We prove common fixed point theorem for coincidentally commuting nonself mappings satisfying generalized contraction condition of Ćirić type in cone metric space. Our results generalize and extend all the recent results related to non-self mappings in the setting of cone metric space.
1. Introduction
Recently, Huang and Zhang [1] introduced the concept of cone metric space by replacing the set of real numbers by an ordered Banach space and obtained some fixed point theorems for mappings satisfying different contractive conditions. The category of cone metric spaces is larger than metric spaces and there are different types of cones. Subsequently, many authors like Abbas and Jungck [2], Abbas and Rhoades [3], Ilić and Rakočević [4], Raja andVaezpour[5] have generalized the results of Huang and Zhang [1] and studied the existence of common fixed points of a pair of self mappings satisfying a contractive type condition in the framework of normal cone metric spaces. However, authors like Janković et al. [6], Jungck et al. [7], Kadelburg et al. [8, 9], Radenović and Rhoades [10], Rezapour and Hamlbarani [11] studied the existence of common fixed points of a pair of self and nonself mappings satisfying a contractive type condition in the situation in which the cone does not need be normal.
The study of fixed point theorems for nonself mappings in metrically convex metric spaces was initiated by Assad and Kirk [12]. Utilizing the induction method of Assad and Kirk [12], many authors like Assad [13], Ćirić [14], Hadžić [15], Hadžić and Gajić [16], Imdad and Kumar [17], Rhoades [18, 19] have obtained common fixed point in metrically convex spaces. Recently, Ćirić and Ume [20] defined a wide class of multivalued nonself mappings which satisfy a generalized contraction condition and proved a fixed point theorem which generalize the results of Itoh [21] and Khan [22].
Very recently, Radenović and Rhoades [10] extended the fixed point theorem of Imdad and Kumar [17] for a pair of nonself mappings to nonnormal cone metric spaces. Janković et al. [6] proved new common fixed point results for a pair of nonself mappings defined on a closed subset of metrically convex cone metric space which is not necessarily normal by adapting Assad-Kirk's method.
The aim of this paper is to prove common fixed point theorems for coincidentally commuting nonself mappings satisfying a generalized contraction condition of Ćirić type in the setting of cone metric spaces. Our results generalize mainly results of Ćirić and Ume [20] and all the recent results related to nonself mappings in the setting of cone metric space.
2. Definitions and Preliminaries
We recall some basic definitions and preliminaries that will be needed in the sequel.
Definition 2.1 (see [1]).
Let 
be a real Banach space. A subset 
of 
is called a Cone if and only if
(1)
is nonempty, closed and 
;
(2)
, 
and 
;
(3)
.
For a given cone 
, a partial ordering is defined as 
on 
with respect to 
by 
, if and only if 
. It is denoted as 
to indicate that 
but 
, while 
will stand for 
, where 
denotes the interior of 
.
The cone 
is called normal, if there is a number 
such that for all 
, 
implies
(21) The least positive number 
satisfying (2.1) is called the normal constant of 
. It is clear that 
. There are nonnormal cones also.
The definition of a cone metric space given by Huang and Zhang [1] is as follows.
Definition 2.2 (see [1]).
Let 
be a nonempty set. Suppose that 
is a real Banach space, 
is a cone with 
and 
is a partial ordering with respect to 
.
If the mapping 
satisfies the following:
(1)
for all 
and 
if and only if 
;
(2)
for all 
;
(3)
for all 
;
then 
is called a cone metric on 
and 
is called a cone metric space.
Example 2.3 (see [1]).
Let 
, 
and 
such that 
, where 
is a constant. Then 
is a cone metric space.
Definition 2.4 (see [1]).
Let 
be a cone metric space and 
a sequence in 
. Then, one has the following.
(1)
converges to 
, if for every 
with 
, there is 
such that for all 
,
(22) It is denoted by 
or 
, 
.
(2)If for any 
, there is a number 
such that for all 
(23) then 
is called a Cauchy sequence in 
.
(3)
is a complete cone metric space, if every Cauchy sequence in 
is convergent.
(4)A self mapping 
is said to be continuous at a point 
, if 
implies that 
for every 
in 
.
The following two lemmas of Huang and Zhang [1] will be required in the sequel.
Lemma 2.5 (see [1]).
Let 
be a cone metric space and 
a normal cone with normal constant 
. A sequence 
in 
converges to 
if and only if 
as 
.
Lemma 2.6 (see [1]).
Let 
be a cone metric space and 
a normal cone with normal constant 
. A sequence 
in 
is a Cauchy sequence if and only if 
as 
.
The following Corollary of Rezapour [23] will be needed in the sequel.
Corollary 2.7 (see [23]).
Let 
, the real Banach space.
(i)If 
and 
, then 
.
(ii)If 
and 
, then 
.
(iii)If 
for each 
, then 
.
The following remarks of Radenović and Rhoades [10] will be needed in the sequel.
Remark 2.8 (see [10]).
If 
, 
and 
, then there exists 
such that for all 
, it follows that 
.
Remark 2.9 (see [10]).
If 
and 
, then 
, where 
is a sequence and 
is a given point in 
.
Remark 2.10 (see [10]).
If 
and 
, 
, then 
for each cone 
.
Remark 2.11 (see [10]).
If 
is a real Banach space with a cone 
and if 
, where 
and 
, then 
.
3. Main Results
In the following, we suppose that 
is a Banach space, 
is a cone in 
with 
and 
is partial ordering with respect to 
.
Theorem 3.1.
Let 
be a complete cone metric space and 
a nonempty closed subset of 
such that for each 
and 
there exists a point 
such that
(31) Suppose that 
are two nonself mappings satisfying for all 
with 
,
(32) and 
are nonnegative real numbers such that
(33) Also assume that
(i)
;
(ii)
;
(iii)
is closed in 
;
Then there exists a coincidence point of 
and 
in 
. Moreover, if 
and 
are coincidentally commuting, then 
and 
have a unique common fixed point in 
.
Proof.
Two sequences 
and 
are constructed in the following way. Let 
. As 
, by (i) there exists a point 
such that 
. Since 
, from (ii) it follows that 
. Let 
be such that 
. Since 
, there exists 
such that 
.
If 
, then 
which implies that there exists a point 
such that 
. Otherwise, if 
, then there exists a point 
such that
(34) Since 
, there exists a point 
such that 
and thus
(35) Assume that 
.
Thus repeating the arguments, two sequences 
and 
are obtained such that
(i)
;
(ii)
;
(iii)
whenever 
, then there exists 
such that
(36) Next, we claim that 
is a Cauchy sequence in 
. The following are derived. Let us denote
(37) Obviously, two consecutive terms cannot lie in 
. Note that, if 
, then 
and 
belong to 
. Now, three cases are distinguished.
Case 1.
If 
, then 
. Now from (3.2),
(38) where
(39) Thus
(310) Now foursubcasesarise.
Subcase 1.1.
If 
and 
, then (3.8) becomes
(311) Subcase 1.2.
If 
and 
, then (3.8) becomes
(312) Subcase 1.3.
If 
and 
, then (3.8) becomes
(313) Subcase 1.4.
If 
and 
, then (3.8) becomes
(314) Combining allSubcases1.1, 1.2, 1.3, and 1.4, it follows that
(315) where 
. Hence
(316) Case 2.
If 
, then 
. Now,
(317) Proceeding as in Case 1,
(318) Case 3.
If 
, then 
, 
and 
. Now,
(319) Thus
(320) where
(321) Thus
(322) Again foursubcasesarise.
Subcase 3.1.
If 
, then (3.20) becomes
(323) Thus
(324) where using Case 2,
(325) Then (3.24) becomes
(326) Subcase 3.2.
If 
, then (3.20) becomes
(327) Proceeding as in Subcase 3.1, it follows that
(328) Subcase 3.3.
If 
, then (3.20) becomes
(329) Thus
(330) where using Case 2,
(331) Then (3.30) becomes
(332) Subcase 3.4.
If 
, then (3.20) becomes
(333) Proceeding as in Subcase 3.1, it follows that
(334) Combining all fourSubcases3.1, 3.2, 3.3, and 3.4, we have
(335) where 
by (3.3). Hence
(336) Now, combining main Cases 1, 2, and 3, it follows that
(337) where
(338) Following the procedure of Assad and Kirk [12], it can be easily shown by induction that for 
,
(339) By triangle inequality, for 
, it follows that
(340) From Remark 2.9, 
, which implies by Definition 2.4(2) that 
is a Cauchy sequence in 
which is a closed subset of the complete cone metric space and hence is complete. Then there exists a point 
such that 
as 
. Thus
(341) Since 
, there exists a point 
such that 
. By the construction of 
, it was seen that there exists a subsequence 
such that
(342) We will prove that 
. Consider
(343) where
(344) Thus
(345) Now again four cases arise.
Case 1.
If 
, then (3.43) becomes
(346) Case 2.
If 
, then (3.43) becomes
(347) Case 3.
If 
, then (3.43) becomes
(348) Case 4.
If 
, then (3.43) becomes
(349) Combining Cases 1, 2, 3, and 4, it follows that
(350) Thus
(351) where 
.
Let 
be given with 
. From 
as 
and Definition 2.4(1),
(352) From 
as 
and by Definition 2.4 (1),
(353) From the definition of convergence in cone metric space and by (3.52) and (3.53), inequality (3.43) becomes
(354) Therefore, 
for each 
. Then by (iii) of Corollary 2.7, we have 
, that is, 
which implies that 
is the coincidence point of 
and 
.
Since 
and 
are coincidentally commuting, 
for 
which implies 
. Consider
(355) where
(356) Thus 
and 
. Two cases arise.
Case 1.
If 
and 
, then (3.55) becomes
(357) Case 2.
If 
and 
, then (3.55) becomes
(358) Combining Cases 1 and 2, it follows that
(359) Since 
by (3.3), it follows from Remark 2.11 that 
which implies that 
. Thus 
.
Uniqueness: if 
is another common fixed point of 
and 
in 
, then 
. Now by (3.2), it follows that
(360) where
(361) Thus 
and 
. Two cases arise.
Case 1.
If 
and 
, then (3.60) becomes 
. Since 
, by Remark 2.11 we have 
which implies that 
is the unique common fixed point of 
and 
.
Case 2.
If 
and 
, then (3.60) becomes
(362) Since 
, by Remark 2.11 we have 
which implies 
is the unique common fixed point of 
and 
. Hence 
is the unique common fixed point of 
and 
in 
.
The following example illustrates Theorem 3.1.
Example 3.2.
Let 
, 
, 
, 
and 
. Define two nonself mappings 
as 
and 
for all 
.
Now let us see that conditions (i)–(iii) in Theorem 3.1 are satisfied.
It may be seen that 
and 
. Then 
and 
. Also, 
as 
. Moreover 
is closed in 
.
Next, we shall see that inequality (3.2) is satisfied by taking 
and 
. It is easy to see that 
.
Now, LHS of inequality (3.2) is 
. Taking 
and 
, it follows that 
.
Next, RHS of inequality (3.2) is 
, where 
, 
and 
. Then RHS of inequality (3.2) is 
if 
and 
. Thus LHS of inequality (3.2) 
RHS of inequality (3.2). Similarly, LHS of inequality (3.2) 
RHS of inequality (3.2) for all possible cases of 
and 
. Thus all the conditions of Theorem 3.1 are satisfied. Hence "0" is the unique common fixed point of 
and 
in 
.
Corollary 3.3.
Let 
be a complete cone metric space and 
a nonempty closed subset of 
such that for each 
and 
there exists a point 
such that
(363) Suppose that 
is a nonself mapping satisfying for all 
with 
,
(364) and 
are nonnegative real numbers such that 
. Also assume that 
. Then there exists a unique fixed point of 
in 
.
Proof.
The proof of this corollary follows by taking 
, the identity mapping of 
in Theorem 3.1.
Remark 3.4.
Our results generalize the results of Radenović and Rhoades [10] and Janković et al. [6] and extend the results of Ćirić and Ume [20] to cone metric space for single valued mappings.
Acknowledgment
The authors would like to thank the referees for their valuable suggestions which lead to the improvement of the presentation of the paper.
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