Abstract
We study sufficient conditions for existence of solutions to the coupled systems of higher order hybrid fractional differential equations with three-point boundary conditions. For this motive, we apply the coupled fixed point theorem of Krasnoselskii type to form adequate conditions for existence of solutions to the proposed system. We finish the paper with suitable illustrative example.
1. Introduction
Fractional calculus is found to be more practical and effective than the classical calculus in the mathematical modeling of several phenomena. Fractional differential equations are very important and significant part of the mathematics and have various applications in viscoelasticity, electroanalytical chemistry, and many physical problems [1–6]. A systematic presentation of the applications of fractional differential equations can be found in the book of Balachandran and Park [7]. In recent years, many works have been devoted to the study of the mathematical aspects of fractional order differential equations [8–12]. There are numerous advanced and efficient methods, which have been focusing on the existence of solution to fractional differential equations. One of the powerful tools for obtaining the existence of solutions to such equations is the fixed point methods. Many authors use fixed point theorems to prove the existence and uniqueness of solution to nonlinear fractional differential equations; see, for example, [13–17].
On the other hand, the study for coupled systems of fractional differential equations is also important as such systems occur in various problems of applied nature, for instance, [18–25]. Additionally, fixed point theory can be used to develop the existence theory for the coupled systems of fractional hybrid differential equations [13, 16, 17]. Bashiri et al. [17] discussed the existence of solution to the following system of fractional hybrid differential equations of order :where , and the functions and satisfy certain conditions. is the R-L fractional derivative of order .
Recently, the existence of solutions for fractional differential equations involving the Caputo fractional derivative was studied in [13, 26–28]. Motivated by the work of Bashiri et al. [17], in this paper we are concerned with the existence of solutions to three-point boundary value problem for a coupled system of hybrid fractional differential equations of order given bywhere , , , , and . is the Caputo fractional derivative of order . Moreover, an example is given to illustrate the validity of the existence result.
2. Preliminaries
Throughout this manuscript , denote the class of continuous functions , and denote the class of functions such that(i)the map is measurable for each ,(ii)the map is continuous for each ,(iii)the map is continuous for each .
We need the following definitions which can be found in [9].
Definition 1. The Riemann-Liouville fractional integral of order of function is defined asprovided that the right side is pointwise defined on .
Definition 2. Let be a positive real number, such that , , and exists, a function of class . Then Caputo fractional derivative of is defined asprovided that the right side is pointwise defined on , where and represents the integer part of .
Definition 3 (see [29]). The mapping has a coupled fixed point if and .
Theorem 4 (see [17]). Let be a nonempty, closed, convex, and bounded subset of the Banach space and . Suppose that and are two operators such that there exists such that, for all , one has is completely continuous,, for all . Then the operator has at least a coupled fixed point in whenever .
Lemma 5. The following result holds for fractional differential equations: for arbitrary , , where and represents the integer part of .
3. Existence Results
Let us set the following notations for convenience:For the forthcoming analysis, we assume that the function is increasing in for all ;there exists such that there exists a continuous function such that
Lemma 6. If and for , then integral representation of the system (2) is given by
Proof. Applying the operator on the first equation of system (2) and using Lemma 5, we obtainApplying the initial conditions , for , we conclude that . Therefore (13) becomesNow, to find the values of and , since , from (14) we have Also, since , from (14), we have Solving (15) and (16) for , we get and using (7), we can write Similarly, solving (15) and (16) and using (7), we obtained thatSubstituting the values of and in (14), we can write which implies that Similarly, repeating the above process with the second equation of system (2), we obtain integral equation (12).
Now, we are in a position to present the existence theorem for the system (2).
Theorem 7. Assume that hypotheses – hold. Then there exists a solution for coupled systems (2) of higher order hybrid FHDEs with three-point boundary conditions.
Proof. Set and a subset of defined by Clearly is a closed, convex, and bounded subset of the Banach space . Now, since is a solution of the FHDEs system (2) if and only if satisfies the system of integral equations in Lemma 6, to show the existence solution of system (2) it is enough to show the existence solution of the integral equations in Lemma 6. For this, define two operators and by Then the operators form of system (2) is We have to show that the operators and satisfy all the conditions of Theorem 4. For this, let ; then we have Taking the supremum over and using (7), we get Thus satisfies condition of Theorem 4 with and .
Next, we show that is compact and continuous operator on . Let be a sequence in such that . Then for all , we have Thus the map is continuous on .
Let ; then we have Taking the supremum over , we get Thus is uniformly bounded on .
Now, let such that ; then for any , we have Since is uniformly continuous on for , for any we can find such that Let ; then, we have Thus is equicontinuous and hence is completely continuous on .
To prove hypothesis of Theorem 4, let and such that ; using (7) and (8), we have This implies Taking the supremum over on , we can write That is, . Thus condition of Theorem 4 holds. Therefore, all the conditions of Theorem 4 are satisfied; hence the operator has a coupled fixed point on . Consequently, system (2) has a solution defined on .
To illustrate Theorem 7, we construct the following example.
Example 8. We discuss the following hybrid fractional differential equations with three-point boundary conditions:where , , and . is the Caputo fractional derivative of order .
Here Therefore,Now, for and , we have Similarly, Next, That is, there exists a continuous function such that Finally, since we can write Thus . It follows that assumptions and are satisfied. Therefore, by Theorem 7 we conclude that problem (36) has a solution.
4. Conclusion
We have successfully developed appropriate conditions for existence of at least one solution to a complicated higher order coupled system of nonlinear hybrid fractional differential equations. The respective conditions have been derived by using coupled fixed point theorem of Krasnoselskii type. The obtained results were also demonstrated by a suitable example.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All the authors have equal contribution. The first two authors designed the problem and the last two authors read and corrected its style and language and prepared the final version.
Acknowledgments
This work was financially supported by the Rajamangala University of Technology Thanyaburi (RMUTT), Pathum Thani, Thailand.