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Krasnosel'skii-Type Fixed-Set Results
Fixed Point Theory and Applications volume 2010, Article number: 394139 (2010)
Abstract
Some new Krasnosel'skii-type fixed-set theorems are proved for the sum , where
is a multimap and
is a self-map. The common domain of
and
is not convex. A positive answer to Ok's question (2009) is provided. Applications to the theory of self-similarity are also given.
1. Introduction
The Krasnosel'skii fixed-point theorem [1] is a well-known principle that generalizes the Schauder fixed-point theorem and the Banach contraction principle as follows.
Krasnosel'skii Fixed-Point Theorem
Let be a nonempty closed convex subset of a Banach space
,
, and
. Suppose that
(a) is compact and continuous;
(b) is a
-contraction;
(c) for every
.
Then there exists such that
.
This theorem has been extensively used in differential and functional differential equations and was motivated by the observation that the inversion of a perturbed differential operator may yield the sum of a continuous compact map and a contraction map. Note that the conclusion of the theorem does not need to hold if the convexity of is relaxed even if
is the zero operator. Ok [2] noticed that the Krasnosel'skii fixed-point theorem can be reformulated by relaxing or removing the convexity hypothesis of
and by allowing the fixed-point to be a fixed-set. For variants or extensions of Krasnosel'skii-type fixed-point results, see [3–9], and for other interesting results see [10–13].
In this paper, we prove several new Krasnosel'skii-type fixed-set theorems for the sum , where
is a multimap and
is a self-map. The common domain of
and
is not convex. Our results extend, generalize, or improve several fixed-point and fixed-set results including that given by Ok [2]. A positive answer to Ok's question [2] is provided. Applications to the theory of self-similarity are also given.
2. Preliminaries
Let be a nonempty subset of a metric space
,
a normed space,
the boundary of
,
the interior of
,
the closure of
,
the set all nonempty subsets of
,
the set of nonempty bounded subsets of
,
the family of nonempty closed subsets of
,
the family of nonempty compact subsets of
,
the set of real numbers, and
. A map
is called the Kuratoswki measure of noncompactness on
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ1_HTML.gif)
for every , where
denotes the diameter of
. Let
and
. We write
. We say that (a)
is a fixed point of
if
, and the set of fixed points of
will be denoted by
; (b)
is nonexpansive if
for all
; (c)
is
-contraction if
for all
and some
; (d)
is
-condensing if it is continuous and, for every
with
,
and
; (e)
is
-set-contractive if it is continuous and, for every
,
, and
; (f)
is compact if
is a compact subset of
.
Definition 2.1.
Let , and let
be either "a nondecreasing map satisfying
for every
'' or "an upper semicontinuous map satisfying
for every
.'' One says that
is a
-contraction if
for all
.
Remark 2.2.
A mapping is said to be a
-contraction in the sense of Garcia-Falset [6] if there exists a function
satisfying either "
is continuous and
for
" or "there exists
with
and nondecreasing such that
" for which the inequality
holds for all
,
. Our definition for
-contraction is different in some sense from that of Garcia-Falset.
Lemma 2.3 (see [2]).
Let be a nonempty closed subset of a normed space
. If
is compact and continuous, then there exists a minimal
such that
.
Theorem 2.4 (see [14]).
Let be a nonempty bounded closed convex subset of a Banach space
. Suppose that
is an
-condensing map. Then
has a fixed point in
.
Let be a complete metric space. If
is a
-contraction, then
has a unique fixed point in
.
Theorem 2.6 (see [14]).
Let be a closed subset of a Banach space
such that
is bounded, open, and containing the origin. Suppose that
is an
-condensing map satisfying
for all
and
. Then
has a fixed point in
.
Theorem 2.7 (see [14]).
Let be a closed subset of a Banach space
such that
is bounded, open, and containing the origin. Suppose that
is a 1-set-contractive map satisfying
for all
and
. If
is closed, then
has a fixed point in
.
3. Fixed-Set Results
We now reformulate the Krasnosel'skii fixed-point theorem by allowing the fixed-point to be a fixed-set and removing the convexity hypothesis of . Under suitable conditions, we look for a nonempty compact subset
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ2_HTML.gif)
or
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ3_HTML.gif)
Theorem 3.1.
Let be a nonempty closed subset of a Banach space
,
, and
. Suppose that
(a) is compact and continuous;
(b) is
-condensing and
is a bounded subset of
;
(c).
Then there exists such that
.
Proof.
Fix . Let
denote the set of closed subsets
of
for which
and
. Note that
is nonempty since
. Take
. As
is closed,
, and
, we have
. Let
. Notice that
is a bounded subset of
containing
. So
is a closed subset of
,
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ4_HTML.gif)
This shows that and
. Since
is a bounded subset of
and
is compact, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ5_HTML.gif)
As is
-condensing, it follows that
. Thus
is a compact subset of
. As the Vietoris topology and the Hausdorff metric topology coincide on
[18, page 17 and page 41],
is compact and hence closed. Define
by
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ6_HTML.gif)
for every . Since
is continuous and
is compact-valued and continuous, both
and
are compact subsets of
and hence
. Moreover, the maps
and
are continuous, so
is continuous. By Lemma 2.3, there exists
such that
since
is compact and hence closed. Let
. As
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ7_HTML.gif)
However is a compact subset of
[18, page 16], so
.
Corollary 3.2 ([2, Theorem  2.4]).
Let be a nonempty closed subset of a Banach space
,
, and
. Suppose that
(a) is compact and continuous;
(b) is compact and continuous;
(c).
Then there exists such that
.
In the following corollary, we assume that whenever
is upper semicontinuous.
Corollary 3.3.
Let be a nonempty closed subset of a Banach space
,
, and
. Suppose that
(a) is compact and continuous;
(b) is a
-contraction and
is bounded;
(c).
Then there exists such that
.
Remark 3.4.
The following statements are equivalent [19]:
(i) is a
-contraction, where
is nondecreasing, right continuous such that
for all
and
;
(ii) is a
-contraction, where
is upper semicontinuous such that
for all
and
.
Note that Corollary 3.3 provides a positive answer to the following question of Ok [2]. We do not know at present if the fixed-set can be taken to be a compact set in the statement of [2, Corollary ].
Theorem 3.5.
Let be a nonempty closed subset of a normed space
,
, and
. Suppose that
(a) is compact and continuous;
(b);
(c) is a continuous single-valued map on
.
Then
(i)there exists a minimal such that
and
;
(ii)there exists a maximal such that
.
Proof.
Let . Then, by (b), there exists
such that
, and, as
is a single-valued map on
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ8_HTML.gif)
So . Note that
is compact-valued and
is a compact subset of
. The continuity of
follows from that of
and
. Moreover,
is a compact subset of
, and hence
is a compact subset of
. By Lemma 2.3, there exists a minimal
such that
. But, since
is continuous and
is compact-valued,
is compact-valued and maps compact sets to compact sets. Then
, is a compact subset of M, so
. Thus
, and hence
.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ9_HTML.gif)
and . Clearly
is nonempty since
. Then
. Take
. It follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ10_HTML.gif)
and hence and
. Thus
.
Theorem 3.6.
Let be a nonempty closed subset of a normed space
,
, and
. Suppose that
(a) is compact and continuous;
(b) is a
-contraction;
(c)if , then (
has a convergent subsequence;
(d).
Then
(i)there exists a minimal such that
and
;
(ii)there exists a maximal such that
.
Proof.
Let . By (b), (d), and the closeness of
, the map
is a
-contraction from
into
. So, by Theorem 2.5, there exists a unique
such that
. Then
, and so
. Since the map
has a unique fixed-point, its fixed-point set
is singleton. So
is a single-valued map. To show that
is continuous, let
be a sequence in
such that
. Define
and
. Then
, and
. We claim that
is convergent. First, notice that
is bounded; otherwise,
has a subsequence
such that
. As
, (c) implies that
has a convergent subsequence, a contradiction. Next, as
is continuous and one-to-one, it follows from (c) that the sequence
converges to
. Therefore,
is continuous. Now the result follows from Theorem 3.5.
In the following result, we assume that whenever
is upper semicontinuous.
Theorem 3.7.
Let be a nonempty compact subset of a Banach space
,
, and
. Suppose that
(a) is continuous;
(b) is a
-contraction;
(c).
Then
(i)there exists a minimal such that
and
;
(ii)there exists a maximal such that
.
(iii)there exists such that
.
Proof.
Parts (i) and (ii) follow from Theorem 3.6. Part (iii) follows from Theorem 3.1.
Theorem 3.8.
Let be a closed subset of a Banach space
such that
is bounded, open, and containing the origin,
, and
. Suppose that
(a) is compact and continuous;
(b) is an
-condensing map satisfying
for all
;
(c) is a continuous single-valued map on
;
(d).
Then
(i)there exists a minimal such that
and
;
(ii)there exists a maximal such that
.
(iii)there exists such that
.
Proof.
Let . As
is
-condensing, part (d) and the closeness of
imply that the map
is an
-condensing self-map of
. Moreover, this map satisfies
for all
and
; otherwise, there are
and
such that
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ11_HTML.gif)
which contradicts the second part of (b). It follows from Theorem 2.6 that there exists such that
. Then
, and so
. Now parts (i) and (ii) follow from Theorem 3.5. Part (iii) follows from Theorem 3.1.
Theorem 3.9.
Let be a closed subset of a Banach space
such that
is bounded, open, and containing the origin,
, and
. Suppose that
(a) is compact and continuous;
(b) is a
-set-contractive map satisfying
for all
;
(c) is closed, and
is a continuous single-valued map on
;
(d).
Then
(i)there exists a minimal such that
and
;
(ii)there exists such that
.
Proof.
Let . As
is 1-set-contractive, part (d) and the closeness of
imply that the map
is a 1-set-contractive self-map of
. Moreover, this map satisfies
for all
and
; otherwise, there are
and
such that
. This implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ12_HTML.gif)
which contradicts the second part of (b). It follows from Theorem 2.7 that there exists such that
. Then
, and so
. Now the result follows from Theorem 3.5.
Definition 3.10 (self-similar sets).
Let be a nonempty closed subset of a Banach space
. If
are finitely many self-maps of
, then the list
is called aniterated function system (IFS). This IFS is continuous (resp., contraction,
-condensing, etc.) if each
is so. A nonempty subset
of
is said to be self-similar with respect to the IFS
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ13_HTML.gif)
Remark 3.11.
It is well known that there exists a unique compact self-similar set with respect to any contractive IFS; see [20].
Example 3.12.
Consider an IFS such that
(a) is a compact and continuous multimap;
(b) for each
.
Then the existence of a compact self-similar set with respect to the IFS is ensured by letting
to be zero in each of the following situations.
(i)Suppose that is an
-condensing map such that
is bounded. Then Theorem 3.1 ensures the existence of a compact subset
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ14_HTML.gif)
(ii)Suppose that is a
-contraction satisfying condition (c) of Theorem 3.6. Then there exists a minimal compact subset
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ15_HTML.gif)
(iii)Suppose that is a closed subset of a Banach space
such that
is bounded, open, and containing the origin,
is an
-condensing map satisfying
for all
, and
is a continuous single-valued map on
. Then Theorem 3.8 ensures the existence of a minimal compact subset
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ16_HTML.gif)
(iv)Suppose that is a closed subset of a Banach space
such that
is bounded, open, and containing the origin,
is a 1-set-contractive map satisfying
for all
,
is closed, and
is a continuous single-valued map on
. Then Theorem 3.9 ensures the existence of a minimal compact subset
of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F394139/MediaObjects/13663_2010_Article_1274_Equ17_HTML.gif)
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Acknowledgments
The authors thank the referee for his valuable suggestions. This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah under project no. 3-017/429.
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Al-Thagafi, M., Shahzad, N. Krasnosel'skii-Type Fixed-Set Results. Fixed Point Theory Appl 2010, 394139 (2010). https://doi.org/10.1155/2010/394139
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DOI: https://doi.org/10.1155/2010/394139