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
be a metric space and
be its non-empty, closed set. For each
and
, put
For each
and non-empty set
, set
For each mapping
, define
Suppose that the following assumption holds:
(A1) For each , the set is compact.
Assume that m is a natural number, , are continuous operators and that the following assumption holds:
(A2) For each
,
,
and
, we have
and
Note that operators satisfying (A2) are called para-contractions [
19].
Assume that for every point
, a non-empty set
is given. In other words,
Suppose that the following assumption holds:
(A3) For each
there exists
such that for each
,
Define
for each
,
and
Denote by Card
the cardinality of a set
D. For each
, set
In the following we suppose that the sum over an empty set is zero.
We study the asymptotic behavior of sequences of iterates
, where
. 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 , and that Then an integer exists such that for each sequence which satisfiesandthe inequalityholds for all integers ,and . The following global convergence result is proven in
Section 5.
Theorem 2. Assume a sequence and that for each integer , Thenand a natural number exist such that for each integer and if an integer satisfies , then 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
.
3. An Auxiliary Result
Lemma 1. Assume that and that satisfies Then exists such that for each and each satisfyingthe inequalityis true. Proof. Let
. It is sufficient to show that
exists such that for each
satisfying (7), Inequality (8) is true. Assume the contrary, then for each integer
, there exists
such that
and
In view of (A1) and (9), extracting a subsequence and re-indexing, we may assume without loss of generality that there exists
From (9)–(12) and the continuity of
,
and
This contradicts (6) and (A2). The contradiction reached proves Lemma 1. □
4. Proof of Theorem 1
Lemma 1 implies that exists such that the following property holds:
(a) for each
and each
satisfying
we have
Assume that
,
and that for each integer
,
Let
be an integer. From (2) and (16),
exists such that
Assumption (A2) and Equations (3), (13) and (17) imply that
Since
t is an arbitrary non-negative integer, Equations (13), (15) and (18) imply that for each integer
,
and
Property (a) and Equations (17), (19) and (20) imply that
Thus, we have shown that the following property holds:
(b) if an integer
satisfies (20), then
Assume that
is an integer. Property (b) and Equations (18)–(20) imply that
Since
n is an arbitrary natural number, we conclude that
Since is any element of , Theorem 1 is proven.
5. Proof of Theorem 2
In view of Theorem 1, the sequence
is bounded. In view of (A1), it has a limit point
and a subsequence
such that
In view of (A3) and (21), we may assume without loss of generality that
and that
exists such that
It follows from Theorem 1, the continuity of
and Equations (21) and (23) that
In view of (24) and (25),
Fix
, such that
Assumption (A3), the continuity of
and (26) imply that
exists such that for each
,
Theorem 1 implies that an integer
exists such that for each integer
,
Assume that
is an integer and that
It follows from (27), (28), (30) and (32) that
and
In view of (33),
exists such that
From (29), (31) and (35),
It follows from (25), (34) and (36) that
Combined with Assumption (A2) and Equations (32) and (35), this implies that
Thus, we have shown that if is an integer and (32) holds, then (33) is true and if and (35) holds, then and .
By induction and (21), we obtain that
for all sufficiently large natural numbers
i. Since
is an arbitrary element of
, we conclude that
and Theorem 2 are proven.
6. Krasnosel’ski-Mann Iterations
Assume that
is a normed space and that
. 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
the identity operator:
,
. Let
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
.
Theorem 3. Assume that , and that Then there exists an integer such that for eachand each sequence which satisfiesandthe inequalityholds for all integers ,and . Theorem 4. Assume thatand that a sequence satisfies (39). Thenand a natural number exist such that for each integer and if an integer satisfiesthen 7. Proof of Theorem 3
Lemma 1 implies that exists such that the following property holds:
(c) for each
and each
satisfying
we have
Assume that (38) holds and that a sequence
satisfies (39) and
Let
be an integer. From (2) and (39),
exists such that
Assumption (A2) and Equations (3), (40) and (43) imply that
is a fixed point of
and that
Since
t is an arbitrary non-negative integer, Equations (40), (42) and (44) imply that for each integer
,
and
It follows from (38), (43) and (45) that
and
Property (c) and Equation (46) imply that
From (38), (43) and (47),
Thus, we have shown that the following property holds:
(d) if an integer
satisfies (45), then
Assume that
is an integer. Property (d) and Equations (40), (42) and (44) imply that
and in view of (41),
Since
n is an arbitrary natural number, we conclude that
Since
is any element of
, we can obtain
Theorem 3 is thus proven.
8. Proof of Theorem 4
In view of Theorem (3), the sequence
is bounded. In view of (A1), it has a limit point
and a subsequence
such that
In view of (A3) and Equations (38), (39) and (49), extracting a subsequence and re-indexing, we may assume without loss of generality that
and that
exists such that
and that there exists
It follows from Theorem (3), the continuity of
and Equations (49), (51) and (52) that
In view of (53) and (54),
Fix
such that
Assumption (A3), the continuity of
and (55) imply that
exists such that for each
,
Theorem (3) implies that an integer
exists such that for each integer
,
Assume that
is an integer and that
It follows from (56), (57), (59) and (61) that
and
In view of (39),
exists such that
From (38), (58) and (64),
and
It follows from (54), (56), (57), (59), (61) and (65) that
Combined with Assumption (A2) and Equations (39), (61) and (64), this implies that
Thus, we have shown that if is an integer and (61) holds, then .
By induction and (49), we can obtain that
for all sufficiently large natural numbers
i. Since
is an arbitrary element of
, we can conclude that
and Theorem (4) are proven.