#### Abstract

We establish common fixed points theorems for two self-mappings satisfying a nonlinear contractive condition of Ćirić type with a *Q*-function. Furthermore, using the scalarization method, we deduce some results of common fixed point in tvs-cone metric spaces with a *c*-distance. As application, we give a positive answer to the question of Ćirić et al. posed in 2012. Our results extend and generalize many recent results.

#### 1. Introduction and Preliminaries

In 1996, Kada et al. [1] for the first time introduced and studied the concept of -distances on a metric space. They gave examples of -distance and improved Caristi’s fixed point theorem, Ekeland’s variational principle, and Takahashi’s nonconvex minimization theorem. Later, Al-Homidan et al. [2] introduced the concept of -functions which generalized the concept of -distances, and they established a generalized Ekeland-type variational principle with a -function.

*Definition 1. *Let be a metric space with metric . Then a function is called a -function on if the following are satisfied:(q1), for any ;(q2)if and is a sequence in such that it converges to a point and for some , then ;(q3)for any , there exists such that and imply .

If condition (q2) is replaced by the following stronger condition:(q2′)for any , is lower semicontinuous, then is called a -distance on .

For some examples of -functions and -distances, the reader can see [1, 2]. The following lemma has been presented in [1, 2].

Lemma 2. *Let be a metric space and let be a Q-function on . Let and be sequences in , let and be sequences in converging to 0, and let . Then the following hold.*(i)*If and for any , then . In particular, if and , then .*(ii)*If and for any , then converges to .*(iii)*If for any with , then is a Cauchy sequence.*(iv)*If for any , then is a Cauchy sequence.*

Ume [3] proved some fixed point theorems in a complete metric space using the concept of a -distance and more general contractive mapping than quasi-contractive mapping. Ilic and Rakocevic [4] established some common fixed points results for maps on metric spaces with -distance and generalized the results of Ume [3]. Recently, di Bari and Vetro [5] obtained common fixed points for maps satisfying a nonlinear contractive condition. As application, using the scalarization method of Du [6], they deduce a result of common fixed point in cone metric space. For more early results in cone metric spaces (or in -metric spaces) one can consider Zabrejko [7] and the references therein.

In 2007, Huang and Zhang [8] reintroduced and studied the concept of cone metric spaces over a real Banach space which is a generalization of metric spaces, and they proved several fixed point theorems in cone metric spaces. Then, many authors have studied fixed point problems in cone metric spaces; see [9] for a survey of fixed point results in these spaces. Recently, Du [6] used the scalarization function and showed that many of the fixed point theorems in metric spaces and in a generalized cone metric space (i.e., a so-called tvs-cone metric space) are equivalent. For more scalarization methods on fixed point problems, see also [10–12].

In the following we suppose that is a real Hausdorff topological vector space (tvs for short) with the zero vector . A nonempty subset of is called a convex cone if and for . A convex cone is said to be pointed if . For a given cone , we can define a partial ordering with respect to by if and only if . will stand for and , while stand for , where denotes the interior of .

Throughout this paper, unless otherwise specified, we always assume that is a tvs, a proper, closed, and convex pointed cone in with , and . Following [6, 8], we give the following notion.

*Definition 3. *Let be a nonempty set. A function is called a tvs-cone metric space if the following conditions hold:(p1) for all and if and only if ;(p2) for all ;(p3) for all .

Let and be a sequence in . Then, it is said that(i) converges to if for every with there exists a natural number such that for all ; we denote it by as ;(ii) is a Cauchy sequence if for every with there exists a such that for all ;(iii) is complete if every Cauchy sequence is convergent in .

Recently, some generalized distance on cone metric spaces is introduced. Cho et al. [13] introduced the concept of a -distance in cone metric spaces and proved some fixed point theorems in ordered cone metric spaces. Ćirić et al. [14] introduce the concept of a -cone distance on tvs-cone metric spaces and proved various fixed point theorems for -cone distance contraction mappings in tvs-cone metric spaces. Let us recall these concepts.

*Definition 4. *Let be a tvs-cone metric space. Then a function is called a -distance on if the following conditions hold:(c1) for all ;(c2) for all ;(c3)for each and , if for some , then whenever is a sequence in converging to a point ;(c4)for all with , there exists with such that and imply .

If condition (c3) is replaced by the following stronger condition:(c3′)for any , is lower semicontinuous, then is called a -cone distance on .

*Remark 5 (see [13]). *(1) does not necessarily hold for all .

(2) is not necessarily equivalent to .

Ćirić et al. [14] posed a question as follows.

*Question 1. *Let be a complete tvs-cone metric space with -cone distance on . Suppose such that, for some constant and for every , there exists , such that . Does there exist a unique fixed point of , and ?

In this paper, we establish common fixed points theorems for two self-mappings satisfying a nonlinear contractive condition of Ćirić type with a -function, which generalize the result of di Bari and Vetro [5]. Furthermore, using the scalarization method, we deduce some results of common fixed points in tvs-cone metric spaces with a -distance. As application, we give a positive answer to Question 1 and extend some recent results presented in [15, 16]. In particular, we will see that the assumption that satisfies, for all , can be completely removed in [15].

#### 2. The Scalarization Method

First, we recall the concept of the nonlinear scalarization function, which is one of the most useful tools to solve problems in vector optimization, control theory, and so forth; see, for example, [17–20].

The function is defined as follows:

Lemma 6 (see [17]). *For each and , the following statements are satisfied:*(i)* ;*(ii)* ;*(iii)* ;*(iv)* ;*(v)* , in particular, , for all ;*(vi)* is positively homogeneous and continuous on ;*(vii)*if , that is, , then ;*(viii)*if , that is, , then ;*(ix)* for all .*

Du [6] used the above scalarization function and obtained the following results.

Theorem 7 (see [6, Theorem 2.1]). *Let be tvs-cone metric spaces. Then defined by is a metric.*

Theorem 8 (see [6, Theorem 2.2]). *Let be a tvs-cone metric space, , and a sequence in . Let be the same as in Theorem 7. Then the following statements hold:*(i)* converges to in if and only if converges to in ;*(ii)* is a Cauchy sequence in if and only if is a Cauchy sequence in ;*(iii)* is complete if and only if is complete.*

* Proof. *Du has proven the necessity of (i) and (ii); see [6, Theorem 2.2]. As the proof of (ii) is similar to that of (i) and (iii) follows from (i) and (ii), it is enough to prove the sufficiency of (i).

Assume now that and with . Then and consequently is a neighborhood of . For , there is such that ; that is, . Since as , there exists a positive integer such that
By (ii) of Lemma 6, we have for all . From this and , we obtain
Thus converges to in .

Now, inspired by the scalarization method of Du [6], we give the following result.

Theorem 9. *Let be a tvs-cone metric space and let be a c-distance on . Then defined by is a Q-function on , where .*

*Proof. *By , , and (iii) of Lemma 6, we have ; that is, , for all . Using (vii) and (ix) of Lemma 6 and (c2), we obtain
That is, satisfies (q1).

Next we prove that satisfies (q2). For this, let , such that in ; that is, as , and for some . According to (i) of Theorem 8, it follows that as in . By and (i) of Lemma 6, we have . It follows from (c3) that . Again using (i) of Lemma 6, we obtain
That is, satisfies (q2).

Finally, we show that satisfies (q3). Let be given. Then implies . As satisfies the condition (c4), there is with such that and imply . Since is a neighborhood of , there exists such that ; that is, . Using (ii) of Lemma 6, when and , we have and , which imply . Again using (ii) of Lemma 6, we obtain . Thus we have shown that satisfies (q3) and is a -function on .

Similarly, we can deduce the following.

Theorem 10. *Let be a tvs-cone metric space and let be a w-cone distance on . Then defined by is a w-distance on , where .*

#### 3. The Results for Maps on Metric Space

Let be the family of functions satisfying the following conditions:(i) is nondecreasing;(ii);(iii);(iv) for all .

It is obvious that if is defined by for some , or , then .

*Remark 11 (see [5, Remark 2.1]). *If , then we have that and for all .

Let be a metric space and be a -function on . For , we define .

Letting be self-mappings, and are a -quasi-contraction if there exists such that
for all .

Let us recall that the self-mappings and on are said to be weakly compatible if they commute at their coincidence point (i.e., whenever ). and are said to be compatible if
whenever is a sequence in such that

If and satisfy and , let us define such that . Having defined , let be such that . We say that is a --sequence of initial point . Define

The following lemma is crucial in this paper.

Lemma 12. *Let be a metric space and let be a Q-function on . Suppose that the self-mappings are a -quasi-contraction with . Let . For , let be a --sequence of initial point . Then one has the following:*(i)*for each and , there exist with such that
*(ii)*for each , there exists such that
*(iii)*for each , is a Cauchy sequence. If converges to some , then
**for all , where .*

*Proof. *(i) Let and . Since and are a -quasi-contraction with , for every , we have
This implies that
for some .

(ii) By property (iii) of the function , for , there is a such that , for all . Therefore, for each ,
Thus,
for all . It follows that

(iii) Define for every . Obviously for all . Consequently, for all with ,
From Lemma 2(iii) and Remark 11, it follows that is a Cauchy sequence. If it converges to , then (19) implies that .

Theorem 13. *Let be a complete metric space and let be a Q-function on . Suppose that the self-mappings are a -quasi-contraction with . Let , and let*(D1)*for every with **If and are weakly compatible, then the mappings and have a unique common fixed point in and .*

*Proof. *Let be fixed. As , we construct a --sequence of initial point . Using Lemma 12(iii), we see that is a Cauchy sequence. Since is complete, there exists such that and also .

Let us prove that . If , then (D1), (13), and (19) imply
This is a contradiction. Hence .

Thus
and so . Similarly, .

Now, we show that is a common fixed point for and . Since and are weakly compatible, we deduce that
From this, we obtain that
From (24) and (25), it follows that and . Using , , and Lemma 2(i), we obtain that . From (23), it follows that .

To prove the uniqueness of the common fixed point of and , let us suppose that there exists and . From the definition of -quasi-contraction, it follows that
Thus . By Lemma 2(i), we conclude that and .

Theorem 14. *Let be a metric space and let be a Q-function on . Suppose that the self-mappings are a -quasi-contraction with . Let , and let*(D2)*for every with **If or is a complete subspace of and and are weakly compatible, then the mappings and have a unique common fixed point in and .*

*Proof. *Let be fixed. As , we construct a --sequence of initial point . Using Lemma 12 (iii), we see that is a Cauchy sequence. Since or is a complete subspace of , there exists such that and also . Let be such that .

Let us prove that . If , then (D2), (13), and (19) imply
This is a contradiction. Hence .

Similar to the proof of Theorem 13, we can conclude that is a unique common fixed point and .

Theorem 15. *Let be a complete metric space and let be a Q-function on . Suppose that the self-mappings are a -quasi-contraction with . Let and let and be continuous mappings. If and are compatible, then the mappings and have a unique common fixed point in and .*

*Proof. *Let be fixed. As , we construct a --sequence of initial point . Using Lemma 12 (iii), we see that is a Cauchy sequence. Since is complete, there exists such that and also . Since and are compatible, we have
Using the continuousness of and , we deduce that and . From (29), it follows that .

Similar to the proof of Theorem 13, we can conclude that is a unique common fixed point and .

*Remark 16. *From the proof of Theorems 13 and 15, we see that the completeness of the space can be replaced by the completeness of the subspace or .

If we take for some in Theorems 13–15, then we have the following results.

Corollary 17. *Let be a complete metric space with metric and let be a Q-function on . Let and be the mappings of into itself satisfying and
**
for all and some .**Assume that either of the following holds:*(i)* and are weakly compatible and satisfy the condition (D1);*(ii)* and are compatible and continuous.**Then the mappings and have a unique common fixed point in and .*

Corollary 18. *Let be a metric space with metric and let be a Q-function on . Let and be the mappings of into itself satisfying and
**
for all and some .**Assume that either of the following holds:*(i)* and are weakly compatible and satisfy the condition (D1);*(ii)* and are weakly compatible and satisfy the condition (D2);*(iii)* and are compatible and continuous.**If or is complete, then the mappings and have a unique common fixed point in and .*

In the results presented by di Bari and Vetro [5], the condition (D1) or (D2) is not required. In fact, the conditions (D1) and (D2) always hold when .

Proposition 19. *Let be a metric space and let the self-mappings be such that . If and are a -quasi-contraction with , then for every with *

*Proof. *Suppose that there exists with and
Then there exists a sequence in such that
It follows that and , and so . Define .

Thus, for sufficiently large, we have
From
we deduce that
Since , we get . Using the property (iv) of the function , we see that
which is a contradiction.

Proposition 20. *Let be a metric space and let the self-mappings be such that . Suppose that and are a weakly compatible pair and a -quasi-contraction with . If or is closed subspace of , then for every with *

*Proof. *Suppose that there exists with and
Then there exists a sequence in such that
It follows that and , and so . Since , and or is closed, we have . Thus there exists such that .

Let us prove that . If , then . Similar to the proof of Proposition 19, we can get a contradiction. Thus .

Since and are weakly compatible, we obtain
which is a contradiction with .

*Remark 21. *Using Theorem 13 and Proposition 20, or Theorem 14 and Proposition 19, we get Theorem 2.2 of di Bari and Vetro [5].

*Remark 22. *In Corollary 17 we use the assumption that and are weakly compatible, which is weaker than the assumption that and commutes in Theorem of Ilic and Rakocevic [4].

#### 4. The Results for Maps on Tvs-Cone Metric Space

We denote with the set of all functions which have the following properties:(i) is nondecreasing, that is, whenever with ;(ii);(iii);(iv) for all .

We denote with the set of all functions which have the following properties (see [5]):(i);(ii) for all ;(iii) for some ;(iv)if , , then there exists and such that for all .

*Remark 23 (see [5, Remark 2.3 and Theorem 3.5]). *If , then the function is defined by

Let be a tvs-cone metric space and let be a -distance on . Let be self-mappings. Then and are called a -quasi-contraction if there exists such that for all
where

Now we give several Lemmas as follows.

Lemma 24. *Let be a tvs-cone metric space and let be a c-distance on . Suppose that the mappings satisfy the following condition:*(C1)*if , there exists such that
**Then for every with **
where .*

*Proof. *Take with . Using (vii) and (ix) of Lemma 6 and the assumption (C1), we have
for all . Using (iv) of Lemma 6, and imply that

Similarly, we can deduce the following.

Lemma 25. *Let be a tvs-cone metric space and let be a c-distance on . Suppose that the mappings satisfy the following condition:*(C2)*if , there exists such that
**Then for every with **
where .*

Lemma 26. *Let be a tvs-cone metric space and let be self-mappings. Then*(i)* is continuous in if and only if so is in ;*(ii)* and are compatible in if and only if so is in .*

*Proof. *From (i) of Theorem 8, the conclusions are obvious.

Lemma 27. *Let be a tvs-cone metric space and let be a c-distance on . Suppose the self-mappings are a -quasi-contraction with . Then there exists such that and are -quasi-contraction, where .*

*Proof. *We choose such that . Define as . From Lemma 6 and the properties of the function , the function has the following properties:(i) is nondecreasing;(ii);(iii);(iv) for all .

Then . Notice that , so
for all . Thus, from where satisfies (45), we deduce that
That is, and are a -quasi-contraction.

Using Lemmas 24, 26, and 27 and Theorems 13 and 15, we obtain the following.

Theorem 28. *Let be a complete tvs-cone metric space and let be a c-distance on . Let and be the mappings of into itself satisfying and a -quasi-contraction with .**Assume that either of the following holds:*(i)* and are weakly compatible and satisfy the condition (C1);*(ii)* and are compatible and continuous.**Then the mappings and have a unique common fixed point in and .*

Using Lemmas 24–27, Theorems 13–15, and Remark 16, we have the following.

Theorem 29. *Let be a tvs-cone metric space and let be a c-distance on . Let and be the mappings of into itself satisfying and a -quasi-contraction with .**Assume that either of the following holds:*(i)* and are weakly compatible and satisfy the condition (C1);*(ii)* and are weakly compatible and satisfy the condition (C2);*(iii)* and are compatible and continuous.**If or is complete, then the mappings and have a unique common fixed point in and .*

Theorem 30. *Let be a complete tvs-cone metric space and let be a c-distance on . Let and be the mappings of into itself satisfying and the following condition:*(a)*for all , where are nonnegative constants such that .**Assume that either of the following holds:*(i)*if , there exists such that
*(ii)* and are compatible and continuous.**Then the mappings and have a unique common fixed point in and .*

*Proof. *Put and . Using Lemma 6 and the assumption (a), we obtain
for all , where . By Theorem 7, Theorem 8, and Theorem 9, we see that is a complete metric space and is a -function on . Applying Corollary 17 in , we deduce that and have a unique common fixed point in . It is easy to see that from the condition (a).

If we take , the identity mapping on , and , then we get the following corollaries.

Corollary 31. *Let be a complete tvs-cone metric space and let be a c-distance on . Suppose that the mapping satisfies that, for some constant and for every , there exists
**
such that
**Assume that either of the following holds:*(i)*if , there exists such that
*(ii)* is continuous.**Then has a unique fixed point in and .*

Corollary 32. *Let be a tvs-cone metric space and let be a c-distance on . Suppose that the mapping satisfies that, for some constant and for every , there exists
**
such that
**
Assume that either of the following holds:*(i)*if , there exists such that
*(ii)*if , there exists such that
*(iii)* is continuous.**If or is complete, then has a unique fixed point in and .*

Corollary 33. *Let be a complete tvs-cone metric space and let be a c-distance on . Suppose that a mapping satisfies the following condition:*(a)*for all , where are nonnegative constants such that .**Assume that either of the following holds:*(i)*if , there exists such that
*(ii)* is continuous.**Then has a unique fixed point in . If , then .*

*Remark 34. *As can be seen, the results of Corollary 31 and Corollary 32 provide an answer to Question 1.

*Remark 35. *Using Corollary 33, we can obtain [15, Theorem 2] and [16, Theorem 3.3]. Moreover, we see that the assumption is weakened to and the assumption is removed in [15, Theorem 2].

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The author is grateful to the anonymous referees for useful comments and suggestions, which greatly improved the paper. This work was supported in part by Scientific Studies of Higher Education Institution of Inner Mongolia (NJZZ13019) and in part by Program of Higher-Level Talents of Inner Mongolia University (30105-125150 and 30105-135117).