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We construct a Green’s function for the three-term fractional differential equation , , where , , and f is continuous, satisfying the boundary conditions , , where . To accomplish this, we first construct a Green’s function for the two-term problem , , satisfying the same boundary conditions. A lemma from spectral theory is integral to our construction. Some limiting properties of the Green’s function for the two-term problem are also studied. Finally, existence results are given for a nonlinear problem.
In this paper, we use a lemma from spectral theory to develop a Green’s function for the fractional differential equation
where and , satisfying the boundary conditions
where . It will be assumed throughout that f is continuous on
We first construct the Green’s function for
This satisfies the boundary condition in Equation (2). The limiting properties of this Green’s function are studied first. Then, by using this Green’s function, we can construct the Green’s function for Equations (1) and (2).
Two-term fractional boundary value problems were first studied by Graef et. al  using spectral theory. These techniques were improved upon in , where the authors were able to write the Green’s function corresponding to the boundary value problem
where as a series of functions. Later, in , these authors studied the boundary value problem
where . They showed that a Green’s function can be constructed in a closed form using generalized Mittag-Leffler functions. Recently , the Green’s function for the boundary value problem
where and a is constant was constructed using alternate methods.
In this paper, we use the techniques from  to first construct the Green’s function corresponding to Equations (2) and (3). This Green’s function will be constructed using generalized Mittag-Leffler functions. Some limiting properties such as are studied. The limiting properties of the Green’s functions were studied for a one-term fractional boundary value problem in . We will see that the results here are similar to the ones in .
The Green’s function constructed for Equations (2) and (3) are then used along with the technique from  to construct the Green’s function for Equations (1) and (2). We believe this is the first paper to study three-term fractional boundary value problems. For more works studying two-term fractional boundary value problems, see, for example, [6,7,8].
For a detailed review of fractional calculus, we refer the reader to the monograph by Diethelm  and the book by Podlubny . The following definitions and properties can be found in these references:
Let and recall the Riemann–Liouville fractional integral of a function u is defined by
provided that the right-hand side exists. Moreover, let n denote a positive integer, and assume . The αth Riemann–Liouville fractional derivative of the function , denoted as , is defined as
provided that the right-hand side exists.
We need a few properties in fractional calculus to construct and analyze the family of the Green’s functions. Recall that
The power rule will be employed, which states that
where it is assumed that is not a positive integer. If is a positive integer, then the right hand side of Equation (5) vanishes. To see this, one can appeal to the convention that if is a positive integer, or one can perform the calculation on the left-hand side and calculate
Moreover, we state and prove the following identities, which will also be employed in Section 3:
Assume , and Assume h is continuous on Then, we have the following:
Consider the cases and independently. We show the details for The details for are similar, the details for are easy to verify, and the details for are trivial. The calculations employ the Euler beta function
If then a similar calculation is performed. The Leibniz rule is sufficient for or , and the calculation for is trivial.
To prove (2), notice that
Since each of and converge uniformly on , then
To prove (3), we see that
Finally, to prove (4), we start by noticing
Since each of
converge uniformly on term by term differentiation is valid, and
The following lemma on spectral theory in Banach spaces will be integral to our construction:
Let X be a Banach space and be a linear operator with the operator norm and spectral radius of . Then, we have the following:
Then, exists, and
where is the identity operator.
The generalized Mittag-Leffler function is defined as
It is known that the generalized Mittag-Leffler function is an entire function as long as . Notice that .
As we shall observe an asymptotic property of the Green’s functions of the boundary value problem in Equations (2) and (3) as functions of and , respectively, we shall also make use of a further generalized Mittag-Leffler type function first studied in a special case by Le Roy  and, we believe, recently introduced by Gerhold :
Again, if , then denotes an entire function.
The following asymptotic result will be useful for obtaining the asymptotic properties of the Green’s function associated with Equations (2) and (3) as functions of and , respectively:
(Theorem 1 ). Let and be arbitrary. Then, for in the sector
we have the asymptotics
3. Green’s Function for the Two-Term Problem
We look to construct the Green’s function for the boundary value problem in Equations (2) and (3). Let be the Banach space of continuous functions with the standard maximum norm . Assume u is a solution to Equations (2) and (3). Then, we have
where , , , and are constants.
We can apply to both sides of Equation (7) to obtain
Since , , and since , , hence
We can define and by
Notice that if , then
Then, by Lemma 2, we obtain
Now, we have
Since the generalized Mittag-Leffler function is entire for the positive parameters and , converges uniformly for , and
Similarly, we write
The convergence of is uniform on the triangle, where , and so the convergence of
We must point out that this result is similar to the result for the one-term problem in  (Theorem 2.5).
4. Green’s Function for the Three-Term Problem
For the remainder of this article, is fixed, and there is no need to specify a Green’s function as a function of or Therefore, in particular, let be the Green’s function for Equations (2) and (3). We define
Assume f is continuous on and define and . Notice that
and, in general, that
Therefore, by assuming , we have
Now, let u be a solution for Equations (1) and (2). Thus, we have
Additionally, u satisfies the boundary conditions in Equation (2). Thus, by Theorem 2, we have
Assume . Let be the Green’s function for Equations (2) and (3). Define
Assume f is continuous on , and assume, where . Then, u satisfies Equations (1) and (2) if and only if
5. The Existence of Solutions
Consider the nonlinear boundary value problem
satisfying the boundary conditions of Equation (2), where a and f satisfy the conditions of Theorem 5. Here, it is assumed that is a continuous function. We define by
Let . Assume a and f satisfy the conditions of Theorem 5, and assume there exists an such that
Then, Equations (2) and (10) have at least one solution u with .
The proof is an application of Schauder’s fixed point theorem. We define the set . Then, for and , we have
Therefore, . Hence, , and is uniformly bounded.
We show that is uniformly bounded. Since , and
Notice that if then and in general, the following is true:
Therefore, since , we have
since . Thus, there exists such that
In addition, if , then , and is uniformly bounded. Thus, is uniformly bounded, equicontinuous, and hence sequentially compact.
By Schauder’s fixed point theorem, T has a fixed point in K. Hence, Equations (2) and (10) have at least one solution u with . □
6. An Example
As an example, consider the case where and . Let and . Here, for , since is increasing as a function of t, then we have
then the unique solution of
satisfying the boundary conditions
is given by
If, for example, , , then in this case, we have
Now, consider the nonlinear boundary value problem
satisfying the boundary conditions
Notice that for , it holds that
By Theorem 6, this boundary value problem has a solution u with .
In this paper, a three-term fractional boundary value problem was studied. Spectral theory was used to calculate the Green’s function for a two-term problem first. The limiting properties of this Green’s function were studied. Then, the Green’s function for the three-term problem was constructed. Finally, this paper considered the existence of solutions to a nonlinear three-term problem, and an example was constructed.
Writing—original draft preparation, P.W.E. and J.T.N. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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