Global Convergence of Algorithms Based on Unions of Non-Expansive Maps

Abstract

In his recent research, M. K. Tam (2018) considered a framework for the analysis of iterative algorithms which can be described in terms of a structured set-valued operator. At each point in the ambient space, the value of the operator can be expressed as a finite union of values of single-valued para-contracting operators. He showed that the associated fixed point iteration is locally convergent around strong fixed points. In the present paper we generalize the result of Tam and show the global convergence of his algorithm for an arbitrary starting point. An analogous result is also proven for the Krasnosel’ski–Mann iterations.

1. Introduction

The study of the fixed point theory of non-expansive operators [1,2,3,4,5,6,7,8,9] has been a rapidly growing area of research since Banach’s classical result [10] on the existence of a unique fixed point for a strict contraction. Numerous developments have taken place in this area including, in particular, studies of feasibility, common fixed point problems and variational inequalities, which find important applications in engineering, medical and the natural sciences. See [1,7,8,9,11,12,13,14,15,16] and the references therein. In [17], a framework was suggested for the analysis of iterative algorithms, determined by a structured set-valued operator. For such algorithms it was shown in [17] that the associated fixed point iteration is locally convergent around strong fixed points. In [18], an analogous result was obtained for Krasnosel’ski–Mann iterations. In the present paper we generalize the main result of [17] and show the global convergence of the algorithm for an arbitrary starting point. An analogous result is also proven for the Krasnosel’ski–Mann iterations.

2. Global Convergence of Iterates

Let $(X,\rho )$ be a metric space and $C\subset X$ be its non-empty, closed set. For each $x\in X$ and $r>0$, put

$$B(x,r)=\{y\in X:\rho (x,y)\le r\}.$$

For each $x\in X$ and non-empty set $D\subset X$, set

$$\rho (x,D)=inf\left\{\rho \right(x,y):y\in D\}.$$

For each mapping $S:C\to C$, define

$$Fix\left(S\right)=\{x\in C:S(x)=x\}.$$

Fix

$$\theta \in C.$$

Suppose that the following assumption holds:

(A1) For each $M>0$, the set $B(\theta ,M)\cap C$ is compact.

Assume that m is a natural number, $T_i:C\to C$, $i=1,\dots ,m$ are continuous operators and that the following assumption holds:

(A2) For each $i\in \{1,\dots ,m\}$, $z\in Fix\left(T_i\right)$, $x\in C$ and $y\in C\setminus Fix\left(T_i\right)$, we have

$$\rho (z,T_i\left(x\right))\le \rho (z,x)$$

and

$$\rho (z,T_i\left(y\right))<\rho (z,y).$$

Note that operators satisfying (A2) are called para-contractions [19].

Assume that for every point $x\in X$, a non-empty set

$$\varphi \left(x\right)\subset \{1,\dots ,m\}$$

(1)

is given. In other words,

$$\varphi :X\to 2^{\{1,\dots ,m\}}\backslash \{⌀\}.$$

Suppose that the following assumption holds:

(A3) For each $x\in C$ there exists $\delta >0$ such that for each $y\in B(x,\delta )\cap C$,

$$\varphi \left(y\right)\subset \varphi \left(x\right).$$

Define

$$T\left(x\right)=\{T_i\left(x\right):i\in \varphi \left(x\right)\}$$

(2)

for each $x\in C$,

$$\overline{F}\left(T\right)=\{z\in C:T_i\left(z\right)=z,i=1,\dots ,m\}$$

(3)

and

$$F\left(T\right)=\{z\in C:z\in T(z\left)\right\}.$$

(4)

Assume that

$$\overline{F}\left(T\right)\ne ⌀.$$

Denote by Card$\left(D\right)$ the cardinality of a set D. For each $z\in R^1$, set

$$\lfloor z\rfloor =max\{i:i\mathrm{is}\mathrm{an}\mathrm{integer}\mathrm{and}i\le z\}.$$

In the following we suppose that the sum over an empty set is zero.

We study the asymptotic behavior of sequences of iterates $x_{t+1}\in T\left(x_t\right)$, where $t=0,1,\dots$. In particular, we are interested in their convergence to a fixed point of T. This iterative algorithm was introduced in [17], also containing its application to sparsity-constrained minimisation.

The following result, which is proven in Section 4, shows that almost all iterates of our set-valued mappings are approximated solutions of the corresponding fixed point problem. Many results of this type are reported in [8,9].

Theorem 1.

Assume that $M>0$, $ϵ\in (0,1)$ and that

$$\overline{F}\left(T\right)\cap B(\theta ,M)\ne ⌀.$$

(5)

Then an integer $Q\ge 1$ exists such that for each sequence ${\left\{x_i\right\}}_{i=0}^{\infty }\subset C$ which satisfies

$$\rho (x_0,\theta )\le M$$

and

$$x_{t+1}\in T\left(x_t\right)foreachintegert\ge 0$$

the inequality

$$\rho (x_t,\theta )\le 3M$$

holds for all integers $t\ge 0$,

$$Card\left(\{t\in \{0,1,\dots ,\}:\rho (x_t,x_{t+1})>ϵ\}\right)\le Q$$

and ${lim}_{t\to \infty }\rho (x_t,x_{t+1})=0$.

The following global convergence result is proven in Section 5.

Theorem 2.

Assume a sequence ${\left\{x_t\right\}}_{t=0}^{\infty }\subset C$ and that for each integer $t\ge 0$,

$$x_{t+1}\in T\left(x_t\right).$$

Then

$$x_{\ast }=\underset{t\to \infty }{lim}{x}_t$$

and a natural number $t_0$ exist such that for each integer $t\ge t_0$

$$\varphi \left(x_t\right)\subset \varphi \left(x_{\ast }\right)$$

and if an integer $i\in \varphi \left(x_t\right)$ satisfies $x_{t+1}=T_i\left(x_t\right)$, then

$$T_i\left(x_{\ast }\right)=x_{\ast }.$$

Theorem (2) generalizes the main result of [17], which establishes a local convergence of the iterative algorithm for iterates starting from a point belonging to a neighborhood of a strong fixed point belonging to the set $\overline{F}\left(T\right)$.

3. An Auxiliary Result

Lemma 1.

Assume that $M,ϵ>0$ and that $z_{\ast }\in C$ satisfies

$$T_i\left(z_{\ast }\right)=z_{\ast },i=1,\dots ,m.$$

(6)

Then $\delta >0$ exists such that for each $s\in \{1,\dots ,m\}$ and each $x\in C\cap B(\theta ,M)$ satisfying

$$\rho (x,T_s\left(x\right))>ϵ$$

(7)

the inequality

$$\rho (z_{\ast },T_s\left(x\right))\le \rho (z_{\ast },x)-\delta$$

(8)

is true.

Proof. 

Let $s\in \{1,\dots ,m\}$. It is sufficient to show that $\delta >0$ exists such that for each $x\in C\cap B(\theta ,M)$ satisfying (7), Inequality (8) is true. Assume the contrary, then for each integer $k\ge 1$, there exists

$$x_k\in C\cap B(\theta ,M)$$

(9)

such that

$$\rho (x_k,T_s\left(x_k\right))>ϵ$$

(10)

and

$$\rho (z_{\ast },T_s\left(x_k\right))>\rho (z_{\ast },x_k)-k^{-1}.$$

(11)

In view of (A1) and (9), extracting a subsequence and re-indexing, we may assume without loss of generality that there exists

$$x_{\ast }=\underset{k\to \infty }{lim}{x}_k.$$

(12)

From (9)–(12) and the continuity of $T_s$,

$$\rho (x_{\ast },\theta )\le M,$$

$$\rho (x_{\ast },T_s\left(x_{\ast }\right))=\underset{k\to \infty }{lim}\rho (x_k,T_s\left(x_k\right))\ge ϵ$$

and

$$\rho (z_{\ast },T_s\left(x_{\ast }\right))\ge \rho (z_{\ast },x_{\ast }).$$

This contradicts (6) and (A2). The contradiction reached proves Lemma 1. □

4. Proof of Theorem 1

From (5), there exists

$$z_{\ast }\in B(\theta ,M)\cap \overline{F}\left(T\right).$$

(13)

Lemma 1 implies that $\delta \in (0,ϵ)$ exists such that the following property holds:

(a) for each $s\in \{1,\dots ,m\}$ and each $x\in C\cap B(z_{\ast },2M)$ satisfying

$$\rho (x,T_s\left(x\right))>ϵ$$

we have

$$\rho (z_{\ast },T_s\left(x\right))\le \rho (z_{\ast },x)-\delta .$$

Choose a natural number

$$Q\ge 2M{\delta }^{-1}.$$

(14)

Assume that ${\left\{x_i\right\}}_{i=0}^{\infty }\subset C$,

$$\rho (x_0,\theta )\le M$$

(15)

and that for each integer $t\ge 0$,

$$x_{t+1}\in T\left(x_t\right).$$

(16)

Let $t\ge 0$ be an integer. From (2) and (16), $s\in \{1,\dots ,m\}$ exists such that

$$x_{t+1}=T_s\left(x_t\right).$$

(17)

Assumption (A2) and Equations (3), (13) and (17) imply that

$$\rho (z_{\ast },x_{t+1})=\rho (z_{\ast },T_s\left(x_t\right))\le \rho (z_{\ast },x_t).$$

(18)

Since t is an arbitrary non-negative integer, Equations (13), (15) and (18) imply that for each integer $i\ge 0$,

$$\rho (z_{\ast },x_i)\le \rho (z_{\ast },x_0)\le 2M$$

(19)

and

$$\rho (x_i,\theta )\le 3M.$$

Assume that

$$\rho (x_{t+1},x_t)>ϵ.$$

(20)

Property (a) and Equations (17), (19) and (20) imply that

$$\rho (z_{\ast },x_{t+1})=\rho (z_{\ast },T_s\left(x_t\right))\le \rho (z_{\ast },x_t)-\delta .$$

Thus, we have shown that the following property holds:

(b) if an integer $t\ge 0$ satisfies (20), then

$$\rho (z_{\ast },x_{t+1})\le \rho (z_{\ast },x_t)-\delta .$$

Assume that $n\ge 1$ is an integer. Property (b) and Equations (18)–(20) imply that

$$2M\ge \rho (z_{\ast },x_0)\ge \rho (z_{\ast },x_0)-\rho (z_{\ast },x_{n+1})$$

$$=\sum _{t=0}^n(\rho (z_{\ast },x_t)-\rho (z_{\ast },x_{t+1}))$$

$$\ge \sum \{\rho (z_{\ast },x_t)-\rho (z_{\ast },x_{t+1}):t\in \{0,\dots ,n\},\rho (x_t,x_{t+1})>ϵ\}$$

$$\ge \delta Card\left(\{t\in \{0,\dots ,n\}:\rho (x_t,x_{t+1})>ϵ\}\right)$$

, and in view of (14),

$$Card\left(\{t\in \{0,\dots ,n\}:\rho (x_t,x_{t+1})>ϵ\}\right)\le 2M{\delta }^{-1}\le Q.$$

Since n is an arbitrary natural number, we conclude that

$$Card\left(\{t\in \{0,1,\dots \}:\rho (x_t,x_{t+1})>ϵ\}\right)\le Q.$$

Since $ϵ$ is any element of $(0,1)$, Theorem 1 is proven.

5. Proof of Theorem 2

In view of Theorem 1, the sequence ${\left\{x_t\right\}}_{t=0}^{\infty }$ is bounded. In view of (A1), it has a limit point $x_{\ast }\in C$ and a subsequence ${\left\{x_{t_k}\right\}}_{k=0}^{\infty }$ such that

$$x_{\ast }=\underset{k\to \infty }{lim}{x}_{t_k}.$$

(21)

In view of (A3) and (21), we may assume without loss of generality that

$$\varphi \left(x_{t_k}\right)\subset \varphi \left(x_{\ast }\right),k=1,2,\dots$$

(22)

and that

$$\widehat{p}\in \varphi \left(x_{\ast }\right)$$

exists such that

$$x_{t_k+1}=T_{\widehat{p}}\left(x_{t_k}\right),k=1,2,\dots .$$

(23)

It follows from Theorem 1, the continuity of $T_{\widehat{p}}$ and Equations (21) and (23) that

$$T_{\widehat{p}}\left(x_{\ast }\right)=\underset{k\to \infty }{lim}{T}_{\widehat{p}}\left(x_{t_k}\right)=\underset{k\to \infty }{lim}{x}_{t_k+1}=\underset{k\to \infty }{lim}{x}_{t_k}=x_{\ast }.$$

(24)

Set

$$I_1=\{i\in \varphi \left(x_{\ast }\right):T_i\left(x_{\ast }\right)=x_{\ast }\},I_2=\varphi \left(x_{\ast }\right)\setminus I_1.$$

(25)

In view of (24) and (25),

$$\widehat{p}\in I_1.$$

Fix ${\delta }_0\in (0,1)$, such that

$$\rho (x_{\ast },T_i\left(x_{\ast }\right))>2{\delta }_0,i\in I_2.$$

(26)

Assumption (A3), the continuity of $T_i,i=1,\dots ,m$ and (26) imply that ${\delta }_1\in (0,{\delta }_0)$ exists such that for each $x\in B(x_{\ast },{\delta }_1)\cap C$,

$$\varphi \left(x\right)\subset \varphi \left(x_{\ast }\right),$$

(27)

$$\rho (x,T_i\left(x\right))>{\delta }_0,i\in I_2.$$

(28)

Theorem 1 implies that an integer $q_1\ge 1$ exists such that for each integer $t\ge q_1$,

$$\rho (x_t,x_{t+1})\le {\delta }_0/2.$$

(29)

Assume that

$$ϵ\in (0,{\delta }_1),$$

(30)

$$t\ge q_1$$

(31)

is an integer and that

$$\rho (x_t,x_{\ast })\le ϵ.$$

(32)

It follows from (27), (28), (30) and (32) that

$$\varphi \left(x_t\right)\subset \varphi \left(x_{\ast }\right)$$

(33)

and

$$\rho (x_t,T_i\left(x_t\right))>{\delta }_0,i\in I_2.$$

(34)

In view of (33),

$$s\in \varphi \left(x_{\ast }\right)$$

exists such that

$$x_{t+1}=T_s\left(x_t\right).$$

(35)

From (29), (31) and (35),

$$\rho (x_t,T_s\left(x_t\right))=\rho (x_t,x_{t+1})\le {\delta }_0/2.$$

(36)

It follows from (25), (34) and (36) that

$$s\in I_1,T_s\left(x_{\ast }\right)=x_{\ast }.$$

Combined with Assumption (A2) and Equations (32) and (35), this implies that

$$\rho (x_{t+1},x_{\ast })=\rho (T_s\left(x_t\right),x_{\ast })\le \rho (x_t,x_{\ast })\le ϵ.$$

Thus, we have shown that if $t\ge q_1$ is an integer and (32) holds, then (33) is true and if $s\in \varphi \left(x_{\ast }\right)$ and (35) holds, then $s\in I_1$ and $\rho (x_{t+1},x_{\ast })\le ϵ$.

By induction and (21), we obtain that

$$\rho (x_i,x_{\ast })\le ϵ$$

for all sufficiently large natural numbers i. Since $ϵ$ is an arbitrary element of $(0,{\delta }_1)$, we conclude that

$$\underset{t\to \infty }{lim}{x}_t=x_{\ast }$$

and Theorem 2 are proven.

6. Krasnosel’ski-Mann Iterations

Assume that $(X,\parallel ·\parallel )$ is a normed space and that $\rho (x,y)=\parallel x-y\parallel ,x,y\in X$. We use the notation, definitions and assumptions introduced in Section 2. In particular, we assume that Assumptions (A1)–(A3) hold. Suppose that the set C is convex and denoted by $Id:X\to X$ the identity operator: $Id\left(x\right)=x$, $x\in X$. Let

$$\kappa \in (0,2^{-1}).$$

We consider the Krasnosel’ski-Mann iteration associated with our set-valued mapping T and obtain the global convergence result (see Theorem (4) below), which generalizes the local convergence result of [18] for iterates starting from a point belonging to a neighborhood of a strong fixed point belonging to the set $\overline{F}\left(T\right)$.

The following result is proven in Section 7.

Theorem 3.

Assume that $M>0$, $ϵ\in (0,1)$ and that

$$\overline{F}\left(T\right)\cap B(\theta ,M)\ne ⌀$$

(37)

Then there exists an integer $Q\ge 1$ such that for each

$${\left\{{\lambda }_t\right\}}_{t=0}^{\infty }\subset (\kappa ,1-\kappa )$$

(38)

and each sequence ${\left\{x_i\right\}}_{i=0}^{\infty }\subset C$ which satisfies

$$\parallel x_0-\theta \parallel \le M$$

and

$$x_{t+1}\in (1-{\lambda }_t)x_t+{\lambda }_{t}T\left(x_t\right)for each integer t\ge 0$$

(39)

the inequality

$$\parallel x_t-\theta \parallel \le 3M$$

holds for all integers $t\ge 0$,

$$Card\left(\right\{t\in \{0,1,\dots ,\}:\parallel x_t-x_{t+1}\parallel >ϵ\left\}\right)\le Q$$

and ${lim}_{t\to \infty }\parallel x_t-x_{t+1}\parallel =0$.

The following result is proven in Section 8.

Theorem 4.

Assume that

$${\left\{{\lambda }_t\right\}}_{t=0}^{\infty }\subset (\kappa ,1-\kappa )$$

and that a sequence ${\left\{x_t\right\}}_{t=0}^{\infty }\subset C$ satisfies (39). Then

$$x_{\ast }=\underset{t\to \infty }{lim}{x}_t$$

and a natural number $t_0$ exist such that for each integer $t\ge t_0$

$$\varphi \left(x_t\right)\subset \varphi \left(x_{\ast }\right)$$

and if an integer $i\in \varphi \left(x_t\right)$ satisfies

$$x_{t+1}={\lambda }_t{T}_i\left(x_t\right)+(1-\lambda )x_t,$$

then

$$T_i\left(x_{\ast }\right)=x_{\ast }.$$

7. Proof of Theorem 3

From (37), there exists

$$z_{\ast }\in B(\theta ,M)\cap \overline{F}\left(T\right).$$

(40)

Lemma 1 implies that $\delta \in (0,ϵ)$ exists such that the following property holds:

(c) for each $s\in \{1,\dots ,m\}$ and each $x\in C\cap B(z_{\ast },2M)$ satisfying

$$\rho (x,T_s\left(x\right))>ϵ$$

we have

$$\rho (z_{\ast },T_s\left(x\right))\le \rho (z_{\ast },x)-\delta .$$

Choose a natural number

$$Q\ge 2M{\delta }^{-1}{\kappa }^{-1}.$$

(41)

Assume that (38) holds and that a sequence ${\left\{x_i\right\}}_{i=0}^{\infty }\subset C$ satisfies (39) and

$$\parallel x_0-\theta \parallel \le M.$$

(42)

Let $t\ge 0$ be an integer. From (2) and (39), $s\in \{1,\dots ,m\}$ exists such that

$$x_{t+1}={\lambda }_t{T}_s\left(x_t\right)+(1-{\lambda }_t)x_t.$$

(43)

Assumption (A2) and Equations (3), (40) and (43) imply that $z_{\ast }$ is a fixed point of $T_s$ and that

$$\parallel x_{t+1}-z_{\ast }\parallel =\parallel {\lambda }_t{T}_s\left(x_t\right)+(1-{\lambda }_t)x_t-z_{\ast }\parallel$$

$$\le {\lambda }_t\parallel T_s\left(x_t\right)-z_{\ast }\parallel +(1-{\lambda }_t)\parallel x_t-z_{\ast }\parallel \le \parallel z_{\ast }-x_t\parallel .$$

(44)

Since t is an arbitrary non-negative integer, Equations (40), (42) and (44) imply that for each integer $i\ge 0$,

$$\parallel z_{\ast }-x_i\parallel \le \parallel z_{\ast }-x_0\parallel \le 2M$$

and

$$\parallel x_i-\theta \parallel \le 3M.$$

Assume that

$$\parallel x_{t+1}-x_t\parallel >ϵ.$$

(45)

It follows from (38), (43) and (45) that

$$ϵ<\parallel x_{t+1}-x_t\parallel =\parallel {\lambda }_t{T}_s\left(x_t\right)+(1-{\lambda }_t)x_t-x_t\parallel ={\lambda }_t\parallel T_s\left(x_t\right)-x_t\parallel$$

and

$$\parallel T_s\left(x_t\right)-x_t\parallel \ge ϵ{\lambda }_t^{-1}\ge ϵ{(1-\kappa )}^{-1}.$$

(46)

Property (c) and Equation (46) imply that

$$\parallel z_{\ast }-T_s\left(x_t\right)\parallel \le \parallel z_{\ast }-x_t\parallel -\delta .$$

(47)

From (38), (43) and (47),

$$\parallel x_{t+1}-z_{\ast }\parallel =\parallel {\lambda }_t{T}_s\left(x_t\right)+(1-{\lambda }_t)x_t-z_{\ast }\parallel$$

$$\le {\lambda }_t\parallel T_s\left(x_t\right)-z_{\ast }\parallel +(1-{\lambda }_t)\parallel x_t-z_{\ast }\parallel$$

$$\le {\lambda }_t(\parallel x_t-z_{\ast }\parallel -\delta )+(1-{\lambda }_t)\parallel x_t-z_{\ast }\parallel$$

$$\le \parallel x_t-z_{\ast }\parallel -{\lambda }_t\delta \le \parallel x_t-z_{\ast }\parallel -\delta \kappa .$$

(48)

Thus, we have shown that the following property holds:

(d) if an integer $t\ge 0$ satisfies (45), then

$$\parallel z_{\ast }-x_{t+1}\parallel \le \parallel z_{\ast }-x_t\parallel -\delta \kappa .$$

Assume that $n\ge 1$ is an integer. Property (d) and Equations (40), (42) and (44) imply that

$$2M\ge \parallel z_{\ast }-x_0\parallel \ge \parallel z_{\ast }-x_0\parallel -\parallel z_{\ast }-x_{n+1}\parallel$$

$$=\sum _{t=0}^n(\parallel z_{\ast }-x_t\parallel -\parallel z_{\ast }-x_{t+1}\parallel )$$

$$\ge \sum \{\parallel z_{\ast }-x_t\parallel -\parallel z_{\ast }-x_{t+1}\parallel :t\in \{0,\dots ,n\},\parallel x_t-x_{t+1}\parallel >ϵ\}$$

$$\ge \delta \kappa Card\left(\right\{t\in \{0,\dots ,n\}:\parallel x_t-x_{t+1}\parallel >ϵ\left\}\right),$$

and in view of (41),

$$Card\left(\right\{t\in \{0,\dots ,n\}:\parallel x_t-x_{t+1}\parallel >ϵ\left\}\right)\le 2M{\left(\delta \kappa \right)}^{-1}\le Q.$$

Since n is an arbitrary natural number, we conclude that

$$Card\left(\right\{t\in \{0,1,\dots \}:\parallel x_t-x_{t+1}\parallel >ϵ\left\}\right)\le Q.$$

Since $ϵ$ is any element of $(0,1)$, we can obtain

$$\underset{t\to \infty }{lim}\parallel x_t-x_{t+1}\parallel =0.$$

Theorem 3 is thus proven.

8. Proof of Theorem 4

In view of Theorem (3), the sequence ${\left\{x_t\right\}}_{t=0}^{\infty }$ is bounded. In view of (A1), it has a limit point $x_{\ast }\in C$ and a subsequence ${\left\{x_{t_k}\right\}}_{k=0}^{\infty }$ such that

$$x_{\ast }=\underset{k\to \infty }{lim}{x}_{t_k}.$$

(49)

In view of (A3) and Equations (38), (39) and (49), extracting a subsequence and re-indexing, we may assume without loss of generality that

$$\varphi \left(x_{t_k}\right)\subset \varphi \left(x_{\ast }\right),k=1,2,\dots$$

(50)

and that

$$\widehat{p}\in \varphi \left(x_{\ast }\right)$$

exists such that

$$x_{t_k+1}={\lambda }_{t_k}{T}_{\widehat{p}}\left(x_{t_k}\right)+(1-{\lambda }_{t_k})x_{t_k},k=1,2,\dots$$

(51)

and that there exists

$${\lambda }_{\ast }=\underset{k\to \infty }{lim}{\lambda }_{t_k}\in [\kappa ,1-\kappa ].$$

(52)

It follows from Theorem (3), the continuity of $T_{\widehat{p}}$ and Equations (49), (51) and (52) that

$${\lambda }_{\ast }{T}_{\widehat{p}}\left(x_{\ast }\right)+(1-{\lambda }_{\ast })x_{\ast }$$

$$=\underset{k\to \infty }{lim}({\lambda }_{t_k}{T}_{\widehat{p}}\left(x_{t_k}\right)+(1-{\lambda }_{t_k})x_{t_k})$$

$$=\underset{k\to \infty }{lim}{x}_{t_k+1}=\underset{k\to \infty }{lim}{x}_{t_k}=x_{\ast }.$$

(53)

Set

$$I_1=\{i\in \varphi \left(x_{\ast }\right):T_i\left(x_{\ast }\right)=x_{\ast }\},I_2=\varphi \left(x_{\ast }\right)\backslash I_1.$$

(54)

In view of (53) and (54),

$$\widehat{p}\in I_1.$$

Fix ${\delta }_0\in (0,1)$ such that

$$\parallel x_{\ast }-T_i\left(x_{\ast }\right)\parallel >2{\delta }_0,i\in I_2.$$

(55)

Assumption (A3), the continuity of $T_i,i=1,\dots ,m$ and (55) imply that ${\delta }_1\in (0,{\delta }_0)$ exists such that for each $x\in B(x_{\ast },{\delta }_1)\cap C$,

$$\varphi \left(x\right)\subset \varphi \left(x_{\ast }\right),$$

(56)

$$\parallel x-T_i\left(x\right)\parallel >{\delta }_0,i\in I_2.$$

(57)

Theorem (3) implies that an integer $q_1\ge 1$ exists such that for each integer $t\ge q_1$,

$$\parallel x_t-x_{t+1}\parallel \le \kappa {\delta }_0/2.$$

(58)

Assume that

$$ϵ\in (0,{\delta }_1),$$

(59)

$$t\ge q_1$$

(60)

is an integer and that

$$\parallel x_t-x_{\ast }\parallel \le ϵ.$$

(61)

It follows from (56), (57), (59) and (61) that

$$\varphi \left(x_t\right)\subset \varphi \left(x_{\ast }\right)$$

(62)

and

$$\parallel x_t-T_i\left(x_t\right)\parallel >{\delta }_0,i\in I_2.$$

(63)

In view of (39),

$$s\in \varphi \left(x_t\right)\subset \varphi \left(x_{\ast }\right)$$

exists such that

$$x_{t+1}={\lambda }_t{T}_s\left(x_t\right)+(1-{\lambda }_t)x_t.$$

(64)

From (38), (58) and (64),

$$\kappa {\delta }_0/2\ge \parallel x_{t+1}-x_t\parallel ={\lambda }_t\parallel T_s\left(x_t\right)-x_t\parallel$$

and

$$\parallel x_t-T_s\left(x_t\right)\parallel \le \kappa {\delta }_0{\left(2{\lambda }_t\right)}^{-1}\le {\delta }_0/2.$$

(65)

It follows from (54), (56), (57), (59), (61) and (65) that

$$s\in I_1,T_s\left(x_{\ast }\right)=x_{\ast }.$$

Combined with Assumption (A2) and Equations (39), (61) and (64), this implies that

$$\parallel x_{t+1}-x_{\ast }\parallel =\parallel {\lambda }_t{T}_s\left(x_t\right)+(1-{\lambda }_t)x_t-x_{\ast }\parallel$$

$$\le {\lambda }_t\parallel T_s\left(x_t\right)-x_{\ast }\parallel +(1-{\lambda }_t)\parallel x_t-x_{\ast }\parallel$$

$$\le \parallel x_t-x_{\ast }\parallel \le ϵ.$$

Thus, we have shown that if $t\ge q_1$ is an integer and (61) holds, then $\parallel x_{t+1}-x_{\ast }\parallel \le ϵ$.

By induction and (49), we can obtain that

$$\parallel x_i-x_{\ast }\parallel \le ϵ$$

for all sufficiently large natural numbers i. Since $ϵ$ is an arbitrary element of $(0,{\delta }_1)$, we can conclude that

$$\underset{t\to \infty }{lim}{x}_t=x_{\ast }$$

and Theorem (4) are proven.