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

Abstract

In this paper, we establish some fixed point results for -admissible mappings embedded in -simulation functions in the context of -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 . Some examples and corollaries are also considered to illustrate the obtained results.

1. 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 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 metric and metric-like spaces by introducing a new space called 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 [5–8]).

Zoto et al. [9] introduced the concept of admissible mappings and provided some fixed point theorems for these mappings under some new conditions of contractivity in the setting of metric-like spaces. Recently, Cho [10] proposed the notion of 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 [12–15].

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 [16–19].

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.

Fractional hybrid differential equations (in short, FHDEs) have been studied using Riemann-Liouville and Caputo fractional derivatives of order in many literatures, see [22–27].

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 in Banach algebra spaces.

Inspired by the above works, we investigate the existence of a unique fixed point for -admissible mapping via -simulation function and control function in more general setting (-metric-like space) than partial metric, 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. where and denote the Caputo fractional derivatives of orders and , respectively, , , , , , , are real constants such that

2. Basic Concepts

In order to fix the framework needed to state our main results, we recall the following notions.

Definition 1 [1]. Let be a nonempty set and be a given real number. A function is a metric if for all , the following conditions are satisfied.
(b₁)
(b₂)
(b₃)
The pair is called a metric space, and is the coefficient of it.

Note that, every metric space is a metric space with coefficient .

Definition 2 [3]. A metric-like space on a nonempty set is a function such that for all :
(σ₁)
(σ₂)
(σ₃)
Then, the pair is called a metric-like space.

It should be noted that satisfies all of the conditions of a metric except that may be positive for .

Definition 3 [4]. A function on a nonempty set is called metric-like if for any , the following conditions hold true.
(σ_b1)
(σ_b2)
(σ_b3)
The pair is called a metric-like space.

Remark 4. The class of metric-like spaces is considerably larger than both metric spaces and metric-like spaces. Since, every metric is a metric-like with same coefficient and zero self-distance. Also, every metric-like is a metric-like with . However, the converse implications do not hold (see for example, [1, 4]).

Example 5. Let and be defined as then, is a metric-like space with parameter .

Example 6. Let and . The function , defined by is a metric-like with constant , and so, is a metric-like space.

Definition 7. Let be a metric-like space and be a sequence in , and . Then, (1)The setis called an open ball with center and radius . Also, the family forms a base of the topology generated by on . (2) is said to converge to w.r.t. if(3) is said to be Cauchy ifexists and is finite. (4) is said to be complete if for every Cauchy sequence in , there exists such that

Lemma 8. Let be a metric-like space with parameter and be a convergent sequence in such that

Then, every subsequence with converges to the same limit .

Proof. Since and , then for a given From and (10), we have Therefore,

Definition 9 [10]. Let be the set of all simulation functions that fulfil:
(ξ₁)
(ξ₂)
(ξ₃) For any two sequences with

In [10], authors used the function defined by Jleli and Samet in [29] to propose the following result.

Definition 10 [29]. Let be the set of all functions that fulfil:
() is non-decreasing
() For any sequence

Theorem 11 [10]. Let be a complete generalized metric space and satisfy

Then, has a unique fixed point, and for every initial point , the Picard sequence converges to that fixed point.

Definition 12 [9]. Let be a metric-like space with parameter , be a function, and and be arbitrary constants. A mapping is admissible if

In addition, is said to be triangular admissible if it is admissible and

Definition 13 [18, 19]. The Riemann-Liouville fractional integral of order of a function is given by

The Caputo fractional derivative of order of is given by where and denote the gamma function, provided that the right side is point-wise defined on .

Lemma 14 [23]. Let and . Then, for all , we have: (1) and (2), for some (3)

3. A Set of Fixed Point Results

Our first main result is the following theorem.

Theorem 15. Let be a complete metric-like space with parameter . Suppose that is a triangular admissible mapping and satisfy

Consider that the following properties hold true (a)If is a sequence in such that as and , then (b)For all , we have , where denotes the set of fixed points of

Moreover, if there exists such that , then has a unique fixed point.

Proof. Starting with that point . We define a sequence by Regarding that is an admissible, then by induction, we get If for some , then , that is, is a fixed point of , and the proof is completed. So, we assume that From (23) and (24), we apply (21) at and to get Hence, the sequence is monotone decreasing and bounded below by 1. Therefore, there exists such that To prove that , suppose the contrary that and obtain a contradiction. From (25), (26), and , we have that is all we need. Thus, and Also, implies Now, we show that Consider the sequence It is easy to verify that Hence, the sequence is decreasing and bounded below by 1. Consequently, there exists such that Assume that , then from (31), we conclude that Taking limit as , together with (33), implies Also, we have Again, taking limit as , together with (33), (35) implies According to (23) and the fact that is a triangular admissible, we derive On account of the above observations, we apply condition (21) and then () to obtain Therefore, we have And hence which is a contradiction, then and Thus, (30) holds true, and the sequence is Cauchy. By the completeness of , there exists such that Now, consider the subsequence of the sequence such that Lemma 8, together with (43), imply that Apply (21), we obtain Hence, From (45), (47), and , we conclude that To see that this fixed point in unique, suppose that is another fixed point of and apply (21) to get the opposite. This is impossible, so , and the fixed point is unique.
Let denote the class of which satisfies the condition

Remark 16. Since metric-like space is a proper extension of partial metric, metric-like, and metric spaces. Then, we can derive our main results in the setting of these spaces.

Corollary 17. Let be a complete metric-like space and be a mapping such that

Then, has a unique fixed point.

Proof. For all with and , condition (51) can be written as Therefore, Corollary 17 follows from Theorem 15 by taking for all , , and .

4. Fractional Hybrid Differential Equations

Here, we place our considered problem (1) in the space with a mapping defined on it as:

It is evident that is a complete metric-like space with coefficient .

For convenience, we define the following functions , :

Lemma 18. Let , then the integral representation of the boundary value problem where , , and , , are real constants such that is given by where is the Green function and is given by

Proof. Applying the operator 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 yields Substituting the values of and into (62), we get Provided the functions are defined as in Eq. (56), we obtain One can easily verify that Thus, for For , we have For , we get Joining the previous three cases together, we get (60).
Define the operator by In view of Lemma 18, fixed points of are solutions of the FHDE (1). Now, we assume the following conditions which allow us to establish the existence and uniqueness results for the solution of the multipoint boundary value problem (1) by applying Theorem 15.
(C₁) The functions and are continuous
(C₂) There exist two functions with bounds and , respectively, such that (C₃) There exist a functions and a continuous nondecreasing function such that (C₄) Let and be equal then,

Theorem 19. If hold true, then the problem (1) has one solution in .

Proof. First, we show that is contraction on . Now, we define the functions , , and as

Consequently, we obtain where . Also, other hypothesis of Theorem 15 is satisfied. Hence, has a unique fixed point, and then, FHDE (1) has on solution in . This completes the proof.

Now, we present an example to support our results.

Example 20. Let us consider the following FHDE with five-point boundary conditions. where . To show the existence of a unique solution of (81), we apply Theorem 19 with By computation, we can show that the hypotheses are satisfied with Therefore, we conclude that problem (81) has one solution.

5. Concluding Remarks and Observations

Our results extend the results of [10, 11] and many others. Indeed, we deal with a class of -admissible -contractions in a larger structure such as a -metric-like space. Also, the problem we used as an application is different from that one appeared in [23] in the sense that we discussed not only existence but also uniqueness for our considered problem under multipoint boundary conditions via Caputo fractional derivatives.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors of this paper declare that they have no conflicts of interest.

Acknowledgments

The authors thank the anonymous referees for their constructive comments and suggestions given for improving the presentation of the present paper.