Fixed point theorems of $(a,b)$-monotone mappings in Hilbert spaces

Abstract

We propose a new class of nonlinear mappings, called $(a,b)$-monotone mappings, and show that this class of nonlinear mappings contains nonspreading mappings, hybrid mappings, firmly nonexpansive mappings, and $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mappings with $a_1<1$. We also give an example to show that a $(a,b)$-monotone mapping is not necessary to be a quasi-nonexpansive mapping. We establish an existence theorem of fixed points and the demiclosed principle for the class of $(a,b)$-monotone mappings. As a special case of our result, we give an existence theorem of fixed points for $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mappings with $a_1<1$. We also consider Mann’s type weak convergence theorem and CQ type strong convergence theorem for $(a,b)$-monotone mappings. We give an example of $(a,b)$-monotone mappings which assures the Mann’s type weak convergence.

1 Introduction

Let H be a real Hilbert space with a nonempty closed convex subset C. Let $T:C\to C$ be a self-mapping defined on C. We denote by $F(T)$ the set of fixed points of T. The mapping T is called quasi-nonexpansive if $F(T)\ne \mathrm{\varnothing }$ and

$$\parallel Tx-Ty\parallel \le \parallel x-y\parallel \text{for all }x\in C\text{ and }y\in F(T).$$

Takahashi et al. [1–7] gave the following definitions of nonlinear mappings and studied the existence and convergence theorems of fixed points for these mappings.

Definition 1.1 A mapping $T:C\to C$ is called

• (i) nonspreading [1] if for every $x,y\in C$,

  $$2{\parallel Tx-Ty\parallel }^2\le {\parallel Tx-y\parallel }^2+{\parallel Ty-x\parallel }^2,$$

• (ii) TY [3] if for every $x,y\in C$,

  $$2{\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+{\parallel Tx-y\parallel }^2,$$

• (iii) hybrid [4] if for every $x,y\in C$,

  $$3{\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+{\parallel Tx-y\parallel }^2+{\parallel Ty-x\parallel }^2,$$

• (iv) λ-hybrid ($\lambda \in \mathbb{R}$) [5] if for every $x,y\in C$,

  $${\parallel Tx-Ty\parallel }^2\le {\parallel x-y\parallel }^2+2\lambda 〈x-Tx,y-Ty〉,$$

• (v) $(\alpha ,\beta )$-generalized hybrid ($\alpha ,\beta \in \mathbb{R}$) [6] if for every $x,y\in C$,

  $$\alpha {\parallel Tx-Ty\parallel }^2+(1-\alpha ){\parallel Ty-x\parallel }^2\le \beta {\parallel Tx-y\parallel }^2+(1-\beta ){\parallel x-y\parallel }^2,$$

• (vi) α-nonexpansive ($\alpha \in (-\mathrm{\infty },1)$) [7] if for every $x,y\in C$,

  $${\parallel Tx-Ty\parallel }^2\le \alpha {\parallel Tx-y\parallel }^2+\alpha {\parallel Ty-x\parallel }^2+(1-2\alpha ){\parallel x-y\parallel }^2.$$

It is obvious that the mappings mentioned in Definition 1.1 are quasi-nonexpansive. Recently, Lin et al. [8] gave the following definition of a new class of nonlinear mappings.

Definition 1.2 [8]

Let $a_1\in [0,1]$, $a_2,a_3\in [0,1)$, $k_1,k_2\in [0,1-a_2)\cap [0,1-a_3)$ and $a_1+a_2+a_3=1$. A mapping $T:C\to C$ is called a $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mapping if for every $x,y\in C$,

$${\parallel Tx-Ty\parallel }^2\le a_1{\parallel x-y\parallel }^2+a_2{\parallel Tx-y\parallel }^2+a_3{\parallel Ty-x\parallel }^2+k_1{\parallel x-Tx\parallel }^2+k_2{\parallel y-Ty\parallel }^2.$$

This class of mappings are not necessary to be quasi-nonexpansive and contains nonexpansive mappings, nonspreading mappings, hybrid mappings, and TY mappings. Lin et al. [8] studied weak and strong convergence theorems of $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mappings, but existence theorems of fixed points for $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mapping are not discussed in [8]. On the other hand, Aoyama and Kohsaka [7] characterized the existence of fixed points of α-nonexpansive mappings in uniformly convex Banach spaces.

Motivated by the literatures above, we study existence theorems of fixed points for the mappings mentioned in Definitions 1.1 and 1.2 in an unified method. Precisely, we propose a new class of nonlinear mappings in Hilbert spaces.

Definition 1.3 Let $a\in (\frac{1}{2},\mathrm{\infty })$ and $b\in (-\mathrm{\infty },a)$. A mapping $T:C\to C$ is called an $(a,b)$-monotone mapping if for every $x,y\in C$,

$$〈x-y,Tx-Ty〉\ge a{\parallel Tx-Ty\parallel }^2+(1-a){\parallel x-y\parallel }^2-b{\parallel x-Tx\parallel }^2-b{\parallel y-Ty\parallel }^2$$

or equivalently,

Remark 1.1 Let C be a nonempty, closed, and convex subset of a Hilbert space, and let $\alpha >0$. Recall that a mapping $T:C\to C$ is called α-inverse strongly monotone if

$$〈x-y,Tx-Ty〉\ge \alpha {\parallel Tx-Ty\parallel }^2\text{for all }x,y\in C.$$

A firmly nonexpansive mapping is an α-inverse strongly monotone mapping with $\alpha =1$. Note that a firmly nonexpansive mapping (1-inverse strongly monotone mapping with $\alpha =1$) is a (1,0)-monotone mapping.

Next, we give an example to show that a $(a,b)$-monotone mapping is not necessary to be a quasi-nonexpansive mapping.

Example 1.1 Let $H={\mathbb{R}}^2$. Let $\varphi :\mathbb{R}\times H\to H$ and $T:H\to H$ be defined by

for all $\alpha \in \mathbb{R}$ and for all $x=(rcos\theta ,rsin\theta )\in H$. Then the following statements hold:

• (i) T is a $(4,3)$-monotone mapping;

• (ii) T is not a quasi-nonexpansive mapping.

Proof It’s obvious that $F(T)=\{0\}$. We first prove part (i). For each $x\in H$,

$$Tx=\frac{5}{4}\varphi (\frac{3}{4}\pi ,x)=\frac{5\sqrt{2}}{8}\varphi (\pi ,x)+\frac{5\sqrt{2}}{8}\varphi (\frac{\pi }{2},x).$$

Then for each $x,y\in H$, we have

• (1) $〈x-y,Tx-Ty〉=-\frac{5\sqrt{2}}{8}{\parallel x-y\parallel }^2$,

• (2) ${\parallel Tx-Ty\parallel }^2=\frac{25}{16}{\parallel x-y\parallel }^2$,(3)

  $${\parallel x-Tx\parallel }^2={\parallel x\parallel }^2+{\parallel Tx\parallel }^2-2\parallel x\parallel \parallel Tx\parallel cos\frac{3}{4}\pi =(\frac{41}{16}+\frac{5\sqrt{2}}{4}){\parallel x\parallel }^2.$$

Then

By parallelogram law, we have

Take $a=4$ and $b=3$. Then

Then T is a $(4,3)$-monotone mapping. Next we want to prove part (ii). Since $\parallel Tx-T0\parallel =\frac{5}{4}\parallel x-0\parallel$, T is not a quasi-nonexpansive mapping. The proof of part (ii) is complete. □

Remark 1.2 Since T in Example 1.1 is not a quasi-nonexpansive mapping, T is not nonspreading, TY, hybrid, λ-hybrid, $(\alpha ,\beta )$-generalized hybrid, and α-nonexpansive. This example shows that an $(a,b)$-monotone mapping is not necessary to be a quasi-nonexpansive mapping, TY mapping, hybrid mapping, λ-hybrid mapping, $(\alpha ,\beta )$-generalized hybrid mapping, and α-nonexpansive mapping.

In this paper, we first show that the class of $(a,b)$-monotone mappings contains nonspreading mappings, hybrid mappings, TY mappings, firmly nonexpansive mappings, and $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mappings with $a_1<1$. We also give an example to show that this class of mappings are not necessary to be quasi-nonexpansive mappings. We establish an existence theorem of fixed points and the demiclosed principle for the class of $(a,b)$-monotone mappings. As a special case of our result, we give an existence theorem of fixed points for $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mappings with $a_1<1$. We also consider Mann’s type weak convergence theorem and CQ type strong convergence theorem for $(a,b)$-monotone mappings. An example of $(a,b)$-monotone mappings is given to show the Mann’s type weak convergence.

2 Preliminaries

In this paper, we use the following notations:

• (i) ⇀ for weak convergence and → for strong convergence.

• (ii) ${\omega }_w(x_n)=\{x:\mathrm{\exists }{x}_{n_j}\rightharpoonup x\}$ denotes the weak ω-limit set of $\{x_n\}$.

Let us recall some known results, which will be used later.

Proposition 2.1 [8]

Let C be a nonempty, closed, and convex subset of a Hilbert space H. A mapping $T:C\to C$ be a mapping.

• (i) If T is a nonexpansive mapping, then T is a $(1,0,0,0,0)$-generalized hybrid mapping;

• (ii) If T is a nonspreading mapping, then T is a $(0,\frac{1}{2},\frac{1}{2},0,0)$-generalized hybrid mapping;

• (iii) If T is a hybrid mapping, then T is a $(\frac{1}{3},\frac{1}{3},\frac{1}{3},0,0)$-generalized hybrid mapping;

• (iv) If T is a TY mapping, then T is a $(\frac{1}{2},\frac{1}{2},0,0,0)$-generalized hybrid mapping;

• (v) If T is an $(\alpha ,\beta )$-generalized hybrid mapping with $\alpha \ge 1\ge \beta \ge 0$ and $\alpha >\beta$, then T is a $(\frac{1-\beta }{\alpha },\frac{\beta }{\alpha },1-\frac{1}{\alpha },0,0)$-generalized hybrid mapping.

Lemma 2.1 [3]

Let C be a nonempty, closed and convex subset of a Hilbert space H. Let $T:C\to C$ be a mapping. Suppose that there exist $x\in C$ and a Banach limit μ such that $\{T^{n}x\}$ is bounded and

$${\mu }_n{\parallel T^{n}x-Ty\parallel }^2\le {\mu }_n{\parallel T^{n}x-y\parallel }^2\mathit{\text{for all}}y\in C.$$

Then T has a fixed point.

Lemma 2.2 [9]

Let H be a real Hilbert space. Let C be a closed convex subset of H, let $w,x,y\in H$ and let a be a real number. The set

$$D:=\{v\in C:{\parallel y-v\parallel }^2\le {\parallel x-v\parallel }^2+〈w,v〉+a\}$$

is closed and convex.

Lemma 2.3 Let K be a closed convex subset of a real Hilbert space H and let $P_K$ be the metric projection from H onto K. Let $x\in H$ and $z\in K$. Then $z=P_{K}x$ if and only if

$$〈x-z,y-z〉\le 0\mathit{\text{for all}}y\in K.$$

Lemma 2.4 [9]

Let K be a closed convex subset of a real Hilbert space H. Let $\{x_n\}$ be a sequence in H and $u\in H$. Let $q=P_{K}u$. Suppose that ${\omega }_w(x_n)\subseteq K$ and

$$\parallel x_n-u\parallel \le \parallel u-q\parallel \mathit{\text{for all}}n\in \mathbb{N}.$$

Then $x_n\to q$.

Lemma 2.5 Let H be a real Hilbert space. Then

$${\parallel u-v\parallel }^2={\parallel u\parallel }^2-{\parallel v\parallel }^2-2〈u-v,v〉,\mathit{\text{for all}}u,v\in H.$$

Theorem 2.1 [10]

Let H be a Hilbert space and let $\{x_n\}$ be a bounded sequence in H. Then $\{x_n\}$ is weakly convergent if and only if each weakly convergent subsequence of $\{x_n\}$ has the same weak limit, that is, for $x\in H$,

$$x_n\rightharpoonup x\iff (x_{n_i}\rightharpoonup y\Rightarrow x=y).$$

3 Fixed point theorem of $(a,b)$-monotone mappings

Proposition 3.1 Let C be a nonempty, closed, and convex subset of a Hilbert space H. If $T:C\to C$ is a $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mapping with $a_1<1$, then T is a $(\frac{1}{1-a_1},\frac{k}{1-a_1}+\frac{1}{2})$-monotone mapping, where $k=max\{k_1,k_2\}$.

Proof If T is an $(a_1,a_2,a_3,k_1,k_2)$-generalized hybrid mapping with $a_1<1$, then for every $x,y\in C$,

and

where $k=max\{k_1,k_2\}$.

Note that

We have

Without loss of generality, we may assume that $a_2\ge a_3$.

Since $k<min\{1-a_2,1-a_3\}$, we have that

$$\begin{array}{rcl}\frac{k}{1-a_1}+\frac{1}{2}-\frac{1}{1-a_1}& =& \frac{1}{2(1-a_1)}(2k+(1-a_1)-2)\\ <& \frac{1}{2(1-a_1)}(2(1-a_2)+(1-a_1)-2)=\frac{1}{2(1-a_1)}(1-a_1-2a_2)\\ \le & \frac{1}{2(1-a_1)}(1-a_1-a_2-a_3)=0,\end{array}$$

that is, $\frac{k}{1-a_1}+\frac{1}{2}<\frac{1}{1-a_1}$. Take $a=\frac{1}{1-a_1}\ge 1>\frac{1}{2}$ and $b=\frac{k}{1-a_1}+\frac{1}{2}<a$, we see that T is an $(a,b)$-monotone mapping. □

The following proposition follows immediately from Propositions 2.1 and 3.1.

Proposition 3.2 Let C be a nonempty, closed, and convex subset of a Hilbert space H. A mapping $T:C\to C$ be a mapping.

• (i) If T is a nonspreading mapping, then T is a $(1,\frac{1}{2})$-monotone mapping;

• (ii) If T is a hybrid mapping, then T is a $(\frac{3}{2},\frac{1}{2})$-monotone mapping;

• (iii) If T is a TY mapping, then T is a $(2,\frac{1}{2})$-monotone mapping;

• (vi) If T is an $(\alpha ,\beta )$-generalized hybrid mapping with $\alpha \ge 1\ge \beta \ge 0$, $\alpha >\beta$ and $\alpha +\beta >1$, then T is an $(\frac{\alpha }{\alpha +\beta -1},\frac{1}{2})$-monotone mapping.

Proposition 3.3 Let C be a closed convex subset of a Hilbert space. Let T be a $(a,b)$-monotone mapping defined on C. Then

$${\parallel x-p\parallel }^2\ge {\parallel Tx-p\parallel }^2+\frac{1-2b}{2a-1}{\parallel x-Tx\parallel }^2\mathit{\text{for all}}x\in C\mathit{\text{and}}p\in F(T).$$

Proof Since T is a $(a,b)$-monotone mapping, we have that for each $x\in C$ and $p\in F(T)$,

$$〈x-p,Tx-p〉\ge a{\parallel Tx-p\parallel }^2+(1-a){\parallel x-p\parallel }^2-b{\parallel x-Tx\parallel }^2,$$

that is,

Then

$$(2a-1){\parallel x-p\parallel }^2\ge (2a-1){\parallel Tx-p\parallel }^2+(1-2b){\parallel x-Tx\parallel }^2,$$

that is,

$${\parallel x-p\parallel }^2\ge {\parallel Tx-p\parallel }^2+\frac{1-2b}{2a-1}{\parallel x-Tx\parallel }^2.$$

 □

Now we give a demiclosed principle of $(a,b)$-monotone mappings:

Theorem 3.1 Let C be a closed convex subset of a Hilbert space. Let T be a $(a,b)$-monotone mapping defined on C. If a sequence $\{x_n\}\subseteq C$ with $x_n\rightharpoonup x^{\ast }$ and $\parallel x_n-T{x}_n\parallel \to 0$. Then $x^{\ast }=T{x}^{\ast }$.

Proof Since T is a $(a,b)$-monotone mapping, we have that

that is,

Since $x_n\rightharpoonup x^{\ast }$ and $\parallel x_n-T{x}_n\parallel \to 0$, $\{x_n\}$ and $\{T{x}_n\}$ are bounded. Taking limit on the inequality above, we have

$$b{\parallel x^{\ast }-T{x}^{\ast }\parallel }^2\ge a{\parallel x^{\ast }-T{x}^{\ast }\parallel }^2.$$

Since $b<a$, we have that $\parallel x^{\ast }-T{x}^{\ast }\parallel =0$, that is, $x^{\ast }=T{x}^{\ast }$. □

Corollary 3.1 [2–4]

Let C be a closed convex subset of a Hilbert space. Let T be a self-mapping defined on C and satisfies one of the following:

• (i) T is a nonspreading mapping;

• (ii) T is a hybrid mapping;

• (iii) T is a TY mapping.

If a sequence $\{x_n\}\subseteq C$ with $x_n\rightharpoonup x^{\ast }$ and $\parallel x_n-T{x}_n\parallel \to 0$, then $x^{\ast }=T{x}^{\ast }$.

Theorem 3.2 Let C be a closed convex subset of a Hilbert space. Let T be a $(a,b)$-monotone mapping defined on C. If $F(T)$ is nonempty, then $F(T)$ is closed and convex.

Proof First, we show that $F(T)$ is closed. For each $x\in \overline{F(T)}$, there exists a sequence ${\{x_n\}}_{n\in \mathbb{N}}\subseteq F(T)$ with $x_n\to x$. Since $x_n\to x$ and $x_n\in F(T)$ for all $n\in \mathbb{N}$, we have that $x_n\rightharpoonup x$ and $\parallel x_n-T{x}_n\parallel =0$ for all $n\in \mathbb{N}$. By Theorem 3.1, $x=Tx$. Next, we want to show that $F(T)$ is a convex subset of C. Take any $u,v\in F(T)$ and $t\in [0,1]$. Let $z_t:=tu+(1-t)v$. By Proposition 3.3, we have

Since $\frac{2b-1}{2a-1}<1$, we have that $z_t=T{z}_t$. □

Theorem 3.3 Let C be a nonempty subset of a Hilbert space H. Let $T:C\to C$ be a $(a,b)$-monotone mapping with $b\in (-\mathrm{\infty },\frac{1}{2}]$. Suppose that $\{T^{n}x\}$ is bounded for some $x\in C$. Then ${\mu }_n{\parallel T^{n}x-Ty\parallel }^2\le {\mu }_n{\parallel T^{n}x-y\parallel }^2$ for all Banach limits μ and for all $y\in C$.

Proof Let μ be a Banach limit and let $y\in C$ be given. Since T is a $(a,b)$-monotone mapping with $b\in (-\mathrm{\infty },\frac{1}{2}]$, we have that

that is,

$$\frac{1}{2}{\parallel T^{n}x-Ty\parallel }^2+\frac{1}{2}\parallel y-T^{n+1}x\parallel \ge a{\parallel T^{n+1}x-Ty\parallel }^2+(1-a){\parallel T^{n}x-y\parallel }^2.$$

Then

$${\mu }_n{\parallel Ty-T^{n}x\parallel }^2+{\mu }_n{\parallel y-T^{n}x\parallel }^2\ge 2a{\mu }_n{\parallel Ty-T^{n}x\parallel }^2+2(1-a){\mu }_n{\parallel y-T^{n}x\parallel }^2.$$

Hence $(2a-1){\mu }_n{\parallel y-T^{n}x\parallel }^2\ge (2a-1){\mu }_n{\parallel Ty-T^{n}x\parallel }^2$. Since $a>\frac{1}{2}$, we have that

$${\mu }_n{\parallel y-T^{n}x\parallel }^2\ge {\mu }_n{\parallel Ty-T^{n}x\parallel }^2.$$

 □

As a direct consequence of Theorem 3.3 and Lemma 2.1, we have the following existence theorem of fixed points for $(a,b)$-monotone mappings.

Theorem 3.4 Let C be a nonempty, closed, and convex subset of a Hilbert space H. Let $T:C\to C$ be a $(a,b)$-monotone mapping with $b\in (-\mathrm{\infty },\frac{1}{2}]$. Then $F(T)\ne \mathrm{\varnothing }$ if and only if there exists $x\in C$ such that $\{T^{n}x\}$ is bounded.

Corollary 3.2 [1, 3, 4]

Let C be a closed convex subset of a Hilbert space. Let T be a self-mapping defined on C and satisfies one of the following:

• (i) T is a nonspreading mapping;

• (ii) T is a hybrid mapping;

• (iii) T is a TY mapping.

Then $F(T)\ne \mathrm{\varnothing }$ if and only if there exists $x\in C$ such that $\{T^{n}x\}$ is bounded.

4 Convergence theorems

In this section, we first prove a weak convergence theorem of Mann’s type for $(a,b)$-monotone mappings in a Hilbert space.

Theorem 4.1 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let $T:C\to C$ be a $(a,b)$-monotone mapping satisfies $F(T)\ne \mathrm{\varnothing }$. If a sequence ${\{{\alpha }_n\}}_{n\in \mathbb{N}}\subseteq (0,1)$ with ${\alpha }_n>\frac{2b-1}{2a-1}$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}(1-{\alpha }_n)({\alpha }_n+\frac{1-2b}{2a-1})>0$, then for each $x_1\in C$, the sequence $\{x_n\}$ with $x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n$ for all $n\in \mathbb{N}$ weakly converges to some fixed point of T.

Proof We first show that there exists a sequence $\{{\alpha }_n\}$ satisfies our assumptions. Since $b<a$, $2a>1$, we have $\frac{2b-1}{2a-1}<1$, there exists a constant $\alpha \in \mathbb{R}$ such that $\frac{2b-1}{2a-1}<\alpha <1$. If we take ${\alpha }_n=\alpha$ for all $n\in \mathbb{N}$, then $\{{\alpha }_n\}\subseteq (0,1)$ such that ${\alpha }_n>\frac{2b-1}{2a-1}$ and ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}(1-{\alpha }_n)({\alpha }_n+\frac{1-2b}{2a-1})>0$. Since T is a $(a,b)$-monotone mapping, by Proposition 3.3, we have that for each $p\in F(T)$ and $x\in C$,

$${\parallel x-p\parallel }^2\ge {\parallel Tx-p\parallel }^2+\frac{1-2b}{2a-1}{\parallel x-Tx\parallel }^2.$$

Since ${\alpha }_n>\frac{2b-1}{2a-1}$, we have that

Then ${lim}_{n\to \mathrm{\infty }}\parallel x_n-p\parallel$ exists and sequence $\{x_n\}$ is bounded. Further, from the inequality above, we have that

$$(1-{\alpha }_n)({\alpha }_n+\frac{1-2b}{2a-1}){\parallel x_n-T{x}_n\parallel }^2\le {\parallel x_n-p\parallel }^2-{\parallel x_{n+1}-p\parallel }^2.$$

Since ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}(1-{\alpha }_n)({\alpha }_n+\frac{1-2b}{2a-1})>0$, we have ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$.

Therefore, ${lim}_{n\to \mathrm{\infty }}\parallel x_{n+1}-x_n\parallel ={lim}_{n\to \mathrm{\infty }}(1-{\alpha }_n)\parallel x_n-T{x}_n\parallel =0$. Since $\{x_n\}$ is bounded, there exist a subsequence $\{x_{n_j}\}$ of $\{x_n\}$ and a point $x^{\ast }\in C$ such that $x_{n_j}\rightharpoonup x^{\ast }$. Since T is a $(a,b)$-monotone mapping, by Theorem 3.1, we have $x^{\ast }=T{x}^{\ast }$.

For each subsequence $\{x_{n_k}\}$ of $\{x_n\}$ with $x_{n_k}\rightharpoonup u$ for some $u\in C$, we follow the same argument as above, we see that $u=Tu$. We have to show that $u=x^{\ast }$. Otherwise, if $u\ne x^{\ast }$, then by Optial condition,

$$\begin{array}{rcl}\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-x^{\ast }\parallel & <& \underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-u\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}\parallel x_n-u\parallel \\ =& \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_k}-u\parallel \\ <& \underset{k\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_k}-x^{\ast }\parallel \\ =& \underset{n\to \mathrm{\infty }}{lim}\parallel x_n-x^{\ast }\parallel =\underset{j\to \mathrm{\infty }}{lim\hspace{0.17em}inf}\parallel x_{n_j}-x^{\ast }\parallel .\end{array}$$

This leads to a contradiction. Therefore $u=x^{\ast }$. By Theorem 2.1, we have that $x_n\rightharpoonup x^{\ast }$. □

Example 4.1 Let H, ϕ, T be the same as in Example 1.1. For any fixed $x_1\in H$, take a sequence $\{x_n\}$ as in Theorem 4.1 with ${\alpha }_n=\frac{3}{4}$ for all $n\in \mathbb{N}$, that is,

$$x_{n+1}=\frac{3}{4}{x}_n+\frac{1}{4}T{x}_n.$$

Then

$$x_{n+1}=\frac{3}{4}{x}_n+\frac{5}{16}(\frac{\sqrt{2}}{2}\varphi (\pi ,x_n)+\frac{\sqrt{2}}{2}\varphi (\frac{\pi }{2},x_n))=(\frac{3}{4}-\frac{5\sqrt{2}}{32})x_n+\frac{5\sqrt{2}}{32}\varphi (\frac{\pi }{2},x_n)$$

and hence

$$\parallel x_{n+1}\parallel \le (\frac{3}{4}-\frac{5\sqrt{2}}{32})\parallel x_n\parallel +\frac{5\sqrt{2}}{32}\parallel x_n\parallel \le \frac{3}{4}\parallel x_n\parallel .$$

Therefore, $x_n\to 0\in F(T)$, and hence $x_n\rightharpoonup 0$.

Corollary 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let $T:C\to C$ be a mapping with $F(T)\ne \mathrm{\varnothing }$ and satisfies one of the following:

• (i) T is a nonspreading mapping;

• (ii) T is a hybrid mapping;

• (iii) T is a TY mapping.

If a sequence ${\{{\alpha }_n\}}_{n\in \mathbb{N}}\subseteq (0,1)$ satisfies ${lim\hspace{0.17em}inf}_{n\to \mathrm{\infty }}(1-{\alpha }_n)({\alpha }_n)>0$, then for each $x_1\in C$, the sequence $\{x_n\}$ with $x_{n+1}={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n$ for all $n\in \mathbb{N}$ weakly converges to some fixed point of T.

Proof Since T is an $(a,b)$-monotone mapping with $b=\frac{1}{2}$, we have $\frac{2b-1}{2a-1}=0$. Then Corollary 4.1 follows from Theorem 4.1. □

Corollary 4.2 Let C be a nonempty, closed, and convex subset of real Hilbert space H. Let $T:C\to C$ be a mapping with $F(T)\ne \mathrm{\varnothing }$ and satisfies one of the following:

• (i) T is a nonspreading mapping;

• (ii) T is a hybrid mapping;

• (iii) T is a TY mapping.

Then for each $x_1\in C$, the sequence $\{x_n\}$ with $x_{n+1}=\frac{1}{2}{x}_n+\frac{1}{2}T{x}_n$ for all $n\in \mathbb{N}$ weakly converges to some fixed point of T.

Proof Take ${\alpha }_n=\frac{1}{2}\text{ for all }n\in \mathbb{N}$. Then Corollary 4.2 follows from Corollary 4.1. □

Next we prove a strong convergence theorem by hybrid method for $(a,b)$-monotone mappings in a Hilbert space.

Theorem 4.2 Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let $T:C\to C$ be a $(a,b)$-monotone mapping with $F(T)\ne \mathrm{\varnothing }$. Suppose that $\{x_n\}$ is a sequence generated by the following scheme:

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}};\hfill \\ y_n=t_n{x}_n+(1-t_n)T{x}_n;\hfill \\ C_n=\{v\in C:{\parallel y_n-v\parallel }^2\le {\parallel x_n-v\parallel }^2+(1-t_n)\frac{2b-1}{2a-1}{\parallel x_n-T{x}_n\parallel }^2\};\hfill \\ Q_n=\{v\in C:〈x_n-v,x_n-x_0〉\le 0\};\hfill \\ x_{n+1}=P_{C_n\cap Q_n}{x}_0,\mathit{\text{where}}{P}_{C_n\cap Q_n}\mathit{\text{is the metric projection from H onto}}{C}_n\cap Q_n.\hfill \end{array}$$

If the sequence ${\{t_n\}}_{n\in \mathbb{N}}\subseteq (0,1)$ satisfies ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}(\frac{2b-1}{2a-1}+t_n)<1$, then $x_n\to P_{F(T)}{x}_0$.

Proof By Lemma 2.2, we see that $C_n$ is closed and convex for all $n\in \mathbb{N}$. For any $p\in F(T)$, by Proposition 3.3, we have

$$\begin{array}{rcl}{\parallel y_n-p\parallel }^2& =& {\parallel t_n(x_n-p)+(1-t_n)(T{x}_n-p)\parallel }^2\\ =& t_n{\parallel x_n-p\parallel }^2+(1-t_n){\parallel T{x}_n-p\parallel }^2-t_n(1-t_n){\parallel x_n-T{x}_n\parallel }^2\\ \le & t_n{\parallel x_n-p\parallel }^2+(1-t_n){\parallel T{x}_n-p\parallel }^2\\ \le & t_n{\parallel x_n-p\parallel }^2+(1-t_n)({\parallel x_n-p\parallel }^2+\frac{2b-1}{2a-1}{\parallel x_n-T{x}_n\parallel }^2)\\ =& {\parallel x_n-p\parallel }^2+(1-t_n)\frac{2b-1}{2a-1}{\parallel x_n-T{x}_n\parallel }^2.\end{array}$$

Hence, $p\in C_n$. Then we have that $F(T)\subseteq C_n$ for all $n\ge 0$.

Next, we show that $F(T)\subseteq Q_n$ for all $n\ge 0$. We prove this by induction. For $n=0$, we have $F(T)\subseteq C=Q_0$. Assume that $F(T)\subseteq Q_n$. Since $x_{n+1}$ is the projection of $x_0$ onto $C_n\cap Q_n$, by Lemma 2.3, we have $〈x_{n+1}-z,x_0-x_{n+1}〉\ge 0$ for all $z\in C_n\cap Q_n$. As $F(T)\subseteq C_n\cap Q_n$ by the induction assumption, the last inequality holds, in particular, for all $z\in F(T)$. This together with the definition of $Q_{n+1}$ implies that $F(T)\subseteq Q_{n+1}$. Hence $F(T)\subseteq Q_n$ for all $n\in \mathbb{N}$. Then the sequence $\{x_n\}$ is well defined.

The definition of $Q_n$ and Lemma 2.3 imply that $x_n=P_{Q_n}{x}_0$, which in turn implies that $\parallel x_n-x_0\parallel \le \parallel p-x_0\parallel$ for all $p\in F(T)$, in particular, $\{x_n\}$ is bounded and $\parallel x_n-x_0\parallel \le \parallel q-x_0\parallel$ with $q=P_{F(T)}{x}_0$.

That $x_{n+1}\in Q_n$ asserts that

$$〈x_n-x_{n+1},x_n-x_0〉\le 0.$$

It follows from Lemma 2.5 and the inequality above that

$$\begin{array}{rcl}{\parallel x_{n+1}-x_n\parallel }^2& =& {\parallel x_{n+1}-x_0\parallel }^2-{\parallel x_n-x_0\parallel }^2-2〈x_{n+1}-x_n,x_n-x_0〉\\ \le & {\parallel x_{n+1}-x_0\parallel }^2-{\parallel x_n-x_0\parallel }^2.\end{array}$$

The last inequality implies that $\{\parallel x_n-x_0\parallel \}$ is increasing. Since $\{x_n\}$ is bounded, we have that ${lim}_{n\to \mathrm{\infty }}\parallel x_n-x_0\parallel$ exists and $\parallel x_{n+1}-x_n\parallel \to 0$. Since $x_{n+1}\in C_n$,

Note that

Then

Then

Without loss of generality, we may assume that $x_n\ne T{x}_n$ for all $n\in \mathbb{N}$. Otherwise, $x_n\in F(T)$ for some $n\in \mathbb{N}$ and we complete the proof. Therefore,

$$\parallel x_n-T{x}_n\parallel -2\parallel x_n-x_{n+1}\parallel \le (\frac{2b-1}{2a-1}+t_n)\parallel x_n-T{x}_n\parallel .$$

Hence,

$$\underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}\parallel x_n-T{x}_n\parallel \le \underset{n\to \mathrm{\infty }}{lim\hspace{0.17em}sup}(\frac{2b-1}{2a-1}+t_n)\parallel x_n-T{x}_n\parallel .$$

By the choice of $\{t_n\}$, ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}(\frac{2b-1}{2a-1}+t_n)<1$. Therefore, ${lim}_{n\to \mathrm{\infty }}\parallel x_n-T{x}_n\parallel =0$. Consequently, ${\omega }_w(x_n)\subseteq F(T)$ by Theorem 3.1. Hence, applying Lemma 2.4 (to $u:=x_0$ and $K:=F(T)$), one can conclude that $x_n\to q$. □

Corollary 4.3 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let $T:C\to C$ be a mapping with $F(T)\ne \mathrm{\varnothing }$ and satisfies one of the following conditions:

• (i) T is a nonspreading mapping;

• (ii) T is a hybrid mapping;

• (iii) T is a TY mapping.

Suppose that $\{x_n\}$ is a sequence generated by the following scheme:

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}};\hfill \\ y_n=t_n{x}_n+(1-t_n)T{x}_n;\hfill \\ C_n=\{v\in C:{\parallel y_n-v\parallel }^2\le {\parallel x_n-v\parallel }^2+(1-t_n)\frac{2b-1}{2a-1}{\parallel x_n-T{x}_n\parallel }^2\};\hfill \\ Q_n=\{v\in C:〈x_n-v,x_n-x_0〉\le 0\};\hfill \\ x_{n+1}=P_{C_n\cap Q_n}{x}_0,\mathit{\text{where}}{P}_{C_n\cap Q_n}\mathit{\text{is the metric projection from H onto}}{C}_n\cap Q_n.\hfill \end{array}$$

If the sequence ${\{t_n\}}_{n\in \mathbb{N}}\subseteq (0,1)$ satisfies ${lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{t}_n<1$. Then $x_n\to P_{F(T)}{x}_0$.

Proof Since T is a $(a,b)$-monotone mapping with $b=\frac{1}{2}$, we have $\frac{2b-1}{2a-1}=0$, then Corollary 4.3 follows from Theorem 4.2. □

Corollary 4.4 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let $T:C\to C$ be a mapping with $F(T)\ne \mathrm{\varnothing }$ and satisfies one of the following conditions:

• (i) T is a nonspreading mapping;

• (ii) T is a hybrid mapping;

• (iii) T is a TY mapping.

Suppose that $\{x_n\}$ is a sequence generated by the following scheme:

$$\{\begin{array}{c}{x}_0\in C\mathit{\text{chosen arbitrarily}};\hfill \\ y_n=\frac{1}{2}{x}_n+\frac{1}{2}T{x}_n;\hfill \\ C_n=\{v\in C:{\parallel y_n-v\parallel }^2\le {\parallel x_n-v\parallel }^2\};\hfill \\ Q_n=\{v\in C:〈x_n-v,x_n-x_0〉\le 0\};\hfill \\ x_{n+1}=P_{C_n\cap Q_n}{x}_0,\mathit{\text{where}}{P}_{C_n\cap Q_n}\mathit{\text{is the metric projection from H onto}}{C}_n\cap Q_n.\hfill \end{array}$$

Then $x_n\to P_{F(T)}{x}_0$.

Proof Take $t_n=\frac{1}{2}$ for all $n\in \mathbb{N}$, then Corollary 4.4 follows from Corollary 4.3. □