On the convergence of hybrid projection algorithms for total quasi-asymptotically pseudo-contractive mapping

Abstract

In this paper, a hybrid projection algorithm for a total quasi-asymptotically pseudo-contractive mapping is introduced in a Hilbert space. A strong convergence theorem of the proposed algorithm to a fixed point of a total quasi-asymptotically pseudo-contractive mapping is proved. Our main result extends and improves many corresponding results.

MSC:47H05, 47H09.

1 Introduction

Throughout this paper, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by $〈\cdot ,\cdot 〉$ and $\parallel \cdot \parallel$. The symbol → is denoted by a strong convergence. Let C be a nonempty closed and convex subset of H, and let $T:C\to C$ be a mapping. In this paper, we denote the fixed point set of T by $\mathcal{F}(T)$, that is, $\mathcal{F}(T):=\{x\in C:Tx=x\}$.

Recall that T is said to be asymptotically nonexpansive if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ such that

$$\parallel T^{n}x-T^{n}y\parallel \le k_n\parallel x-y\parallel ,\mathrm{\forall }n\ge 1,\mathrm{\forall }x,y\in C.$$

(1.1)

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings.

T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}(\parallel T^{n}x-T^{n}y\parallel -\parallel x-y\parallel )\le 0.$$

(1.2)

Noticing that if we define

$${\rho }_n=max\{0,\underset{x,y\in C}{sup}(\parallel T^{n}x-T^{n}y\parallel -\parallel x-y\parallel )\},$$

(1.3)

then ${\rho }_n\to 0$ as $n\to \mathrm{\infty }$. It follows that (1.2) is reduced to

$$\parallel T^{n}x-T^{n}y\parallel \le \parallel x-y\parallel +{\rho }_n,\mathrm{\forall }n\ge 1,\mathrm{\forall }x,y\in C.$$

(1.4)

The class of mappings, which are asymptotically nonexpansive in the intermediate sense, was introduced by Bruck et al. [2] (see also [3]). It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.

Recall that T is said to be asymptotically pseudocontractive if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ such that

$$〈T^{n}x-T^{n}y,x-y〉\le k_n{\parallel x-y\parallel }^2,\mathrm{\forall }x,y\in C.$$

(1.5)

It is not hard to see that (1.5) is equivalent to

$${\parallel T^{n}x-T^{n}y\parallel }^2\le (2k_n-1){\parallel x-y\parallel }^2+{\parallel x-y-(T^{n}x-T^{n}y)\parallel }^2,\mathrm{\forall }n\ge 1,x,y\in C.$$

(1.6)

The class of an asymptotically pseudocontractive mapping was introduced by Schu [4] (see also [5]). In [6], Rhoades gave an example to show that the class of asymptotically pseudocontractive mappings contains properly the class of asymptotically nonexpansive mappings, see [6] for more details. Zhou [7] showed that every uniformly Lipschitz and asymptotically pseudocontractive mapping, which is also uniformly asymptotically regular, has a fixed point.

T is said to be an asymptotically pseudocontractive mapping in the intermediate sense if there exists a sequence $\{k_n\}\subset [1,\mathrm{\infty })$ with $k_n\to 1$ as $n\to \mathrm{\infty }$ such that

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\underset{x,y\in C}{sup}(〈T^{n}x-T^{n}y,x-y〉-k_n{\parallel x-y\parallel }^2)\le 0.$$

(1.7)

Put

$${\tau }_n=max\{0,\underset{x,y\in C}{sup}(〈T^{n}x-T^{n}y,x-y〉-k_n{\parallel x-y\parallel }^2)\}.$$

(1.8)

It follows that ${\tau }_n\to 0$ as $n\to \mathrm{\infty }$. Then, (1.8) is reduced to the following:

$$〈T^{n}x-T^{n}y,x-y〉\le k_n{\parallel x-y\parallel }^2+{\tau }_n,\mathrm{\forall }n\ge 1,x,y\in C.$$

(1.9)

The class of asymptotically pseudocontractive mappings in the intermediate sense was introduced by Qin et al. [8].

Recall that T is said to be total asymptotically pseudocontractive if there exist sequences $\{k_n\},\{{\nu }_n\}\subset [0,\mathrm{\infty })$ with $k_n,{\nu }_n\to 0$ as $n\to \mathrm{\infty }$ such that

$$〈T^{n}x-T^{n}y,x-y〉\le {\parallel x-y\parallel }^2+k_n\varphi (\parallel x-y\parallel )+{\nu }_n,\mathrm{\forall }n\ge 1,x,y\in C,$$

(1.10)

where $\varphi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a continuous and strictly increasing function with $\varphi (0)=0$. The class of a total asymptotically pseudocontractive mapping was introduced by Qin [9].

It is easy to see that (1.10) is equivalent to the following: for all $n\ge 1$, $x,y\in C$,

$${\parallel T^{n}x-T^{n}y\parallel }^2\le {\parallel x-y\parallel }^2+2k_n\varphi (\parallel x-y\parallel )+{\parallel x-y-(T^{n}x-T^{n}y)\parallel }^2+2{\nu }_n.$$

(1.11)

If $\varphi (\lambda )={\lambda }^2$, then (1.10) is reduced to

$$〈T^{n}x-T^{n}y,x-y〉\le (1+k_n){\parallel x-y\parallel }^2+{\nu }_n,\mathrm{\forall }n\ge 1,x,y\in C.$$

(1.12)

Put

$${\nu }_n=max\{0,\underset{x,y\in C}{sup}(〈T^{n}x-T^{n}y,x-y〉-(1+k_n){\parallel x-y\parallel }^2)\}.$$

(1.13)

If $\varphi (\lambda )={\lambda }^2$, then the class of total asymptotically pseudocontractive mappings is reduced to the class of asymptotically pseudocontractive mappings in the intermediate sense.

In this paper, we introduce and study the following mapping.

Definition 1.1 A mapping $T:C\to C$ is said to be total quasi-asymptotically pseudo-contractive if $\mathcal{F}(T)\ne \mathrm{\varnothing }$, and there exist sequences $\{{\mu }_n\}\subset [0,\mathrm{\infty })$ and $\{{\xi }_n\}\subset [0,\mathrm{\infty })$ with ${\mu }_n\to 0$ and ${\xi }_n\to 0$ as $n\to \mathrm{\infty }$ such that

$$〈T^{n}x-p,x-p〉\le {\parallel x-p\parallel }^2+{\mu }_n\varphi (\parallel x-p\parallel )+{\xi }_n,\mathrm{\forall }n\ge 1,x\in C,p\in \mathcal{F}(T),$$

(1.14)

where $\varphi :[0,\mathrm{\infty })\to [0,\mathrm{\infty })$ is a continuous and strictly increasing function with $\varphi (0)=0$.

It is easy to see that (1.14) is equivalent to the following:

$$\begin{array}{rl}{\parallel T^{n}x-p\parallel }^2\le & {\parallel x-p\parallel }^2+2{\mu }_n\varphi (\parallel x-p\parallel )\\ +{\parallel x-T^{n}x\parallel }^2+2{\xi }_n,\mathrm{\forall }n\ge 1,x\in C,p\in \mathcal{F}(T).\end{array}$$

(1.15)

Remark 1 It is clear that every total asymptotically pseudo-contractive mapping with $F(T)\ne \mathrm{\varnothing }$ is total quasi-asymptotically pseudo-contractive, but the converse maybe not true.

Remark 2 If $\varphi (\lambda )={\lambda }^2$, the (1.14) is reduced to

$$〈T^{n}x-p,x-p〉\le (1+{\mu }_n){\parallel x-p\parallel }^2+{\xi }_n,\mathrm{\forall }n\ge 1,x\in C,p\in \mathcal{F}(T).$$

(1.16)

Remark 3 Put

$${\xi }_n=max\{0,\underset{x,y\in C}{sup}(〈T^{n}x-p,x-p〉-(1+{\mu }_n){\parallel x-p\parallel }^2)\}.$$

(1.17)

If $\varphi (\lambda )={\lambda }^2$, then the class of total quasi-asymptotically pseudo-contractive mappings is reduced to the class of quasi-asymptotically pseudo-contractive mappings in the intermediate sense.

Recently, the iterative approximation of fixed points for asymptotically pseudo-contractive mappings, total asymptotically pseudo-contractive mappings in Hilbert, or Banach spaces has been studied extensively by many authors, see, for example, [7, 9–13]. In this paper, we shall consider and study a total quasi-asymptotically pseudo-contractive mapping as a generalization of (total) asymptotically pseudo-contractive mappings. Furthermore, we shall introduce an iterative algorithm for finding a fixed point of a total quasi-asymptotically pseudo-contractive mapping.

2 Preliminaries

A mapping $T:C\to C$ is said to be uniformly L-Lipschitzian if there exists some $L>0$ such that

$$\parallel T^{n}x-T^{n}y\parallel \le L\parallel x-y\parallel ,\mathrm{\forall }x,y\in C,n\ge 1.$$

(2.1)

Let C be a nonempty closed convex subset of a real Hilbert space H. For every point $x\in H$, there exists a unique nearest point in C, denoted by $P_{C}x$, such that $\parallel x-P_{C}x\parallel \le \parallel x-y\parallel$ holds for all $y\in C$, where $P_C$ is said to be the metric projection of H onto C.

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

Lemma 2.1 [14]

Let C be a nonempty closed convex subset of a real Hilbert space H and let $P_C$ be the metric projection from H onto C (i.e., for $x\in H$, $P_C$ is the only point in C such that $\parallel x-P_{C}x\parallel =inf\{\parallel x-z\parallel :z\in C\}$). Given $x\in H$ and $z\in C$, then $z=P_{C}x$ if and only if the relation

$$〈x-z,y-z〉\le 0,\mathrm{\forall }y\in C$$

(2.2)

holds.

Lemma 2.2 Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a uniformly L-Lipschitzian and total quasi-asymptotically pseudo-contractive mapping with $\mathcal{F}(T)\ne \mathrm{\varnothing }$. Suppose there exist positive constants M and $M^{\ast }$ such that $\varphi (\zeta )\le M^{\ast }{\zeta }^2$ for all $\zeta >M$. Then $\mathcal{F}(T)$ is a closed convex subset of C.

Proof Since ϕ is an increasing function, it follows that $\varphi (\zeta )\le \varphi (M)$ if $\zeta \le M$ and $\varphi (\zeta )\le M^{\ast }{\zeta }^2$ if $\zeta \ge M$. In either case, we can always obtain that

$$\varphi (\zeta )\le \varphi (M)+M^{\ast }{\zeta }^2.$$

(2.3)

Since T is uniformly L-Lipschitzian continuous, $\mathcal{F}(T)$ is closed. We need to show that $\mathcal{F}(T)$ is convex. To this end, let $p_i\in \mathcal{F}(T)$ ($i=1,2$), and write $p=t{p}_1+(1-t)p_2$ for $t\in (0,1)$. We take $\alpha \in (0,\frac{1}{1+L})$, and define $y_{\alpha ,n}=(1-\alpha )p+\alpha T^{n}p$ for each $n\in \mathbb{N}$. Then, for all $z\in \mathcal{F}(T)$, we have from (2.3) that

$$\begin{array}{rl}{\parallel p-T^{n}p\parallel }^2=& 〈p-T^{n}p,p-T^{n}p〉\\ =& \frac{1}{\alpha }〈p-y_{\alpha ,n},p-T^{n}p〉\\ =& \frac{1}{\alpha }〈p-y_{\alpha ,n},p-T^{n}p-(y_{\alpha ,n}-T^n{y}_{\alpha ,n})〉\\ +\frac{1}{\alpha }〈p-y_{\alpha ,n},y_{\alpha ,n}-T^n{y}_{\alpha ,n}〉\\ \le & \frac{1+L}{\alpha }{\parallel p-y_{\alpha ,n}\parallel }^2+\frac{1}{\alpha }〈p-z,y_{\alpha ,n}-T^n{y}_{\alpha ,n}〉\\ +\frac{1}{\alpha }〈z-y_{\alpha ,n},y_{\alpha ,n}-T^n{y}_{\alpha ,n}〉\\ =& \frac{1+L}{\alpha }{\parallel p-y_{\alpha ,n}\parallel }^2+\frac{1}{\alpha }〈p-z,y_{\alpha ,n}-T^n{y}_{\alpha ,n}〉\\ +\frac{1}{\alpha }〈z-y_{\alpha ,n},y_{\alpha ,n}-z+z-T^n{y}_{\alpha ,n}〉\\ \le & \frac{1+L}{\alpha }{\parallel p-y_{\alpha ,n}\parallel }^2+\frac{1}{\alpha }〈p-z,y_{\alpha ,n}-T^n{y}_{\alpha ,n}〉\\ +\frac{1}{\alpha }\{{\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n\}\\ =& \alpha (1+L){\parallel p-T^{n}p\parallel }^2+\frac{1}{\alpha }〈p-z,y_{\alpha ,n}-T^n{y}_{\alpha ,n}〉\\ +\frac{1}{\alpha }\{{\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n\}.\end{array}$$

This implies that

$$\begin{array}{rl}\alpha [1-\alpha (1+L)]{\parallel p-T^{n}p\parallel }^2\le & 〈p-z,y_{\alpha ,n}-T^n{y}_{\alpha ,n}〉\\ +{\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n.\end{array}$$

(2.4)

Now, we take $z=p_i$ ($i=1,2$) in (2.4), multiplying t and $(1-t)$ on the both sides of the above inequality (2.4), respectively, and adding up, and we can get

$$\alpha [1-\alpha (1+L)]{\parallel p-T^{n}p\parallel }^2\le {\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n.$$

(2.5)

Letting $n\to \mathrm{\infty }$ in (2.5), we obtain $T^{n}p\to p$. Since T is continuous, we have $T^{n+1}p\to Tp$ as $n\to \mathrm{\infty }$, therefore, $p=Tp$. This proves that $\mathcal{F}(T)$ is a closed convex subset of C. □

3 Main results

In this section, we shall give our main results of this paper.

Theorem 3.1 Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a uniformly L-Lipschitzian and total quasi-asymptotically pseudo-contractive mapping with $\mathcal{F}(T)\ne \mathrm{\varnothing }$. Suppose that there exist positive constants M and $M^{\ast }$ such that $\varphi (\zeta )\le M^{\ast }{\zeta }^2$ for all $\zeta >M$. Let $\{x_n\}$ be a sequence generated by the following iterative scheme:

$$\{\begin{array}{c}{x}_1\in C\mathit{\text{chosen arbitrarily}},\hfill \\ C_1=C,Q_1=C,\hfill \\ y_n=(1-{\alpha }_n)x_n+{\alpha }_n{T}^n{x}_n,\hfill \\ C_{n+1}=\{z\in C_n:{\alpha }_n[1-{\alpha }_n(1+L)]{\parallel x_n-T^n{x}_n\parallel }^2\le 〈x_n-z,y_n-T^n{y}_n〉+{\theta }_n\},\hfill \\ Q_{n+1}=\{z\in Q_n:〈x_n-z,x_1-x_n〉\ge 0\},\hfill \\ x_{n+1}=P_{C_{n+1}\cap Q_{n+1}}{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

(3.1)

where ${\theta }_n={\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n$, $\{{\alpha }_n\}$ is a sequence in $[a,b]$ with $a,b\in (0,\frac{1}{1+L})$. Then the sequence $\{x_n\}$ converges strongly to a point $P_{\mathcal{F}(T)}{x}_1$, where $P_{\mathcal{F}(T)}$ is the projection from C onto $\mathcal{F}(T)$.

Proof We split the proof into seven steps.

Step 1. Show that $P_{\mathcal{F}(T)}{x}_1$ is well defined for every $x_1\in C$.

By Lemma 2.2, we know that $\mathcal{F}(T)$ is a closed and convex subset of C. Therefore, in view of the assumption of $\mathcal{F}(T)\ne \mathrm{\varnothing }$, $P_{\mathcal{F}(T)}{x}_1$ is well defined for every $x_1\in C$.

Step 2. Show that $C_n$ and $Q_n$ are closed and convex for all $n\ge 1$.

From the definitions of $C_n$ and $Q_n$, it is obvious that $C_n$ and $Q_n$ are closed and convex for all $n\ge 1$. We omit the details.

Step 3. Show that $\mathcal{F}(T)\subset C_n\cap Q_n$ for all $n\ge 1$.

To this end, we first prove that $\mathcal{F}(T)\subset C_n$ for all $n\ge 1$. This can be proved by induction on n. It is obvious that $\mathcal{F}(T)\subset C_1=C$. Assume that $\mathcal{F}(T)\subset C_n$ for some $n\in \mathbb{N}$. Then, using the uniform L-Lipschitzian continuity of T, the total quasi-asymptotic pseudo-contractiveness of T and (2.3), we have for any $w\in \mathcal{F}(T)$ that

$$\begin{array}{rl}{\parallel x_n-T^n{x}_n\parallel }^2=& 〈x_n-T^n{x}_n,x_n-T^n{x}_n〉\\ =& \frac{1}{{\alpha }_n}〈x_n-y_n,x_n-T^n{x}_n〉\\ =& \frac{1}{{\alpha }_n}〈x_n-y_n,x_n-T^n{x}_n-(y_n-T^n{y}_n)〉+\frac{1}{{\alpha }_n}〈x_n-y_n,y_n-T^n{y}_n〉\\ =& \frac{1}{{\alpha }_n}〈x_n-y_n,x_n-T^n{x}_n-(y_n-T^n{y}_n)〉\\ +\frac{1}{{\alpha }_n}〈x_n-w+w-y_n,y_n-T^n{y}_n〉\\ \le & \frac{1+L}{{\alpha }_n}{\parallel x_n-y_n\parallel }^2+\frac{1}{{\alpha }_n}〈x_n-w,y_n-T^n{y}_n〉\\ +\frac{1}{{\alpha }_n}〈w-y_n,y_n-T^n{y}_n〉\\ =& \frac{1+L}{{\alpha }_n}{\parallel x_n-y_n\parallel }^2+\frac{1}{{\alpha }_n}〈x_n-w,y_n-T^n{y}_n〉\\ +\frac{1}{{\alpha }_n}〈w-y_n,y_n-w+w-T^n{y}_n〉\\ =& \frac{1+L}{{\alpha }_n}{\parallel x_n-y_n\parallel }^2+\frac{1}{{\alpha }_n}〈x_n-w,y_n-T^n{y}_n〉\\ -\frac{1}{{\alpha }_n}{\parallel w-y_n\parallel }^2+\frac{1}{{\alpha }_n}〈w-y_n,w-T^n{y}_n〉\\ \le & \frac{1+L}{{\alpha }_n}{\parallel x_n-y_n\parallel }^2+\frac{1}{{\alpha }_n}〈x_n-w,y_n-T^n{y}_n〉\\ +\frac{1}{{\alpha }_n}\{{\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n\}\\ =& (1+L){\alpha }_n{\parallel x_n-T^n{x}_n\parallel }^2+\frac{1}{{\alpha }_n}〈x_n-w,y_n-T^n{y}_n〉\\ +\frac{1}{{\alpha }_n}\{{\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n\},\end{array}$$

which implies that

$${\alpha }_n[1-{\alpha }_n(1+L)]{\parallel x_n-T^n{x}_n\parallel }^2\le 〈x_n-w,y_n-T^n{y}_n〉+{\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n,$$

which shows that $w\in C_{n+1}$. By the mathematical induction principle, $\mathcal{F}(T)\subset C_n$ for all $n\ge 1$.

Next, we prove $\mathcal{F}(T)\subset Q_n$ for all $n\ge 1$. By induction, for $n=1$, we have $\mathcal{F}(T)\subset C=Q_1$. Assume that $\mathcal{F}(T)\subset Q_n$ for some $n\in \mathbb{N}$. Since $x_n$ is the projection of $x_1$ onto $C_n\cap Q_n$, by Lemma 2.1, we have

$$〈x_n-z,x_1-x_n〉\ge 0,\mathrm{\forall }z\in C_n\cap Q_n.$$

(3.2)

Since $\mathcal{F}(T)\subset C_n\cap Q_n$, we easily see that

$$〈x_n-w,x_1-x_n〉\ge 0,\mathrm{\forall }w\in \mathcal{F}(T),$$

(3.3)

which implies that $\mathcal{F}(T)\subset Q_{n+1}$. This proves that $\mathcal{F}(T)\subset C_n\cap Q_n$ for all $n\ge 1$.

Step 4. Show that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_1\parallel$ exists.

In view of (3.1) and Lemma 2.1, we have $x_n=P_{Q_n}{x}_1$ and $x_{n+1}\in Q_n$, which implies

$$\parallel x_n-x_1\parallel \le \parallel x_{n+1}-x_1\parallel ,\mathrm{\forall }n\ge 1.$$

On the other hand, since $\mathcal{F}(T)\subset Q_n$, we also have

$$\parallel x_n-x_1\parallel \le \parallel w-x_1\parallel ,\mathrm{\forall }w\in \mathcal{F}(T),\mathrm{\forall }n\ge 1.$$

Therefore, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_1\parallel$ exists and $\{x_n\}$ is bounded.

Step 5. Show that $\{x_n\}$ is a Cauchy sequence.

Noticing the construction of $C_n$, one has $C_m\subset C_n$ and $x_m=P_{C_m}{x}_1\in C_n$ for any positive integer $m>n$. From (3.2), we have

$$〈x_n-x_{n+m},x_1-x_n〉\ge 0.$$

It follows that

$$\begin{array}{rl}{\parallel x_n-x_{n+m}\parallel }^2& ={\parallel x_n-x_1+x_1-x_{n+m}\parallel }^2\\ ={\parallel x_n-x_1\parallel }^2+{\parallel x_1-x_{n+m}\parallel }^2-2〈x_1-x_n,x_1-x_{n+m}〉\\ ={\parallel x_n-x_1\parallel }^2+{\parallel x_1-x_{n+m}\parallel }^2-2〈x_1-x_n,x_1-x_n+x_n-x_{n+m}〉\\ \le {\parallel x_1-x_{n+m}\parallel }^2-{\parallel x_n-x_1\parallel }^2-〈x_1-x_n,x_n-x_{n+m}〉\\ \le {\parallel x_1-x_{n+m}\parallel }^2-{\parallel x_n-x_1\parallel }^2.\end{array}$$

(3.4)

Letting $n\to \mathrm{\infty }$ in (3.4), one has ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_{n+m}\parallel =0$, $\mathrm{\forall }m\ge n$. Hence, $\{x_n\}$ is a Cauchy sequence. Since H is a Hilbert space and C is closed and convex, one can assume that $x_n\to q\in C$ as $n\to \mathrm{\infty }$.

Step 6. Show that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$.

It follows from $x_{n+1}\in C_n$ and (3.1) that

$$\begin{array}{rl}{\alpha }_n[1-{\alpha }_n(1+L)]{\parallel x_n-T^n{x}_n\parallel }^2& \le 〈x_n-x_{n+1},y_n-T^n{y}_n〉+{\theta }_n\\ \le \parallel x_n-x_{n+1}\parallel \parallel y_n-T^n{y}_n\parallel +{\theta }_n.\end{array}$$

(3.5)

Since $\{y_n\}$ is bounded, $\{T^n{y}_n\}$ is bounded, ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$ and ${\alpha }_n\in (a,b)$, we have from (3.5) that

$$\underset{n\to \mathrm{\infty }}{lim}\parallel x_n-T^n{x}_n\parallel =0.$$

On the other hand, we notice that

$$\begin{array}{rl}\parallel x_n-T{x}_n\parallel \le & \parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-T^{n+1}{x}_{n+1}\parallel \\ +\parallel T^{n+1}{x}_{n+1}-T^{n+1}{x}_n\parallel +\parallel T^{n+1}{x}_n-T{x}_n\parallel \\ \le & (1+L)\parallel x_n-x_{n+1}\parallel +\parallel x_{n+1}-T^{n+1}{x}_{n+1}\parallel +L\parallel T^n{x}_n-x_n\parallel .\end{array}$$

From ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel =0$ and ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T^n{x}_n\parallel =0$, we have

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

It follows that $T{x}_n\to q$ as $n\to \mathrm{\infty }$. Since T is continuous, one has that q is a fixed point of T; that is, $q\in \mathcal{F}(T)$.

Step 7. Finally, we prove $q=P_{\mathcal{F}(T)}{x}_1$.

By taking the limit in (3.3), we have

$$〈q-w,x_1-q〉\ge 0,\mathrm{\forall }w\in \mathcal{F}(T),$$

which implies that $q=P_{\mathcal{F}(T)}{x}_1$ by using Lemma 2.1. This completes the proof. □

Since every total asymptotically pseudo-contractive mapping with $\mathcal{F}(T)\ne \mathrm{\varnothing }$ is total quasi-asymptotically pseudo-contractive, we immediately obtain the following corollary:

Corollary 3.2 Let C be a nonempty bounded and closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a uniformly L-Lipschitzian and total asymptotically pseudo-contractive mapping with $\mathcal{F}(T)\ne \mathrm{\varnothing }$. Suppose there exist positive constants M and $M^{\ast }$ such that $\varphi (\zeta )\le M^{\ast }{\zeta }^2$ for all $\zeta >M$. Let $\{x_n\}$ be a sequence generated by the following iterative scheme:

$$\{\begin{array}{c}{x}_1\in C\mathit{\text{chosen arbitrarily}},\hfill \\ C_1=C,Q_1=C,\hfill \\ y_n=(1-{\alpha }_n)x_n+{\alpha }_n{T}^n{x}_n,\hfill \\ C_{n+1}=\{z\in C_n:{\alpha }_n[1-{\alpha }_n(1+L)]{\parallel x_n-T^n{x}_n\parallel }^2\le 〈x_n-z,y_n-T^n{y}_n〉+{\theta }_n\},\hfill \\ Q_{n+1}=\{z\in Q_n:〈x_n-z,x_1-x_n〉\ge 0\},\hfill \\ x_{n+1}=P_{C_{n+1}\cap Q_{n+1}}{x}_1,\mathrm{\forall }n\ge 1,\hfill \end{array}$$

where ${\theta }_n={\mu }_n[\varphi (M)+M^{\ast }{(diamC)}^2]+{\xi }_n$, $\{{\alpha }_n\}$ is a sequence in $[a,b]$ with $a,b\in (0,\frac{1}{1+L})$. Then the sequence $\{x_n\}$ converges strongly to a point $P_{\mathcal{F}(T)}{x}_1$, where $P_{\mathcal{F}(T)}$ is the projection from C onto $\mathcal{F}(T)$.

Remark 3.3 Since the class of the total quasi-asymptotically pseudo-contractive mappings includes the class of asymptotically pseudocontractive mappings, the class of asymptotically pseudocontractive mappings in the intermediate sense, the class of the total asymptotically pseudo-contractive mappings, the class of quasi-asymptotically pseudo-contractive mappings in the intermediate sense as special cases, Theorem 3.1 improves the corresponding results in Zhou [7], Qin et al. [9], Chang [10] and Qin et al. [12].