^{1}

^{*}

^{1}

^{*}

In this paper, we use the mappings with quasi-contractive conditions, defined on a partially ordered set with cone metric structure, to construct convergent sequences and prove that the limits of the constructed sequences are the unique (common) fixed point of the mappings, and give their corollaries. The obtained results improve and generalize the corresponding conclusions in references.

Huang and Zhang [

Let E be a real Banach space. A subset P_{0} of E is called a cone if and only if:

i) P_{0} is closed, nonempty, and

ii)

iii)

Given a cone_{0} by _{0}).

The cone P_{0} is called normal if there is a number

The least positive number K satisfying the above is called the normal constant of P_{0}. It is clear that

In the following we always suppose that E is a real Banach space, P_{0} is a cone in E with _{0}.

Let X be a nonempty set. Suppose that the mapping

d1)

d2)

d3)

Then d is called a cone metric on X, and

Let

e) Cauchy sequence if for every

g) convergent sequence if for every

Let _{0} if for any sequence

Lemma 1 [

1) if

2) if

Lemma 2 [_{0} and

At first, we give an example to show that there exists a self-map f on a partially ordered set

Example Let

Then obviously, for each

Theorem 1 Let

i) there exist A, B, C, D, E ≥ 0 with

ii) for each

Then f has a fixed point

Proof Take any

For any fixed

so

Let

Repeating this process,

Let

Obviously,

If

Hence

Another version of Theorem 1 is following:

Theorem 2 Let

i) there exist

ii) for each

Then f has a fixed point

Proof Take

From now, we give common fixed point theorems for a pare of maps.

Theorem 3 Let

i) there exist A, B, C, D, E ≥ 0 with

ii) for each

Then f and g have a common fixed point

Proof Take any element

For any

hence

where

hence

where

Let

For any

where

So for any

Obviously,

Suppose that f is continuous, then

So

If

so

Modifying the idea of Zhang [

Corollary 1 The conditions of A, B, C, D, E in i) of Theorem 3 can be replaced by the following:

i') there exist A, B, C, D, E ≥ 0 and

Proof Since

hence

Corollary 2 The conditions of A, B, C, D, E in i) of Theorem 3 can be replaced by the following:

i'') there exist A, B, C, D, E ≥ 0 such that A + B + C + D + E = 1, C > B and D > E or C < B and D < E.

Proof Take

Obviously, A', B, C, D, E satisfy i') in Corollary 1.

Corollary 3 The conditions of

i''') there exist A, B, C, D, E ≥ 0 such that

Proof Since

or

which implies that

or

If

The following is a non-continuous version of Theorem 3.

Theorem 4 Let

iii) if an increasing sequence

iv) if an increasing sequence

Then f and g have a common fixed point

Proof By i) and ii) in Theorem 3, we construct a sequence

Case I: Suppose iv) holds, then

so we obtain

where

So

For

so

Case II: Suppose iii) holds, then

so we obtain

where

For

so

So in any case,

Remark 1 We can also modify Corollary 1 - 3 to give the corresponding corollaries of Theorem 4, but we omit the part.

Remark 2 In this paper, we discuss the common fixed point problems for mappings with quasi-contractive type (i.e., expansive type) on partially ordered cone metric spaces, but some authors in references discussed the same problems for contractive or Lipschitz type. So our results improve and generalize the corresponding conclusions.

This paper is supported by the NNSF of China (No. 11361063, No. 11361064).