Various Fixed Point Theorems in Complex Valued -Metric Spaces

Abstract

We prove some common fixed point results for a pair of mappings which satisfy generalized contractive conditions with rational expressions having point-dependent control functions as coefficients in complex valued -metric spaces. The results of this paper generalize and extend the several known results in complex valued -metric spaces. Finally, examples are provided to verify the effectiveness and to usability of our main results.

1. Introduction and Preliminaries

The concept of complex valued metric space was introduced by Azam et al. [1], proving some fixed point results for mappings satisfying a rational inequality in complex valued metric spaces. Since then, several papers have dealt with fixed point theory in complex valued metric spaces (see [2–10] and references therein). Rao et al. [11] initiated the studying of fixed point results on complex valued -metric spaces, which was more general than the complex valued metric spaces [1]. Following this paper, a number of authors have proved several fixed point results for various mapping satisfying a rational inequalities in the context of complex valued -metric spaces (see [12–15]) and the related references therein.

Recently, Sintunavarat et al. [8, 9], Sitthikul and Saejung [10], and Singh et al. [7] obtained common fixed point results by replacing the constant of contractive condition to control functions in complex valued metric spaces. In a continuation of [7, 10, 14, 16], in this paper, we establish some common fixed point results for a pair of mappings satisfying more general contractive conditions involving rational expressions having point-dependent control functions as coefficients in complex valued -metric spaces.

Consistent with Rao et al. [11], the following definitions and results will be needed in the sequel.

Let be the set of complex numbers and . Define a partial order on as follows:

if and only if

Consequently, one can infer that if one of the following conditions is satisfied:(i)(ii)(iii)(iv)

In particular, we write if and one of (i), (ii), and (iii) is satisfied and we write if only (iii) is satisfied. Notice that(a)if , then ;(b)if and , then ;(c)if and , then for all

The following definition is recently introduced by Rao et al. [11].

Definition 1 (see [11]). Let be a nonempty set and let be a given real number. A function is called a complex valued -metric on if for all the following conditions are satisfied: (i) and if and only if (ii)(iii) The pair is called a complex valued -metric space.

Example 2 (see [11]). If , define a mapping by , for all Then, is complex valued -metric space with

Definition 3 (see [11]). Let be a complex valued -metric space. (i)A point is called interior point of a set whenever there exists such that (ii)A point is called a limit point of a set whenever for every (iii)A subset is called an open set whenever each element of is an interior point of .(iv)A subset is called closed set whenever each limit point of belongs to .(v)The family is a subbasis for a Hausdorff topology on .

Definition 4 (see [11]). Let be a complex valued -metric space, and let be a sequence in and (i)If for every , with , there is such that for all , then is said to be convergent and converges to . We denote this by or (ii)If for every , with there is such that for all , where , then is said to be a Cauchy sequence.(iii)If every Cauchy sequence in is convergent in , then is said to be a complete complex valued -metric space.

Lemma 5 (see [11]). Let be a complex valued -metric space and let be a sequence in . Then, converges to if and only if as

Lemma 6 (see [11]). Let be a complex valued -metric space and let be a sequence in . Then, is Cauchy sequence if and only if as , where .

2. Main Result

Throughout this paper, let be a complete complex valued -metric space and be mappings. In our results, we will use the following family of functions.

Let be a complete complex valued -metric space with the coefficient and let be mappings. Let be the family of all functions such that for all and for fixed ,(F1);(F2)

We start this section with the following observation.

Proposition 7. Let be a complex valued -metric space and let be mappings. Let and define the sequence by Assume that there exists a mapping for all and for a fixed element and . Then, and .

Proof. Let and . Then, we have Similarly, we have

Lemma 8 (see [10]). Let be a sequence in and If satisfies , for all , then is a Cauchy sequence.

Now, we proceed to establish common fixed point theorems for the general contraction conditions in complex valued -metric space.

Theorem 9. Let be a complete complex valued -metric space with the coefficient and let be mappings. If there exist mappings such that for all and for fixed , Then, and have a unique common fixed point.

Proof. Let , from (5) we have which implies thatSince , In a similar way, by setting in (5), we haveLet and the sequence be defined by (1). We show that is a Cauchy sequence. From Proposition 7 and for all , we obtainwhich yields thatSimilarly, one can obtainLet
Since , thus we have and , or in factThus, by Lemma 8 we get that this sequence is Cauchy sequence in Since is complete, there exists some such that as . Let, on contrary, ; thenSo by using the triangular inequality and (5), we getThis implies thatLetting , it follows thata contradiction, and so ; that is, It follows similarly that This implies that is a common fixed point of and
We now prove that this is unique:Therefore, we haveSince , we have .
Thus, , which proves the uniqueness of common fixed point in This concludes the theorem.

Remark 10. If we replace by , with for all and so , then we get the result of Theorem 3.1 of Sintunavarat and Kumam [8] (complex valued -metric space version).

Remark 11. If we set mappings as and , where such that and for all , we get Theorem 4 of Azam et al. [1] (complex valued -metric space version).

Next theorem is presented for single mapping satisfying slightly different conditions.

Theorem 12. Let be a complete complex valued -metric space with the coefficient and let be a mapping. If there exist mappings such that for all and for fixed ,then has a unique fixed point.

Proof. Let and the sequence be defined by , where . Now we show that is a Cauchy sequence. From condition (21), we haveTherefore,Let ; then,By Lemma 8, this sequence is Cauchy sequence in Since is complete, there exists some such that as Next, we show that is a fixed point of
From (21), we haveThis implies thatwhich on making reduces toa contradiction, and so ; that is, This implies that is a fixed point of
Uniqueness of fixed point is an easy consequence of condition (22). This completes the proof.

Corollary 13. Let be a complete complex valued -metric space with the coefficient and let be a mapping. If there exist mappings such that for all and for some fixed ,then has a unique fixed point.

Proof. By Theorem 12, there exists such that . Then,and so . So Therefore, the fixed point of is unique.

Example 14. Let and be defined by for all Then, is a complex valued -metric space with the coefficient Now we define self-mappings by and . Further, for all and for fixed , we define the functions byClearly for all and for a fixed

Now considerThat is, for all and for a fixed . NowThat is, for all and for a fixed . Similarly, we can show thatNow for the verification of inequality (5), it is sufficient to show that .

ConsiderThat is, for all and for fixed Therefore, all the conditions of Theorem 9 are satisfied, also remains fixed under and and is indeed unique.

Competing Interests

The authors declare that they have no competing interests.