Related Fixed Point Theorems via General Approach of Simulations Functions

. In this work, we extend and complement some results in view of general and wider structures, such as b − metric spaces. By considering existing classes of Ζ − contractions and Ψ − simulating functions with a solid impact in database results of ﬁxed point theory, we introduce a new general class of simulating functions, called as Ψ − s simulation functions, and also types of κ ψ − s − contractions in a more general framework. This approach covers, extends, and uniﬁes several published works in the early and late literature.


Introduction
Some of the significant generalizations of metric fixed point theory are related with the well-known Banach Contraction Principle [1] and classical contractions such as Boyd and Wong, Geraghty, Browder, and Ciric.In recent years, the theory of fixed points has attracted widespread attention and has been rapidly growing.It was massively studied by many researchers giving new results by using classes of implicit functions defining new and large contractive conditions.Recently, Khojasteh et al. [2] presented the notion of Ζ−contractions involving a new class of simulation functions that has been used and improved by many authors in various spaces, see .Authors in [19] proposed new notion Ψ−simulation functions and established the type of Ζ ψ −contractions.
Inspired by the above works, in this paper we introduce a new class of general type of Ψ − s simulation functions, defined in the setting of b−metric-like spaces.
is class generalizes further and complements some results given in the framework of b−metric spaces.

Preliminaries
Definition 1 (see [6]).Let X be a nonempty set and s ≥ 1 be a given real number.A mapping d: X × X ⟶ [0, +∞) is called a b−metric-like if for all x, y, z ∈ X, the following conditions are satisfied: (1) e pair (X, d) is called a b−metric-like space.In a b−metric-like space (X, d), if x, y ∈ X, and d(x, y) � 0, then x � y; however, the converse need not be true, and d(x, x)may be positive for x ∈ X. Definition 2 (see [6]).Let (X, d) be a b−metric-like space with parameter s ≥ 1 and let x q   be any sequence in X and x ∈ X. en, we have the following: (a) x q   is said to be convergent to x if lim q⟶+∞ d(x q , x) � d(x, x) (b) x q   is said to be a Cauchy sequence in (X, d) if lim q,p⟶+∞ d(x q , x p ) exists and is finite (c) e pair (X, d) is called a complete b−metric-like space if, for every Cauchy sequence x q   in X, there is x ∈ X such that lim q,p⟶+∞ d(x q , x p ) � lim q⟶+∞ d(x q , x) � d(x, x) Lemma 1 (see [6,29,30]).Let x q   and y q   be two sequences in (X, d) that converge to x and y, respectively.en, we have In particular, d(x, y) � 0⟹ lim q⟶+∞ d(x q , y q ) � 0. Also, for each z ∈ X, the above inequality becomes In particular, if, d(x, x) � 0, then Lemma 2 (see [23]).Let x q   be a sequence in theb-metriclike space (X, d) with parameter s ≥ 1, such that If lim q,p⟶+∞ d(x q , x p ) ≠ 0, then there are ε > 0 and two sequences of natural numbers p(k), q(k) with q k > p k > k, (positive integers) such that Note: in the continuous section of the paper, we will use (X * , d, s) (resp.(X, d, s)) to denote that the space with parameter s ≥ 1 is complete (resp.noncomplete).

Main Results
Let (X * , d, s) be a b-metric-like space and Ψ([0, +∞)) represent the collection of continuous functions ψ: [0, +∞) ⟶ [0, +∞) with the following properties: If in the definition above we take s � 1, then we obtain the definition of a Ψ− simulation function.
If we take ψ as the identity function, then we get a definition of an s− simulation function.If we take s � 1 and ψ(v) � v, then we get the definition of a simulation function.
We denote by K ψ−s the set of all Ψ − s simulation functions.In the following example, we give such a kind of functions.
Example 1.Let κ: [0, +∞) 2 ⟶ R be defined by (1) where ϕ: for all x, y ∈ X, where A(x, y), is defined as in (8), then the self-map f has a unique fixed point in X.
Proof.Let x 0 ∈ X be an arbitrary element.Define a sequence x q   in X such that ∀q ∈ N ∪ 0 { }, x q+1 � f(x q ).If d(x q , x q+1 ) � 0 for some q ∈ N ∪ 0 { }, that is, x q+1 � x q and x q � x q+1 � f(x q ); therefore, x q is a fixed point of f. us, suppose that d(x q , x q+1 ) > 0 for all q ∈ N ∪ 0 { }.Considering the set A(x, y), we have we obtain using (10), By the supposition d(x q , x q+1 ) > 0 and ( 12), we get en, applying condition (9) and property κ 1 , we have for all q ∈ N 0 at is, a contradiction.erefore, From ( 9) and using ( 14), we obtain In view of property of (ψ 1 ), the above inequality gives d(x q , x q+1 ) < d(x q−1 , x q ) for all q ∈ N. Hence, d(x q , x q+1 )   is a decreasing sequence of nonnegative reals, so there is l ≥ 0 so that d(x q , x q+1 ) ⟶ l.Also, by (14), Suppose that l > 0, then lim q⟶+∞ d(x q , x q+1 ) � lim q⟶+∞ A(x q−1 , x q ) � l > 0. By property (κ 2 ), we have which is a contradiction.erefore, l � 0. Hence, Next, we show that lim q,p⟶+∞ d(x q , x p ) � 0. Suppose, to the contrary, that is, lim q,p⟶+∞ d(x q , x p ) > 0, then by Lemma 2, there are ε > 0 and sequences p k   and q k   of positive integers with q k > p k > k such that From the definition of A(x, y), we have By the upper limit k ⟶ +∞ in (20) and keeping in mind (18)(19), we obtain Also, from condition κ 1 , we have which by property of (ψ 1 ) implies By taking upper limit on both sides of (23) in view of ( 19) and (21), it follows that εs λ < εs, (24) which contradicts ε > 0. us, lim q,p⟶+∞ d(x q , x p ) � 0 and the sequence For elements ω and x q , we consider 4 Journal of Mathematics By Lemma 1 together with ( 18) and ( 25), it follows by passing in the upper limit of (26): Now, using the κ 1 condition, we have which implies Taking the limit superior in (29) and by Lemma 1 and inequality ( 27), we obtain By (30), it follows that d(ω, fω) � 0, and so fω � ω.
Suppose ω, y ∈ X are two different fixed points of f.By ( 14 From condition (9) and property κ 1 , we have which is a contradiction.erefore, d(ω, y) � 0 and ω � y. us, there is a unique fixed point of f.
e pair (X, d) is a b− metric-like space with coefficient s � 2. We claim that the mapping f satisfies the contraction type condition (8): Case 2. For x � y � 1, we note Here, 0 is the unique fixed point of f.Some applications of eorem 1 are the following corollaries.
Corollary 1.Let f: X ⟶ X be a mapping on a b−metriclike space (X * , d, s).Suppose that there are ψ ∈ Ψ and λ ≥ 1 such that for all x, y ∈ X, where A(x, y) is defined as in (8).en, the self-map f has a unique fixed point in X.
Proof.In eorem 1, take into account the function  A(x, y)) for all x, y ∈ X, where A(x, y) is defined as in (8).en, the self-mapf admits a unique fixed point in X.
Proof.In eorem 1, take into account the function □ Corollary 3. Let f: X ⟶ X be a mapping on a b−metriclike space (X * , d, s).Suppose that there are ψ ∈ Ψ, α ∈ (0, 1) and λ ≥ 1 such that for all x, y ∈ X, where A(x, y) is defined as in (8).en, the self-map f has a unique fixed point in X.

□
Corollary 4. Let f: X ⟶ X be a mapping on a b−metriclike space (X * , d, s).Suppose that there are ψ ∈ Ψ, λ ≥ 1, and ϕ: R for all x, y ∈ X, where A(x, y) is defined as in (8).en, the self-mapf has a unique fixed point in X.

□
Corollary 5. Let f: X ⟶ X be a mapping on a b−metriclike space (X * , d, s).Suppose that there are ψ ∈ Ψ, λ ≥ 1, F: R + × R + ⟶ R a C-class function and φ: R + ⟶ R + a continuous function, such that for all x, y ∈ X, where A(x, y) is defined as in (8).en, the self-map f has a unique fixed point in X.
Proof.In eorem 1, take into account the function □ Remark 2. Corollary 5 is much wider because condition (43) includes many other contractive conditions.Corollary 6.Let f: X ⟶ X be a mapping on a b−metriclike space (X * , d, s).Suppose that there exist a function φ: [0, +∞) ⟶ [0, +∞) with lim inf t⟶v φ(t) > 0 for all v > 0, and some constant λ ≥ 1 such that for all x, y ∈ X, where A(x, y) is defined as in (8).en, the self-map f has a unique fixed point in X.
In the following result, we include two mappings f and g in the set for all x, y ∈ X, where E(x, y) is denoted by (45); then, the mappings f and g have a unique common fixed point in X.
en, from Lemma 2, there are ε > 0 and two subsequences p k   and q k   of positive integers, with From (45), we note Hence, by (54)-(56), and Lemma 2, we have By (46) and using properties (ψ 1 ), (κ 1 ), we have Journal of Mathematics which leads to Hence, by (55), (57), and (58) and taking the upper limit, we obtain which implies that ε � 0, a contradiction with ε > 0. It remains that lim q,p⟶+∞ d(x q , x p ) � 0; therefore, x q   is a Cauchy sequence in X.Since (X * , d, s) is a complete b-metric-like space, there is ω ∈ X such that x q   is convergent to ω, that is, which implies Taking the upper limit as q ⟶ +∞ and using Lemma 1 and (64), we have s λ− 1 d(ω, gω) < d(ω, gω), that is, d(ω, gω) � 0 and ω is a fixed point of g.Similarly, we can get d(fω, ω) � 0 and so ω is a common fixed point for mappings f and g.Suppose ω, δ ∈ X are two different common fixed points of f and g such that d(ω, δ) > 0. en,  for all x, y ∈ X, where E(x, y) is defined as in (45).
en, the self-mappings f and g have a unique common fixed point in X.

□ Remark 3.
e above theorem reduces to a one mapping if we put g � f.Further corollaries can be stated for s � 1, either by taking the function ψ as an identity function or by taking different functions κ ∈ K ψ−s as listed in Corollary 1-6.

Conclusion
In this work, we established common fixed point results for one and two mappings on a b−metric-like space which overcomes and unifies classical and previous results developed in papers [19][20][21][22][23][24][25][26][27][28].e considered set of generalized contractive mappings contains the families of many contractions as a proper subset.We remark based on Example 2/(4) which are functions of C−class used by many researchers and taken as a special case of Ψ − s simulation functions.By using additional set of functions Ψ, ϕ, coefficient λ, and parameter s, the rich class of Ψ − s simulation functions make it possible to collect, extend, and complement previously existing results related to a variety types of contractions.In terms of Ψ − s simulating functions, many classical and still recent contractions take a simple form as κ(d(fx, fy), A(x, y)) ≥ 0 not including other additional symbols and long formulas.
is wide approach reflects a wide work and an unifying power for more general theorems made on the theory of fixed points.