Fixed Point Results via 
 G
 -Function over the Complete Partial 
 b
 -Metric Space

In this paper, we consider an auxiliary function 
 
 G
 
 to combine and unify several existing fixed point theorems in the setting of the complete partial 
 
 b
 
 -metric space. We consider also some examples to support the observed main results.


Introduction and Preliminaries
The notion of the distance has been investigated and improved from the beginning of the mathematics sciences. The first formal definition was given by Hausdorff and Frechet under the name of metric spaces. The formal definition was extended, improved, and generalized in several ways. In this paper, we shall consider the combination of notions of partial metric space and b-metric space. Partial metric space, defined by Matthews [1,2] is the most economical way to calculate the distance in computer science. So, it is important in the setting of theoretical computer science. On the other hand, b-metric is the most interesting and real generalization of metric spaces; in this case, the triangle inequality is replaced by a modified version of triangle inequality.For more details on the advances of fixed point theory in the setting of b-metric spaces, see e.g. [13]- [27].
In this paper, we shall propose a fixed point theorem by using an auxiliary function G to combine, generalize, and unify several fixed point results in the setting of the complete partial b-metric spaces.
In [3], the authors proposed a new fixed point theorem in the setting of metric spaces.
Let X be a nonempty set.
for a given real number s ≥ 1 and for all x, y, z ∈ X the following conditions hold: The triplet ðX, b, s ≥ 1Þ is called a b-metric space.
(ii) A function ρ : X × X ⟶ ½0, ∞Þ is a partial metric on X if for all x, y, z ∈ X the following conditions hold: The pair ðX, ρÞ is said to be a partial metric space. Combining these two concepts, Shukla [4] introduced the notion of partial b-metric space as follows.
(iii) A function ρ b : X × X ⟶ ½0, ∞Þ is a partial b-metric on X if for all x, y, z ∈ X the following conditions hold: The triplet ðX, ρ b , s ≥ 1Þ is said to be a partial b-metric space.
On a partial b-metric space ðX, ρ b , s ≥ 1Þ a sequence fx n g is said to be (i) convergent to x ∈ X if lim n→∞ ρ b ðx n , xÞ = bðx, xÞ (the limit of a convergent sequence is not necessarily unique) (ii) Cauchy if lim n,m→∞ ρ b ðx n , x p Þ exists and its finite Moreover, the partial b-metric space is complete if for every Cauchy sequence fax n g there exists x ∈ X such that lim n,p→∞ Let ðX, ρ b , s ≥ 1Þ be a partial b-metric space. We say that a self-mapping T on X is continuous if for every sequence fx n g in X which converges to a point x ∈ X we have In [5], the authors introduced the following new notions.
Moreover, they proved that if the partial b-metric space For a better understanding of the connections between these spaces (partial metric space, b-metric space, and partial b-metric space), we mention some papers that can be consulted [6][7][8][9][10][11][12].

Main Results
The following is the main result of the paper.
Journal of Function Spaces for every x, y ∈ X . If T is continuous or ρ b is continuous, then T has a unique fixed point.
Proof. Starting with a point x 0 ∈ X, we consider the sequence fxg defined by x n = Tx n−1 , n ∈ ℕ. Without losing the generality, we can assume that for any n ∈ ℕ, we have bðx n , x n+1 Þ > 0. Indeed, on the contrary, if there exists a positive integer j 0 such that x j 0 = x j 0 +1 , we get that x j 0 is a fixed point of T, because due to the way the sequence was fxg defined, it follows that x j 0 = Tx j 0 . Moreover, using this remark, we can easily see that Again supposing that which is a contradiction. Taking x = x n and y = x n+1 in (8) we get There are two possibilities, namely, which leads us (since ϕðuÞ < u for any u > 0) to But, this is a contradiction, and then Therefore, by (11) and taking into account ð f 3 Þ, we have Consequently, for every n ∈ ℕ, we obtain Let p, m ∈ ℕ such that p < m. By applying the (triangletype inequality) ðρ b4 Þ, we have 3 Journal of Function Spaces and (17) leads us to where S n = ∑ n i=0 s i ϕ i ðGðρ b ðx 0 , x 1 Þ, γðx 0 Þ, γðx 1 ÞÞÞ. Keeping in mind ðϕ 2 Þ, we deduce that there exists S n ⟶ S as n ⟶ ∞, and from (18), we get Consequently, fx n g is a 0-Cauchy sequence in a 0complete partial b-metric space, and then there exists ς ∈ X such that lim p,m→∞ Moreover, by ð f 3 Þ together with (16), we have  (20), We claim that this point ς is in fact a fixed point of the mapping T. If the mapping T is continuous, then by (6), we have Thus, applying the triangle inequality ðρ 4 Þ, and together with (20) and (24), letting n ⟶ ∞, we get ρ b ðς, TςÞ = 0, that is, ς is a fixed point of T. Let assume now that ρ b is continuous, that is, lim n→∞ ρ b ðx n , TςÞ = ρ b ðς, TςÞ Replacing x by x n and y by ς in (8), we have (for every n ∈ ℕ) Letting n ⟶ ∞ and taking into account ð f 1 Þ, we have Consequently, Gðρ b ðς, TςÞ, 0, γðTςÞÞ = 0: But, taking ð f 3 Þ into account, we get which means ρ b ðς, TςÞ = 0: Thus, Tς = ς: As a last step, we claim that ς is the unique fixed point of T. Supposing on the contrary, that there exists another point υ ∈ X such that Tς = ς ≠ υ = Tυ. First of all, applying (8) with which implies that γðυÞ = γðTυÞ = 0. Let now x = ς and y = υ in (8). We have This is a contradiction. Therefore, ρ b ðς, υÞ = 0, that is, T admits a unique fixed point.

4
Journal of Function Spaces Theorem 3. Let ðX, ρ b , s ≥ 1Þ be a 0-complete partial b -metric space, a function γ ∈ Γ, G ∈ F , and a self-mapping T : X ⟶ X . If there exists ϕ ∈ Φ b such that for every x, y ∈ X , then T has a unique fixed point.
Proof. Of course, since the function Gðτ, υ, ωÞ = τ + υ + ω ∈ F, by Theorem 2, we have that the sequence fx n g defined as x n = Tx n−1 is convergent to a point ς ∈ X, and moreover, (22) and (23) hold. We claim that this point ς is a fixed point of T. For this purpose, by (31), for x = ς and y = ς, we get Letting n ⟶ ∞, in the above inequality and keeping in mind (19), (22), and (23), we get which is a contradiction. Therefore, ρ b ðς, TςÞ = 0, that is, Tς = ς: As in the previous theorem, supposing that there exists υ, another fixed point of T, by (31), we have which is a contradiction. Thus, γðυÞ = 0 and taking x = ς and y = υ in (31), we have But, this is a contraction, so ρ b ðς, υÞ = 0 which proves the uniqueness of the fixed point.
Let the mapping T : X ⟶ X be defined as Choosing ϕðuÞ = u/2 and γðuÞ = u, we have (We considered here max fx, yg = x: The case max fx, yg = y is similar.) Consequently, by Theorem 3, the mapping T admits a unique fixed point.

Conclusion
In this paper, we investigate the uniqueness and the existence of a fixed point for certain contraction via the G-function in one of the most general frames and complete the partial b -metric space. Regarding that the metric fixed point theory has a key role in the solution of not only differential equations and fractional differential equations but also integral equations, our results can be applied in these problems.

Data Availability
No data were used to support this study.

Conflicts of Interest
The authors declare that they have no competing interests.