The Existence of Fixed Points for a Different Type of Contractions on Partial b -Metric Spaces

The aim of this study is to present ﬁxed point results in the setting of partial b -metric spaces. A diﬀerent type of contractions is used to prove ﬁxed point results in the given space, which are real generalization of many well-known results. The readers are also provided with some very interesting examples to illustrate the feasibility of the proposed work.


Introduction and Preliminaries
e notion of metric space was initiated in 1906 by French Mathematician Frechet [1]. In metric fixed point theory, the Banach contraction principle [2] is one of the most fundamental tools to investigate the existence and uniqueness of solutions for contraction maps in a complete metric space. Since the inception of this principle, many authors studied fixed point theory vividly and enriched this field with different ideas. is classical result was generalized in different spaces, and different structures were attained using this topic and one may recall the existing notions, partial b-metric spaces [3], R-partial b-metric spaces [4], fuzzy cone b-metric spaces [5], G b -metric spaces [6], orthogonal partial b-metric spaces [7], orthogonal m-metric spaces [8], and several others. More details can be found in [9][10][11][12][13].
One important and extensively used generalization of metric space is the notion of b-metric spaces, which was introduced by Czerwik [14] and is defined as follows.
Definition 1 (see [14]). Let I be a nonempty set and s ≥ 1 be a given real number. A function b: I × I ⟶ [0, ∞) is referred as a b-metric if the following conditions hold for all σ, υ, ς ∈ I: e triplet (I, b, s) is called a b-metric space.
Example 1 (see [15]). Let I � 1, 2, 3, . . . { } ∪ ∞ and define b: Matthews [16], while working on networking, observed that a self-distance may not be zero, i.e., d(σ, σ) ≠ 0. A loop is a good example of this case. He not only generalized the classical Banach fixed point theorem but also established some convergence criterion in this setting to ensure existence of a fixed point. His further investigations led him to the introduction of partial metric space, which is defined as follows.
Definition 3 (see [16]). Let (I, ℷ) be a partial metric space and σ κ be a sequence in I. en, (ii) σ κ is referred as a Cauchy sequence if lim κ,j⟶∞ ℷ(σ κ , σ j ) exists and is finite. (iii) A partial metric space is complete if each Cauchy sequence converges in I satisfying e concept of partial metric space was further extended to partial b-metric space by Shukla [3] in 2014 by combining the partial metric space and b-metric space.
Definition 4 (see [3]). Let I be a nonempty set and s ≥ 1 be a given real number. A mapping ℷ b : I × I ⟶ [0, ∞) is referred as a partial b-metric if, for all σ, ς, υ ∈ I, Definition 5 (see [3]). Let (I, ℷ b , s) be a partial b-metric space and σ κ be a sequence in I. en, Remark 1 e converse implication does not hold in general. (ii) A partial b-metric is a generalization of a partial metric.

Preliminary Results on Partial b-Metric Spaces
Following [17], we state the following.
We claim that ψ is an For this, let σ � σ κ+k and σ � σ κ . en, On the contrary, ψ is not a Banach contraction. Indeed, In [11], the authors introduced the following assumption: Go back to Example 5. We proceed as follows. Let Definition 8. Let a n be a sequence in (0, ∞) and b n be a sequence in [0, ∞). We call b n ∈ O(a n ) if there exists C > 0 satisfying b n ≤ Ca n , for all n ∈ N.
Then, σ d is a Cauchy sequence.

On (γ, F)-Weak Contractions
We shall investigate the existence of a fixed point for (c, F)-weak contraction mappings in partial b-metric spaces. We shall also provide some examples in support of main results.
Definition 14. Let (I, ℷ b , s) be a partial b-metric space and c be defined above. e partial b-metric space is c-complete iff every Cauchy sequence σ κ in I satisfying c(σ κ , σ κ+1 ) ≥ 1 is convergent to σ ∈ I, for all κ ∈ N.
Remark 4. If (I, ℷ b , s) is a complete partial b-metric space, then (I, ℷ b , s) is a c-complete partial b-metric space, but not conversely.