∎

11institutetext: Alexander J. Zaslavski 22institutetext: Department of
Mathematics
The Technion – Israel Institute of Technology
Haifa, 32000, Israel
<EMAIL_ADDRESS>
# Global Convergence of Algorithms Based on Unions of Nonexpansive Maps

Alexander J. Zaslavski

(Received: date / Accepted: date)

###### 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
paracontracting operators. He showed that the associated fixed point iteration
is locally convergent around strong fixed points. This result generalizes a
theorem due to Bauschke and Noll (2014). 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 proved for the
Krasnosel’ski-Mann iterations.

###### Keywords:

Convergence analysis Fixed point Nonexpansive mapping Paracontracting operator

###### MSC:

47H04 47H10

††journal: Optimization Letters

##  1 Introduction

For more than sixty years now, there has been a lot of research activity in
the study of the fixed point theory of nonexpansive operators alwa ; g17 ;
goeb90a ; gr84a ; khak15 ; kop12 ; rz14a ; qin ; z16a ; z18a . The starting
point of this study is Banach’s celebrated theorem ban22a  concerning the
existence of a unique fixed point for a strict contraction. It also concerns
the convergence of (inexact) iterates of a nonexpansive mapping to one of its
fixed points. Since that seminal result, many developments have taken place in
this field including, in particular, studies of feasibility, common fixed
point problems and variational inequalities, which find important applications
in mathematical analysis, optimization theory, and in engineering, medical and
the natural sciences baubor96a ; cen18a ; g17 ; gr17 ; qin ; txy ; z16a ; z18a
. In particular in tam18a , it was considered a framework for the analysis of
iterative algorithms, which can be described in terms of a structured set-
valued operator. Namely, at every point in the ambient space, it is assumed
that the value of the operator can be expressed as a finite union of values of
single-valued paracontracting operators. For such algorithms it was shown in
tam18a  that the associated fixed point iteration is locally convergent around
strong fixed points. In dao  an analogous result was obtained for
Krasnosel’ski-Mann iterations. The result of tam18a  generalizes a theorem due
to Bauschke and Noll baus14a . In the present paper we generalize the main
result of tam18a  and show the global convergence of the algorithm for an
arbitrary starting point. An analogous result is also proved for the
Krasnosel’ski-Mann iterations.

##  2 Global convergence of iterates

Suppose that $(X,\rho)$ is a metric space and that $C\subset X$ is its
nonempty, closed set. For every point $x\in X$ and every positive number $r$
define

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

For each $x\in X$ and each nonempty set $D\subset X$ set

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

For every operator $S:C\to C$ set

$\mbox{Fix}(S)=\\{x\in C:\;S(x)=x\\}.$

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 every natural number $i\in\\{1,\dots,m\\}$, every point
$z\in\mbox{Fix}(T_{i})$, every point $x\in C$ and every $y\in
C\setminus\mbox{Fix}(T_{i})$, we have

$\rho(z,T_{i}(x))\leq\rho(z,x)$

and

$\rho(z,T_{i}(y))<\rho(z,y).$

Note that operators satisfying (A2) are called paracontractions els .

Assume that for every point $x\in X$, a nonempty set

$\phi(x)\subset\\{1,\dots,m\\}$ $None$

is given. In other words,

$\phi:X\to 2^{\\{1,\dots,m\\}}\setminus\\{\emptyset\\}.$

Suppose that the following assumption holds:

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

$\phi(y)\subset\phi(x).$

Define

$T(x)=\\{T_{i}(x):\;i\in\phi(x)\\}$ $None$

for each $x\in C$,

$\bar{F}(T)=\\{z\in C:\;T_{i}(z)=z,\;i=1,\dots,m\\}$ $None$

and

$F(T)=\\{z\in C:\;z\in T(z)\\}.$ $None$

Assume that

$\bar{F}(T)\not=\emptyset.$

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

$\lfloor z\rfloor=\inf\\{i:\;i\mbox{ is an integer and }i\leq z\\}.$

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

We study the asymptotic behavior of sequences of iterates $x_{t+1}\in
F(x_{t})$, $t=0,1,\dots$. In particular we are interested in their convergence
to a fixed point of $T$. This iterative algorithm was introduced in tam18a
which also contains its application to sparsity constrained minimisation.

The following result, which is proved 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 collected in
z16a ; z18a .

###### Theorem 2.1

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

$\bar{F}(T)\cap B(\theta,M)\not=\emptyset.$ $None$

Then there exists an integer $Q\geq 1$ such that for each sequence
$\\{x_{i}\\}_{i=0}^{\infty}\subset C$ which satisfy

$\rho(x_{0},\theta)\leq M$

and

$x_{t+1}\in F(x_{t})\mbox{ for each integer }t\geq 0$

the inequality

$\rho(x_{t},\theta)\leq 3M$

holds for all integers $t\geq 0$,

$\mbox{Card}(\\{t\in\\{0,1,\dots,\\}:\;\rho(x_{t},x_{t+1})>\epsilon\\})\leq Q$

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

The following global convergence result is proved in Section 5.

###### Theorem 2.2

Assume that a sequence $\\{x_{t}\\}_{t=0}^{\infty}\subset C$ and that for each
integer $t\geq 0$,

$x_{t+1}\in F(x_{t}).$

Then there exist

$x_{*}=\lim_{t\to\infty}x_{t}$

and a natural number $t_{0}$ such that for each integer $t\geq t_{0}$

$\phi(x_{t})\subset\phi(x_{*})$

and if an integer $i\in\phi(x_{t})$ satisfies $x_{t+1}=T_{i}(x_{t})$, then

$T_{i}(x_{*})=x_{*}.$

Theorem 2.2 generalizes the main result of tam18a  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
$\bar{F}(T)$.

##  3 An auxiliary result

###### Lemma 1

Assume that $M,\epsilon>0$ and that $z_{*}\in C$ satisfies

$T_{i}(z_{*})=z_{*},\;i=1,\dots,m.$ $None$

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

$\rho(x,T_{s}(x))>\epsilon$ $None$

the inequality

$\rho(z_{*},T_{s}(x))\leq\rho(z_{*},x)-\delta$ $None$

is true.

###### Proof

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

$x_{k}\in C\cap B(\theta,M)$ $None$

such that

$\rho(x_{k},T_{s}(x_{k}))>\epsilon$ $None$

and

$\rho(z_{*},T_{s}(x_{k}))>\rho(z_{*},x_{k})-k^{-1}.$ $None$

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

$x_{*}=\lim_{k\to\infty}x_{k}.$ $None$

By (9)-(12) and the continuity of $T_{s}$,

$\rho(x_{*},\theta)\leq M,$
$\rho(x_{*},T_{s}(x_{*}))=\lim_{k\to\infty}\rho(x_{k},T_{s}(x_{k}))\geq\epsilon$

and

$\rho(z_{*},T_{s}(x_{*}))\geq\rho(z_{*},x_{*}).$

This contradicts (6) and (A2). The contradiction we have reached proves Lemma
3.1.

##  4 Proof of Theorem 2.1

By (5), there exists

$z_{*}\in B(\theta,M)\cap\bar{F}(T).$ $None$

Lemma 3.1 implies that there exists $\delta\in(0,\epsilon)$ such that the
following property holds:

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

$\rho(x,T_{s}(x))>\epsilon$

we have

$\rho(z_{*},T_{s}(x))\leq\rho(z_{*},x)-\delta.$

Choose a natural number

$Q\geq 2M\delta^{-1}.$ $None$

Assume that $\\{x_{i}\\}_{i=0}^{\infty}\subset C$,

$\rho(x_{0},\theta)\leq M$ $None$

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

$x_{t+1}\in F(x_{t}).$ $None$

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

$x_{t+1}=T_{s}(x_{t}).$ $None$

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

$\rho(z_{*},x_{t+1})=\rho(z_{*},T_{s}(x_{t}))\leq\rho(z_{*},x_{t}).$ $None$

Since $t$ is an arbitrary nonnegative integer equations (13), (15) and (18)
imply that for each integer $i\geq 0$,

$\rho(z_{*},x_{i})\leq\rho(z_{*},x_{0})\leq 2M$ $None$

and

$\rho(x_{i},\theta)\leq 3M.$

Assume that

$\rho(x_{t+1},x_{t})>\epsilon.$ $None$

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

$\rho(z_{*},x_{t+1})=\rho(z_{*},T_{s}(x_{t}))\leq\rho(z_{*},x_{t})-\delta.$

Thus we have shown that the following property holds:

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

$\rho(z_{*},x_{t+1})\leq\rho(z_{*},x_{t})-\delta.$

Assume that $n\geq 1$ is an integer. Property (b) and equations (18)-(20)
imply that

$2M\geq\rho(z_{*},x_{0})\geq\rho(z_{*},x_{0})-\rho(z_{*},x_{n+1})$
$=\sum_{t=0}^{n}(\rho(z_{*},x_{t})-\rho(z_{*},x_{t+1}))$
$\geq\sum\\{\rho(z_{*},x_{t})-\rho(z_{*},x_{t+1}):\;t\in\\{0,\dots,n\\},\;\rho(x_{t},x_{t+1})>\epsilon\\}$
$\geq\delta\mbox{Card}(\\{t\in\\{0,\dots,n\\}:\;\rho(x_{t},x_{t+1})>\epsilon\\})$

and in view of (14),

$\mbox{Card}(\\{t\in\\{0,\dots,n\\}:\;\rho(x_{t},x_{t+1})>\epsilon\\})\leq
2M\delta^{-1}\leq Q.$

Since $n$ is an arbitrary natural number we conclude that

$\mbox{Card}(\\{t\in\\{0,1,\dots\\}:\;\rho(x_{t},x_{t+1})>\epsilon\\})\leq Q.$

Since $\epsilon$ is any element of $(0,1)$ Theorem 2.1 is proved.

##  5 Proof of Theorem 2.2

In view of Theorem 2.1, the sequence $\\{x_{t}\\}_{t=0}^{\infty}$ is bounded.
In view of (A1), it has a limit point $x_{*}\in C$ and a subsequence
$\\{x_{t_{k}}\\}_{k=0}^{\infty}$ such that

$x_{*}=\lim_{k\to\infty}x_{t_{k}}.$ $None$

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

$\phi(x_{t_{k}})\subset\phi(x_{*}),\;k=1,2,\dots$ $None$

and that there exists

$\widehat{p}\in\phi(x_{*})$

such that

$x_{t_{k}+1}=T_{\widehat{p}}(x_{t_{k}}),\;k=1,2,\dots.$ $None$

It follows from Theorem 2.1, the continuity of $T_{\widehat{p}}$ and equations
(21) and (23) that

$T_{\widehat{p}}(x_{*})=\lim_{k\to\infty}T_{\widehat{p}}(x_{t_{k}})=\lim_{k\to\infty}x_{t_{k}+1}=\lim_{k\to\infty}x_{t_{k}}=x_{*}.$
$None$

Set

$I_{1}=\\{i\in\phi(x_{*}):\;T_{i}(x_{*})=x_{*}\\},\;I_{2}=\phi(x_{*})\setminus
I_{1}.$ $None$

In view of (24) and (25),

$\widehat{p}\in I_{1}.$

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

$\rho(x_{*},T_{i}(x_{*}))>2\delta_{0},\;i\in I_{2}.$ $None$

Assumption (A3), the continuity of $T_{i},\;i=1,\dots,m$ and (26) imply that
there exists $\delta_{1}\in(0,\delta_{0})$ such that for each $x\in
B(x_{*},\delta_{1})\cap C$,

$\phi(x)\subset\phi(x_{*}),$ $None$ $\rho(x,T_{i}(x))>\delta_{0},\;i\in
I_{2}.$ $None$

Theorem 2.1 implies that there exists an integer $q_{1}\geq 1$ such that for
each integer $t\geq q_{1}$,

$\rho(x_{t},x_{t+1})\leq\delta_{0}/2.$ $None$

Assume that

$\epsilon\in(0,\delta_{1}),$ $None$ $t\geq q_{1}$ $None$

is an integer and that

$\rho(x_{t},x_{*})\leq\epsilon.$ $None$

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

$\phi(x_{t})\subset\phi(x_{*})$ $None$

and

$\rho(x_{t},T_{i}(x_{t}))>\delta_{0},\;i\in I_{2}.$ $None$

In view of (33), there exists

$s\in\phi(x_{*})$

such that

$x_{t+1}=T_{s}(x_{t}).$ $None$

. By (29), (31) and (35),

$\rho(x_{t},T_{s}(x_{t}))=\rho(x_{t},x_{t+1})\leq\delta_{0}/2.$ $None$

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

$s\in I_{1},\;T_{s}(x_{*})=x_{*}.$

Combined with assumption (A2) and equations (32) and (35) this implies that

$\rho(x_{t+1},x_{*})=\rho(T_{s}(x_{t}),x_{*})\leq\rho(x_{t},x_{*})\leq\epsilon.$

Thus we have shown that if $t\geq q_{1}$ is an integer and (32) holds, then
(33) is true and if $s\in\phi(x_{*})$ and (35) holds, then $s\in I_{1}$ and
$\rho(x_{t+1},x_{*})\leq\epsilon$. By induction and (21), we obtain that

$\rho(x_{i},x_{*})\leq\epsilon$

for all sufficiently large natural numbers $i$. Since $\epsilon$ i an
arbitrary element of $(0,\delta_{1})$ we conclude that

$\lim_{t\to\infty}x_{t}=x_{*}$

and Theorem 2.2 is proved.

##  6 Kraasnosel’ski-Mann iterations

Assume that $(X,\|\cdot\|)$ is a normed space and that
$\rho(x,y)=\|x-y\|,\;x,y\in X$. 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 denote by $Id:X\to X$
the identity operator: $Id(x)=x$, $x\in X$. Let

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

We consider Kraasnosel’ski-Mann iteration associated with our set-valued
mapping $T$ and obtain the global convergence result (see Theorem 6.2 below)
which generalizes the local convergence result of dao  for iterates starting
from a point belonging to a neighborhood of a strong fixed point belonging to
the set $\bar{F}(T)$.

The following result is proved in Section 7.

###### Theorem 6.1

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

$\bar{F}(T)\cap B(\theta,M)\not=\emptyset$ $None$

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

$\\{\lambda_{t}\\}_{t=0}^{\infty}\subset(\kappa,1-\kappa)$ $None$

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

$\|x_{0}-\theta\|\leq M$

and

$x_{t+1}\in(1-\lambda_{t})x_{t}+\lambda_{t}T(x_{t})\mbox{ for each integer
}t\geq 0$ $None$

the inequality

$\|x_{t}-\theta\|\leq 3M$

holds for all integers $t\geq 0$,

$\mbox{Card}(\\{t\in\\{0,1,\dots,\\}:\;\|x_{t}-x_{t+1}\|>\epsilon\\})\leq Q$

and $\lim_{t\to\infty}\|x_{t}-x_{t+1}\|=0$.

The following result is proved in Section 8.

###### Theorem 6.2

Assume that

$\\{\lambda_{t}\\}_{t=0}^{\infty}\subset(\kappa,1-\kappa)$

and that a sequence $\\{x_{t}\\}_{t=0}^{\infty}\subset C$ satisfies (39). Then
there exist

$x_{*}=\lim_{t\to\infty}x_{t}$

and a natural number $t_{0}$ such that for each integer $t\geq t_{0}$

$\phi(x_{t})\subset\phi(x_{*})$

and if an integer $i\in\phi(x_{t})$ satisfies

$x_{t+1}=\lambda_{t}T_{i}(x_{t})+(1-\lambda)x_{t},$

then

$T_{i}(x_{*})=x_{*}.$

##  7 Proof of Theorem 6.1

By (37), there exists

$z_{*}\in B(\theta,M)\cap\bar{F}(T).$ $None$

Lemma 3.1 implies that there exists $\delta\in(0,\epsilon)$ such that the
following property holds:

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

$\rho(x,T_{s}(x))>\epsilon$

we have

$\rho(z_{*},T_{s}(x))\leq\rho(z_{*},x)-\delta.$

Choose a natural number

$Q\geq 2M\delta^{-1}\kappa^{-1}.$ $None$

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

$\|x_{0}-\theta\|\leq M.$ $None$

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

$x_{t+1}=\lambda_{t}T_{s}(x_{t})+(1-\lambda_{t})x_{t}.$ $None$

Assumption (A2) and equations (3), (40) and (43) imply that

$\|x_{t+1}-z_{*}\|=\|\lambda_{t}T_{s}(x_{t})+(1-\lambda_{t})x_{t}-z_{*}\|$
$\leq\lambda_{t}\|T_{s}(x_{t})-z_{*}\|+(1-\lambda_{t})\|x_{t}-z_{*}\|\leq\|z_{*}-x_{t}\|.$
$None$

Since $t$ is an arbitrary nonnegative integer equations (40), (42) and (44)
imply that for each integer $i\geq 0$,

$\|z_{*}-x_{i}\|\leq\|z_{*}-x_{0}\|\leq 2M$

and

$\|x_{i}-\theta\|\leq 3M.$

Assume that

$\|x_{t+1}-x_{t}\|>\epsilon.$ $None$

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

$\epsilon<\|x_{t+1}-x_{t}\|=\|\lambda_{t}T_{s}(x_{t})+(1-\lambda_{t})x_{t}-x_{t}\|=\lambda_{t}\|T_{s}(x_{t})-x_{t}\|$

and

$\|T_{s}(x_{t})-x_{t}\|\geq\epsilon\lambda_{t}^{-1}\geq\epsilon(1-\kappa)^{-1}.$
$None$

Property (c) and equation (46) imply that

$\|z_{*}-T_{s}(x_{t})\|\leq\|z_{*}-x_{t}\|-\delta.$ $None$

By (38), (43) and (47),

$\|x_{t+1}-z_{*}\|=\|\lambda_{t}T_{s}(x_{t})+(1-\lambda_{t})x_{t}-z_{*}\|$
$\leq\lambda_{t}\|T_{s}(x_{t})-z_{*}\|+(1-\lambda_{t})\|x_{t}-z_{*}\|$
$\leq\lambda_{t}(\|x_{t}-z_{*}\|-\delta)+(1-\lambda_{t})\|x_{t}-z_{*}\|$
$\leq\|x_{t}-z_{*}\|-\lambda_{t}\delta\leq\|x_{t}-z_{*}\|-\delta\kappa.$
$None$

Thus we have shown that the following property holds:

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

$\|z_{*}-x_{t+1}\|\leq\|z_{*}-x_{t}\|-\delta\kappa.$

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

$2M\geq\|z_{*}-x_{0}\|\geq\|z_{*}-x_{0}\|-\|z_{*}-x_{n+1}\|$
$=\sum_{t=0}^{n}(\|z_{*}-x_{t}\|-\|z_{*}-x_{t+1}\|)$
$\geq\sum\\{\|z_{*}-x_{t}\|-\|z_{*}-x_{t+1}\|:\;t\in\\{0,\dots,n\\},\;\|x_{t}-x_{t+1}\|>\epsilon\\}$
$\geq\delta\kappa\mbox{Card}(\\{t\in\\{0,\dots,n\\}:\;\|x_{t}-x_{t+1}\|>\epsilon\\})$

and in view of (41),

$\mbox{Card}(\\{t\in\\{0,\dots,n\\}:\;\|x_{t}-x_{t+1}\|>\epsilon\\})\leq
2M(\delta\kappa)^{-1}\leq Q.$

Since $n$ is an arbitrary natural number we conclude that

$\mbox{Card}(\\{t\in\\{0,1,\dots\\}:\;\|x_{t}-x_{t+1}\|>\epsilon\\})\leq Q.$

Since $\epsilon$ is any element of $(0,1)$ we obtain that

$\lim_{t\to\infty}\|x_{t}-x_{t+1}\|=0.$

Theorem 6.1 is proved.

##  8 Proof of Theorem 6.2

In view of Theorem 6.1, the sequence $\\{x_{t}\\}_{t=0}^{\infty}$ is bounded.
In view of (A1), it has a limit point $x_{*}\in C$ and a subsequence
$\\{x_{t_{k}}\\}_{k=0}^{\infty}$ such that

$x_{*}=\lim_{k\to\infty}x_{t_{k}}.$ $None$

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

$\phi(x_{t_{k}})\subset\phi(x_{*}),\;k=1,2,\dots$ $None$

and that there exists

$\widehat{p}\in\phi(x_{*})$

such that

$x_{t_{k}+1}=\lambda_{t_{k}}T_{\widehat{p}}(x_{t_{k}})+(1-\lambda_{t_{k}})x_{t_{k}},\;k=1,2,\dots$
$None$

and that there exists

$\lambda_{*}=\lim_{k\to\infty}\lambda_{t_{k}}\in[\kappa,1-\kappa].$ $None$

It follows from Theorem 6.1, the continuity of $T_{\widehat{p}}$ and equations
(49), (51) and (52) that

$\lambda_{*}T_{\widehat{p}}(x_{*})+(1-\lambda_{*})x_{*}$
$=\lim_{k\to\infty}(\lambda_{t_{k}}T_{\widehat{p}}(x_{t_{k}})+(1-\lambda_{t_{k}})x_{t_{k}})$
$=\lim_{k\to\infty}x_{t_{k}+1}=\lim_{k\to\infty}x_{t_{k}}=x_{*}.$ $None$

Set

$I_{1}=\\{i\in\phi(x_{*}):\;T_{i}(x_{*})=x_{*}\\},\;I_{2}=\phi(x_{*})\setminus
I_{1}.$ $None$

In view of (53) and (54),

$\widehat{p}\in I_{1}.$

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

$\|x_{*}-T_{i}(x_{*})\|>2\delta_{0},\;i\in I_{2}.$ $None$

Assumption (A3), the continuity of $T_{i},\;i=1,\dots,m$ and (55) imply that
there exists $\delta_{1}\in(0,\delta_{0})$ such that for each $x\in
B(x_{*},\delta_{1})\cap C$,

$\phi(x)\subset\phi(x_{*}),$ $None$ $\|x-T_{i}(x)\|>\delta_{0},\;i\in I_{2}.$
$None$

Theorem 6.1 implies that there exists an integer $q_{1}\geq 1$ such that for
each integer $t\geq q_{1}$,

$\|x_{t}-x_{t+1}\|\leq\kappa\delta_{0}/2.$ $None$

Assume that

$\epsilon\in(0,\delta_{1}),$ $None$ $t\geq q_{1}$ $None$

is an integer and that

$\|x_{t}-x_{*}\|\leq\epsilon.$ $None$

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

$\phi(x_{t})\subset\phi(x_{*})$ $None$

and

$\|x_{t}-T_{i}(x_{t})\|>\delta_{0},\;i\in I_{2}.$ $None$

In view of (39), there exists

$s\in\phi(x_{t})\subset\phi(x_{*})$

such that

$x_{t+1}=\lambda_{t}T_{s}(x_{t})+(1-\lambda_{t})x_{t}.$ $None$

. By (38), (58) and (64),

$\kappa\delta_{0}/2\geq\|x_{t+1}-x_{t}\|=\lambda_{t}\|T_{s}(x_{t})-x_{t}\|$

and

$\|x_{t}-T_{s}(x_{t})\|\leq\kappa\delta_{0}(2\lambda_{t})^{-1}\leq\delta_{0}/2.$
$None$

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

$s\in I_{1},\;T_{s}(x_{*})=x_{*}.$

Combined with assumption (A2) and equations (39), (61) and (64) this implies
that

$\|x_{t+1}-x_{*}\|=\|\lambda_{t}T_{s}(x_{t})+(1-\lambda_{t})x_{t}-x_{*}\|$
$\leq\lambda_{t}\|T_{s}(x_{t})-x_{*}\|+(1-\lambda_{t})\|x_{t}-x_{*}\|$
$\leq\|x_{t}-x_{*}\|\leq\epsilon.$

Thus we have shown that if $t\geq q_{1}$ is an integer and (61) holds, then
$\|x_{t+1}-x_{*}\|\leq\epsilon$. By induction and (49), we obtain that

$\|x_{i}-x_{*}\|\leq\epsilon$

for all sufficiently large natural numbers $i$. Since $\epsilon$ i an
arbitrary element of $(0,\delta_{1})$ we conclude that

$\lim_{t\to\infty}x_{t}=x_{*}$

and Theorem 6.2 is proved.

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