Fixed Points for Discontinuous Monotone Operators
© Y. Cui and X. Zhang. 2010
Received: 24 September 2009
Accepted: 21 November 2009
Published: 24 December 2009
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.
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  and Chen  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 ).
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 ).
Definition 1.3 (see ).
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 ).
2. Main Results
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 .
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 .
It is similar to the proof of Theorem 2.4; so we omit it.
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 .
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 .
We list for convenience the following assumptions.
Consider the differential equation
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.
In the proof of Theorem 3.1, we obtain the uniformly convergence of the monotone sequences without the compactness condition.
The project supported by the National Science Foundation of China (10971179).
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