Some new theorems of expanding mappings without continuity in cone metric spaces
© Han and Xu; licensee Springer 2013
Received: 28 December 2011
Accepted: 14 December 2012
Published: 7 January 2013
In this paper, fixed point theorems for one mapping and common fixed point theorems for two mappings satisfying generalized expansive conditions are obtained. The mappings are not necessarily continuous and the cone is not normal. These results improve and generalize several well-known comparable results in (Aage and Salunke, in Acta Mathematica Sinica, English Series 27(6):1101-1106, 2011). Moreover, examples are given to support our new results.
Keywordscone metric space expanding mapping common fixed point
1 Introduction and preliminaries
Recently, Huang and Zhang  introduced the concept of a cone metric space as a generalization of a metric space. They proved the properties of sequences in cone metric spaces and obtained various fixed point theorems for contractive mappings. Afterwords, Abbas and Jungck  established the common fixed points for two mappings without exploiting the notion of continuity. Since then, common fixed point theorems in cone metric spaces have been proved for mappings satisfying different contractive conditions by many authors (see [3–8]). But there are few results about expanding mappings. Chintaman and Jagannath  introduced several meaningful fixed point theorems for one expanding mapping. However, the mapping depended strongly on continuity. In this paper, we delete the continuity of the mappings and obtain some fixed point theorems for one expanding mapping and introduce common fixed point theorems for two expanding mappings, which satisfy generalized expansive conditions in nonnormal cone metric spaces. These results improve and generalize some important known results in [9, 10].
P is closed, nonempty and ,
, , ,
The least positive number satisfying the above inequality is called the normal constant of P.
Definition 1.1 ()
for all and if and only if ;
for all ;
for all .
Then d is called a cone metric on X and is called a cone metric space. It is clear that the cone metric space is more general than a metric space.
Definition 1.2 ()
a Cauchy sequence if for every with , there is N such that for all , ;
a convergent sequence if for every with , there is N such that for all , for some fixed x in X.
A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
Lemma 1.3 ()
The limit of a convergent sequence in a cone metric space is unique.
2 Main results
In this section, we prove some fixed point theorems for expanding mappings without continuity in the following theorems.
for all , where () satisfies and . Then f has a fixed point.
Therefore, is a Cauchy sequence in .
Therefore, . From Lemma 1.3, we see . The conclusion is true. □
Taking some particular value of () in Theorem 2.1, we obtain several new results in the following.
for all , where and . Then f has a fixed point.
for all , where k, l are constants and . Then f has a fixed point.
Remark 2.4 Obviously, in our theorem and its corollaries above, we delete the continuity of the mappings which is essential in the results of . Moreover, in Corollary 2.2 we delete , , which is essential in Theorem 2.6 in . In Corollary 2.3 we delete , which is essential in Theorem 2.5 in . Theorem 2.3 in  is a special case of Theorem 2.1 with , , and f is continuous.
Now, we introduce some common fixed point theorems for two expanding mappings which satisfy generalized expansive conditions without continuity of the mappings.
for all , where () satisfies and , . Then f and g have a common fixed point.
where as .
Therefore, is a Cauchy sequence in .
As in the previous proof, it is not difficult to get , i.e., . Therefore, . □
for all , where and . Then f and g have a common fixed point.
for all , where is a constant. Then f and g have a unique common fixed point.
for all , where p, q are positive integers and is a constant. Then f has a unique fixed point.
Proof Let , . Since f is an onto mapping, , are onto mappings, the conditions of Corollary 2.7 are satisfied. □
Remark 2.9 In Corollary 2.8, we obtain Corollary 2.2 in  when we take .
Now, we present the following examples. In Example 1, we gain a fixed point for one expanding mapping of the situation when Corollary 2.2 can be applied, while the results in  cannot. In Example 2, we obtain the common fixed point for two expanding mappings in a cone metric space.
Therefore, we can apply Corollary 2.2 and conclude that f has a (unique) fixed point 0 in X. Since f is not continuous in X and , Theorem 2.6 in  is not applicable. Hence, our theorems have improved and generalized the main results in .
for all by taking , , , , . Thus, all the conditions of Theorem 2.5 are fulfilled. Then f and g have a unique common fixed point 1 in X.
Remark 2.10 Obviously, in the above two examples, we obtain the (common) fixed point which essentially needs the structure of a cone metric and not an ordinary metric on X. Then the results in a metric space in  cannot be applied to these examples.
The authors thank the editor and the referees for their valuable comments and suggestions which improve greatly the quality for this paper. The research was supported by the Foundation of Education Ministry, Hubei Province, China (No: D20102502).
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