L
 -Simulation Functions over 
 b
 -Metric-Like Spaces and Fractional Hybrid Differential Equations

<jats:p>In this paper, we establish some fixed point results for <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3">
                        <msub>
                           <mrow>
                              <mi>α</mi>
                           </mrow>
                           <mrow>
                              <msup>
                                 <mrow>
                                    <mi>q</mi>
                                    <mi>s</mi>
                                 </mrow>
                                 <mrow>
                                    <mi>p</mi>
                                 </mrow>
                              </msup>
                           </mrow>
                        </msub>
                     </math>
                  </jats:inline-formula>-admissible mappings embedded in <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4">
                        <mi mathvariant="script">L</mi>
                     </math>
                  </jats:inline-formula>-simulation functions in the context of <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5">
                        <mi>b</mi>
                     </math>
                  </jats:inline-formula>-metric-like spaces. As an application, we discuss the existence of a unique solution for fractional hybrid differential equation with multipoint boundary conditions via Caputo fractional derivative of order <jats:inline-formula>
                     <math xmlns="http://www.w3.org/1998/Math/MathML" id="M6">
                        <mn>1</mn>
                        <mo><</mo>
                        <mi>α</mi>
                        <mo>≤</mo>
                        <mn>2</mn>
                     </math>
                  </jats:inline-formula>. Some examples and corollaries are also considered to illustrate the obtained results.</jats:p>


Introduction
Fixed point theory has received much attention due to its applications in pure mathematics and applied sciences. Generalization of this theory depends on generalizing the metric type space or the contractive type mapping. The concept of metric spaces has been extended in various directions by reducing or modifying the metric axioms. Since, losing or weakening some of the metric axioms causes loss of some topological properties, hence bringing obstacles in proving some fixed point theorems. These obstacles force researchers to develop new techniques in the development of fixed point theory in order to resolve more real concrete applications.
In 1989, Bakhtin [1] (and also Czerwik [2]) introduced the concept of b − metric spaces and presented a generalization of Banach contraction principle. Amini-Harandi [3] introduced the notion of metric-like spaces which play an important role in topology and logical programming. In 2013, Alghamdi et al. [4] generalized the notions of b − metric and metric-like spaces by introducing a new space called b − metric-like space and proved some related fixed point results. Recently, many results of fixed point of mappings under certain contractive conditions in such spaces have been obtained (see [5][6][7][8]).
Zoto et al. [9] introduced the concept of α qs p − admissible mappings and provided some fixed point theorems for these mappings under some new conditions of contractivity in the setting of b − metric-like spaces. Recently, Cho [10] proposed the notion of θL − contractions and confine his fixed point results for such contractions to generalized metric spaces. Aydi et al. [11] proved that those results are also valid in partial metric spaces.
Fractional calculus is a field of mathematics that deals with the derivatives and integrals of arbitrary order. Indeed, it is found to be more realistic in describing and modeling several natural phenomena than the classical one. In recent years, many researchers have focused on joining fixed point theory with fractional calculus, see for example [12][13][14][15].
The study of differential equations with fractional order has attracted many authors because of its intensive development of fractional calculus itself and its applications in various fields of science and engineering, see [16][17][18][19].
On the other hand, hybrid differential equations have attracted much attention after the pioneering works appeared in [20,21] which discussed main aspects about first-order hybrid differential equations with perturbations of 1st and 2nd types, respectively.
In [23], Derbazi et al. applied Dhage hybrid fixed point theorem [28] to provide sufficient conditions that guarantee the existence only of solutions for a class of FHDEs with three-point boundary conditions due to Caputo fractional derivative of order 1 < α ≤ 2 in Banach algebra spaces.
Inspired by the above works, we investigate the existence of a unique fixed point for α qs p -admissible mapping via L -simulation function ξ : ½1,∞Þ × ½1,∞Þ → ℝ and control function θ : ð0,∞Þ → ð1,∞Þ in more general setting (b-metric-like space) than partial metric, b − metric and metric-like spaces. Also, as an application, we provide appropriate conditions that guarantee the existence of a unique solution to the following FHDE.

Basic Concepts
In order to fix the framework needed to state our main results, we recall the following notions.
Definition 1 [1]. Let X be a nonempty set and s ≥ 1 be a given real number. A function d : for all x, y, z ∈ X, the following conditions are satisfied.
The pair ðX, dÞ is called a b − metric space, and s is the coefficient of it.
Note that, every metric space is a b − metric space with coefficient s = 1.
It should be noted that σ satisfies all of the conditions of a metric except that σðx, xÞ may be positive for x ∈ X.
Definition 3 [4]. A function σ b : X × X → ½0,∞Þ on a nonempty set X is called b − metric-like if for any x, y, z ∈ X, the following conditions hold true.
Remark 4. The class of b − metric-like spaces is considerably larger than both b − metric spaces and metric-like spaces.
Since, every b − metric is a b − metric-like with same coefficient and zero self-distance. Also, every metric-like is a b − metric-like with s = 1. However, the converse implications do not hold (see for example, [1,4]).
Example 5. Let X = R and σ b : X × X → ½0,∞Þ be defined as Definition 7. Let ðX, σ b Þ be a b − metric-like space and fx n g be a sequence in X, and x ∈ X. Then, is called an open ball with center x and radius r. Also, the family forms a base of the topology τ σ b generated by σ b on X.
exists and is finite.
Then, every subsequence fx n k g with n k ≥ k ∈ ℕ converges to the same limit x ∈ X.
Definition 10 [29]. Let Θ be the set of all functions θ : ð0,∞ Þ → ð1,∞Þ that fulfil: (θ 1 ) θ is non-decreasing (θ 2 ) For any sequence fu n g ∈ ð0,∞Þ Theorem 11 [10]. Let ðX, dÞ be a complete generalized metric space and T : X → X satisfy Then, T has a unique fixed point, and for every initial point x 0 ∈ X, the Picard sequence fT n x 0 g converges to that fixed point.
Definition 12 [9]. Let ðX, σ b Þ be a b − metric-like space with parameter s ≥ 1, α : X × X → ½0,∞Þ be a function, and q ≥ 1 and p ≥ 2 be arbitrary constants. A mapping T : In addition, T is said to be triangular α qs p − admissible if it is α qs p − admissible and α x, y ð Þand α y, Ty ð Þ≥ qs p ⇒ α x, Ty ð Þ≥ qs p ,∀x, y ∈ X: ð18Þ Definition 13 [18,19]. The Riemann-Liouville fractional integral of order α > 0 of a function x : ½0,∞Þ → ℝ is given by The Caputo fractional derivative of order α of x is given by where n = ½α + 1 and Γ denote the gamma function, provided that the right side is point-wise defined on ½0, ∞Þ.

A Set of Fixed Point Results
Our first main result is the following theorem.
Journal of Function Spaces α qs p − admissible mapping and satisfy Consider that the following properties hold true (a) If fx n g is a sequence in X such that x n → x ∈ X as n → ∞ and αðx n , x n+1 Þ ≥ qs p , then αðx n , xÞ ≥ qs p , ∀n ∈ ℕ (b) For all x, y ∈ FixðTÞ, we have αðx, yÞ ≥ qs p , where Fi xðTÞ denotes the set of fixed points of T Moreover, if there exists x 0 ∈ X such that αðx 0 , Tx 0 Þ ≥ qs p , then T has a unique fixed point.
Proof. Starting with that point x 0 ∈ X : αðx 0 , Tx 0 Þ ≥ qs p . We define a sequence fx n g ⊂ X by Regarding that T is an α qs p − admissible, then by induction, we get If σ b ðx n , x n+1 Þ = 0 for some n, then x n = x n+1 , that is, x n is a fixed point of T, and the proof is completed. So, we assume that From (23) and (24), we apply (21) at x = x n−1 and y = x n to get Hence, the sequence fθðσ b ðx n , x n+1 ÞÞg is monotone decreasing and bounded below by 1. Therefore, there exists m ≥ 1 such that To prove that m = 1, suppose the contrary that m > 1 and obtain a contradiction. From (25), (26), and ðξ 3 Þ, we have that is all we need. Thus, m = 1 and Also, ðθ 2 Þ implies Now, we show that lim n,m→∞ Consider the sequence It is easy to verify that Hence, the sequence fR k g is decreasing and bounded below by 1. Consequently, there exists r ≥ 1 such that Assume that r > 1, then from (31), we conclude that Taking limit as k → ∞, together with (33), implies Also, we have Again, taking limit as k → ∞, together with (33), (35) implies According to (23) and the fact that T is a triangular α qs p − admissible, we derive On account of the above observations, we apply 4 Journal of Function Spaces condition (21) and then (ξ 3 ) to obtain Therefore, we have And hence which is a contradiction, then r = 1 and lim n,m→∞ Thus, (30) holds true, and the sequence fx n g is σ b − Cauchy. By the completeness of ðX, σ b Þ, there exists x ∈ X such that Now, consider the subsequence fx n k g of the sequence fx n g such that Lemma 8, together with (43), imply that Apply (21), we obtain Hence, From (45), (47), and ðσ 3 Þ, we conclude that To see that this fixed point in unique, suppose that y ∈ X is another fixed point of T and apply (21) to get the opposite.
This is impossible, so x = y, and the fixed point is unique.
Let R denote the class of β : ½0,∞Þ → ½0, 1Þ which satisfies the condition Remark 16. Since b − metric-like space is a proper extension of partial metric, metric-like, and b − metric spaces. Then, we can derive our main results in the setting of these spaces.

Corollary 17.
Let ðX, σÞ be a complete metric-like space and T : X → X be a mapping such that Then, T has a unique fixed point.
Proof. For all x, y ∈ X with x ≠ y and Tx ≠ Ty, condition (51) can be written as Therefore, Corollary 17 follows from Theorem 15 by taking for all x, y ∈ X, t ∈ ð0,∞Þ, and u, v ∈ ð1,∞Þ.

Fractional Hybrid Differential Equations
Here, we place our considered problem (1) in the space Journal of Function Spaces with a mapping σ b : X × X → ℝ + ∪ f0g defined on it as: For convenience, we define the following functions ρ i : J → ℝ, i = 1,2,3,4: Lemma 18. Let h ∈ CðJÞ, then the integral representation of the boundary value problem where 1 < α ≤ 2, 0 < β ≤ 1, 0 < η i < T and ζ i , i = 1; 2; 3, ⋯, m + 1, m ∈ ℕ are real constants such that is given by where Gðt, sÞ is the Green function and is given by Proof. Applying the operator I α on both sides of (57) and using Lemma 14, we have Using the boundary conditions (58), we get Note that, Solving (63) and (64) for c 0 , c 1 yields Journal of Function Spaces Substituting the values of c 0 and c 1 into (62), we get Provided the functions ρ i ðtÞ, i = 1,2,3,4 are defined as in Eq. (56), we obtain One can easily verify that Thus, for 0 ≤ t ≤ η 3