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Fixed Points for Discontinuous Monotone Operators
Fixed Point Theory and Applications volume 2010, Article number: 926209 (2009)
Abstract
We obtain some new existence theorems of the maximal and minimal fixed points for discontinuous monotone operator on an order interval in an ordered normed space. Moreover, the maximal and minimal fixed points can be achieved by monotone iterative method under some conditions. As an example of the application of our results, we show the existence of extremal solutions to a class of discontinuous initial value problems.
1. Introduction
Let be a Banach space. A nonempty convex closed set
is said to be a cone if it satisfies the following two conditions: (i)
,
implies
; (ii)
,
implies
, where
denotes the zero element. The cone
defines an ordering in
given by
if and only if
. Let
be an ordering interval in
, and
an increasing operator such that
,
. It is a common knowledge that fixed point theorems on increasing operators are used widely in nonlinear differential equations and other fields in mathematics ([1–7]).
But in most well-known documents, it is assumed generally that increasing operators possess stronger continuity and compactness. Recently, there have been some papers that considered the existence of fixed points of discontinuous operators. For example, Krasnosel'skii and Lusnikov [8] and Chen [9] discussed the fixed point problems for discontinuous monotonically compact operator. They called an operator A to be a monotonically compact operator if (
) implies that
converges to some
in norm and that
(
). A monotonically compact operator is referred to as an MMC-operator. A is said to be
-monotone if
implies
, where
,
, and
. They proved the following theorem.
Theorem 1.1 (see [8]).
Let be an
-monotone MMC-operator with
. Then
has at least one fixed point
possessing the property of
-continuity.
Motivated by the results of [3, 8, 9], in this paper we study the existence of the minimal and maximal fixed points of a discontinuous operator , which is expressed as the form
. We do not assume any continuity on
. It is only required that
(or
) is an MMC-operator and
(or
) possesses the quasiseparability, which are satisfied naturally in some spaces. As an example for application, we applied our theorem to study first order discontinuous nonlinear differential equation to conclude our paper.
We give the following definitions.
Definition 1.2 (see [3]).
Let be an Hausdorff topological space with an ordering structure.
is called an ordered topological space if for any two sequences
and
in
,
and
,
imply
.
Definition 1.3 (see [3]).
Let be an ordered topological space,
is said to be a quasi-separable set in
if for any totally ordered set
in
, there exists a countable set
such that
is dense in
(i.e., for any
, there exists
such that
).
Obviously, the separability implies the quasi-separability.
Definition 1.4 (see [3]).
Let be two ordered topological spaces. An operator
is said to be a monotonically compact operator if
(
) implies that
converges to some
in norm and that
.
Remark 1.5.
The definition of the MMC-operator is slightly different from that of [8, 9].
2. Main Results
Theorem 2.1.
Let be an ordered topological space, and
an order interval in
. Let
be an operator. Assume that
(i)there exist ordered topological space , increasing operator
, and increasing operator
such that
;
(ii) is quasiseparable and
is an MMC-operator;
(iii),
.
Then has at least one fixed point in
.
Proof.
It follows from the monotonicity of and condition (iii) that
. Set
. Since
,
is nonempty. Suppose that
is a totally ordered set in
. We now show that
has an upper bound in
.
Since , by condition (ii) there exists a countable subset
of
such that
is dense in
. Consider the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ1_HTML.gif)
Since is a totally ordered set,
makes sense and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ2_HTML.gif)
By condition (ii), and Definition 1.4, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ3_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ4_HTML.gif)
and hence make sense.
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ5_HTML.gif)
Using (2.1) and (2.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ6_HTML.gif)
Since is dense in
, for any
there exists a subsequence
of
such that
(
). By (2.6) and Definition 1.2, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ7_HTML.gif)
Hence , therefore
is an upper bound of
.
Now we show . By virtue of (2.4) and condition (iii)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ8_HTML.gif)
Thus and hence
. By (2.7) and condition (ii), we get
and hence
. By (2.3) and Definition 1.2, we get
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ9_HTML.gif)
Hence , and therefore
.
This shows that is an upper bound of
in
. It follows from Zorn's lemma that
has maximal element
. Thus
. And so
, which implies that
and
. As
is a maximal element of
,
; that is,
is a fixed point of
.
Theorem 2.2.
Let be an ordered topological space, and
an order interval in
. Let
be an operator. Assume that
(i)there exist ordered topological space , increasing operator
, and increasing operator
such that
;
(ii) is quasiseparable and
is an MMC-operator;
(iii),
.
Then has at least one fixed point in
.
Proof.
Let ,
. By the conditions (i) and (iii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ10_HTML.gif)
Since is increasing, for any
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ11_HTML.gif)
that is, ; therefore the quasiseparability of
implies that
is quasiseparable. Applying Theorem 2.1, the operator
has at least one fixed point
in
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ12_HTML.gif)
Set . Since
is increasing, by (2.12), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ13_HTML.gif)
that is, is a fixed point of the operator
in
.
Theorem 2.3.
If the conditions in Theorem 2.1 are satisfied, then has the minimal fixed point
and the maximal fixed point
in
; that is,
and
are fixed points of
, and for any fixed point
of
in
, one has
.
Proof.
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ14_HTML.gif)
By Theorem 2.1, . Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ15_HTML.gif)
Since is increasing, for any
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ16_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ17_HTML.gif)
therefore , and thus
. An order of
is defined by the inclusion relation, that is, for any
,
, and if
, then we define
. We show that
has a minimal element. Let
be a totally subset of
and
. Obviously,
is a totally ordered set in
. Since
is quasiseparable, it follows from
that there exists a countable subset
of
such that
is dense in
. Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ18_HTML.gif)
Since is a totally ordered set,
makes sense and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ19_HTML.gif)
Then there exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ20_HTML.gif)
Using the same method as in Theorem 2.1, we can prove that makes sense,
(where
) is an upper bound of
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ21_HTML.gif)
Since (for all
), for any
, we have
, for all
. Since
,
. By (2.20),
, and hence
, for all
, and therefore
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ22_HTML.gif)
Consider . Similarly, we can prove that there exists
such that
is a lower bound of
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ23_HTML.gif)
By (2.22) and (2.23), . Set
. By virtue of (2.21), (2.22), and (2.23),
. It is easy to see that
is a lower bound of
in
. It follows from Zorn's lemma that
has a minimal element.
Let be a minimal element of
. Therefore,
,
, and
. Obviously,
is a fixed point of
. In fact, on the contrary,
and
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ24_HTML.gif)
Since is an increasing operator, this implies that
and
includes properly
. This contradicts that
is the minimal element of
. Similarly,
is a fixed point of
. Since
,
is the minimal fixed point of
and
is the maximal fixed point of
.
Theorem 2.4.
If the conditions in Theorem 2.2 are satisfied, then has the minimal fixed point
and the maximal fixed point
in
; that is,
and
are fixed points of
, and for any fixed point
of
in
, one has
.
Proof.
It is similar to the proof of Theorem 2.4; so we omit it.
Theorem 2.5.
Let be an ordered topological space, and
an order interval in
. Let
be an operator. Assume that
(i)there exist ordered topological space , increasing operator
, and increasing operator
such that
;
(ii) is an continuous operator;
(iii) is a demicontinuous MMC-operator;
(iv),
.
Then has both the minimal fixed point
and the maximal fixed point
in
, and
and
can be obtained via monotone iterates:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ25_HTML.gif)
with and
.
Proof.
We define the sequences
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ26_HTML.gif)
and conclude from the monotonicity of operator and the condition (iv) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ27_HTML.gif)
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ28_HTML.gif)
Since is increasing,
by (2.27). By the condition (iii), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ29_HTML.gif)
By (2.29) and Definition 1.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ30_HTML.gif)
and hence makes sense. Set
, then
. Since
is continuous,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ31_HTML.gif)
By the condition (iii), , that is,
. Note that
; we have
; hence
; that is,
is a fixed point of
. Similarly, there exists
such that
and
is a fixed point of
. By the routine standard proof, it is easy to prove that
is the minimal fixed point of
and
is the maximal fixed point of
in
.
3. Applications
As some simple applications of Theorem 2.5, we consider the existence of extremal solutions for a class of discontinuous scalar differential equations.
In the following, stands for the set of real numbers and
a compact real interval. Let
be the class of continuous functions on
.
is a normed linear space with the maximum norm and partially ordered by the cone
.
is a normal cone in
.
For any , set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ32_HTML.gif)
Then is a Banach space by the norm
.
A function is said to be a Carathéodory function if
is measurable as a function of
for each fixed
and continuous as a function of
for a.a. (almost all)
.
We list for convenience the following assumptions.
(H1),
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ33_HTML.gif)
(H2) is a Carathéodory function.
(H3)There exists such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ34_HTML.gif)
(H4)There exists such that
is nondecreasing for a.a.
.
Consider the differential equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ35_HTML.gif)
where . It is a common knowledge that the initial value problem (3.4) is equivalent to the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ36_HTML.gif)
if is continuous. Therefore, when
is not continuous, we define the solution of the integral equation (3.5) as the solution of the equation (3.4).
Theorem 3.1.
Under the hypotheses (H1)–(H4), the IVP (3.4) has the minimal solution and maximal solution
in
. Moreover, there exist monotone iteration sequences
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ37_HTML.gif)
where and
satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ38_HTML.gif)
Proof.
For any , we consider the linear integral equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ39_HTML.gif)
where . Obviously,
is a linear completely continuous operator. By direct computation, the operator equation
has only zero solution; then by Fredholm theorem, for any
, the operator equation (3.8) has a unique solution in
. We definition the mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ40_HTML.gif)
where is the unique solution of (3.8) corresponding to
. Obviously
is a linear continuous operator; now we show that
is increasing. Suppose that
,
. Set
. By the definition of the operator
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ41_HTML.gif)
This integral inequality implies (for all
); that is,
is an increasing operator. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ42_HTML.gif)
Obviously, is an increasing continuous operator. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ43_HTML.gif)
By (H2), maps element of
into measurable functions. For any
, by (H3) and (H4) we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ44_HTML.gif)
This implies . Hence
maps
into
and
is an increasing operator. Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ45_HTML.gif)
By above discussions we know that and
are all increasing. Thus conditions (i) and (ii) in Theorem 2.5 are satisfied.
Let such that
in
; by (H2) we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ46_HTML.gif)
For any (
), by (2.29), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ47_HTML.gif)
and hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ48_HTML.gif)
where . By (H3),
; thus
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ49_HTML.gif)
where . Applying the Lebesgue dominated convergence theorem, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ50_HTML.gif)
This implies that in
; that is,
is a demicontinuous operator. Since the cone in
is regular, it is easy to see that
is an MMC-operator. Thus condition (iii) in Theorem 2.5 is satisfied.
We now show that condition (iv) in Theorem 2.5 is fulfilled. By (H1) and (3.5), and noting the definition of operator , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F926209/MediaObjects/13663_2009_Article_1366_Equ51_HTML.gif)
This implies that for all
, that is,
. Similarly we can show that
.
Since all conditions in Theorem 2.5 are satisfied, by Theorem 2.5, has the maximal fixed point and the minimal fixed point in
. Observing that fixed point of
is equivalent to solutions of (3.5), and (3.5) is equivalent to (3.4), the conclusions of Theorem 3.1 hold.
Remark 3.2.
In the proof of Theorem 3.1, we obtain the uniformly convergence of the monotone sequences without the compactness condition.
References
Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, New York, NY, USA; 1988:viii+275.
Amann H: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review 1976,18(4):620–709. 10.1137/1018114
Sun J, Zhao Z: Fixed point theorems of increasing operators and applications to nonlinear integro-differential equations with discontinuous terms. Journal of Mathematical Analysis and Applications 1993,175(1):33–45. 10.1006/jmaa.1993.1149
Heikkilä S, Lakshmikantham V: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 181. Marcel Dekker, New York, NY, USA; 1994:xii+514.
Lakshmikantham V, Leela S: Differential and Integral Inequalities. Academic Press, New York, NY, USA; 1969.
Klin-eam C, Suantai S: Strong convergence of monotone hybrid method for maximal monotone operators and hemirelatively nonexpansive mappings. Fixed Point Theory and Applications 2009, 2009:-14.
Plubtieng S, Sriprad W: An extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces. Fixed Point Theory and Applications 2009, 2009:-16.
Krasnosel'skii MA, Lusnikov AB: Regular fixed points and stable invariant sets of monotone operators. Applied Functional Analysis 1996,30(3):174–183. 10.1007/BF02509504
Chen Y-Z: Fixed points for discontinuous monotone operators. Journal of Mathematical Analysis and Applications 2004,291(1):282–291. 10.1016/j.jmaa.2003.11.003
Acknowledgment
The project supported by the National Science Foundation of China (10971179).
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Cui, Y., Zhang, X. Fixed Points for Discontinuous Monotone Operators. Fixed Point Theory Appl 2010, 926209 (2009). https://doi.org/10.1155/2010/926209
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DOI: https://doi.org/10.1155/2010/926209