A new approach in the context of ordered incomplete partial b-metric spaces

Abstract The main purpose of this paper is to find some fixed point results with a new approach, particularly in those cases where the existing literature remains silent. More precisely, we introduce partial completeness, f̄-orbitally completeness, a new type of contractions and many other notions. We also ensure the existence of fixed points for non-contraction maps in the class of incomplete partial b-metric spaces. We have reported some examples in support of our results.


Introduction
The Banach contraction principle plays a significant role in the development of fixed point theory. Banach [1] explained the uniqueness of a fixed point for a contraction map in complete metric spaces. The Banach contraction principle has been generalized by many researchers either by generalizing the distance space or modifying the contraction. In 1989, Bakhtin [2] generalized metric spaces by introducing the concept of b-metric spaces. In 1994, Matthews [3] introduced the concept of a partial metric space, in which a selfdistance of any point may not be zero.  generalized the Banach contraction principle in setting of ordered metric spaces. The key feature in the Ran-Reurings theorem is that the contractive condition on the nonlinear map is only assumed to hold on the comparable elements instead of the whole space as in the Banach contraction principle. In 2005, Nieto and Rodríguez-López [5] proved a fixed point theorem by relaxing some conditions in . In 2008, Suzuki [6] proved a fixed point theorem by assuming a contraction condition on those elements which satisfy the given condition. Recently in 2013, Shukla [7] generalized both b-metrics and partial metrics by introducing a partial bmetric. There is a bulk of literature on fixed point in all these spaces, see . Almost all the existing fixed point results involve the completeness of the underlying space. The main aim of this paper is to ensure the existence of fixed points either without assuming the completeness of the space or the contraction condition is not satisfied. In this direction, we give a brief introduction of our newly introduced concepts, especially partial completeness, a new contraction condition and many other concepts to tackle the problem as pointed out above. The contraction condition is not assumed to hold on the whole space, but we choose a subset with some given properties, on which the contraction condition is assumed to hold. Particularly, we prove fixed point results for non-contraction maps in the setting of ordered partial incomplete b-metric spaces. We give some examples in support of our obtained theorems. Before going to the main result, we recall some definitions.
be a function verifying: Let X be a nonempty set and × → [ ∞) p X X : 0 , be a function such that: . Then, p is called a partial metric on X and ( ) X p , is called a partial metric space.
be a function such that: ). It is neither a partial metric nor a b-metric.
exists and is finite.
n m n m n n , Here, the limit of a convergent sequence may not be unique.
, is a partial b-metric space with an arbitrary coefficient ≥ s 1. If = ζ 1 n for each ∈ n , then = →∞ ζ η lim n n , for each ≥ η 1.
be a partially ordered set and d be a partial b-metric on X. Then, the triplet is called an ordered partial b-metric space.
. For each ∈ θ X and ≥ n 1, take is said to be f -orbitally complete with respect to A, if every strictly increasing Cauchy sequence contained in for some ∈ θ A converges in A and the limit of the sequence is a strict upper bound of the sequence, i.e., if { } ζ n is a strictly increasing Cauchy sequence contained in We present some examples in support of Definition 2.2.
2, 3 be endowed with usual metric and ≼ be defined as the natural ordering ≤.
, , is f -orbitally complete with respect to A.
X , 1 be endowed with the usual metric and ≼ be the natural ordering ≤. Take , , is f -orbitally complete with respect to A.
Before going to the main results of this section, we present some examples to support our claim of ensuring the existence of a fixed point in cases where known results are not applicable.
0, 2 be endowed with the usual metric and ≼ be the natural ordering ≤. Define → f X X : by Clearly, X is not f-orbitally complete. It can be verified by taking = ζ 2 3 and the sequence be endowed with the usual metric and ≼ be the natural ordering ≤. Define → f X X : by , then X is f-orbitally complete with respect to A.
The following result is useful enough in the context of partial b-metric spaces. for all ∈ n m , with < n m.
Continuing the process, we get a strictly increasing sequence { } ζ n in A so that . For this, take ∈ n m , with < n m. From Lemma 2.1, one writes Now, by using (5) and (4), we get That is, 3 Fixed point theorems in partially complete spaces Still in the direction that the ordered partial b-metric space is not complete, we introduce the concept of partial completeness with respect to a subset A.
is said partially complete with respect to A, if every strictly increasing Cauchy sequence in A has a strict upper bound in A, i.e., if { } ζ n is a strictly increasing Cauchy sequence in A, then there exists ∈ ζ A such that ≺ ζ ζ, n for each ∈ n .
Example 3.1. Endow = X with the usual metric of and the natural order ≤. Note that ( ≤) X d , , is an ordered partial b-metric space (with = s 1), but it is not complete. However, if we take is partially complete with respect to A. Here, 3 is the strict upper bound for every strictly increasing Cauchy sequence in A.
The main objective of this concept is to present some fixed point results in noncomplete metric spaces. Here, we have dropped the contraction condition prescribed by Banach.
X, be a totally ordered set. Then, every ordered complete metric space ( ≼) X d , , is partially complete with respect to A, where A is any closed subset of X. The converse of the above statement is not true in general. It can be seen by taking = ( ) X 0, 5 and = ( ] A 1, 3 . With the usual metric and the natural ordering ≤, X is partially complete with respect to A, but it is not complete. Also, if we take 1, 3 , then X is still not complete, but it is partially complete with respect to A.
Theorem 3.1. Let f be a self-map on an ordered partial b-metric space is partially complete with respect to A, then f has a fixed point in A.
Proof. Going through the same lines of proof of Theorem 2.1, we get a strictly increasing Cauchy sequence { } ζ n in A such that , .
n n n 0 0 is partially complete with respect to A, there is ∈ ζ A such that ≺ ζ ζ n , for each ∈ n . Thus, from (9) and (11), we get is an ordered partial b-metric space (with = s 2), which is not complete. Consider → f X X : by Since X is not a complete partial b-metric space, we cannot apply the previous results. Take is partially complete with respect to A. Therefore, it remains to prove that (9) is satisfied. Let ∈ ζ ξ A , be such that < ζ ξ. We  d ξ f ξ  ζ  ξ  ξ  ζ  ,  ,  1 3  4   1 3  2   6  3  1  4 0.
Hence, all the conditions of Theorem 3.1 hold and f has a fixed point in A, which is, = u 1.
m i n , and ≼ be the natural order ≤. Note that ( ≼) X d , , is an ordered partial b-metric space (with = s 2), which is not complete. Take A a a n a : , 1 50 is partially complete with respect to A. Therefore, it remains to prove that (9) is satisfied. Let ∈ ζ ξ A , such that < ζ ξ. We have ( ( )) = − d ζ f ζ ,

Conclusion
We proved some fixed point theorems with a new approach. In this direction, we introduced partial completeness, f -orbitally completeness, a new contraction type and other notions. The aim was to find the existence of a fixed point in the cases where many known results cannot work. The proof for uniqueness of a fixed point needs further explorations. It would be very interesting to analyze the existing literature in light of the results discussed in this paper. Particularly, if we replace Banach contraction by other newly introduced contractions, we may get further interesting results.