Abstract

We establish some unique fixed point theorems in complete partial metric spaces for generalized weakly -contractive mappings, containing two altering distance functions under certain assumptions. Also, we discuss some examples in support of our main results.

1. Introduction and Preliminaries

An abstract metric space was first introduced and studied by the French mathematician Frechet [1] in 1906. Many researchers have generalized the concept of metric space as cone metric space, semimetric space, quasimetric space, and so forth, along with the generalization of contraction mappings with applications (see [2–7]). The best approximations of functions in locally convex spaces were discussed by Mishra et al. [8] and Mishra [9]. The degree of approximation of signals in Lp-space is established in [10].

Matthews [11, 12] initiated the concept of partial metric space as another generalization of metric space to study the denotational semantics of dataflow networks. Also, Matthews [11] generalized the Banach contraction principle to the class of partial metric spaces as follows: let be a complete partial metric space, and then a self-mapping on , satisfying where , has a unique fixed point.

After the Matthews [11] historical contribution, several researchers have established some more fixed point theorems in partial metric spaces and also discussed its topological properties (see [13–15] and references therein).

First, we recall some useful definitions and results, which is useful throughout the paper.

Definition 1 (see [11, 12]). Let be a nonempty set, and a mapping satisfying the following conditions is called a partial metric space on :, ,,,for all , and the pair is called a partial metric space. In the rest of the paper, represents a partial metric space equipped with a partial metric , unless otherwise stated. Let be a partial metric space and then let a function be defined as which is a metric on . Consider the function such that then is a metric on , and both of the above metrics and are equivalent [16].

Remark 2 (see [17]). In a partial metric space , (1) but if , then may not be zero,(2) for all , for all .

Example 3 (see [16]). Consider a mapping such that for all . Then will satisfy all the property of partial metric, and hence is a partial metric space but fails to be the condition of for all nonzero . Therefore is not a metric space.

Example 4 (see [16, 18]). Let    be three mappings and for any arbitrary mapping such that  ,  ,  ,for all , where and are a metric space and a partial metric space, respectively. Then each is the partial metric on .

Definition 5 (see [19]). In a partial metric space , (1) a sequence is said to be convergent to a point if and only if .
(2) A sequence is called Cauchy sequence if and only if is finite.
(3) If every Cauchy sequence converges to a point such that then is known as complete partial metric space.

Definition 6 (see [20, 21]). A self-mapping on a positive real number is said to be an altering distance function, if it holds for all such that(1) is continuous and nondecreasing,(2) .

The generalization of contractive mappings into -contractive mappings has been introduced by Chatterjea [6].

Definition 7 (see [2, 21]). A self-mapping on a metric space , satisfying for all and is a continuous mapping with if and only if is called weakly -contractive mapping or a weak -contraction.

Shukla and Tiwari [3] have introduced the concept of weakly -contractive mappings.

Definition 8 (see [3]). A self-mapping on a complete metric space is said to be weakly -contractive mapping or a weak -contraction, if the following inequality holds: for all and is a continuous function with if and only if .

Lemma 9 (see [7, 14]). In a partial metric space , if a sequence is convergent to a point , then for all . Also, if , then

Lemma 10 (see [13]). If is not a Cauchy sequence in and two sequences and of positive integers such that , then the four sequences tend to , when .

Lemma 11 (see [13, 16]). a partial metric space :(1)a sequence is a Cauchy if and only if it is a Cauchy in ,(2) is complete if and only if it is complete in .
In addition, if and only if

If is a Cauchy sequence in the metric space , we haveand therefore, by definition of , we have

2. Main Results

Theorem 12. Let be a complete partial metric space and and be two altering distance functions such that . Then the self-continuous nondecreasing mapping on , satisfying the conditionfor all and is a continuous function such that if and only if , has a unique fixed point in .

Proof. First we prove that if fixed point of exists, then it will be unique. On the contrary, we consider two fixed points of such that . Then by (12), we have By the property of , we obtain Using Remark 2, we obtain , which is a contradiction with respect to . Thus, we conclude that has a unique fixed point in .
Next, we show that the mappings , satisfying (12), have a fixed point. We choose an arbitrary point in . If , then the theorem follows trivially. Now, we suppose that and we choose such that . Since is a nondecreasing function, then we have . Again, let . Then we get Proceeding with this work, we obtained a sequence in such that and Supposing that for some , then by Remark 2 we have Again, we suppose that . Firstly, we prove that the sequence is nonincreasing. Suppose this is not true, and thenPutting and in (12) and using , we have Using above, we get Using (18) above, we have which contradicts our assumption that for all . Thus, we deduce that is a nonincreasing sequence. ThereforeSince is a monotonically decreasing and bounded below sequence in , then there exists such thatUsing (23) and letting in (20), we get Then (23) reduces toNow, we have required proving that the sequence is a Cauchy sequence in the metric space and so in by Lemma 11. On the contrary, that is, the sequence not being a Cauchy sequence in , sequences in Lemma 10 tend to , when . Now, we put and in (12). We have Taking and applying Lemma 10 in the above inequality, we have which is a contradiction with respect to . Thus is a Cauchy sequence in and so in . Since is complete, is also complete (by Lemma 11). Therefore, the Cauchy sequence converges in ; that is, ; then by Lemma 11, we haveBy Lemma 11, we get . So, by definition of , we get Using (24) and taking in the above inequality, we obtainFrom (28) and (30), we getBy , we obtain Taking and using (31), (24), and Lemma 9 in the above inequality, we haveFrom , we haveBy (33) and (34), we getFrom (35) and (12), we obtain Using (31) and property of in the above inequality, we obtain Thus, is a unique fixed point of in .

Example 13. Let be a complete partial metric space defined by   ∀. Consider a self-map on such that . Also, we define such that , , respectively, and such that .

If , then We observe that, for all , Similarly, we can show the result for . Thus, (12) holds for all and satisfies all the requirements of Theorem 12. So, is the unique fixed point of .