Some New Observations on Generalized Contractive Mappings and Related Results in b -Metric-Like Spaces

In this paper, we consider, discuss, complement, improve, generalize, and enrich some ﬁxed point results obtained for ( β − ψ 1 − ψ 2 )− contractive conditions in ordered b -metric-like spaces. By using our new approach for the proof that one Picard’s sequence is b bl − Cauchy in the context of b -metric-like spaces, we get much shorter proofs than the ones mentioned in the recent papers. Also, by the use of our method, we complement and enrich some common ﬁxed point results for β s, ψ q, ϕ − contraction mappings. Our approach in this paper generalizes and modiﬁes several comparable results in the existing literature.


Introduction
Fixed point theory is one of the most important areas of nonlinear analysis. At the beginning of the development, this part of analysis was related to the use of successive approximation in order to prove the existence and uniqueness of the solution of differential and integral equations. Later on, it is applied in various fields such as economics, physics, chemistry, differential and integral equations, partial differential equations, numerical analysis, and many others. Banach's contraction principle in metric spaces [1] is one of the most important results in fixed point theory and nonlinear analysis in general. In 1922, when Stefan Banach formulated the concept of contraction and proved the famous theorem, scientists around the world started publishing new results that are related either to the generalization of the contractive mapping such as Kannan, Chatterjea, Hardy-Rogers,Ćirić, and many others or by generalizing space itself. By changing some axioms of ordinary metric space, new classes of so-called generalized metric spaces were obtained such as partial metric space, metric-like space, b−metric space, b−metric-like space, and others. For more details about fixed point theory in metric as well as generalized metric spaces, we encourage readers to see [2][3][4][5][6][7][8][9].
In each of them, Banach's well-known theorem is true in b-metric and b−metric-like spaces regardless of the magnitude of the coefficient s in the triangle relation to each.
In [10], Matthews introduced the notion of a partial metric space where nonzero self-distance is considered, which has found great application in computer science. e second important generalization of metric spaces is so-called b−metric spaces. is concept was introduced by Bakhtin [11] and Czerwik [12] where the third axiom of metric spaces, referring to triangular inequality, weakened.
Furthermore, Amini Harandi [13] introduced the notion of metric-like space, as a generalization of a partial metric space, where all of the axioms of a metric is satisfied except that self-distance may be positive.
In [14], the concept of b−metric-like space which generalizes the notions of partial metric space, metric-like space, and b-metric space is introduced.
Relations of the metric spaces and mentioned generalizations are illustrated as follows [15,16]:
Definition 1 (see [13]). Let X be a nonempty set. A mapping b ml : X × X ⟶ [0, +∞) is said to be metric-like if the following conditions hold for all x, y, z ∈ X: In this case, the pair (X, b ml ) is called a metric-like space.
Definition 2 (see [14]). Let X be a nonempty set and s ≥ 1 be a given real number. A b−metric-like space on a nonempty set X is a function b bl : X × X ⟶ [0, +∞) if the following conditions hold for all x, y, z ∈ X: In this case, the pair (X, b bl ) is called a b−metric-like space with the coefficient constant s ≥ 1.
It is clear that each metric-like space is b−metric-like space, while the converse is not true. For more such examples and details see [9,13,14,20,[24][25][26][27]. Also, for various metrics, but in the context of complex domain, see [28,29].
Now, we give the definition of convergence of the sequences in b-metric-like space.
Definition 4 (see [14]). Let x n be a sequence in a b−metriclike space (X, b bl ) with the coefficient s.
Properties such as convergence, completeness, and Cauchyness are introduced in the same way for metric and b-metric spaces. And, in these two types of spaces, the limits of the sequence if exists, is unique, as well as the convergent sequence is a Cauchy. Otherwise, for the other 4 types of space (partial metric, partial b-metric, metric-like, and bmetric like), this is not the case.

Remark 1.
In a b−metric-like space, the limit of a sequence need not be unique and a convergent sequence need not be a b bl −Cauchy sequence (see Example 7 in [20]). However, if the sequence (1) Definition 5. Let X be a nonempty set and suppose f: X ⟶ X and β: e next definition and the corresponding proposition are important in the context of fixed point theory.
e self-mappings f, g: Proposition 1 (see [31]). Let f and g be weakly compatible self-maps of a nonempty set X. If they have a unique point of coincidence w � f(u) � g(u), then w is the unique common fixed point of f and g.
In this paper, we shall use the following result for the proof that some Picard's sequence is b bl −Cauchy. e proof is completely identical with the corresponding in [32] (see also [33][34][35]).
Otherwise, many authors for the proof that some sequence in b−metric-like space is b bl −Cauchy use the next lemma.
Lemma 2 (see [26]). Let (X, b bl ) be a b−metric-like space with the coefficient s > 1 and assume that u n ⟶ u and v n ⟶ v as n ⟶ ∞. en, we have In the case that the coefficient s � 1, the given b−metric like space (X, b bl ) becomes the metric-like space (X, b ml ). If x n is a given sequence in the metric-like space (X, b ml ), then we have the following very useful result.
Lemma 3 (see [23,36]). Let (X, b ml ) be a metric-like space and let x n be a sequence in it such that lim n⟶∞ b ml (x n , x n+1 ) � 0. If x n is not a b ml −Cauchy sequence in (X, b ml ), then there exist ε > 0 and two sequences m k and n k of positive integers such that m k > n k > k and the following four sequences tend to ε + when k ⟶ ∞:

Main Results
In [37], the authors introduced two new types of contractive mappings, namely, (β − ψ 1 − ψ 2 ) contractive mappings of type-I and of type-II in ordered b−metric-like spaces.

contractive mapping of type-I. Suppose that the following conditions hold:
(1) f is β−admissible and L β −admissible (or R β -admissible) (2) ere exists u 1 ∈ X such that u 1 ≾ f(u 1 ) and β(u 1 , f(u 1 )) ≥ 1 (3) f is nondecreasing with respect to ≾ (4) If u n is a sequence in X such that u n ≾ u n+1 and β(u n , u n+1 ) ≥ 1 for all n ∈ N, and u n ⟶ u ∈ X, as n ⟶ ∞, then u n ≾ u and β(u n , u) ≥ 1 for all n ∈ N en, f has a fixed point.
Similar to eorem 2, the authors proved the following result.
If s � 1, then the given contractive conditions in all eorems in [37] imply that the corresponding Picard's sequence is b ml −Cauchy according to Lemma 1. In [20], the authors introduced the so-called Definition 9 (see [20], Definition 5). Let (X, b bl ) be a b−metric-like space with the coefficient s ≥ 1. Let the constant q ≥ 2 and β ∈ [0, 1). e nonlinear self-mappings f, g: X ⟶ X are called β s,ψ q,ϕ −contraction mappings if for all x, y ∈ X.

Journal of Mathematics
Theorem 5 (see [20], eorem 1). Let (X, b bl ) be a b bl −complete b−metric-like space with the coefficient s ≥ 1, and f, g: X ⟶ X be mapping satisfying the following conditions: pair (f, g) is a β s,ψ q,ϕ −contraction (iii) en, f and g have a point of coincidence in X (iv) Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X Remark 3. Some important remarks regarding the previous definition and theorem, namely, it is worth to notice that, for the set M bl (x, y) of real numbers from (11), we have M bl (x, y) ≤ max b bl (g(x), g(y)), b bl (g(x), f(x)), b bl (g(y), f(y)) . (13) Indeed, this follows from the known estimation: erefore, instead of the set M bl (x, y) in [20], we introduce the next set: Now, we give the new formulation and the proof of eorem 1 from [20] taking the set N bl (x, y) instead of M bl (x, y). First, suppose that the pair (f, g) has at least one point of coincidence ω 1 . If it has other point of coincidence for example ω 2 ≠ ω 1 , this means that there are two points u 1 ≠ u 2 from X such that ω 1 � fu 1 � gu 1 and ω 2 � fu 2 � gu 2 .
Step 2. Existence of the point of coincidence. Let x 0 be an arbitrary point in X. Since f(X) ⊂ g(X), there exists x 1 ∈ X such that fx 0 � gx 1 . By continuing this process inductively, we get two sequences x n and z n in X such that ) is a unique point of coincidence for the pair (f, g). Furthermore, let b bl (z n , z n+1 ) > 0, for all n ∈ N. Now, we shall prove that b bl z n , z n+1 ≤ λb bl z n−1 , z n , (19) for some λ ∈ (0, (1/s)). Indeed, according to (11) where instead of M bl , we take N bl , if x � x n , y � x n+1 , we have Journal of Mathematics b bl z n , z n+1 ≤ 2s q b bl z n , z n+1 ≤ N bl x n , x n+1 , (20) where N bl x n , x n+1 � max b bl g x n , g x n+1 , b bl g x n , f x n , b bl g x n+1 , f x n+1 , b bl g x n , f x n+1 + b bl g x n+1 , f x n 4s � max b bl z n−1 , z n , b bl z n−1 , z n , b bl z n , z n+1 , b bl z n−1 , z n+1 + 0 4s � max b bl z n−1 , z n , b bl z n , z n+1 , b bl z n−1 , z n+1 4s ≤ max b bl z n−1 , z n , b bl z n , z n+1 , b bl z n−1 , z n + b bl z n , z n+1 4 ≤ max b bl z n−1 , z n , b bl z n , z n+1 , b bl z n−1 , z n + b bl z n , z n+1 2 ≤ max b bl z n−1 , z n , b bl z n , z n+1 .
Let, for example, the subset f(X) be closed. e proof if g(X) is closed is similar. en, in the first case, there is a unique point u ∈ f(X) such that z n converges to u. Since f(X) ⊂ g(X), there exists v ∈ X such that g(v) � u. We shall show that f(v) � g(v) � u. For this proof, we firstly have 1 s b bl (u, f(v)) ≤ b bl u, z n + b bl f x n , f(v) ≤ b bl u, z n + 1 2s q N bl x n , v , N bl x n , v � max b bl g x n , g(v) , b bl g x n , f x n , b bl (g(v), f(v)), b bl g(v), f x n + b bl g x n , f(v) 4s � max b bl z n−1 , u , b bl z n−1 , z n , b bl (u, f(v)), b bl u, z n + b bl z n−1 , f(v) 4s ≤ max b bl z n−1 , u , b bl z n−1 , z n , b bl (u, f(v)), b bl u, z n + sb bl z n−1 , u + sb bl (u, f(v)) 4s ⟶ b bl (u, f(v)) as n ⟶ ∞.
Letting the limit in (23) as n ⟶ ∞, we obtain 1 is a unique point of coincidence. e result further follows by Proposition 1 from [31].
is completes the proof of eorem 6.
As corollaries of our eorem 6, we obtain the next results: Corollary 1. Let (X, b bl ) be a b bl −complete b−metric-like space with the coefficient s ≥ 1, and f, g: X ⟶ X be a weakly compatible mappings satisfying the following conditions: (i) f(X) ⊂ g(X) and at least one of f(X), g(X) is a closed subset in the space (X, b b ) (ii) Assume that ψ ∈ Ψ, ϕ ∈ Φ, β ∈ [0, 1) and q ≥ 2 such that the condition ψ 2s q b bl (f(x), f(y)) ≤ β ψ N bl (x, y) 1 + ϕ N bl (x, y) , (26) holds for all x, y ∈ X en, f and g have a unique common fixed point in X.

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