Global Attractivity Results for Mixed-Monotone Mappings in Partially Ordered Complete Metric Spaces
© Dž. Burgić et al. 2009
Received: 28 October 2008
Accepted: 9 February 2009
Published: 24 February 2009
We prove fixed point theorems for mixed-monotone mappings in partially ordered complete metric spaces which satisfy a weaker contraction condition than the classical Banach contraction condition for all points that are related by given ordering. We also give a global attractivity result for all solutions of the difference equation , where satisfies mixed-monotone conditions with respect to the given ordering.
1. Introduction and Preliminaries
The following results were obtained first in  and were extended to the case of higher-order difference equations and systems in [2–6]. For the sake of completeness and the readers convenience, we are including short proofs.
from which the result follows.
is a continuous function satisfying the following properties:
(b)the difference equation (1.3) has no solutions of minimal period two in . Then (1.3) has a unique equilibrium and every solution of (1.3) converges to .
from which the result follows.
These results have been very useful in proving attractivity results for equilibrium or periodic solutions of (1.3) as well as for higher-order difference equations and systems of difference equations; see [2, 7–12]. Theorems 1.1 and 1.2 have attracted considerable attention of the leading specialists in difference equations and discrete dynamical systems and have been generalized and extended to the case of maps in , see , and maps in Banach space with the cone see [4–6]. In this paper, we will extend Theorems 1.1 and 1.2 to the case of monotone mappings in partially ordered complete metric spaces.
On the other hand, there has been recent interest in establishing fixed point theorems in partially ordered complete metric spaces with a contractivity condition which holds for all points that are related by partial ordering; see [13–20]. These fixed point results have been applied mainly to the existence of solutions of boundary value problems for differential equations and one of them, namely , has been applied to the problem of solving matrix equations. See also , where the application to the boundary value problems for integro-differential equations is given and  for application to some classes of nonexpansive mappings and  for the application of the Leray-Schauder theory to the problems of an impulsive boundary value problem under the condition of non-well-ordered upper and lower solutions. None of these results is global result, but they are rather existence results. In this paper, we combine the existence results with the results of the type of Theorems 1.1 and 1.2 to obtain global attractivity results.
2. Main Results: Mixed Monotone Case I
We prove the following theorem.
every pair of elements has either a lower or an upper bound.
First, we prove that the fixed point is unique. Condition (iv) is equivalent to the following. For every there exists that is comparable to See .
We consider two cases.
We now estimate the right-hand side of (2.35).
3. Main Results: Mixed Monotone Case II
The following two theorems have similar proofs to the proofs of Theorems 2.1 and 2.3, respectively, and so their proofs will be skipped. Significant parts of these results have been included in  and applied successfully to some boundary value problems in ordinary differential equations.
every pair of elements has either a lower or an upper bound.
Theorems 3.1 and 3.2 generalize and extend the results in . The new feature of our results is global attractivity part that extends Theorems 1.1 and 1.2. Most of presented ideas were presented for the first time in .
The authors are grateful to the referees for pointing out few fine details that improved the presented results.
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