Constructed nets with perturbations for equilibrium and fixed point problems

Abstract

In this paper, an implicit net with perturbations for solving the mixed equilibrium problems and fixed point problems has been constructed and it is shown that the proposed net converges strongly to a common solution of the mixed equilibrium problems and fixed point problems. Also, as applications, some corollaries for solving the minimum-norm problems are also included.

MSC:47J05, 47J25, 47H09.

1 Introduction

The present paper is devoted to solving the following mixed equilibrium problem: Find $u\in C$ such that

$$F(u,v)+〈Au,v-u〉\ge 0$$

(1.1)

for all $v\in C$, where C is a nonempty closed convex subset of a real Hilbert space H, $F:C\times C\to R$ is a bifunction and $A:C\to H$ is a nonlinear operator. The solution set of (1.1) is denoted by S(MEP).

This problem (1.1) includes optimization problems, variational inequalities, minimax problems, and Nash equilibrium problems in noncooperative games as special cases.

Case 1. If $A=0$ in (1.1), then (1.1) reduces to the following equilibrium problem: Find $u\in C$ such that

$$F(u,v)\ge 0$$

(1.2)

for all $v\in C$. The solution of (1.2) is denoted by S(EP).

Case 2. If $F=0$ in (1.1), then (1.1) reduces to the variational inequality problem: Find $z\in C$ such that

$$〈Au,v-u〉\ge 0$$

(1.3)

for all $v\in C$. The solution of (1.2) is denoted by S(VI).

In the literature, there are a large number of references associated with some equilibrium problems and variational inequality problems (see, for instance, [1–32]).

The main purpose of the present paper is to construct the following implicit net with perturbations for solving the mixed equilibrium (1.1) and the fixed point problem:

$$F(z_t,y)+\frac{1}{\lambda }〈y-z_t,z_t-(t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t)〉\ge 0$$

for all $y\in C$. Also, it is shown that the proposed net $\{z_t\}$ converges strongly to a common solution of the mixed equilibrium problems and fixed point problems. As applications, some corollaries for solving the minimum-norm problems are also included.

2 Preliminaries

Let H be a real Hilbert space with an inner product $〈\cdot ,\cdot 〉$ and a norm $\parallel \cdot \parallel$, respectively, and C be a nonempty closed convex subset of a real Hilbert space H.

• (1) A mapping $T:C\to C$ is said to be nonexpansive if

  $$\parallel Tu-Tv\parallel \le \parallel u-v\parallel$$

for all $u,v\in C$. $F(T)$ denotes the set of fixed points of T.

• (2) A mapping $A:C\to H$ is said to be α-inverse-strongly monotone if there exists a positive real number $\alpha >0$ such that

  $$〈Au-Av,u-v〉\ge \alpha {\parallel Au-Av\parallel }^2$$

for all $u,v\in C$. It is clear that any α-inverse-strongly monotone mapping is monotone and $\frac{1}{\alpha }$-Lipschitz continuous.

Throughout this paper, we assume that a bifunction $F:C\times C\to R$ satisfies the following conditions:

(C1) $F(u,u)=0$ for all $u\in C$;

(C2) F is monotone, i.e., $F(u,v)+F(v,u)\le 0$ for all $u,v\in C$;

(C3) for each $u,v,w\in C$, ${lim}_{t\downarrow 0}F(tw+(1-t)u,v)\le F(u,v)$;

(C4) for each $u\in C$, $v\mapsto F(u,v)$ is convex and lower semicontinuous.

In fact, some efforts to construct the algorithms for solving the equilibrium problems have been carried out. For instance, Moudafi [15] presented an iterative algorithm and proved a weak convergence theorem for solving the mixed equilibrium problem (1.1). Takahashi and Takahashi [24] constructed the following iterative algorithm:

$$\{\begin{array}{c}F(z_n,y)+〈A{x}_n,y-z_n〉+\frac{1}{{\lambda }_n}〈y-z_n,z_n-x_n〉\ge 0,\hfill \\ x_{n+1}={\beta }_n{x}_n+T({\alpha }_{n}u+(1-{\beta }_n)z_n)\hfill \end{array}$$

(2.1)

for all $y\in C$ and $n\ge 0$, where $T:C\to C$ is a nonexpansive mapping. They proved that the sequence $\{x_n\}$ generated by (2.1) converges strongly to $z={Proj}_{F(T)\cap S(\mathit{MEP})}(u)$.

Plubtieng and Punpaeng [20] introduced and considered the following two iterative schemes for finding a common element of the set of solutions of the problem (1.2) and the set of fixed points of a nonexpansive mapping in a Hilbert space H.

Implicit iterative algorithm $\{x_n\}$:

$$\{\begin{array}{c}F(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\hfill \\ x_n={\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T{u}_n\hfill \end{array}$$

(2.2)

for all $y\in H$ and $n\ge 1$.

Explicit iterative algorithm $\{x_n\}$:

$$\{\begin{array}{c}F(u_n,y)+\frac{1}{r_n}〈y-u_n,u_n-x_n〉\ge 0,\hfill \\ x_{n+1}={\alpha }_n\gamma f(x_n)+(I-{\alpha }_{n}A)T{u}_n\hfill \end{array}$$

(2.3)

for all $y\in H$ and $n\ge 1$.

They proved that, under certain conditions, the sequences $\{x_n\}$ generated by (2.2) and (2.3) converge strongly to the unique solution of the variational inequality:

$$〈(A-\gamma f)z,x-z〉\ge 0$$

for all $x\in F(T)\cap S(\mathit{EP})$, which is the optimality condition for the minimization problem:

$$\underset{x\in F(T)\cap S(\mathit{EP})}{min}\frac{1}{2}〈Ax,x〉-h(x),$$

where h is a potential function for γf.

We know that there are perturbations always occurring in the iterative processes because the manipulations are inaccurate. Recently, Chuang et al. ([8]) constructed the following iteration process with perturbations for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points for a quasi-nonexpansive mapping with perturbation: $q_1\in H$ and

$$\{\begin{array}{c}{x}_n\in C\text{such that}F(x_n,y)+\frac{1}{{\lambda }_n}〈y-x_n,x_n-q_n〉\ge 0,\hfill \\ q_{n+1}={\alpha }_n{u}_n+(1-{\alpha }_n)({\beta }_n{x}_n+(1-{\beta }_n)T{x}_n)\hfill \end{array}$$

for all $y\in C$ and $n\ge 0$. They shown that the sequence $\{q_n\}$ converges strongly to ${Proj}_{F(T)\cap S(\mathit{EP})}$.

Now, we need the following useful lemmas for our main results.

Lemma 2.1 [11]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let $F:C\times C\to R$ be a bifunction which satisfies the conditions (C1)-(C4). Let $r>0$ and $x\in H$. Then there exists $z\in C$ such that

$$F(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0$$

for all $y\in C$. Further, if

$$T_r(x)=\{z\in C:F(z,y)+\frac{1}{r}〈y-z,z-x〉\ge 0,\mathrm{\forall }y\in C\},$$

then the following hold:

• (1) $T_r$ is single-valued and $T_r$ is firmly nonexpansive, i.e., for any $x,y\in H$,

  $${\parallel T_{r}x-T_{r}y\parallel }^2\le 〈T_{r}x-T_{r}y,x-y〉;$$

• (2) $S(\mathit{EP})$ is closed and convex and $S(\mathit{EP})=F(T_r)$.

Lemma 2.2 [24]

Let C, H, F, and $T_{r}x$ be as in Lemma  2.1. Then the following holds:

$${\parallel T_{s}x-T_{t}x\parallel }^2\le \frac{s-t}{s}〈T_{s}x-T_{t}x,T_{s}x-x〉$$

for all $s,t>0$ and $x\in H$.

Lemma 2.3 [24]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let a mapping $A:C\to H$ be α-inverse-strongly monotone and $r>0$ be a constant. Then we have

$${\parallel (I-rA)x-(I-rA)y\parallel }^2\le {\parallel x-y\parallel }^2+r(r-2\alpha ){\parallel Ax-Ay\parallel }^2$$

for all $x,y\in C$. In particular, if $0\le r\le 2\alpha$, then $I-rA$ is nonexpansive.

Lemma 2.4 [33]

Let C be a closed convex subset of a real Hilbert space H and let $T:C\to C$ be a nonexpansive mapping. Then the mapping $I-T$ is demiclosed, that is, if $\{x_n\}$ is a sequence in C such that $x_n\to u$ weakly and $(I-T)x_n\to v$ strongly, then $(I-T)u=v$.

3 Convergence results

In this section, first, we give our main results.

Part I: Assumptions on the setting of C, F, A, and T:

(A1) C is a nonempty closed convex subset of a real Hilbert space H;

(A2) $F:C\times C\to R$ is a bifunction satisfying the conditions (C1)-(C4);

(A3) $A:C\to H$ is an α-inverse-strongly monotone mapping;

(A4) $T:C\to C$ is a nonexpansive mapping.

Part II: Parameter restrict:

λ is a constant satisfying $a\le \lambda \le b$, where $[a,b]\subset (0,2\alpha )$.

Part III: Perturbations:

$\{u_t\}\subset H$ is a net satisfying ${lim}_{t\to 0+}{u}_t=u\in H$.

Algorithm 3.1 For any $t\in (0,1-\frac{\lambda }{2\alpha })$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$F(z_t,y)+\frac{1}{\lambda }〈y-z_t,z_t-(t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t)〉\ge 0$$

(3.1)

for all $y\in C$.

Remark 3.2 We show that the net $\{z_t\}$ is well defined. Next, we prove that (3.1) can be rewritten as

$$z_t=T_{\lambda }(t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t)$$

(3.2)

for all $t\in (0,1-\frac{\lambda }{2\alpha })$.

In fact, for any $t\in (0,1-\frac{\lambda }{2\alpha })$, $u_t\in H$, and $x\in H$, we find z such that, for all $y\in C$,

$$F(z,y)+\frac{1}{\lambda }〈y-z,z-(t{u}_t+(1-t)Tx-\lambda ATx)〉\ge 0.$$

From Lemma 2.1, we get immediately

$$z=T_{\lambda }(t{u}_t+(1-t)Tx-\lambda ATx).$$

Now, we can define a mapping

$${\vartheta }_t:=T_{\lambda }(t{u}_t+(1-t)T-\lambda AT)$$

for all $t\in (0,1-\frac{\lambda }{2\alpha })$. Again, from Lemma 2.1, we know that $T_{\lambda }$ is nonexpansive. Thus, for any $x,y\in C$, we have

$$\begin{array}{r}\parallel {\vartheta }_{t}x-{\vartheta }_{t}y\parallel \\ =\parallel T_{\lambda }(t{u}_t+(1-t)Tx-\lambda ATx)-T_{\lambda }(t{u}_t+(1-t)Ty-\lambda ATy)\parallel \\ \le \parallel (t{u}_t+(1-t)Tx-\lambda ATx)-(t{u}_t+(1-t)Ty-\lambda ATy)\parallel \\ =(1-t)\parallel (I-\frac{\lambda }{1-t}A)Tx-(I-\frac{\lambda }{1-t}A)Ty\parallel .\end{array}$$

(3.3)

From Lemma 2.3, $I-\frac{\lambda }{1-t}A$ is nonexpansive for all $t\in (0,1-\frac{\lambda }{2\alpha })$. Note that T is also nonexpansive. By (3.3), we deduce

$$\parallel {\vartheta }_{t}x-{\vartheta }_{t}y\parallel \le (1-t)\parallel x-y\parallel$$

for all $x,y\in C$. This indicates that ${\vartheta }_t$ is a contraction on C and so it has a unique fixed point, denoted by $z_t$, in C. That is, $z_t=T_{\lambda }(t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t)$. Hence $\{z_t\}$ is well defined.

Theorem 3.3 Suppose that $F(T)\cap S(\mathit{MEP})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (3.1) converges strongly as $t\to 0+$ to $P_{F(T)\cap S(\mathit{MEP})}(u)$.

Proof Pick up $z\in F(T)\cap S(\mathit{MEP})$. It is obvious that $z=T_{\lambda }(z-\lambda Az)$ for all $\lambda >0$. So, we have

$$z=Tz=T_{\lambda }(z-\lambda Az)=T_{\lambda }(Tz-\lambda ATz)=T_{\lambda }(tTz+(1-t)(Tz-\frac{\lambda }{1-t}ATz))$$

for all $t\in (0,1-\frac{\lambda }{2\alpha })$. Then we have

$$\begin{array}{r}{\parallel z_t-z\parallel }^2\\ ={\parallel T_{\lambda }(t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t)-z\parallel }^2\\ ={\parallel T_{\lambda }(t{u}_t+(1-t)(T{z}_t-\frac{\lambda }{1-t}AT{z}_t))-T_{\lambda }(tz+(1-t)(Tz-\frac{\lambda }{1-t}ATz))\parallel }^2\\ \le {\parallel (t{u}_t+(1-t)(T{z}_t-\frac{\lambda }{1-t}AT{z}_t))-(tz+(1-t)(Tz-\frac{\lambda }{1-t}ATz))\parallel }^2\\ ={\parallel (1-t)((T{z}_t-\frac{\lambda }{1-t}AT{z}_t)-(Tz-\frac{\lambda }{1-t}ATz))+t(u_t-z)\parallel }^2.\end{array}$$

(3.4)

Using the convexity of $\parallel \cdot \parallel$ and the inverse-strong monotonicity of A, we derive

$$\begin{array}{r}{\parallel (1-t)((T{z}_t-\frac{\lambda }{1-t}AT{z}_t)-(Tz-\frac{\lambda }{1-t}ATz))+t(u_t-z)\parallel }^2\\ \le (1-t){\parallel (T{z}_t-\frac{\lambda }{1-t}AT{z}_t)-(Tz-\frac{\lambda }{1-t}ATz)\parallel }^2+t{\parallel u_t-z\parallel }^2\\ =(1-t){\parallel (T{z}_t-Tz)-\lambda (AT{z}_t-ATz)/(1-t)\parallel }^2+t{\parallel u_t-z\parallel }^2\\ =(1-t)({\parallel T{z}_t-Tz\parallel }^2-\frac{2\lambda }{1-t}〈AT{z}_t-ATz,T{z}_t-Tz〉\\ +\frac{{\lambda }^2}{{(1-t)}^2}{\parallel AT{z}_t-ATz\parallel }^2)+t{\parallel u_t-z\parallel }^2\\ \le (1-t)({\parallel T{z}_t-Tz\parallel }^2-\frac{2\alpha \lambda }{1-t}{\parallel AT{z}_t-ATz\parallel }^2+\frac{{\lambda }^2}{{(1-t)}^2}{\parallel AT{z}_t-ATz\parallel }^2)\\ +t{\parallel u_t-z\parallel }^2\\ =(1-t)({\parallel T{z}_t-Tz\parallel }^2+\frac{\lambda }{{(1-t)}^2}(\lambda -2(1-t)\alpha ){\parallel AT{z}_t-ATz\parallel }^2)+t{\parallel u_t-z\parallel }^2\\ \le (1-t)({\parallel z_t-z\parallel }^2+\frac{\lambda }{{(1-t)}^2}(\lambda -2(1-t)\alpha ){\parallel AT{z}_t-ATz\parallel }^2)+t{\parallel u_t-z\parallel }^2.\end{array}$$

(3.5)

By the assumption, we have $\lambda -2(1-t)\alpha \le 0$ for all $t\in (0,1-\frac{\lambda }{2\alpha })$. Then, from (3.4) and (3.5), it follows that

$$\begin{array}{r}{\parallel z_t-z\parallel }^2\\ \le (1-t)({\parallel z_t-z\parallel }^2+\frac{\lambda }{{(1-t)}^2}(\lambda -2(1-t)\alpha ){\parallel AT{z}_t-ATz\parallel }^2)+t{\parallel u_t-z\parallel }^2\\ \le (1-t){\parallel z_t-z\parallel }^2+t{\parallel u_t-z\parallel }^2\end{array}$$

(3.6)

and so

$$\parallel z_t-z\parallel \le \parallel u_t-z\parallel .$$

(3.7)

Since ${lim}_{t\to 0+}{u}_t=u$, there exists a positive constant $M>0$ such that ${sup}_t\{\parallel u_t\parallel \}\le M$. Then, from (3.7), we deduce that $\{z_t\}$ is bounded. Hence $\{T{z}_t\}$ and $\{AT{z}_t\}$ are also bounded.

From (3.4) and (3.5), we obtain

$${\parallel z_t-z\parallel }^2\le (1-t){\parallel z_t-z\parallel }^2+\frac{\lambda }{(1-t)}(\lambda -2(1-t)\alpha ){\parallel AT{z}_t-Az\parallel }^2+t{\parallel u_t-z\parallel }^2$$

and so

$$\frac{\lambda }{(1-t)}(2(1-t)\alpha -\lambda ){\parallel AT{z}_t-Az\parallel }^2\le t{\parallel u_t-z\parallel }^2\to 0.$$

This implies that

$$\underset{t\to 0+}{lim}\parallel AT{z}_t-Az\parallel =0.$$

(3.8)

Next, we show $\parallel z_t-T{z}_t\parallel \to 0$. Since $T_{\lambda }$ is firmly nonexpansive (see Lemma 2.1), we have

$$\begin{array}{rl}{\parallel z_t-z\parallel }^2=& {\parallel T_{\lambda }(t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t)-z\parallel }^2\\ =& {\parallel T_{\lambda }(t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t)-T_{\lambda }(Tz-\lambda ATz)\parallel }^2\\ \le & 〈t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t-(Tz-\lambda ATz),z_t-z〉\\ =& \frac{1}{2}({\parallel t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t-(Tz-\lambda ATz)\parallel }^2+{\parallel z_t-z\parallel }^2\\ -{\parallel t{u}_t+(1-t)T{z}_t-\lambda (AT{z}_t-\lambda ATz)-z_t\parallel }^2).\end{array}$$

Since $I-\lambda A/(1-t)$ is nonexpansive, we have

$$\begin{array}{r}{\parallel t{u}_t+(1-t)T{z}_t-\lambda AT{z}_t-(Tz-\lambda ATz)\parallel }^2\\ ={\parallel (1-t)((T{z}_t-\lambda AT{z}_t/(1-t))-(Tz-\lambda ATz/(1-t)))+t(u_t-z)\parallel }^2\\ \le (1-t){\parallel (T{z}_t-\lambda AT{z}_t/(1-t))-(Tz-\lambda ATz/(1-t))\parallel }^2+t{\parallel u_t-z\parallel }^2\\ \le (1-t){\parallel T{z}_t-Tz\parallel }^2+t{\parallel u_t-z\parallel }^2\\ \le (1-t){\parallel z_t-z\parallel }^2+t{\parallel u_t-z\parallel }^2.\end{array}$$

Thus we have

$$\begin{array}{rl}{\parallel z_t-z\parallel }^2\le & \frac{1}{2}((1-t){\parallel z_t-z\parallel }^2+t{\parallel u_t-z\parallel }^2+{\parallel z_t-z\parallel }^2\\ -{\parallel t{u}_t+(1-t)T{z}_t-z_t-\lambda (AT{z}_t-ATz)\parallel }^2).\end{array}$$

It follows that

$$\begin{array}{rl}0\le & t{\parallel u_t-z\parallel }^2-{\parallel t{u}_t+(1-t)T{z}_t-z_t-\lambda (AT{z}_t-ATz)\parallel }^2\\ =& t{\parallel u_t-z\parallel }^2-{\parallel t{u}_t+(1-t)T{z}_t-z_t\parallel }^2\\ +2\lambda 〈t{u}_t+(1-t)T{z}_t-z_t,AT{z}_t-ATz〉-{\lambda }^2{\parallel AT{z}_t-ATz\parallel }^2\\ \le & t{\parallel u_t-z\parallel }^2-{\parallel t{u}_t+(1-t)T{z}_t-z_t\parallel }^2\\ +2\lambda \parallel t{u}_t+(1-t)T{z}_t-z_t\parallel \parallel AT{z}_t-ATz\parallel \end{array}$$

and so

$${\parallel t{u}_t+(1-t)T{z}_t-z_t\parallel }^2\le t{\parallel u_t-z\parallel }^2+2\lambda \parallel t{u}_t+(1-t)T{z}_t-z_t\parallel \parallel AT{z}_t-Az\parallel .$$

Since $\parallel AT{z}_t-Az\parallel \to 0$ by (3.8), we deduce

$$\underset{t\to 0+}{lim}\parallel t{u}_t+(1-t)T{z}_t-z_t\parallel =0.$$

Therefore, we have

$$\underset{t\to 0+}{lim}\parallel z_t-T{z}_t\parallel =0.$$

(3.9)

From (3.4), it follows that

$$\begin{array}{r}{\parallel z_t-z\parallel }^2\\ \le {\parallel (1-t)((T{z}_t-\frac{\lambda }{1-t}AT{z}_t)-(Tz-\frac{\lambda }{1-t}ATz))+t(u_t-z)\parallel }^2\\ ={(1-t)}^2{\parallel (T{z}_t-\frac{\lambda }{1-t}AT{z}_t)-(Tz-\frac{\lambda }{1-t}ATz)\parallel }^2\\ +2t(1-t)〈u_t-z,(T{z}_t-\frac{\lambda }{1-t}AT{z}_t)-(Tz-\frac{\lambda }{1-t}ATz)〉+t^2{\parallel u_t-z\parallel }^2\\ \le {(1-t)}^2{\parallel z_t-z\parallel }^2+2t(1-t)〈u_t-z,T{z}_t-\frac{\lambda }{1-t}(AT{z}_t-Az)-z〉+t^2{\parallel u_t-z\parallel }^2\\ =(1-2t){\parallel z_t-z\parallel }^2+2t\{(1-t)〈u_t-z,T{z}_t-z-\frac{\lambda }{1-t}(AT{z}_t-Az)〉\\ +t^2({\parallel u_t-z\parallel }^2+{\parallel z_t-z\parallel }^2)\},\end{array}$$

which implies that

$$\begin{array}{rl}{\parallel z_t-z\parallel }^2\le & 〈u_t-z,T{z}_t-z-\frac{\lambda }{1-t}(AT{z}_t-Az)〉+\frac{t}{2}({\parallel u_t-z\parallel }^2+{\parallel z_t-z\parallel }^2)\\ +t\parallel u_t-z\parallel \parallel T{z}_t-z-\frac{\lambda }{1-t}(AT{z}_t-Az)\parallel \\ \le & 〈z-u,z-T{z}_t〉+\frac{\lambda }{1-t}\parallel u_t-z\parallel \parallel AT{z}_t-Az\parallel +(t+\parallel u_t-u\parallel )M_1,\end{array}$$

(3.10)

where $M_1$ is a constant such that

$$sup\{{\parallel u_t-z\parallel }^2+{\parallel z_t-z\parallel }^2+\parallel u_t-z\parallel \parallel T{z}_t-z-\frac{\lambda }{1-t}(AT{z}_t-Az)\parallel :t\in (0,1-\frac{\lambda }{2\alpha })\}\le M_1.$$

Next, we show that $\{z_t\}$ is relatively norm-compact as $t\to 0+$. Assume that $\{t_n\}\subset (0,1)$ is a sequence such that $t_n\to 0+$ as $n\to \mathrm{\infty }$. Put $z_n:=z_{t_n}$. From (3.10), it follows that

$$\begin{array}{r}{\parallel z_n-z\parallel }^2\\ \le 〈z-u,z-T{z}_n〉+\frac{\lambda }{1-t_n}\parallel u_n-z\parallel \parallel AT{z}_n-Az\parallel +(t_n+\parallel u_n-u\parallel )M_1\end{array}$$

(3.11)

for all $z\in F(T)\cap S(\mathit{MEP})$. Since $\{z_n\}$ is bounded, without loss of generality, we may assume that $z_n\rightharpoonup \tilde{x}\in C$. From (3.9), we have

$$\underset{n\to \mathrm{\infty }}{lim}\parallel z_n-T{z}_n\parallel =0.$$

(3.12)

We can use Lemma 2.4 to (3.12) to deduce $\tilde{x}\in F(T)$. Further, we show that $\tilde{x}$ is also in $S(\mathit{MEP})$. Since $z_n=T_{\lambda }(t_n{u}_n+(1-t_n)T{z}_n-\lambda AT{z}_n)$ for any $y\in C$, we have

$$F(z_n,y)+〈AT{z}_n,y-z_n〉+\frac{1}{\lambda }〈y-z_n,z_n-(t_n{u}_n+(1-t_n)T{z}_n)〉\ge 0.$$

From (C2), it follows that

$$〈AT{z}_n,y-z_n〉+\frac{1}{\lambda }〈y-z_n,z_n-(t_n{u}_n+(1-t_n)T{z}_n)〉\ge F(y,z_n).$$

(3.13)

Put $x_t=ty+(1-t)\tilde{x}$ for all $t\in (0,1-\frac{\lambda }{2\alpha })$ and $y\in C$. Then we have $x_t\in C$ and so, from (3.13), it follows that

$$\begin{array}{rl}〈x_t-z_n,A{x}_t〉\ge & 〈x_t-z_n,A{x}_t〉-〈x_t-z_n,AT{z}_n〉\\ -\frac{1}{\lambda }〈x_t-z_n,z_n-(t_n{u}_n+(1-t_n)T{z}_n)〉+F(x_t,z_n)\\ =& 〈x_t-z_n,A{x}_t-A{z}_n〉+〈x_t-z_n,A{z}_n-AT{z}_n〉\\ -\frac{1}{\lambda }〈x_t-z_n,z_n-(t_n{u}_n+(1-t_n)T{z}_n)〉+F(x_t,z_n).\end{array}$$

Since $\parallel z_n-T{z}_n\parallel \to 0$, we have $\parallel A{z}_n-AT{z}_n\parallel \to 0$. Further, since A is monotone, we have $〈x_t-z_n,A{x}_t-A{z}_n〉\ge 0$. So, from (C4), it follows that, as $n\to \mathrm{\infty }$,

$$〈x_t-\tilde{x},A{x}_t〉\ge F(x_t,\tilde{x}).$$

(3.14)

Also, it follows from (C1), (C4), and (3.14) that

$$\begin{array}{rl}0& =F(x_t,x_t)\\ \le tF(x_t,y)+(1-t)F(x_t,\tilde{x})\\ \le tF(x_t,y)+(1-t)〈x_t-\tilde{x},A{x}_t〉\\ =tF(x_t,y)+(1-t)t〈y-\tilde{x},A{x}_t〉\end{array}$$

and hence

$$0\le F(x_t,y)+(1-t)〈y-\tilde{x},A{x}_t〉.$$

Letting $t\to 0$, we have

$$0\le F(\tilde{x},y)+〈y-\tilde{x},A\tilde{x}〉$$

for all $y\in C$. This implies $\tilde{x}\in \mathit{EP}$. Therefore, we can substitute $\tilde{x}$ for z in (3.8) to get

$$\begin{array}{rl}{\parallel z_n-\tilde{x}\parallel }^2\le & 〈\tilde{x}-u,\tilde{x}-T{z}_n〉+\frac{\lambda }{1-t_n}\parallel u_n-\tilde{x}\parallel \parallel AT{z}_n-A\tilde{x}\parallel \\ +(t_n+\parallel u_n-\tilde{x}\parallel )M_1\end{array}$$

for all $\tilde{x}\in F(T)\cap S(\mathit{MEP})$. By (3.5), we know that $\parallel AT{z}_n-Az\parallel \to 0$ for any $z\in F(T)\cap S(\mathit{MEP})$. Then we get $\parallel AT{z}_n-A\tilde{x}\parallel \to 0$. Consequently, the weak convergence of $\{z_n\}$ (and $\{T{z}_n\}$) to $\tilde{x}$ actually implies that $z_n\to \tilde{x}$. This proves the relative norm-compactness of the net $\{z_t\}$ as $t\to 0+$.

Now, we return to (3.11) and take the limit as $n\to \mathrm{\infty }$ to get

$${\parallel \tilde{x}-z\parallel }^2\le 〈z-u,z-\tilde{x}〉$$

for all $z\in F(T)\cap S(\mathit{MEP})$. Equivalently, we have

$$〈u-\tilde{x},z-\tilde{x}〉\le 0$$

for all $z\in F(T)\cap S(\mathit{MEP})$. This clearly implies that

$$\tilde{x}=P_{F(T)\cap S(\mathit{MEP})}(u).$$

Therefore, $\tilde{x}$ is the unique cluster point of the net $\{z_t\}$. Hence the whole net $\{z_t\}$ converges strongly to $\tilde{x}=P_{F(T)\cap S(\mathit{MEP})}(u)$. This completes the proof. □

4 Induced algorithms and corollaries

• (I) Taking $T=I$ in (3.1), we get the following.

Algorithm 4.1 For any $t\in (0,1-\frac{\lambda }{2\alpha })$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$F(z_t,y)+\frac{1}{\lambda }〈y-z_t,z_t-(t{u}_t+(1-t)z_t-\lambda A{z}_t)〉\ge 0$$

(4.1)

for all $y\in C$.

Corollary 4.2 Suppose that $S(\mathit{MEP})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (4.1) converges strongly as $t\to 0+$ to $P_{S(\mathit{MEP})}(u)$.

• (II) Taking $F=0$ in (4.1), we get the following.

Algorithm 4.3 For any $t\in (0,1-\frac{\lambda }{2\alpha })$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$〈y-z_t,z_t-(t{u}_t+(1-t)z_t-\lambda A{z}_t)〉\ge 0$$

(4.2)

for all $y\in C$.

Corollary 4.4 Suppose that $S(\mathit{VI})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (4.2) converges strongly as $t\to 0+$ to $P_{S(\mathit{VI})}(u)$.

• (III) Taking $A=0$ in (4.1), we get the following.

Algorithm 4.5 For any $t\in (0,1)$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$F(z_t,y)+\frac{t}{\lambda }〈y-z_t,z_t-u_t〉\ge 0$$

(4.3)

for all $y\in C$.

Corollary 4.6 Suppose that $S(\mathit{EP})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (4.3) converges strongly as $t\to 0+$ to $P_{S(\mathit{EP})}(u)$.

5 Minimum-norm solutions

In many problems, one needs to find a solution with the minimum norm. In an abstract way, we may formulate such problems as finding a point $x^{†}$ with the property:

$$x^{†}\in C,{\parallel x^{†}\parallel }^2=\underset{x\in C}{min}{\parallel x\parallel }^2,$$

where C is a nonempty closed convex subset of a real Hilbert space H. In other words, $x^{†}$ is the (nearest point or metric) projection of the origin onto C, that is,

$$x^{†}=P_C(0),$$

where $P_C$ is the metric (or nearest point) projection from H onto C.

A typical example is the least-squares solution of the constrained linear inverse problem:

$$\{\begin{array}{c}Ax=b,\hfill \\ x\in C,\hfill \end{array}$$

where A is a bounded linear operator from H to another real Hilbert space $H_1$ and b is a given point in $H_1$. The least-squares solution is the least-norm minimizer of the minimization problem:

$$\underset{x\in C}{min}{\parallel Ax-b\parallel }^2.$$

Motivated by the above least-squares solution of the constrained linear inverse problems, we study the general case of finding the minimum-norm solutions for the mixed equilibrium problem (1.1), the equilibrium problem (1.2), the variational inequality (1.3), and the fixed point problem.

Now, we state our algorithms which can be inducted from the above section.

• (I) Taking $u_t=0$ for all t in (3.1), we get the following.

Algorithm 5.1 For any $t\in (0,1-\frac{\lambda }{2\alpha })$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$F(z_t,y)+\frac{1}{\lambda }〈y-z_t,z_t-((1-t)T{z}_t-\lambda AT{z}_t)〉\ge 0$$

(5.1)

for all $y\in C$.

Corollary 5.2 Suppose that $F(T)\cap S(\mathit{MEP})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (5.1) converges strongly as $t\to 0+$ to $P_{F(T)\cap S(\mathit{MEP})}(0)$, which is the minimum-norm element in $F(T)\cap S(\mathit{MEP})$.

• (II) Taking $u_t=0$ for all t in (4.1), we get the following.

Algorithm 5.3 For any $t\in (0,1-\frac{\lambda }{2\alpha })$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$F(z_t,y)+\frac{1}{\lambda }〈y-z_t,z_t-((1-t)z_t-\lambda A{z}_t)〉\ge 0$$

(5.2)

for all $y\in C$.

Corollary 5.4 Suppose that $S(\mathit{MEP})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (5.2) converges strongly as $t\to 0+$ to $P_{S(\mathit{MEP})}(0)$, which is the minimum-norm element in $S(\mathit{MEP})$.

• (III) Taking $u_t=0$ for all t in (4.2), we get the following.

Algorithm 5.5 For any $t\in (0,1-\frac{\lambda }{2\alpha })$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$〈y-z_t,z_t-((1-t)z_t-\lambda A{z}_t)〉\ge 0$$

(5.3)

for all $y\in C$.

Corollary 5.6 Suppose that $S(\mathit{VI})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (5.3) converges strongly as $t\to 0+$ to $P_{S(\mathit{VI})}(0)$, which is the minimum-norm element in $S(\mathit{VI})$.

• (IV) Taking $A=0$ in (5.1), we get the following.

Algorithm 5.7 For any $t\in (0,1)$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$F(z_t,y)+\frac{1}{\lambda }〈y-z_t,z_t-(1-t)T{z}_t〉\ge 0$$

(5.4)

for all $y\in C$.

Corollary 5.8 Suppose that $F(T)\cap S(\mathit{EP})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (5.4) converges strongly as $t\to 0+$ to $P_{F(T)\cap S(\mathit{EP})}(0)$, which is the minimum-norm element in $F(T)\cap S(\mathit{EP})$.

• (V) Taking $A=0$ in (5.2), we get the following.

Algorithm 5.9 For any $t\in (0,1)$, define a net $\{z_t\}\subset C$ by the implicit manner:

$$F(z_t,y)+\frac{t}{\lambda }〈y-z_t,z_t〉\ge 0$$

(5.5)

for all $y\in C$.

Corollary 5.10 Suppose that $S(\mathit{EP})\ne \mathrm{\varnothing }$. Then the net $\{z_t\}$ defined by (5.5) converges strongly as $t\to 0+$ to $P_{S(\mathit{EP})}(0)$, which is the minimum-norm element in $S(\mathit{EP})$.