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

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.

Let be a compact interval of real numbers, and assume that

(1.1)

is a continuous function satisfying the following properties:

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

• (b) If is a solution of the system

  

  (1.2)

then .

Then

(1.3)

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

Proof.

Set

(1.4)

and for set

(1.5)

Now observe that for each ,

(1.6)

Set

(1.7)

Then

(1.8)

and by the continuity of ,

(1.9)

Therefore in view of (b),

(1.10)

from which the result follows.

Theorem 1.2.

Let be an interval of real numbers and assume that

(1.11)

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.

Set

(1.12)

and for set

(1.13)

Now observe that for each ,

(1.14)

Set

(1.15)

Then clearly (1.8) holds and by the continuity of ,

(1.16)

In view of (b),

(1.17)

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.

For , we have

(2.1)

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

Let be a metric on defined as follows:

(2.2)

Clearly

(2.3)

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.

(i)There exists with

(2.4)

• (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

Then we have the following.

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

  

  (2.6)

If in condition (2.5), then If in addition , then converge to the equilibrium of the equation

(2.7)

• (b) In particular, every solution of

  

  (2.8)

such that converges to the equilibrium of (2.8).

• (c) The following estimates hold:

  

  (2.9)

(2.10)

Proof.

Let and Since and for we have

(2.11)

Now, we have

(2.12)

For we let

(2.13)

By using the monotonicity of , we obtain

(2.14)

that is

(2.15)

We claim that for all the following inequalities hold:

(2.16)

(2.17)

Indeed, for using ,??, and (2.3), we obtain

(2.18)

Assume that (2.16) holds. Using the inequalities

(2.19)

and the contraction condition (2.4), we have

(2.20)

Similarly,

(2.21)

This implies that and are Cauchy sequences in

Indeed,

(2.22)

Since we have

(2.23)

Using (2.23), we conclude that is a Cauchy sequence. Similarly, we conclude that is a Cauchy sequence. Since is a complete metric space, then there exist such that

(2.24)

Using the continuity of which follows from contraction condition (2.4), the equations

(2.25)

imply (2.6).

Assume that Then, in view of the monotonicity of

(2.26)

By using induction, we can show that for all Assume that Then, in view of the monotonicity of , we have

(2.27)

Continuing in a similar way we can prove that for all By using condition (iii) we conclude that whenever exists we must have

(2.28)

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.

Assume that along with conditions (i) and (ii) of Theorem 2.1, the following condition is satisfied:

• (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.

If is comparable to then for all is comparable to We have to prove that

(2.29)

Indeed, using (2.2), we obtain

(2.30)

We estimate , and

First, by using contraction condition (2.4), we have

(2.31)

Now, by using (2.31) and (2.30), we have

(2.32)

which implies that

(2.33)

Case 2.

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

Therefore, we have

(2.34)

Now, we obtain

(2.35)

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

First, by using

(2.36)

we have

(2.37)

Similarly,

(2.38)

So,

(2.39)

Using induction, we obtain

(2.40)

Using (2.40), relation (2.35) becomes

(2.41)

So,

(2.42)

Finally, we prove that We will consider two cases.

Case A.

If is comparable to then is comparable to Now, we obtain

(2.43)

since this implies

(2.44)

Case B.

If is not comparable to then there exists an upper bound or alower bound of and , that is, there exists such that Then by using monotonicity character of we have

(2.45)

Now,

(2.46)

that is

(2.47)

Furthermore,

(2.48)

that is

(2.49)

Similarly,

(2.50)

that is

(2.51)

and

(2.52)

By using induction, we have

(2.53)

Since is a fixed point, we obtain

(2.54)

Using the contractivity condition (2.4) on we have

(2.55)

Now, we estimate the terms on the right-hand side

(2.56)

Now, we have

(2.57)

Continuing this process, we obtain

(2.58)

Using the contractivity of we have

(2.59)

That is

(2.60)

So,

(2.61)

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.

For , we have

(3.1)

Let be a metric on defined as follows:

(3.2)

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.

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

• (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

Then we have the following.

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

  

  (3.5)

If in condition (3.4), then If in addition , then converge to the equilibrium of the equation

(3.6)

• (b) In particular, every solution of

  

  (3.7)

such that converges to the equilibrium of (3.7).

• (c) The following estimates hold:

  

  (3.8)

Theorem 3.2.

Assume that along with conditions (i) and (ii) of Theorem 3.1, the following condition is satisfied:

• (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].