On Generalized Nonexpansive Maps in Banach Spaces

Abstract

We introduce a very general class of generalized non-expansive maps. This new class of maps properly includes the class of Suzuki non-expansive maps, Reich–Suzuki type non-expansive maps, and generalized $\alpha$-non-expansive maps. We establish some basic properties and demiclosed principle for this class of maps. After this, we establish existence and convergence results for this class of maps in the context of uniformly convex Banach spaces and compare several well known iterative algorithms.

1. Introduction

Let X be a Banach space, E be a nonempty subset of X and $P:E\to E$ be a selfmap. An element $z\in E$ is called a fixed point of P if $z=P(z)$. From now on, we will denote the set of all fixed points of P by the notation $F(P)$. The map P is called non-expansive if $||P(a)-P(b)||\le ||a-b||$ for all $a,b\in E$ and it is called quasi-non-expansive if $||P(z)-P(a)||\le ||z-a||$ for all $a\in E$ and $z\in F(P)$. In 1965, Kirk [1], Browder [2], and Gohde [3] independently proved that every non-expansive map has a fixed point if E is closed bounded convex and X is uniformly convex. Fixed point theory of non-expansive and generalized non-expansive maps in an appropriate domain is an important research area on its own and has applications in image recovery and signal processing (see, e.g., [4,5,6,7,8] and references cited therein).

In 2008, Suzuki [9] suggested a weaker notion of non-expansive maps as follows:

Definition 1.

[9] A selfmap P on a subset E of a Banach space is said to satisfy condition $(C)$ (or said to be Suzuki non-expansive) if for each two elements $a,b\in E$,

$$\frac{1}{2}||a-P(a)||\le ||a-b||\Rightarrow ||P(a)-P(b)||\le ||a-b||.$$

Remark 1.

It is clear that every non-expansive map is Suzuki non-expansive. However, an example in [9] shows that there exists maps which are Suzuki non-expansive but not non-expansive.

In 2011, Aoyama and Kohsaka [10] proposed the class of $\alpha$-non-expansive maps as follows:

Definition 2.

[10] A selfmap P on a subset E of a Banach space is said to be α-non-expansive if one can find a real number $\alpha \in [0,1)$ such that for each two elements $a,b\in E$,

$$||P(a)-{P(b)||}^2\le \alpha ||a-{P(b)||}^2+\alpha ||b-{P(a)||}^2+(1-2\alpha )||a-{b||}^2.$$

In 2017, Pant and Shukla [11] proposed the class of generalized $\alpha$-non-expansive maps as follows:

Definition 3.

[11] A selfmap P on a subset E of a Banach space is said to be generalized α-non-expansive if one can find a real number $\alpha \in [0,1)$ such that for each two elements $a,b\in E$,

$$\frac{1}{2}||a-P(a)||\le ||a-b||\Rightarrow$$

$$||P(a)-P(b)||\le \alpha ||b-P(a)||+\alpha ||a-P(b)||+(1-2\alpha )||a-b||.$$

Remark 2.

It is clear that every Suzuki non-expansive map is generalized 0-non-expansive. However, an example in [11] shows that there exist maps which are generalized α-non-expansive but not Suzuki non-expansive.

In 2019, Pant and Pandey [12] proposed the class of Reich–Suzuki type non-expansive maps as follows:

Definition 4.

[12] A selfmap P on a subset E of a Banach space is said to be β-Reich–Suzuki type non-expansive if one can find a real number $\beta \in [0,1)$ such that for each two elements $a,b\in E$,

$$\frac{1}{2}||a-P(a)||\le ||a-b||\Rightarrow$$

$$||P(a)-P(b)||\le \beta ||a-P(a)||+\beta ||b-P(b)||+(1-2\beta )||a-b||.$$

Remark 3.

It is clear that every Suzuki non-expansive map is 0-Reich–Suzuki type non-expansive. However an example in [12] shows that there exists maps which are β-Reich–Suzuki type non-expansive but not Suzuki non-expansive.

Motivated by the above definitions, in this paper, we introduce a new class of generalized non-expansive maps which is properly larger than the the class of Suzuki non-expansive maps, generalized $\alpha$-non-expansive maps and Reich–Suzuki type non-expansive maps. We also establish some basic results for this class. In this way, we improve and extend many well known corresponding results of the metric fixed point theory.

2. Preliminaries

A Banach space X is called uniformly convex [13] if, for any real number $\epsilon \in [0,1)$, one can find a real number $\delta \in (0,\infty )$ such that, $\frac{||a+b||}{2}\le (1-\delta ),$ whenever $||a||\le 1$, $||b||\le 1$ and $||a-b||\ge \epsilon$ for each $a,b\in E$. X is called strict convex if, for any $a,b\in X$ satisfying $||a||=||b||=1$ and $a\ne b$, it follows that $||a+b||<2$.

A Banach space X is said to satisfy Opial condition [14], if, for every weakly convergent sequence $\{a_n\}\subseteq X$ with weak limit say $w\in X$, it follows that

$$\underset{m\to \infty }{lim\; inf}||a_m-w||<\underset{m\to \infty }{lim\; inf}||a_m-w^{\prime }||\mathrm{for}\mathrm{all}{w}^{\prime }\in X-\{w\}.$$

Let E be a nonempty subset of a Banach space X and $\{a_m\}$ a bounded sequence in X. For each $a\in E$, define:

• asymptotic radius of $\{a_m\}$ at a by $A_r(a,\{a_m\}):={lim\; sup}_{m\to \infty }||a-a_m||$;
• asymptotic radius of $\{a_m\}$ relative to E by $A_r(E,\{a_m\})=inf\{A_r(a,\{a_m\}):a\in E\}$;
• asymptotic center of $\{a_m\}$ relative to E by $A_c(E,\{a_m\})=\{a\in E:A_r(a,\{a_m\})=A_r(E,\{a_m\})\}$.

When the space X is uniformly convex [13], then the set $A_c(E,\{a_m\})$ is always singleton. Notice also that the set $A_c(E,\{a_m\})$ is convex as well as nonempty provided that E is weakly compact convex (see, e.g., [15,16]).

The following result is a characterization of uniform convexity, which can be found in [17].

Lemma 1.

Let X be a uniformly convex Banach space and $0<s\le k_m\le t<1$ for every $m\ge 1$. If $\{y_m\}$ and $\{z_m\}$ are two sequences in X such that ${lim\; sup}_{m\to \infty }||y_m||\le \gamma$, ${lim\; sup}_{m\to \infty }||z_m||\le \gamma$ and ${lim}_{m\to \infty }||k_m{y}_m+(1-k_m)z_m||=\gamma$ for some $\gamma \ge 0$, then ${lim}_{m\to \infty }||y_m-z_m||=0$.

3. Generalized $(\mathit{\alpha },\mathit{\beta })$-Non-Expansive Mappings

Inspired by above and [18], we suggest a two parametric class of nonlinear maps.

Definition 5.

A selfmap P on a subset E of a Banach space is said to be generalized $(\alpha ,\beta )$-non-expansive if there exists real numbers $\alpha ,\beta \in {\mathbb{R}}^{+}$ satisfying $\alpha +\beta <1$ such that, for all $a,b\in E$,

$$\frac{1}{2}||a-P(a)||\le ||a-b||\Rightarrow ||P(a)-P(b)||\le \alpha ||a-P(b)||+\alpha ||b-P(a)||+\beta ||a-P(a)||$$

$$+\beta ||b-P(b)||+(1-2\alpha -2\beta )||a-b||.$$

The following proposition gives many examples of generalized $(\alpha ,\beta )$-non-expansive maps.

Proposition 1.

Let P be a selfmap on a subset E of a Banach space. Then, the following hold:

(i) 

If P is Suzuki non-expansive, then P is generalized $(0,0)$-non-expansive.

(ii) 

If P is generalized α-non-expansive, then P is generalized $(\alpha ,0)$-non-expansive.

(iii) 

If P is β-Reich–Suzuki type non-expansive, then P is generalized $(0,\beta )$-non-expansive.

We prove a key lemma.

Lemma 2.

Let P be a selfmap on a subset E of a Banach space. If P is generalized $(\alpha ,\beta )$-non-expansive with a fixed point z, then P is quasi-non-expansive.

Proof. 

Let $z\in F(P)$. Since $\frac{1}{2}||z-P(z)||=0\le ||a-z||$, we have

$$\begin{array}{ll}||z-P(a)||& =||P(z)-P(a)||\\ & \le \alpha ||a-P(z)||+\alpha ||z-P(a)||+\beta ||z-P(z)||+\beta ||a-P(a)||+(1-2\alpha -2\beta )||a-z||\\ & =\alpha ||a-z||+\alpha ||z-P(a)||+\beta ||a-P(a)||+(1-2\alpha -2\beta )||a-z||\\ & \le \alpha ||a-z||+\alpha ||z-P(a)||+\beta (||a-z||+||z-P(a)||)+(1-2\alpha -2\beta )||a-z||\\ & =\alpha ||z-P(a)||+\beta ||z-P(a)||+(1-\alpha -\beta )||a-z||.\end{array}$$

It follows that

$$(1-\alpha -\beta )||z-P(a)||\le (1-\alpha -\beta )||z-a||.$$

Since $(1-\alpha -\beta )>0$, we obtain our desired result. □

From Lemma 2, we obtain the following.

Lemma 3.

Let P be a selmap on a subset E of a Banach space X. If P is generalized $(\alpha ,\beta )$-non-expansive, then the set $F(P)$ is closed. Moreover, $F(P)$ is convex provided that E is convex and X is strictly convex.

We now prove the following facts.

Lemma 4.

Let P be a selfmap on a subset E of a Banach space. If P is generalized $(\alpha ,\beta )$-non-expansive, then for each $a,b\in E$:

(i) 

$||P(a)-P^2(a)||\le ||a-P(a)||$.

(ii) 

Either$\frac{1}{2}||a-P(a)||\le ||a-b||$or$\frac{1}{2}||P(a)-P^2(a)||\le ||P(a)-b||$.

(iii) 

Either$||P(a)-P(b)||\le \alpha ||a-P(b)||+\alpha ||b-P(a)||+\beta ||a-P(a)||+\beta ||b-P(b)||+(1-2\alpha -2\beta )||a-b||$or$||P^2(a)-P(b)||\le \alpha ||P(a)-P(b)||+\alpha ||b-P^2(a)||+\beta ||P(a)-P^2(a)||+\beta ||b-P(b)||+(1-2\alpha -2\beta )||P(a)-b||$.

Proof. 

Since $\frac{1}{2}||a-P(a)||\le ||a-P(a)||$, we have

$$\begin{array}{ll}||P(a)-P^2(a)||& \le \alpha ||a-P^2(a)||+\alpha ||P(a)-P(a)||+\beta ||a-P(a)||+\beta ||P(a)-P^2(a)||+(1-\\ & 2\alpha -2\beta )||a-P(a)||\\ & =\alpha ||a-P^2(a)||+\beta ||a-P(a)||+\beta ||P(a)-P^2(a)||+(1-2\alpha -2\beta )||a-P(a)||\\ & \le \alpha (||a-P(a)||+||P(a)-P^2(a)||)+\beta ||a-P(a)||+\beta ||P(a)-P^2(a)||+(1-2\alpha \\ & -2\beta )||a-P(a)||.\end{array}$$

It follows that

$$(1-\alpha -\beta )||P(a)-P^2(a)||\le (1-\alpha -\beta )||a-P(a)||.$$

Since $(1-\alpha -\beta )>0$, we obtain our desired result.

Now, to establish (ii), we assume the contrary, that is,

$$\frac{1}{2}||a-P(a)||>||a-b||\mathrm{and}\frac{1}{2}||P(a)-P^2(a)||>||P(a)-b||.$$

Using (i),

$$\begin{array}{ll}||a-P(a)||& \le ||a-b||+||b-P(a)||\\ & <\frac{1}{2}||a-P(a)||+\frac{1}{2}||P(a)-P^2(a)||\\ & =||a-P(a)||,\end{array}$$

this contradiction proves (ii). The condition (iii) directly follows from (ii). □

Lemma 5.

Let P be a selfmap on a subset E of a Banach space. If P is generalized $(\alpha ,\beta )$-non-expansive, then for all $a,b\in E$, we have $||a-P(b)||\le \left(\frac{3+\alpha +\beta }{1-\alpha -\beta }\right)||a-P(a)||+||a-b||.$

Proof. 

By Lemma 4(iii), for all $a,b\in E$, either

$$\begin{array}{ll}||P(a)-P(b)||& \le \alpha ||(a)-P(b)||+\alpha ||b-P(a)||+\beta ||a-P(a)||+\beta ||b-P(b)||\\ & +(1-2\alpha -2\beta )||a-b||,\end{array}$$

or

$$\begin{array}{ll}||P^2(a)-P(b)||& \le \alpha ||P(a)-P(b)||+\alpha ||b-P^2(a)||+\beta ||P(a)-P^2(a)||+\beta ||b\\ & -P(a)||+(1-2\alpha -2\beta )||P(a)-b||\end{array}$$

holds. In the first case, we have

$$\begin{array}{ll}||a-P(b)||& \le ||a-P(a)||+||P(a)-P(b)||\\ & \le ||a-P(a)||+\alpha ||a-P(b)||+\alpha ||b-P(a)||+\beta ||a-P(a)||+\beta ||b\\ & -P(b)||+(1-2\alpha -2\beta )||a-b||\\ & \le ||a-P(a)||+\alpha ||a-P(b)||+\alpha (||b-a||+||a-P(a)||)+\beta ||a-\\ & P(a)||+\beta (||b-a||+||a-P(b)||)+(1-2\alpha -2\beta )||a-b||.\end{array}$$

It follows that

$$||a-P(b)||\le \left(\frac{1+\alpha +\beta }{1-\alpha -\beta }\right)||a-P(a)||+||a-b||.$$

In the second case (also using (i)),

$$\begin{array}{ll}||a-P(b)||& \le ||a-P(a)||+||P(a)-P^2(a)||+||P^2(a)-P(b)||\\ & \le 2||a-P(a)||+||P^2(a)-P(b)||\\ & \le 2||a-P(a)||+\alpha ||P(a)-P(b)||+\alpha ||b-P^2(a)||+\beta ||P(a)-\\ & P^2(a)||+\beta ||b-P(b)||+(1-2\alpha -2\beta )||P(a)-b||\\ & \le 2||a-P(a)||+\alpha (||P(a)-a||+||a-P(b)||)+\alpha (||b-P(a)||\\ & +||P(a)-P^2(a)||)+\beta ||P(a)-P^2(a)||+\beta (||b-P(a)||+||P(a)\\ & -a||+||a-P(b)||)+(1-2\alpha -2\beta )(||P(a)-b||)\\ & \le 2||a-P(a)||+\alpha (||P(a)-a||+||a-P(b)||)+\alpha (||b-P(a)||+\\ & ||a-P(a)||)+\beta ||a-P(a)||+\beta (||b-P(a)||+||P(b)-a||+||\\ & a-P(b)||)+(1-2\alpha -2\beta )(||P(a)-b||).\end{array}$$

Thus,

$$\begin{array}{ll}(1-\alpha -\beta )||a-P(b)||& \le (2+\alpha +\beta )||a-P(a)||+\alpha (||b-P(a)||+||a-P(a\\ & )||)+\beta (||b-P(a)||+||P(a)-a||)+(1-2\alpha -2\beta )\\ & (||P(a)-b||)\\ & =(2+\alpha +\beta )||a-P(a)||+\alpha ||a-P(a)||+\beta ||P(a)-\\ & a||+(1-\alpha -\beta )(||P(a)-b||)\\ & \le (2+\alpha +\beta )||a-P(a)||+\alpha ||a-P(a)||+\beta ||P(a)-\\ & a||+(1-\alpha -\beta )(||P(a)-a||+||a-b||).\end{array}$$

It follows that

$$\begin{array}{l}(1-\alpha -\beta )||a-P(b)||\le (3+\alpha +\beta )||a-P(a)||+(1-\alpha -\beta )(||a-b||).\end{array}$$

Since $(1-\alpha -\beta )>0$, we have

$$||a-P(b)||\le \left(\frac{3+\alpha +\beta }{1-\alpha -\beta }\right)||a-P(a)||+||a-b||.$$

Hence, we have obtained the required result in both the cases. □

We finish this section by proving the demiclosed principle.

Lemma 6.

Let P be a selfmap on a subset E of a Banach space having Opial’s property. If P is generalized $(\alpha ,\beta )$-non-expansive, then the following holds:

$$\{a_m\}\subseteq E,a_m\rightharpoonup w,||a_m-P(a_m)||\to 0⟹P(w)=w.$$

Proof. 

From Lemma 5, we have

$$||a_m-P(w)||\le \left(\frac{3+\alpha +\beta }{1-\alpha -\beta }\right)||a_m-P(a_m)||+||a_m-w||.$$

It follows that

$$\underset{m\to \infty }{lim\; inf}||a_m-P(w)||\le \underset{m\to \infty }{lim\; inf}||a_m-w||.$$

By Opial’s property, we must have $P(w)=w$. Hence, the conclusions can be reached. □

4. Convergence Theorems in Uniformly Convex Banach Spaces

In this section, we prove some weak and strong convergence results for the newly introduced class of maps in the context of uniformly convex Banach spaces. From now on, the letter X will stand for the uniformly convex Banach space. Now, it is our purpose to prove some weak and strong convergence for the newly introduced class of maps through a faster iterative algorithm. Let P be a selfmap on a closed convex subset E of X and ${\mu }_m,{\xi }_m,{\varrho }_m\in (0,1)$ for all $m\ge 1$. The well known Mann [19], Ishikawa [20], S [21], Noor [22], Abbas [8], Thakur [23], and K [24] iterative algorithms read as follows:

$$\left\{\begin{array}{l}{a}_1=a\in E,\\ a_{m+1}=(1-{\mu }_m)a_m+{\mu }_{m}P(a_m),\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}{a}_1=a\in E,\\ b_m=(1-{\xi }_m)a_m+{\xi }_{m}P(a_m),\\ a_{m+1}=(1-{\mu }_m)a_m+{\mu }_{m}P(b_m),\end{array}\right.$$

(2)

$$\left\{\begin{array}{l}{a}_1=a\in E,\\ c_m=(1-{\varrho }_m)a_m+{\varrho }_{m}P(a_m),\\ b_m=(1-{\xi }_m)a_m+{\xi }_{m}P(c_m),\\ a_{m+1}=(1-{\mu }_m)a_m+{\mu }_{m}P(b_m),\end{array}\right.$$

(3)

$$\left\{\begin{array}{l}{a}_1=a\in E,\\ b_m=(1-{\xi }_m)a_m+{\xi }_{m}P(a_m),\\ a_{m+1}=(1-{\mu }_m)P(a_m)+{\mu }_{m}P(b_m),\end{array}\right.$$

(4)

$$\left\{\begin{array}{l}{a}_1=a\in E,\\ c_m=(1-{\varrho }_m)a_m+{\varrho }_{m}P(a_m),\\ b_m=(1-{\xi }_m)P(a_m)+{\xi }_{m}P(b_m),\\ a_{m+1}=(1-{\mu }_m)P(b_m)+{\mu }_{m}P(c_m),\end{array}\right.$$

(5)

$$\left\{\begin{array}{l}{a}_1=a\in E,\\ c_m=(1-{\xi }_m)a_m+{\xi }_{m}P(a_m),\\ b_m=P\left((1-{\mu }_m)a_m+{\mu }_m{c}_m\right),\\ a_{m+1}=P(b_m),\end{array}\right.$$

(6)

and

$$\left\{\begin{array}{l}{a}_1=a\in E,\\ c_m=(1-{\xi }_n)a_m+{\xi }_{m}P(a_m),\\ b_m=P((1-{\mu }_m)P(a_m)+{\mu }_{m}P(c_m)),\\ a_{m+1}=P(b_m).\end{array}\right.$$

(7)

In [21], Agarwal et al. proved that the iterative algorithm (4) is better than the Mann iterative algorithm (1) for contraction maps. In addition, in [8], Abbas and Nazir proved that the iterative algorihm (5) is better than the iterative algorihms (1)–(4) for non-expansive maps. Moreover, in [23], Thakur et al. proved that the iterative algorithm (6) is better than the iterative algorithms (1)–(5) for Suzuki maps. Very recently, in [24], Hussian et al. proved that the iterative algorithm (7) is better than all of the iterative algorithms (1)–(6) for Suzuki maps.

In this article, we present some weak and strong convergence results using K iterative algorithm for the class of genralized $(\alpha ,\beta )$-non-expansive maps. Similar results for the algorithms (1)–(6) can be proved on the same line of proofs.

Lemma 7.

Let P be selfmap on a closed convex subset E of X. If P is a generalized $(\alpha ,\beta )$-non-expansive with $F(P)\ne \varnothing$ and $\{a_m\}$ is a sequence generated by the algorithm (7), then ${lim}_{m\to \infty }||a_m-z||$ exists for each $z\in F(P)$.

Proof. 

Let $z\in F(P)$. By Lemma 2, we have

$$\begin{array}{ll}||c_m-z||& =||(1-{\xi }_m)a_m+{\xi }_{m}P(a_m)-z||\\ & \le (1-{\xi }_m)||a_m-z||+{\xi }_m||P(a_m)-z||\\ & \le (1-{\xi }_m)||a_m-z||+{\xi }_m||a_m-z||\\ & =||a_m-z||,\end{array}$$

which implies that

$$\begin{array}{ll}||a_{m+1}-z||& =||P(b_m)-z||\le ||b_m-z||\\ & =||P((1-{\mu }_m)P(a_m)+{\mu }_{m}P(c_m))-z||\\ & \le ||(1-{\mu }_m)P(a_m)+{\mu }_{m}P(c_m)-z||\\ & \le (1-{\mu }_m)||P(a_m)-z||+{\mu }_m||P(c_m)-z||\\ & \le (1-{\mu }_m)||a_m-z||+{\mu }_m||c_m-z||\\ & \le (1-{\mu }_m)||a_m-z||+{\mu }_m||a_m-z||\\ & =||a_m-z||.\end{array}$$

Thus, $\{||a_m-z||\}$ is bounded below and nonincreasing, which implies that ${lim}_{m\to \infty }||a_m-z||$ exists for each $z\in F(P)$. □

Now, we give the necessary and sufficient condition for the existence of a fixed point of self generalized $(\alpha ,\beta )$-non-expansive map on a closed convex subset E of X.

Theorem 1.

Let P be a selfmap on a closed convex subset E of X. If P is a generalized $(\alpha ,\beta )$-non-expansive and $\{a_m\}$ is a sequence generated by the algorithm (7), then $F(P)\ne \varnothing$ if and only if $\{a_m\}$ is bounded and ${lim}_{m\to \infty }||P(a_m)-a_m||=0$.

Proof. 

Suppose that $F(P)\ne \varnothing$ and $z\in F(P)$. Then, by Lemma 7, ${lim}_{m\to \infty }||a_m-z||$ exists and $\{a_m\}$ is bounded. Put

$$\underset{m\to \infty }{lim}||a_m-z||=\gamma .$$

(8)

By the proof of Lemma 7 together with (8), we have

$$\underset{m\to \infty }{lim\; sup}||c_m-z||\le \underset{m\to \infty }{lim\; sup}||a_m-z||=\gamma .$$

(9)

By Lemma 2, we have

$$\underset{m\to \infty }{lim\; sup}||P(a_m)-z||\le \underset{m\to \infty }{lim\; sup}||a_m-z||=\gamma .$$

(10)

Again, by the proof of Lemma 7, we have

$$||a_{m+1}-z||\le (1-{\mu }_m)||a_m-z||+{\mu }_m||c_m-z||.$$

It follows that

$$||a_{m+1}-z||-||a_m-z||\le \frac{||a_{m+1}-z||-||a_m-z||}{{\mu }_m}\le ||c_m-z||-||a_m-z||.$$

Thus, we can get $||a_{m+1}-z||\le ||c_m-z||$. Therefore,

$$\gamma \le \underset{m\to \infty }{lim\; inf}||c_m-z||.$$

(11)

From (9) and (11), we have

$$\gamma =\underset{m\to \infty }{lim}||c_m-z||.$$

(12)

From (12), we have

$$\begin{array}{l}\gamma ={\displaystyle \underset{m\to \infty }{lim}}||c_m-z||={\displaystyle \underset{m\to \infty }{lim}}||(1-{\xi }_m)(a_m-z)+{\xi }_m(P(a_m)-z)||.\end{array}$$

Since $0<{\xi }_m<1$ for all $m\ge 1$, by Lemma 1, we have

$$\underset{m\to \infty }{lim}||P(a_m)-a_m||=0.$$

Conversely, we assume that $\{a_m\}$ is bounded and $\underset{m\to \infty }{lim}||P(a_m)-a_m||=0$. Let $z\in A_c(E,\{a_m\})$. By Lemma 5, we have

$$\begin{array}{ll}{A}_r(P(z),\{a_m\})& ={\displaystyle \underset{m\to \infty }{lim\; sup}}||a_m-P(z)||\\ & \le \left(\frac{3+\alpha +\beta }{1-\alpha -\beta }\right){\displaystyle \underset{m\to \infty }{lim\; sup}}||P(a_m)-a_m||+{\displaystyle \underset{m\to \infty }{lim\; sup}}||a_m-z||\\ & ={\displaystyle \underset{m\to \infty }{lim\; sup}}||a_m-z||\\ & =A_r(z,\{a_m\}).\end{array}$$

It follows that $P(z)\in A_c(E,\{a_m\})$. Since X is uniformly convex, we have $A_c(E,\{a_m\})$ consists of a unique element. Thus, we have $P(z)=z$. □

The following result is based on the compactness of the domain.

Theorem 2.

Let P be a selfmap on compact convex subset E of X. Let P be a generalized $(\alpha ,\beta )$-non-expansive with $F(P)\ne \varnothing$, then $\{a_m\}$ generated by the algorithm (7) converges strongly to an element of $F(P)$.

Proof. 

By Theorem 1, ${lim}_{m\to \infty }||P(a_m)-a_m||=0$. By compactness assumption, one can find a strongly convergent subsequence namely $\{a_{m_k}\}$ of $\{a_m\}$ such that $a_{m_k}⟶s$ for some $s\in E$. By Lemma 5, we have

$$||a_{m_k}-P(s)||\le \left(\frac{3+\alpha +\beta }{1-\alpha -\beta }\right)||a_{m_k}-P(a_{m_k})||+||a_{m_k}-s||⟶0.$$

Since in Banach spaces, convergent sequence has a unique limit, we have $P(s)=s$. By Lemma 7, ${lim}_{m\to \infty }||a_m-s||$ exists. Hence, s is the strong limit of $\{a_m\}$. □

We state the following result without the proof because its proof is elementary.

Theorem 3.

Let P be a selfmap on a closed convex subset E of X. Let P be a generalized $(\alpha ,\beta )$-non-expansive and $\{a_m\}$ is a sequence generated by the algorithm (7). If $F(P)\ne \varnothing$ and ${lim\; inf}_{m\to \infty }dist(a_m,F(P))=0$, then $\{a_m\}$ converges strongly to an element of $F(P)$.

The next result is based on Condition I.

Definition 6.

[25] A selfmap P on a subset E of a Banach space is said to satisfy condition I if there is a nondecreasing function $h:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ with the property $h(0)=0$ and $h(u)>0$ for all $u\in (0,\infty )$ such that $||a-P(a)||\ge h(dist(a,F(P)))$ for all $a\in E$.

Theorem 4.

Let P be a selfmap on a closed convex subset E of X. Let P be generalized $(\alpha ,\beta )$-non-expansive with $F(P)\ne \varnothing$. If P satisfies condition I, then $\{a_m\}$ generated by the algorithm (7) converges strongly to an element of $F(P)$.

Proof. 

From Theorem 1, it follows that

$$\underset{m\to \infty }{lim\; inf}||P(a_m)-a_m||=0.$$

(13)

Since T satisfies condition I, we have

$$||a_m-P(a_m)||\ge h(\mathrm{dist}(a_m,F(P))).$$

From (13), we get

$$\underset{m\to \infty }{lim\; inf}h((\mathrm{dist}(a_m,F(P)))=0.$$

Since the function $h:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is nondecreasing with the property $h(0)=0$ and $h(u)>0$ for each $u\in (0,\infty )$, we have

$$\underset{m\to \infty }{lim\; inf}\mathrm{dist}(a_m,F(P))=0.$$

The conclusion follows from Theorem 3. □

The following result is based on the Opial condition.

Theorem 5.

Let P be a selfmap on a closed convex subset E of X having Opial property. If P is generalized $(\alpha ,\beta )$-non-expansive with $F(P)\ne \varnothing$, then, $\{a_m\}$ generated by the algorithm (7) converges weakly to an element of $F(P)$.

Proof. 

By Theorem 1, $\{a_m\}$ is bounded and ${lim}_{m\to \infty }||P(a_m)-a_m||=0$. Since X is uniformly convex, X is reflexive. Hence, one can find a weakly convergent subsequence $\{a_{m_j}\}$ of $\{a_m\}$ with weak limit say $v_1\in E$. By Lemma 6, we have $v_1\in F(P)$. It is sufficient to show that $v_1$ is the weak limit $\{a_m\}$. If $v_1$ is not the weak limit of $\{a_m\}$, then one can find another weakly convergent subsequence $\{a_{m_k}\}$ of $\{a_m\}$ with a weak limit, say $v_2\in E$ and $v_2\ne v_1$. Again, by Lemma 6, $v_2\in F(P)$. By Lemma 7 and Opial condition, we have

$$\begin{array}{ll}{\displaystyle \underset{m\to \infty }{lim}}||a_m-v_1||& ={\displaystyle \underset{j\to \infty }{lim}}||a_{m_j}-v_1||<{\displaystyle \underset{j\to \infty }{lim}}||a_{m_j}-v_2||\\ & ={\displaystyle \underset{m\to \infty }{lim}}||a_m-v_2||={\displaystyle \underset{k\to \infty }{lim}}||a_{m_k}-v_2||\\ & <{\displaystyle \underset{k\to \infty }{lim}}||a_{m_k}-v_1||={\displaystyle \underset{m\to \infty }{lim}}||a_m-v_1||.\end{array}$$

This is a contradiction. Hence, the conclusions can be reached. □

5. Example

The following example shows that there exist maps which are generalized $(\alpha ,\beta )$-non-expansive but neither generalized $\alpha$ non-expansive nor $\beta$-Reich–Suzuki type.

Example 1.

Define $P:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ by $P(a)=\frac{a}{2}$ if $\frac{1}{2}<a<\infty$ and $P(a)=0$ when $0\le a\le \frac{1}{2}$. We shall prove that P is generalized $(\frac{1}{4},\frac{1}{4})$-non-expansive.

We shall divide the proof into three cases.

(i)

If $0\le a,b\le \frac{1}{2}$, then we have

$$\begin{array}{l}\frac{1}{4}|a-P(b)|+\frac{1}{4}|b-P(a)|+\frac{1}{4}|a-P(a)|+\frac{1}{4}|b-P(b)|\ge 0=|P(a)-P(b)|.\end{array}$$

(ii)

If $\frac{1}{2}<a,b<\infty$, then we have

$$\begin{array}{ll}\frac{1}{4}|a-P(b)|+\frac{1}{4}|b-P(a)|+\frac{1}{4}|a-P(a)|+\frac{1}{4}|b-P(b)|& =\frac{1}{4}|a-\frac{b}{2}|+\frac{1}{4}|b-\frac{a}{2}|\\ & +\frac{1}{4}|a-\frac{a}{2}|+\frac{1}{4}|b-\frac{b}{2}|\\ & \ge \frac{1}{4}|\frac{3a}{2}-\frac{3b}{2}|+\frac{1}{4}|\frac{a}{2}-\frac{b}{2}|\\ & \ge \frac{1}{4}|\frac{4a}{2}-\frac{4b}{2}|\\ & =\frac{1}{2}|a-b|\\ & =|P(a)-P(b)|.\end{array}$$

(iii)

If $\frac{1}{2}<a<\infty$ and $0\le b\le \frac{1}{2}$, then we have

$$\begin{array}{ll}\frac{1}{4}|a-P(b)|+\frac{1}{4}|b-P(a)|+\frac{1}{4}|a-P(a)|+\frac{1}{4}|b-P(b)|& =\frac{1}{4}|a|+\frac{1}{4}|b-\frac{a}{2}|+\frac{1}{4}|a-\\ & \frac{a}{2}|+\frac{1}{4}|b|\\ & =\frac{1}{4}|a|+\frac{1}{4}|b-\frac{a}{2}|+\frac{1}{4}|\frac{a}{2}|+\frac{1}{4}|b|\\ & \ge \frac{1}{4}|\frac{4a}{2}|=\frac{1}{2}|a|=|P(a)-P(b)|.\end{array}$$

Hence, P is generalized $(\frac{1}{4},\frac{1}{4})$-non-expansive. However, for $a=\frac{1}{2}$ and $b=\frac{4}{5}$, we have $\frac{1}{2}|a-P(a)|<|a-b|$. However,

(i).

$|P(a)-P(b)|>|a-b|$.

(ii).

$|P(a)-P(b)|>\frac{1}{4}|a-P(b)|+\frac{1}{4}|b-P(a)|+(1-2(\frac{1}{4}))|a-b|$.

(iii).

$|P(a)-P(b)|>\frac{1}{4}|a-P(a)|+\frac{1}{4}|b-P(b)|+(1-2(\frac{1}{4}))|a-b|$.

Hence, P is neither generalized $\frac{1}{4}$-non-expansive nor $\frac{1}{4}$-Reich–Suzuki type. We obtained the influence of initial point for the K iterative algorithm (7) by ${\mu }_m=0.90$, ${\xi }_m=0.65$, ${\varrho }_m=0.90$ in the below table.

Remark 4.

In

Table 1. Influence of initial points for various iterative algorithms.

$\mathit{Number}\mathit{of}\mathit{Iterations}\mathit{Required}\mathit{to}\mathit{Obtain}\mathit{Fixed}\mathit{Point}.$
Initial Points	Mann	Ishikawa	Noor	S	Abbas	Thakur	K
5	32	31	30	4	3	2	2
150	40	36	35	7	5	4	3
500	43	38	36	9	6	5	4
1000	45	39	37	9	6	6	4
5000	48	42	39	11	8	7	5
10000	50	43	40	12	8	7	6

and

Table 2. Influence of parameters: comparison of various iterative algorithms.

Iterations	Initial Points
	10	${10}^2$	${10}^3$	${10}^4$	${10}^5$	${10}^6$
For ${\mu }_m=\frac{m}{{(m+1)}^{\frac{10}{9}}},{\xi }_m=\frac{1}{{(m+3)}^{\frac{2}{3}}},{\varrho }_m=\frac{m}{(3m+1)}$
Mann	39	45	51	58	64	70
Ishikawa	37	43	49	55	60	67
Noor	37	42	48	54	60	66
S	5	8	11	14	17	21
Abbas	4	6	9	11	13	16
Thakur	3	5	6	8	9	11
K	2	3	4	5	6	7
for ${\mu }_m=\frac{m}{{(m+7)}^{\frac{17}{14}}},{\xi }_m=\frac{m}{(m+2)},{\varrho }_m=m^{-\frac{1}{3}}$
Mann	95	107	120	133	146	159
Ishikawa	89	97	105	113	121	130
Noor	88	96	103	111	119	126
S	6	8	11	14	17	19
Abbas	3	5	6	8	10	11
Thakur	3	5	6	8	9	11
K	2	3	4	5	7	8
for ${\mu }_m=1-{(\frac{1}{5m+3})}^{\frac{1}{2}},{\xi }_m=m^{-3},{\varrho }_m=1-\frac{1}{(2m+5)}$
Mann	23	26	30	33	37	40
Ishikawa	22	25	29	32	36	40
Noor	22	25	29	32	36	39
S	5	8	12	15	18	22
Abbas	3	5	7	8	10	12
Thakur	3	4	6	8	9	11
K	2	3	4	5	6	8
for ${\mu }_m=\frac{2m}{(9m+8)},{\xi }_m=1-\frac{1}{{(m+7)}^2},{\varrho }_m=1-\frac{6m}{{(7m+3)}^4}$
Mann	166	185	205	224	244	265
Ishikawa	155	168	181	194	206	219
Noor	153	164	174	185	196	206
S	5	8	11	14	17	20
Abbas	3	5	6	8	9	11
Thakur	3	5	6	8	9	11
K	2	3	4	5	6	8
for ${\mu }_m=\frac{m}{(m+5)},{\xi }_m=\frac{m}{{(36m^2+1)}^{\frac{1}{2}}},{\varrho }_m={\left(\frac{2m}{(4m+5)}\right)}^{\frac{1}{2}}$
Mann	32	35	39	42	45	49
Ishikawa	31	34	37	40	44	47
Noor	31	34	37	40	43	46
S	6	9	12	15	19	22
Abbas	4	5	7	9	11	13
Thakur	3	5	6	8	10	11
K	2	3	4	6	7	8

, the items in bold show that the K iterative algorithm (7) converges faster than other algorithms for the class of generalized $(\alpha ,\beta )$-non-expansive maps.

6. Conclusions

In this article, we have presented a new wider class of generalized non-expansive maps, namely, the class of generalized $(\alpha ,\beta )$-non-expansive maps. We have also established fundamental properties of these maps in Banach spaces. We have proved that K iterative algorithm of Hussain et al. [24] converges faster to a fixed point of a map in this class. Since the class of generalized $(\alpha ,\beta )$-non-expansive maps properly includes the classes of non-expansive, Suzuki non-expansive, Reich–Suzuki type non-expansive, and generalized $\alpha$-non-expansive maps, our results extend the corresponding work proved and discussed in [1,2,3,8,11,23,24,26,27,28].