- Research Article
- Open Access

# 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

https://doi.org/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.

## Keywords

- Periodic Solution
- Difference Equation
- Initial Point
- Fixed Point Theorem
- Nonexpansive Mapping

## 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)
- (b)

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.

- (a)

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.

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)

- (a)

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

## References

- Kulenović MRS, Ladas G, Sizer WS:
**On the recursive sequence .***Mathematical Sciences Research Hot-Line*1998,**2**(5):1–16.MathSciNetMATHGoogle Scholar - Kulenović MRS, Ladas G:
*Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjecture*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2002:xii+218.MATHGoogle Scholar - Kulenović MRS, Merino O:
**A global attractivity result for maps with invariant boxes.***Discrete and Continuous Dynamical Systems. Series B*2006,**6**(1):97–110.MathSciNetMATHGoogle Scholar - Nussbaum RD:
**Global stability, two conjectures and Maple.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(5):1064–1090. 10.1016/j.na.2006.01.005MathSciNetView ArticleMATHGoogle Scholar - Smith HL:
**The discrete dynamics of monotonically decomposable maps.***Journal of Mathematical Biology*2006,**53**(4):747–758. 10.1007/s00285-006-0004-3MathSciNetView ArticleMATHGoogle Scholar - Smith HL:
**Global stability for mixed monotone systems.***Journal of Difference Equations and Applications*2008,**14**(10–11):1159–1164. 10.1080/10236190802332126MathSciNetView ArticleMATHGoogle Scholar - Camouzis E, Ladas G:
*Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Advances in Discrete Mathematics and Applications*.*Volume 5*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2008:xxii+554.MATHGoogle Scholar - Gibbons CH, Kulenović MRS, Ladas G:
**On the recursive sequence .***Mathematical Sciences Research Hot-Line*2000,**4**(2):1–11.MathSciNetMATHGoogle Scholar - Gibbons CH, Kulenović MRS, Ladas G, Voulov HD:
**On the trichotomy character of .***Journal of Difference Equations and Applications*2002,**8**(1):75–92. 10.1080/10236190211940MathSciNetView ArticleMATHGoogle Scholar - Grove EA, Ladas G:
*Periodicities in Nonlinear Difference Equations, Advances in Discrete Mathematics and Applications*.*Volume 4*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2005:xiv+379.MATHGoogle Scholar - Kulenović MRS, Merino O:
*Discrete Dynamical Systems and Difference Equations with Mathematica*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2002:xvi+344.View ArticleMATHGoogle Scholar - Kulenović MRS, Nurkanovic M:
**Asymptotic behavior of a system of linear fractional difference equations.***Journal of Inequalities and Applications*2005,**2005**(2):127–143. 10.1155/JIA.2005.127MathSciNetMATHGoogle Scholar - Agarwal RP, El-Gebeily MA, O'Regan D:
**Generalized contractions in partially ordered metric spaces.***Applicable Analysis*2008,**87**(1):109–116. 10.1080/00036810701556151MathSciNetView ArticleMATHGoogle Scholar - Gnana Bhaskar T, Lakshmikantham V:
**Fixed point theorems in partially ordered metric spaces and applications.***Nonlinear Analysis: Theory, Methods & Applications*2006,**65**(7):1379–1393. 10.1016/j.na.2005.10.017MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, Rodríguez-López R:
**Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations.***Acta Mathematica Sinica*2007,**23**(12):2205–2212. 10.1007/s10114-005-0769-0MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, Rodríguez-López R:
**Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations.***Order*2005,**22**(3):223–239. 10.1007/s11083-005-9018-5MathSciNetView ArticleMATHGoogle Scholar - Nieto JJ, Pouso RL, Rodríguez-López R:
**Fixed point theorems in ordered abstract spaces.***Proceedings of the American Mathematical Society*2007,**135**(8):2505–2517. 10.1090/S0002-9939-07-08729-1MathSciNetView ArticleMATHGoogle Scholar - O'Regan D, Petruşel A:
**Fixed point theorems for generalized contractions in ordered metric spaces.***Journal of Mathematical Analysis and Applications*2008,**341**(2):1241–1252. 10.1016/j.jmaa.2007.11.026MathSciNetView ArticleMATHGoogle Scholar - Petruşel A, Rus IA:
**Fixed point theorems in ordered -spaces.***Proceedings of the American Mathematical Society*2006,**134**(2):411–418.MathSciNetView ArticleMATHGoogle Scholar - Ran ACM, Reurings MCB:
**A fixed point theorem in partially ordered sets and some applications to matrix equations.***Proceedings of the American Mathematical Society*2004,**132**(5):1435–1443. 10.1090/S0002-9939-03-07220-4MathSciNetView ArticleMATHGoogle Scholar - Ahmad B, Nieto JJ:
**The monotone iterative technique for three-point second-order integrodifferential boundary value problems with -Laplacian.***Boundary Value Problems*2007,**2007:**-9.Google Scholar - Su Y, Wang D, Shang M:
**Strong convergence of monotone hybrid algorithm for hemi-relatively nonexpansive mappings.***Fixed Point Theory and Applications*2008,**2008:**-8.Google Scholar - Xian X, O'Regan D, Agarwal RP:
**Multiplicity results via topological degree for impulsive boundary value problems under non-well-ordered upper and lower solution conditions.***Boundary Value Problems*2008,**2008:**-21.Google Scholar - Kulenović MRS, Merino O:
**Competitive-exclusion versus competitive-coexistence for systems in the plane.***Discrete and Continuous Dynamical Systems. Series B*2006,**6**(5):1141–1156.MathSciNetView ArticleMATHGoogle Scholar - Smith HL:
**Planar competitive and cooperative difference equations.***Journal of Difference Equations and Applications*1998,**3**(5–6):335–357. 10.1080/10236199708808108MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.