Fractional Difference and Differential Operators and their Applications in Nonlinear SystemsView this Special Issue
Fractional Hybrid Differential Equations and Coupled Fixed-Point Results for -Admissible Contractions in Metric Spaces
In this paper, we investigate the existence of a unique coupled fixed point for admissible mapping which is of contraction in the context of metric space. We have also shown that the results presented in this paper would extend many recent results appearing in the literature. Furthermore, we apply our results to develop sufficient conditions for the existence and uniqueness of a solution for a coupled system of fractional hybrid differential equations with linear perturbations of second type and with three-point boundary conditions.
Fixed-point theory is an outstanding source which gives responsible techniques for the existence of fixed points for self-mappings under different conditions. One of the newest branches of fixed-point theory concerned with the study of coupled fixed points, brought by Guo and Lakshmikantham . In , Bhaskar and Lakshmikantham established some fixed and coupled fixed-point theorems for contractions in two variables defined on partially ordered metric spaces with applications to ordinary differential equations. Thereafter, these results were extended by several authors (see [3–6]).
Inspired by the notion of partial metric (or, metric) which is one of the vital generalizations of the standard metric, Asadi et al.  proposed the concept of metric which refines the metric and produces useful basic topological concepts. For some fixed-point results and various contractive definitions that have been employed in metric space, we refer the reader to [8–12].
Theorem 1. Let be a complete metric space and be an admissible mapping. Suppose that the following condition is satisfied:for all and , where , is an altering distance function, and is an ultra-altering distance function. Suppose that either(a) is continuous(b)If is a sequence in such that , , , then If there exists such that , then has a fixed point.
Hybrid differential equations have been of great interest as they include several dynamic systems as special cases. The papers [17, 18] discussed the existence and uniqueness results and some fundamental differential inequalities for first-order hybrid differential equations with perturbations of 1st and 2nd type, respectively.
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 fact, fractional differential equations (FDEs) play a major role in modeling many real-life problems such as physical phenomena, computer networking, medicine (the modeling of human tissue), mechanics (theory of viscoelasticity), electrical engineering (transmission of ultrasound waves) and many others (see [19–21]).
Fractional hybrid differential equations (FHDEs) can be employed in modeling and describing nonhomogenous physical phenomena that take place in their form. FHDEs have been studied using a Riemann–Liouville differential operator of order in many literature studies (see [22–26]).
In line with the above studies, our purpose in this paper is to introduce the notion of admissible mapping with two variables and generalize Theorem 1 to coupled fixed-point version. Then, we apply our main results to prove the existence and uniqueness of a solution to the following system of FHDEs involving Riemann–Liouville fractional derivative:for all , , , , , , , and .
In 1994, Matthews  introduced the notion of a metric space as a part of the study of denotational semantics of dataflow networks. In metric spaces, self-distance of an arbitrary point need not be equal to zero.
Definition 1. (see ). A metric on a nonempty set is a mapping such that, for all , Then, is called a metric space.
Notice that, every metric space can be defined to be metric space with zero self-distance. After that, Asadi et al. generalized the above definition by relaxing the axiom as follows.
Definition 2. (see ). For a nonempty set , a function is called an metric if it fulfils the following: , where Then, the pair is called an metric space.
Lemma 1. (see ). Every metric is an metric.
Here, we give an example to show that the converse might not be held.
Example 1. (see ). Let and defineSo is metric but it is not metric for . Also, is not metric for self-distances are not zero.
Thus, the class of metric spaces is effectively larger than that of both ordinary metric and metric spaces.
Notation 1. Let be an metric space; then defineHence, is an ordinary metric induced by the metric .
Each metric on generates a topology on formed by the setwhereThe notions of convergent sequence, Cauchy sequence, and complete metric space are given as follows:(1)A sequence in converges to a point if(2)A sequence in is called Cauchy if exist and are finite.(3) is said to be complete if every Cauchy sequence in it converges, with respect to , to a point , and
Lemma 2 (see ). Let be an metric space; then,(1) is a Cauchy sequence in if and only if it is Cauchy sequence in the metric space .(2) is complete if and only if is complete. Furthermore,
Lemma 3 (see ). Assume that and in an metric space ; then,
As a consequence of Lemma 3, we have
Definition 3. (see ). A mapping is called a class function if it is continuous and satisfies the following axioms:(1)(2) implies that either or for all Let denote the class functions.
Lemma 4 (see ). Riemann–Liouville fractional integral and derivative have the following properties:(1) and , for all , (2), where and (3),
3. Fixed-Point Results
First, we introduce the following concepts that generalize the corresponding ones used in  and will be beneficial in the sequel.
Definition 6. Let and ; then, is called an admissible mapping ifNote that, if equation (16) holds, then we have too. Consider the following classes of functions:
Theorem 2. Let be a complete metric space and be an admissible mapping for which there exist , , , and such that and for all with ; we havewhereSuppose that either(a)T is continuous.(b)For a convergent sequence in , we haveIf there exist such that and , then has a coupled fixed point.
Proof. Starting with , define the sequences byBy induction methodology for , we shall prove thatIndeed, we have and . Suppose that (22) holds for some and we are going to prove it for . Since is admissible mapping, then by (21), we obtain and . Thus, (22) holds for all . From (18)–(22), we havewhereHence,where . Similarly, we haveAdding (25) and (26) and using properties on and , we obtainSince is strictly increasing, then Hence, the sequence is monotone decreasing and bounded as follows. Therefore, there exist some such thatNow, we shall prove that . Assume that . Using the properties of and letting in (27) yield thatwhich is contradiction. Thus, andIn what follows, we prove that and are Cauchy sequences in . Since we havethenThat is,On the other hand, we haveTherefore, (33) and (34) imply that is an Cauchy sequence. In a similar way, we can show that is also a Cauchy sequence. By the completeness of the space , there exist such thatWith respect to the sequence , we obtainbutThus, the uniqueness of the limit implies thatNow, suppose that holds. According to Lemma 2, since and are Cauchy sequences in a complete metric space , then they converge to some in the metric space . Also, as is continuous, converges to in , that is, which is equivalent toAlso, we havebutThus, the uniqueness of the limit implies thatBy Lemma 3, we obtainCompared with (39), we obtainFrom (38), (42), and (44), we obtainProceeding as above, one can obtainSuppose that holds, then and . Setting and in (18), we obtainwhereThat is,In a similar way, one can obtainAdding (49) and (50) and using properties on and , we obtainTaking limits at yieldsTherefore, we have . Again from (18) and taking into account that , we obtain that . Consequently, and .
For the uniqueness of the coupled fixed point in Theorem 2, we consider the following condition:
Proof. Theorem 2 asserts that has at least one coupled fixed point. Assume that and are two coupled fixed points of , then or . Now, we apply (18) and use the properties of to obtainHence, we haveSimilarly, we haveHence, by () i.e., is the unique coupled fixed point of .
If we define andthen we get the following corollary which is a generalization of the main results in .
Corollary 1. Let be an ordered complete metric space and be an increasing mapping for which there exist , , and such that and for all with and ; we havewhereSuppose that either(a)T is continuous.(b)For a convergent sequence in , we haveIf there exist such that and , then has a coupled fixed point.
Now, we introduce the following classes of functions and byIf we consider , for some and for some , then we obtain an extension of the main result in .
Corollary 2. Let be a complete metric space and be an admissible mapping such thatfor all with , where , , and . Suppose that either(a)T is continuous.(b)For a convergent sequence in , we haveIf there exist such that and , then has a coupled fixed point.
Remark 1. Notice that in [32, 33], it was shown that each coupled fixed-point theorem can be observed from the analogue of single/standard fixed-point theorems. On the other hand, for the usage of it in application, the coupled fixed-point theorem can be used to handle the problem. Therefore, in this paper, we consider the coupled fixed-point results, Theorem 2 and Theorem 3.
4. Fractional Differential Equations
In this section, we present sufficient conditions for the existence and uniqueness of the solution of coupled systems (2) and (3). Before starting and proving the main results, we need to fix the analytical framework of our considered problem.
Consider the complete metric space , where and is defined by
In addition, define the operator aswhere
Now, we claim that whenever is a coupled fixed point of the operator , it follows that and solve (2) and (3).
Lemma 5. Let , , , and ; then, the boundary value problemhas the integral representation of solution
Proof. Applying the operator on both sides of (67) and using Lemma 4, we obtainAlso, we haveHence, we obtainAt and , we haveBy subtracting (73) from (72), we obtainConsequently, the general solution of (67) is