}âˆ’{s}_{n}\right){âˆ¥Bxâˆ’Byâˆ¥}^{2}\\ â‰¤{âˆ¥xâˆ’yâˆ¥}^{2},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆC.\end{array}$

This implies from (2.5) that

$\begin{array}{rl}{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}& â‰¤{\mathrm{Î±}}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{v}_{n}âˆ’pâˆ¥}^{2}\\ ={\mathrm{Î±}}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{J}_{{s}_{n}}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)âˆ’{J}_{{s}_{n}}\left(pâˆ’{s}_{n}Bp\right)âˆ¥}^{2}\\ â‰¤{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’\left(1âˆ’{\mathrm{Î±}}_{n}\right){s}_{n}\left(2\mathrm{Î²}âˆ’{s}_{n}\right){âˆ¥B{x}_{n}âˆ’Bpâˆ¥}^{2}.\end{array}$

It follows that

$\begin{array}{rl}\left(1âˆ’{\mathrm{Î±}}_{n}\right){s}_{n}\left(2\mathrm{Î²}âˆ’{s}_{n}\right){âˆ¥B{x}_{n}âˆ’Bpâˆ¥}^{2}& â‰¤{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}\\ â‰¤âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥\left(âˆ¥{x}_{n}âˆ’pâˆ¥+âˆ¥{y}_{n}âˆ’pâˆ¥\right).\end{array}$

In view of restrictions (b) and (c), we find from (2.8) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥B{x}_{n}âˆ’Bpâˆ¥=0.$
(2.9)

Since ${J}_{{s}_{n}}$ is firmly nonexpansive, we find that

$\begin{array}{rl}{âˆ¥{v}_{n}âˆ’pâˆ¥}^{2}=& {âˆ¥{J}_{{s}_{n}}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)âˆ’{J}_{{s}_{n}}\left(pâˆ’{s}_{n}Bp\right)âˆ¥}^{2}\\ â‰¤& ã€ˆ{v}_{n}âˆ’p,\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)âˆ’\left(pâˆ’{s}_{n}Bp\right)ã€‰\\ =& \frac{1}{2}\left({âˆ¥{v}_{n}âˆ’pâˆ¥}^{2}+{âˆ¥\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)âˆ’\left(pâˆ’{s}_{n}Bp\right)âˆ¥}^{2}\\ âˆ’{âˆ¥\left({v}_{n}âˆ’p\right)âˆ’\left(\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)âˆ’\left(pâˆ’{s}_{n}Bp\right)\right)âˆ¥}^{2}\right)\\ â‰¤& \frac{1}{2}\left({âˆ¥{v}_{n}âˆ’pâˆ¥}^{2}+{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{n}âˆ’{x}_{n}+{s}_{n}\left(B{x}_{n}âˆ’Bp\right)âˆ¥}^{2}\right)\\ =& \frac{1}{2}\left({âˆ¥{v}_{n}âˆ’pâˆ¥}^{2}+{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥}^{2}âˆ’{s}_{n}^{2}{âˆ¥B{x}_{n}âˆ’Bpâˆ¥}^{2}\\ âˆ’2{s}_{n}ã€ˆ{v}_{n}âˆ’{x}_{n},B{x}_{n}âˆ’Bpã€‰\right)\\ â‰¤& \frac{1}{2}\left({âˆ¥{v}_{n}âˆ’pâˆ¥}^{2}+{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥}^{2}+2{s}_{n}âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥âˆ¥B{x}_{n}âˆ’Bpâˆ¥\right).\end{array}$

This in turn implies that

${âˆ¥{v}_{n}âˆ’pâˆ¥}^{2}â‰¤{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥}^{2}+2{s}_{n}âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥âˆ¥B{x}_{n}âˆ’Bpâˆ¥.$
(2.10)

Combining (2.5) with (2.10), we arrive at

$\begin{array}{rl}{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}& â‰¤{\mathrm{Î±}}_{n}{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}+\left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{v}_{n}âˆ’pâˆ¥}^{2}\\ â‰¤{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’\left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥}^{2}+2{s}_{n}âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥âˆ¥B{x}_{n}âˆ’Bpâˆ¥.\end{array}$

It follows that

$\begin{array}{rl}\left(1âˆ’{\mathrm{Î±}}_{n}\right){âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥}^{2}& â‰¤{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}+2{s}_{n}âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥âˆ¥B{x}_{n}âˆ’Bpâˆ¥\\ â‰¤âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥\left(âˆ¥{x}_{n}âˆ’pâˆ¥+âˆ¥{y}_{n}âˆ’pâˆ¥\right)+2{s}_{n}âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥âˆ¥B{x}_{n}âˆ’Bpâˆ¥.\end{array}$

In view of (2.8) and (2.9), we see from restriction (c) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥=0.$
(2.11)

On the other hand, we find from (2.5) that

$\begin{array}{rl}\left(1âˆ’{\mathrm{Î±}}_{n}\right)\left(1âˆ’{r}_{n}^{2}{\mathrm{Î±}}^{2}\right){âˆ¥{v}_{n}âˆ’{z}_{n}âˆ¥}^{2}& â‰¤{âˆ¥{x}_{n}âˆ’pâˆ¥}^{2}âˆ’{âˆ¥{y}_{n}âˆ’pâˆ¥}^{2}\\ â‰¤âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥\left(âˆ¥{x}_{n}âˆ’pâˆ¥+âˆ¥{y}_{n}âˆ’pâˆ¥\right).\end{array}$

In view of restrictions (a) and (c), we obtain from (2.8) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{v}_{n}âˆ’{z}_{n}âˆ¥=0.$
(2.12)

Notice that

$\begin{array}{rl}{âˆ¥{u}_{n}âˆ’{z}_{n}âˆ¥}^{2}& ={âˆ¥{P}_{C}\left({v}_{n}âˆ’{r}_{n}A{z}_{n}\right)âˆ’{P}_{C}\left({v}_{n}âˆ’{r}_{n}A{v}_{n}\right)âˆ¥}^{2}\\ â‰¤{âˆ¥\left({v}_{n}âˆ’{r}_{n}A{z}_{n}\right)âˆ’\left({v}_{n}âˆ’{r}_{n}A{v}_{n}\right)âˆ¥}^{2}\\ â‰¤{r}_{n}^{2}{\mathrm{Î±}}^{2}{âˆ¥{z}_{n}âˆ’{v}_{n}âˆ¥}^{2}.\end{array}$

Thanks to (2.12), we arrive at

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{u}_{n}âˆ’{z}_{n}âˆ¥=0.$
(2.13)

Notice that

$\begin{array}{rl}âˆ¥{x}_{n}âˆ’S{x}_{n}âˆ¥& â‰¤âˆ¥{x}_{n}âˆ’S{u}_{n}âˆ¥+âˆ¥S{u}_{n}âˆ’S{x}_{n}âˆ¥\\ â‰¤\frac{âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥}{1âˆ’{\mathrm{Î±}}_{n}}+âˆ¥{u}_{n}âˆ’{x}_{n}âˆ¥\\ â‰¤\frac{âˆ¥{x}_{n}âˆ’{y}_{n}âˆ¥}{1âˆ’{\mathrm{Î±}}_{n}}+âˆ¥{u}_{n}âˆ’{z}_{n}âˆ¥+âˆ¥{z}_{n}âˆ’{v}_{n}âˆ¥+âˆ¥{v}_{n}âˆ’{x}_{n}âˆ¥.\end{array}$

In view of (2.8), (2.11), (2.12), and (2.13), we find from restriction (c) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’S{x}_{n}âˆ¥=0.$

Since $\left\{{x}_{n}\right\}$ is bounded, we find that there exists a subsequence $\left\{{x}_{{n}_{i}}\right\}$ of $\left\{{x}_{n}\right\}$ such that ${x}_{{n}_{i}}â‡€qâˆˆC$. From Lemma 1.2, we easily conclude that $qâˆˆF\left(S\right)$.

Now, we are in a position to show that $xâˆˆVI\left(C,A\right)$. Define

$Wx=\left\{\begin{array}{cc}Ax+{N}_{C}x,\hfill & xâˆˆC,\hfill \\ \mathrm{âˆ\dots },\hfill & xâˆ‰C.\hfill \end{array}$

For any given $\left(x,y\right)âˆˆG\left(W\right)$, we have $yâˆ’Axâˆˆ{N}_{C}x$. Since ${u}_{n}âˆˆC$, we see from the definition of ${N}_{C}$

$ã€ˆxâˆ’{u}_{n},yâˆ’Axã€‰â‰¥0.$
(2.14)

In view of ${u}_{n}={P}_{C}\left({v}_{n}âˆ’{r}_{n}A{z}_{n}\right)$, we obtain that

$ã€ˆxâˆ’{u}_{n},{u}_{n}+{r}_{n}A{z}_{n}âˆ’{v}_{n}ã€‰â‰¥0$

and hence

$ã€ˆxâˆ’{u}_{n},\frac{{u}_{n}âˆ’{v}_{n}}{{r}_{n}}+A{z}_{n}ã€‰â‰¥0.$
(2.15)

In view of (2.14) and (2.15), we find that

$\begin{array}{rl}ã€ˆxâˆ’{u}_{{n}_{i}},yã€‰â‰¥& ã€ˆxâˆ’{u}_{{n}_{i}},Axã€‰\\ â‰¥& ã€ˆxâˆ’{u}_{{n}_{i}},Axã€‰âˆ’ã€ˆxâˆ’{u}_{{n}_{i}},\frac{{u}_{{n}_{i}}âˆ’{v}_{{n}_{i}}}{{r}_{{n}_{i}}}+A{z}_{{n}_{i}}ã€‰\\ =& ã€ˆxâˆ’{u}_{{n}_{i}},Axâˆ’A{u}_{{n}_{i}}ã€‰+ã€ˆxâˆ’{u}_{{n}_{i}},A{u}_{{n}_{i}}âˆ’A{z}_{{n}_{i}}ã€‰\\ âˆ’ã€ˆxâˆ’{u}_{{n}_{i}},\frac{{u}_{{n}_{i}}âˆ’{v}_{{n}_{i}}}{{r}_{{n}_{i}}}ã€‰\\ â‰¥& ã€ˆxâˆ’{u}_{{n}_{i}},A{u}_{{n}_{i}}âˆ’A{z}_{{n}_{i}}ã€‰âˆ’ã€ˆxâˆ’{u}_{{n}_{i}},\frac{{u}_{{n}_{i}}âˆ’{v}_{{n}_{i}}}{{r}_{{n}_{i}}}ã€‰.\end{array}$
(2.16)

Notice that

$âˆ¥{x}_{n}âˆ’{u}_{n}âˆ¥â‰¤âˆ¥{x}_{n}âˆ’{v}_{n}âˆ¥+âˆ¥{v}_{n}âˆ’{z}_{n}âˆ¥+âˆ¥{z}_{n}âˆ’{u}_{n}âˆ¥.$

It follows from (2.11), (2.12), and (2.13) that

$\underset{nâ†’\mathrm{âˆž}}{lim}âˆ¥{x}_{n}âˆ’{u}_{n}âˆ¥=0.$

Since A is Lipschitz continuous, we find from (2.16) that

$ã€ˆxâˆ’q,yã€‰â‰¥0.$

Since W is maximal monotone, we conclude that $qâˆˆ{W}^{âˆ’1}\left(0\right)$. This proves that $qâˆˆVI\left(C,A\right)$.

Finally, we prove that $qâˆˆ{\left(B+M\right)}^{âˆ’1}\left(0\right)$. Notice that

${x}_{n}âˆ’{s}_{n}B{x}_{n}âˆˆ{v}_{n}+{s}_{n}M{v}_{n};$

that is,

$\frac{{x}_{n}âˆ’{v}_{n}}{{s}_{n}}âˆ’B{x}_{n}âˆˆM{v}_{n}.$
(2.17)

Let $\mathrm{Î¼}âˆˆM\mathrm{Î½}$. Since M is monotone, we find from (2.17)

$ã€ˆ\frac{{x}_{n}âˆ’{v}_{n}}{{s}_{n}}âˆ’B{x}_{n}âˆ’\mathrm{Î¼},{v}_{n}âˆ’\mathrm{Î½}ã€‰â‰¥0.$

In view of the restriction (b), we see that

$ã€ˆâˆ’Bqâˆ’\mathrm{Î¼},qâˆ’\mathrm{Î½}ã€‰â‰¥0.$

This implies that $âˆ’BqâˆˆMq$, that is, $qâˆˆ{\left(B+M\right)}^{âˆ’1}\left(0\right)$. This completes $qâˆˆ\mathcal{F}$. Assume that there exists another subsequence $\left\{{x}_{{n}_{j}}\right\}$ of $\left\{{x}_{n}\right\}$ which converges weakly to ${q}^{â€²}âˆˆ\mathcal{F}$. We can easily conclude from Opialâ€™s condition that $q={q}^{â€²}$.

Finally, we show that $q={Proj}_{\mathcal{F}}{x}_{1}$ and $\left\{{x}_{n}\right\}$ converges strongly to q. This completes the proof of Theorem 2.1. In view of the weak lower semicontinuity of the norm, we obtain from (2.6) that

$\begin{array}{rl}âˆ¥{x}_{1}âˆ’{Proj}_{\mathcal{F}}{x}_{1}âˆ¥& â‰¤âˆ¥{x}_{1}âˆ’qâˆ¥â‰¤\underset{nâ†’\mathrm{âˆž}}{limâ€‰inf}âˆ¥{x}_{1}âˆ’{x}_{n}âˆ¥\\ â‰¤\underset{nâ†’\mathrm{âˆž}}{limâ€‰sup}âˆ¥{x}_{1}âˆ’{x}_{n}âˆ¥â‰¤âˆ¥{x}_{1}âˆ’{Proj}_{\mathcal{F}}{x}_{1}âˆ¥,\end{array}$

which yields that ${lim}_{nâ†’\mathrm{âˆž}}âˆ¥{x}_{1}âˆ’{x}_{n}âˆ¥=âˆ¥{x}_{1}âˆ’{Proj}_{\mathcal{F}}{x}_{1}âˆ¥=âˆ¥{x}_{1}âˆ’qâˆ¥$. It follows that $\left\{{x}_{n}\right\}$ converges strongly to ${Proj}_{\mathcal{F}}{x}_{1}$. This completes the proof.â€ƒâ–¡

If $B=0$, then Theorem 2.1 is reduced to the following.

Corollary 2.2 Let C be a nonempty, closed, and convex subset of H. Let$S:Câ†’C$be a nonexpansive mapping with a nonempty fixed point set and$A:Câ†’H$be an Î±-Lipschitz continuous and monotone mapping. Let$M:Hâ‡‰H$be a maximal monotone operator such that$D\left(M\right)âŠ‚C$. Assume that$\mathcal{F}:=F\left(S\right)âˆ©{M}^{âˆ’1}\left(0\right)âˆ©VI\left(C,A\right)$is not empty. Let$\left\{{x}_{n}\right\}$be a sequence generated by the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {C}_{1}=C,\hfill \\ {z}_{n}={Proj}_{C}\left({J}_{{s}_{n}}{x}_{n}âˆ’{r}_{n}A{J}_{{s}_{n}}{x}_{n}\right),\hfill \\ {y}_{n}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{Proj}_{C}\left({J}_{{s}_{n}}{x}_{n}âˆ’{r}_{n}A{z}_{n}\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}nâ‰¥0,\hfill \end{array}$

where${J}_{{s}_{n}}={\left(I+{s}_{n}M\right)}^{âˆ’1}$, $\left\{{r}_{n}\right\}$is a sequence in$\left(0,\frac{1}{\mathrm{Î±}}\right)$, $\left\{{s}_{n}\right\}$is a sequence in$\left(0,+\mathrm{âˆž}\right)$, and$\left\{{\mathrm{Î±}}_{n}\right\}$is a sequence in$\left(0,1\right)$. Assume that the following restrictions are satisfied:

1. (a)

$0;

2. (b)

$0;

3. (c)

$0â‰¤{\mathrm{Î±}}_{n}â‰¤d<1$,

where a, b, c, and d are real constants. Then the sequence$\left\{{x}_{n}\right\}$converges strongly to${Proj}_{\mathcal{F}}{x}_{1}$.

If $M=0$, then ${J}_{{s}_{n}}=I$. Corollary 2.2 is reduced to the following.

Corollary 2.3 Let C be a nonempty, closed, and convex subset of H. Let$S:Câ†’C$be a nonexpansive mapping with a nonempty fixed point set and$A:Câ†’H$be an Î±-Lipschitz continuous and monotone mapping. Assume that$\mathcal{F}:=F\left(S\right)âˆ©VI\left(C,A\right)$is not empty. Let$\left\{{x}_{n}\right\}$be a sequence generated by the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {C}_{1}=C,\hfill \\ {z}_{n}={Proj}_{C}\left({x}_{n}âˆ’{r}_{n}A{x}_{n}\right),\hfill \\ {y}_{n}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{Proj}_{C}\left({x}_{n}âˆ’{r}_{n}A{z}_{n}\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}nâ‰¥0,\hfill \end{array}$

where$\left\{{r}_{n}\right\}$is a sequence in$\left(0,\frac{1}{\mathrm{Î±}}\right)$, and$\left\{{\mathrm{Î±}}_{n}\right\}$is a sequence in$\left(0,1\right)$. Assume that the following restrictions are satisfied:

1. (a)

$0;

2. (b)

$0â‰¤{\mathrm{Î±}}_{n}â‰¤c<1$,

where a, b, and c are real constants. Then the sequence$\left\{{x}_{n}\right\}$converges strongly to${Proj}_{\mathcal{F}}{x}_{1}$.

If $A=0$, then Theorem 2.1 is reduced to the following.

Corollary 2.4 Let C be a nonempty, closed, and convex subset of H. Let$S:Câ†’C$be a nonexpansive mapping with a nonempty fixed point set and$B:Câ†’H$be a Î²-inverse-strongly monotone mapping. Let$M:Hâ‡‰H$be a maximal monotone operator such that$D\left(M\right)âŠ‚C$. Assume that$\mathcal{F}:=F\left(S\right)âˆ©{\left(B+M\right)}^{âˆ’1}\left(0\right)$is not empty. Let$\left\{{x}_{n}\right\}$be a sequence generated by the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {C}_{1}=C,\hfill \\ {y}_{n}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{J}_{{s}_{n}}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}nâ‰¥0,\hfill \end{array}$

where${J}_{{s}_{n}}={\left(I+{s}_{n}M\right)}^{âˆ’1}$, $\left\{{s}_{n}\right\}$is a sequence in$\left(0,2\mathrm{Î²}\right)$, and$\left\{{\mathrm{Î±}}_{n}\right\}$is a sequence in$\left(0,1\right)$. Assume that the following restrictions are satisfied:

1. (a)

$0;

2. (b)

$0â‰¤{\mathrm{Î±}}_{n}â‰¤c<1$,

where a, b, and c are real constants. Then the sequence$\left\{{x}_{n}\right\}$converges strongly to${Proj}_{\mathcal{F}}{x}_{1}$.

Let $f:Hâ†’\left(âˆ’\mathrm{âˆž},+\mathrm{âˆž}\right]$ be a proper convex lower semicontinuous function. Then the subdifferential âˆ‚f of f is defined as follows:

$\mathrm{âˆ‚}f\left(x\right)=\left\{yâˆˆH:f\left(z\right)â‰¥f\left(x\right)+ã€ˆzâˆ’x,yã€‰,\phantom{\rule{0.2em}{0ex}}zâˆˆH\right\},\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}xâˆˆH.$

From Rockafellar [34], we know that âˆ‚f is maximal monotone. It is not hard to verify that $0âˆˆ\mathrm{âˆ‚}f\left(x\right)$ if and only if $f\left(x\right)={min}_{yâˆˆH}f\left(y\right)$.

Let ${I}_{C}$ be the indicator function of C, i.e.,

${I}_{C}\left(x\right)=\left\{\begin{array}{cc}0,\hfill & xâˆˆC,\hfill \\ +\mathrm{âˆž},\hfill & xâˆ‰C.\hfill \end{array}$

Since ${I}_{C}$ is a proper lower semicontinuous convex function on H, we see that the subdifferential $\mathrm{âˆ‚}{I}_{C}$ of ${I}_{C}$ is a maximal monotone operator. It is clear that ${J}_{s}x={P}_{C}x$, $\mathrm{âˆ€}xâˆˆH$. Notice that ${\left(B+\mathrm{âˆ‚}{I}_{C}\right)}^{âˆ’1}\left(0\right)=VI\left(C,B\right)$. Indeed,

$\begin{array}{r}xâˆˆ{\left(B+\mathrm{âˆ‚}{I}_{C}\right)}^{âˆ’1}\left(0\right)\phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}0âˆˆBx+\mathrm{âˆ‚}{I}_{C}x\\ \phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}âˆ’Bxâˆˆ\mathrm{âˆ‚}{I}_{C}x\\ \phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}ã€ˆBx,yâˆ’xã€‰â‰¥0\\ \phantom{\rule{1em}{0ex}}âŸº\phantom{\rule{1em}{0ex}}xâˆˆVI\left(C,B\right).\end{array}$
(2.18)

In the light of the above, the following is not hard to derive from Corollary 2.4.

Corollary 2.5 Let C be a nonempty, closed, and convex subset of H. Let$S:Câ†’C$be a nonexpansive mapping with a nonempty fixed point set and$B:Câ†’H$be a Î²-inverse-strongly monotone mapping. Assume that$\mathcal{F}:=F\left(S\right)âˆ©VI\left(C,B\right)$is not empty. Let$\left\{{x}_{n}\right\}$be a sequence generated by the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {C}_{1}=C,\hfill \\ {y}_{n}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{P}_{C}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right),\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}nâ‰¥0,\hfill \end{array}$

where$\left\{{s}_{n}\right\}$is a sequence in$\left(0,2\mathrm{Î²}\right)$, and$\left\{{\mathrm{Î±}}_{n}\right\}$is a sequence in$\left(0,1\right)$. Assume that the following restrictions are satisfied:

1. (a)

$0;

2. (b)

$0â‰¤{\mathrm{Î±}}_{n}â‰¤c<1$,

where a, b, and c are real constants. Then the sequence$\left\{{x}_{n}\right\}$converges strongly to${Proj}_{\mathcal{F}}{x}_{1}$.

3 Applications

First, we consider the problem of finding a minimizer of a proper convex lower semicontinuous function.

Theorem 3.1 Let$f:Hâ†’\left(âˆ’\mathrm{âˆž},+\mathrm{âˆž}\right]$be a proper convex lower semicontinuous function such that${\left(\mathrm{âˆ‚}f\right)}^{âˆ’1}\left(0\right)$is not empty. Let$\left\{{x}_{n}\right\}$be a sequence generated by the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆH,\hfill \\ {C}_{1}=H,\hfill \\ {y}_{n}=arg{min}_{sâˆˆH}\left\{f\left(z\right)+\frac{{âˆ¥zâˆ’{x}_{n}âˆ¥}^{2}}{2{s}_{n}}\right\},\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}nâ‰¥0,\hfill \end{array}$

where$\left\{{s}_{n}\right\}$is a positive sequence such that$0, where a is a real constant. Then the sequence$\left\{{x}_{n}\right\}$converges strongly to${Proj}_{{\left(\mathrm{âˆ‚}f\right)}^{âˆ’1}\left(0\right)}{x}_{1}$.

Proof Putting $A=B=0$, $S=I$, and ${\mathrm{Î±}}_{n}â‰¡0$, we can immediately draw the desired conclusion from Theorem 2.1.â€ƒâ–¡

Second, we consider the problem of approximating a common fixed point of a pair of nonexpansive mappings.

Theorem 3.2 Let C be a nonempty, closed, and convex subset of H. Let$S:Câ†’C$and$T:Câ†’C$be a pair of nonexpansive mappings with a nonempty common fixed point set. Let$\left\{{x}_{n}\right\}$be a sequence generated by the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {C}_{1}=C,\hfill \\ {z}_{n}=\left(1âˆ’{s}_{n}\right){x}_{n}+{s}_{n}T{x}_{n},\hfill \\ {y}_{n}={\mathrm{Î±}}_{n}{x}_{n}+\left(1âˆ’{\mathrm{Î±}}_{n}\right)S{z}_{n},\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}nâ‰¥0,\hfill \end{array}$

where$\left\{{s}_{n}\right\}$, and$\left\{{\mathrm{Î±}}_{n}\right\}$are sequences in$\left(0,1\right)$. Assume that the following restrictions are satisfied:

1. (a)

$0;

2. (b)

$0â‰¤{\mathrm{Î±}}_{n}â‰¤c<1$,

where a, b, and c are real constants. Then the sequence$\left\{{x}_{n}\right\}$converges strongly to${Proj}_{F\left(S\right)âˆ©F\left(T\right)}{x}_{1}$.

Proof Putting $A=0$, $M=\mathrm{âˆ‚}{I}_{C}$, and $B=Iâˆ’T$, we see that B is $\frac{1}{2}$-inverse-strongly monotone. We also have $F\left(T\right)$=$VI\left(C,B\right)$ and ${P}_{C}\left({x}_{n}âˆ’{s}_{n}B{x}_{n}\right)=\left(1âˆ’{s}_{n}\right){x}_{n}+{s}_{n}T{x}_{n}$. In view of (2.18), we can immediately obtain the desired result.â€ƒâ–¡

Let F be a bifunction of $CÃ—C$ into $\mathbb{R}$, where $\mathbb{R}$ denotes the set of real numbers. Recall the following equilibrium problem in the terminology of Blum and Oettli [35] (see also Fan [36]):

$\text{Find}\phantom{\rule{0.25em}{0ex}}xâˆˆC\phantom{\rule{0.25em}{0ex}}\text{such}\phantom{\rule{0.25em}{0ex}}\text{that}\phantom{\rule{0.25em}{0ex}}F\left(x,y\right)â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆC.$
(3.1)

To study the equilibrium problem (3.1), we may assume that F satisfies the following conditions:

(A1) $F\left(x,x\right)=0$ for all $xâˆˆC$;

(A2) F is monotone, i.e., $F\left(x,y\right)+F\left(y,x\right)â‰¤0$ for all $x,yâˆˆC$;

(A3) for each $x,y,zâˆˆC$,

$\underset{tâ†“0}{limâ€‰sup}F\left(tz+\left(1âˆ’t\right)x,y\right)â‰¤F\left(x,y\right);$

(A4) for each $xâˆˆC$, $yâ†¦F\left(x,y\right)$ is convex and lower semi-continuous.

Putting $F\left(x,y\right)=ã€ˆAx,yâˆ’xã€‰$ for every $x,yâˆˆC$, we see that the equilibrium problem (3.3) is reduced to the variational inequality (1.1).

The following lemma can be found in [35] and [37].

Lemma 3.3 Let C be a nonempty, closed, and convex subset of H and$F:CÃ—Câ†’\mathbb{R}$be a bifunction satisfying (A1)-(A4). Then, for any$s>0$and$xâˆˆH$, there exists$zâˆˆC$such that

$F\left(z,y\right)+\frac{1}{s}ã€ˆyâˆ’z,zâˆ’xã€‰â‰¥0,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}yâˆˆC.$

Further, define

${T}_{s}x=\left\{zâˆˆC:F\left(z,y\right)+\frac{1}{s}ã€ˆyâˆ’z,zâˆ’xã€‰â‰¥0,\mathrm{âˆ€}yâˆˆC\right\}$
(3.2)

for all$s>0$and$xâˆˆH$. Then, the following hold:

1. (a)

${T}_{s}$ is single-valued;

2. (b)

${T}_{s}$ is firmly nonexpansive; that is,

${âˆ¥{T}_{s}xâˆ’{T}_{s}yâˆ¥}^{2}â‰¤ã€ˆ{T}_{s}xâˆ’{T}_{s}y,xâˆ’yã€‰,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}x,yâˆˆH;$
3. (c)

$F\left({T}_{s}\right)=EP\left(F\right)$;

4. (d)

$EP\left(F\right)$ is closed and convex.

Lemma 3.4[30]

Let C be a nonempty, closed, and convex subset of H, F be a bifunction from$CÃ—C$to$\mathbb{R}$which satisfies (A1)-(A4), and${A}_{F}$be a multivalued mapping of H into itself defined by

${A}_{F}x=\left\{\begin{array}{cc}\left\{zâˆˆH:F\left(x,y\right)â‰¥ã€ˆyâˆ’x,zã€‰,\mathrm{âˆ€}yâˆˆC\right\},\hfill & xâˆˆC,\hfill \\ \mathrm{âˆ\dots },\hfill & xâˆ‰C.\hfill \end{array}$
(3.3)

Then${A}_{F}$is a maximal monotone operator with the domain$D\left({A}_{F}\right)âŠ‚C$, $EP\left(F\right)={A}_{F}^{âˆ’1}\left(0\right)$, where$FP\left(F\right)$stands for the solution set of (3.1), and

${T}_{s}x={\left(I+s{A}_{F}\right)}^{âˆ’1}x,\phantom{\rule{1em}{0ex}}\mathrm{âˆ€}xâˆˆH,r>0,$

where${T}_{s}$is defined as in (3.3).

Finally, we consider finding a solution of the equilibrium problem.

Theorem 3.5 Let C be a nonempty, closed, and convex subset of H. Let$F:CÃ—Câ†’\mathbb{R}$be a bifunction satisfying (A1)-(A4) such that. Let$\left\{{x}_{n}\right\}$be a sequence generated by the following iterative process:

$\left\{\begin{array}{c}{x}_{1}âˆˆC,\hfill \\ {C}_{1}=C,\hfill \\ {y}_{n}={\left(I+{s}_{n}{A}_{F}\right)}^{âˆ’1}{x}_{n},\hfill \\ {C}_{n+1}=\left\{vâˆˆ{C}_{n}:âˆ¥{y}_{n}âˆ’vâˆ¥â‰¤âˆ¥{x}_{n}âˆ’vâˆ¥\right\},\hfill \\ {x}_{n+1}={Proj}_{{C}_{n+1}}{x}_{1},\phantom{\rule{1em}{0ex}}nâ‰¥0,\hfill \end{array}$

where${A}_{F}$is defined as (3.3), and$\left\{{s}_{n}\right\}$is a positive sequence such that$0, where a is a real constant Then the sequence$\left\{{x}_{n}\right\}$converges strongly to${Proj}_{FP\left(F\right)}{x}_{1}$.

Proof Putting $A=B=0$, $S=I$ and ${\mathrm{Î±}}_{n}â‰¡0$, we immediately reach the desired conclusion from Lemma 3.4.â€ƒâ–¡

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Acknowledgements

The first author was supported by the National Natural Science Foundation of China (11071169, 11271105), the Natural Science Foundation of Zhejiang Province (Y6110287, Y12A010095).

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Correspondence to Jing Lu.

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The authors declare that they have no competing interests.

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Both authors contributed equally to this work. Both authors read and approved the final manuscript.

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Gu, F., Lu, J. Projection methods of iterative solutions in Hilbert spaces. Fixed Point Theory Appl 2012, 162 (2012). https://doi.org/10.1186/1687-1812-2012-162