# Global Attractivity Results for Mixed-Monotone Mappings in Partially Ordered Complete Metric Spaces

- Dž Burgić
^{1}, - S. Kalabušić
^{2}and - M. R. S. Kulenović
^{3}Email author

**2009**:762478

**DOI: **10.1155/2009/762478

© Dž. Burgić et al. 2009

**Received: **28 October 2008

**Accepted: **9 February 2009

**Published: **24 February 2009

## Abstract

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 [1] 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.

Theorem 1.1.

- (a)
is nondecreasing in for each , and is nonincreasing in for each ;

- (b)If is a solution of the system(1.2)

then .

has a unique equilibrium and every solution of (1.3) converges to .

Proof.

from which the result follows.

Theorem 1.2.

is a continuous function satisfying the following properties:

(a) is nonincreasing in for each , and is nondecreasing in for each ;

(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 .

Proof.

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 [3], 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 [20], has been applied to the problem of solving matrix equations. See also [21], where the application to the boundary value problems for integro-differential equations is given and [22] for application to some classes of nonexpansive mappings and [23] 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

Let be a partially ordered set and let be a metric on such that is a complete metric space. Consider We will use the following partial ordering.

This partial ordering is well known as "south-east ordering" in competitive systems in the plane; see [5, 6, 12, 24, 25].

We prove the following theorem.

Theorem 2.1.

Let be a map such that is nonincreasing in for all and nondecreasing in for all Suppose that the following conditions hold.

- (ii)There exists such that the following condition holds:(2.5)

- (iii)
If is a nondecreasing convergent sequence such that , then , for all and if is a nonincreasing convergent sequence such that , then , for all ; if for every , then

- (a)For every initial point such that condition (2.5) holds, , where satisfy(2.6)

- (b)In particular, every solution of(2.8)

- (c)The following estimates hold:(2.9)

Proof.

This implies that and are Cauchy sequences in

imply (2.6).

which in the case when implies

By letting in (2.23), we obtain the estimate (2.9).

Remark 2.2.

Property (iii) is usually called closedness of the partial ordering, see [6], and is an important ingredient of the definition of an ordered -space; see [17, 19].

Theorem 2.3.

- (iv)
every pair of elements has either a lower or an upper bound.

Then, the fixed point is unique and

Proof.

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 [16].

Let and be two fixed points of the map .

We consider two cases.

Case 1.

We estimate , and

Case 2.

If is not comparable to then there exists an upper bound or a lower bound of and Then, is comparable to and

We now estimate the right-hand side of (2.35).

Finally, we prove that We will consider two cases.

Case A.

Case B.

## 3. Main Results: Mixed Monotone Case II

Let be a partially ordered set and let be a metric on such that is a complete metric space. Consider We will use the following partial order.

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 [14] and applied successfully to some boundary value problems in ordinary differential equations.

Theorem 3.1.

- (i)There exists with(3.3)

- (ii)There exists such that the following condition holds:(3.4)

- (iii)
If is a nondecreasing convergent sequence such that , then , for all and if is a nonincreasing convergent sequence such that , then , for all ; if for every , then

- (a)For every initial point such that the condition (3.2) holds, , where satisfy(3.5)

- (b)In particular, every solution of(3.7)

- (c)The following estimates hold:(3.8)

Theorem 3.2.

- (iv)
every pair of elements has either a lower or an upper bound.

Then, the fixed point is unique and

Remark 3.3.

Theorems 3.1 and 3.2 generalize and extend the results in [14]. 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 [14].

## Declarations

### Acknowledgment

The authors are grateful to the referees for pointing out few fine details that improved the presented results.

## Authors’ Affiliations

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