Fractional Boundary Value Problems and Lyapunov-type Inequalities with Fractional Integral Boundary Conditions

We discuss boundary value problems for Riemann–Liouville fractional differential equations with certain fractional integral boundary conditions. Such boundary conditions are different from the widely considered pointwise conditions in the sense that they allow solutions to have singularities, and different from other conditions given by integrals with a singular kernel since they arise from well-defined initial value problems. We derive Lyapunov-type inequalities for linear fractional differential equations and apply them to establish nonexistence, uniqueness, and existence-uniqueness of solutions for certain linear fractional boundary value problems. Parallel results are also obtained for sequential fractional differential equations. An example is given to show how computer programs and numerical algorithms can be used to verify the conditions and to apply the results. 1 Fractional integral boundary conditions Boundary value problems (BVPs) for fractional differential equations are important in applications and have been studied extensively by many authors, and the references cited therein. A lot of work has been done on fractional BVPs consisting of a fractional differential equation in the form (D α a + x) (t) = f (t, x) on (a, b) (1.1) with α > 0, and a pointwise boundary condition (BC) at the end points; in particular, the Dirichlet BC x(a) = x(b) = 0 (1.2)


Fractional integral boundary conditions
Boundary value problems (BVPs) for fractional differential equations are important in applications and have been studied extensively by many authors, see [3,6,[10][11][12][13]16,19,20,22,27,29,34] and the references cited therein.A lot of work has been done on fractional BVPs consisting of a fractional differential equation in the form with α > 0, and a pointwise boundary condition (BC) at the end points; in particular, the Dirichlet BC x(a) = x(b) = 0 (1.2) for 1 < α ≤ 2.Here with α > 0 and t > a, is the αth-order left-sided Riemann-Liouville fractional integral of x(t) at a, and D α a + x (t) denotes the αth-order left-sided Riemann-Liouville fractional derivative of x(t) at a defined as where n = α + 1 with α the integer part of α and Γ(α) = ∞ 0 t α−1 e −t dt is the gamma function.In particular, when α = i ∈ N 0 , then (1.5) In the following, for the consistence of notations for BCs, we also denote D −α a + x (t) := I α a + x (t) for 0 < α < 1.
We note that with Riemann-Liouville fractional derivative involved, any solution of Eq. (1.1) with a pointwise BC such as (1.2), if it exists, must be bounded on [a, b].However, unlike integer-order differential equations, the majority of the solutions of Eq. (1.1) is unbounded at the left endpoint a no matter how good the right-hand function f (t, x) is.This can be seen from [18, (2.1.39)]that every solution of Eq. (1.1) satisfies with c j ∈ R for j = 1, . . ., n, which shows that either x(a) = 0 or x(t) is unbounded at a. Consequently, we should not expect that any BVP consisting Eq. (1.1) and a pointwise BC to have any solution unless the BC includes or implies the condition x(a) = 0.This is the reason why fractional BVPs have been studied mainly with the Dirichlet BC at a so far.More specifically, any pointwise BC including one of the following is ill-posed: In fact, the BC in (i) violates (1.6); and the BCs in (ii) and (iii) are each equivalent to one of the two sets of conditions: x(a) = x (a) = 0 and x(a) = x (i) (b) = 0, and hence does not agree with the number requirement for well-posed BCs.
In this paper, we use fractional integral BCs to allow and "smoothen" the singularity of solutions at a.This idea is motivated by the initial conditions for Cauchy problems associated with Eq. (1.1) given in [18, (3.1.2)]: where n = α + 1 for α / ∈ N and n = α for α ∈ N, and Note that α − n = 0 for α ∈ N. By (1.5) We notice that the existence and uniqueness of solutions have been established for the Cauchy problem (1.1), (1.7) with any b k ∈ R. From this point of view, a more reasonable BC should involve D α−k a + x (a + ) rather than x (k) (a) for k = 1, 2, . . ., n.In particular, for Eq.(1.1) with 1 < α ≤ 2, we may assign a homogeneous linear separated BC as (1.9) and a coupled BC as where c ij ∈ R and K ∈ R 2×2 such that det K = 0. Nonhomogeneous BCs can be defined accordingly.Such BCs permit solutions unbounded at a and hence are more general than pointwise BCs.It is easy to see from (1.8) that when α = 2, BCs (1.9) and (1.respectively.Therefore, (1.9) and (1.10) are natural extensions of the self-adjoint BCs for second-order linear differential equations to fractional differential equations with 1 < α ≤ 2. We point out that the BCs considered in this paper are different from the general integral BCs with a singular kernel in the sense that they originate from the fractional initial conditions with which the existence-uniqueness results are derived.Problems with such BCs can be investigated in many approaches based on results on initial value problems.For instance, using the Fredholm alternative method to study the existence and uniqueness of boundary value problems, as shown in our Theorems 5.3 and 5.7.Such approaches are not allowed for general integral BCs.
In this paper, we consider the fractional BVP consisting of the linear equation .11)and the BC Lyapunov-type inequalities are derived and used to establish the existence and uniqueness for solutions of this BVP.Parallel results are also obtained for certain sequential fractional BVPs.Further discussions on higher order and nonlinear fractional BVPs will be given in forthcoming papers.
This paper is organized as follows: after this section, we briefly review the existing results on Lyapunov-type inequalities in Section 2. In Sections 3 and 4, we derive new Lyapunov-type inequalities for fractional differential equations and sequential fractional differential equations, respectively.Finally in Section 5, we apply the obtained Lyapunov-type inequalities to establish the existence and uniqueness for solutions of some fractional BVPs.We also give an example to show how computer programs and numerical algorithms can be used to verify the conditions and to apply the results.

Existing results on Lyapunov-type inequalities
For the second-order linear differential equation with q ∈ L([a, b], R), the following result is known as the Lyapunov inequality, see [1,21].It was first noticed by Wintner [30] and later by several other authors that inequality (2.2) can be improved by replacing |q(t)| by q + (t) := max{q(t), 0}, the positive part of q(t), to become Inequality (2.3) was further generalized to a more general form of second-order differential equations by Hartman [15, Chapter XI], and improved by Brown and Hinton [2] and Harris and Kong [14] later on.
With a simple modification in the theorem, we can easily obtain a variation of Theorem 2.2.
These results are derived using the Green's function for BVP (1.11), (1.2) obtained in [8], which is an extension of the one given in [3] for the case that a = 0 and b = 1.

Fractional Lyapunov-type inequalities
In this section, we let −∞ < a < b < ∞ and consider fractional differential equation where 1 < α ≤ 2 and q ∈ L(a, b).To present our main results, we need the concept of γ-th right-sided Riemann-Liouville fractional derivative of a function u(t) at b defined as where γ ≥ 0 and n = γ + 1.In particular, when 0 ≤ γ < 1, (3.2) reduces to With the left-sided and right-sided fractional derivatives given in (1.4) and (3.3), we have the following fractional integration by parts formula, see [29, (2.64) where φ ∈ L p (a, b) and ψ ∈ L r (a, b) such that p −1 + r −1 ≤ 1 + γ.
In the following we define and let D 2−α b − [G(t, s)q(s)] be the right-sided fractional derivative of G(t, s)q(s) with respect to s and D ]. Now we present our main result on fractional Lyapunov-type inequalities.
In fact, for 1 < α < 2, from (1.4) we have and (3.10) holds clearly when α = 2 since y(t) = x(t).Then it follows that BVP (3.1), (3.6) becomes the second-order linear BVP Hence the solution y(t) satisfies where G(t, s), given in (3.5), is the Green's function for BVP (3.11).For a fixed t ∈ [a, b], By taking the absolute value on both sides we have ≡ m on J and y(t) < m a.e. on [a, b]\J.Then for t ∈ J, y (t) = 0 and From (3.11), q(t) ≡ 0 on J.This implies that for any which also leads to (3.7).
(b) From the proof of Part (a) we see that (3.13) holds.By the assumption, y(t) = 0 on (a, b).Without loss of generality, we assume that y(t) > 0 on (a, b).Then it follows that Now, a similar argument as in Part (a) leads to (3.9).
The corollary below is a special case of Theorem 3.1. (3.16) Proof.By Theorem 3.1 we see that (3.7) holds.By the assumption and the definition of (3.17 Using the facts that q(t) we see that

Sequential fractional Lyapunov-type inequalities
Here we let −∞ < a < b < ∞ and consider the sequential fractional differential equation where q ∈ L([a, b], R), and 0 < α, β ≤ 1.In the following, we define and let D 1−α b − [G(t, s)q(s)] and D 1−α b − [G(t, s)q(s)] + be defined in the same way as in Section 3. Now we present Lyapunov-type inequalities for Eq.(4.1).
Using the fact that y(a) = 0 and differentiating both sides with respect to t we have Thus BVP (4.7) becomes The rest of the proof is essentially the same as the proof of Theorem 3.1.We omit the details.
The following corollary is a special case of Theorem 4.1.
Proof.The proof is similar to that of Corollary 3.2.By Theorem 4.1 we see that (4.4) holds.
From the assumption and the definition of Using the facts that q(t) we see that Substituting (4.12) in (4.11) we see that (4.9) holds.

Applications to boundary value problems
In the last section, we apply the results on the Lyapunov-type inequalities obtained in Sections 2 and 3 to study the nonexistence, uniqueness, and existence-uniqueness of solutions of related fractional-order linear BVPs.We first consider the BVP consisting of the equation and the BC The following result is on the nonexistence of solutions of BVP (5.1), (5.2).
(b) The proof is similar to Part (a) and hence is omitted.
Next we consider the fractional-order nonhomogeneous linear BVP consisting of the equation with 1 < α ≤ 2 and q, w ∈ L((a, b), R); and the BC where k 1 , k 2 ∈ R. Based on Theorem 3.1, we obtain a criterion for BVP (5.5), (5.6) to have a unique solution and a relation among solutions if the problem has more than one solution.Then BVP (5.5), (5.6) has a unique solution on (a, b) for any k 1 , k 2 ∈ R.
Remark 5.5.We note from Section 1 that the BVP consisting of Eq. (5.5) and the pointwise BC does not have a solution unless k 1 = 0.Even for the case with k 1 = 0, the existence and uniqueness of solutions of BVP (5.5), (5.8) cannot be established by the Fredholm alternative method.This is due to the fact that Eq. (5.5) with a pointwise initial condition may not have a unique solution.
For the case with k 1 = k 2 = 0 and w(t) ≡ 0, from Theorem 2.2, we can easily derive the following result: Assume (5.9) Then BVP (5.5), (5.8) has only the zero solution.
We observe that this result has been improved by Corollary 5.4 for α = 2 since BVPs (5.5), (5.6) and (5.5), (5.8) become the same second-order homogeneous linear problem.When 1 < α < 2, we compare the two results by comparing the right-hand numbers of (5.7) and (5.9) (under the assumption that D 2−α b − [G(t, s)q(s)] ≥ 0 for BVP (5.5), (5.6)).We claim that and H(α) → ∞ as α → 1 + .In fact, Then (5.10) follows from the fact that α/(α − 1) > 4 1−1/α for 1 < α < 2. This shows that condition (5.7) is weaker than condition (5.9), and much weaker when α is close to 1; which is reasonable since BC (5.6) allows the solution x(t) to have a singularity at a, while BC (5.8) requires the solution to be bounded.Now, we state the results for the sequential fractional BVPs which are parallel to Theorem 5.3 and Corollary 5.4.We omit the proofs since they are essentially in the same way.Consider the BVP consisting of the equation and the BC (5.12) The following result is on the nonexistence of solutions of BVP (5.11), (5.12).Then BVP (5.11), (5.12) has no I-positive solution.
Next we consider the sequential nonhomogeneous linear BVPs consisting of the equation where 0 < α, β ≤ 1 and q, w ∈ L((a, b)), R), and the BC where k 1 , k 2 ∈ R. Now we present a criterion for BVP (5.13), (5.14) to have a unique solution and a relation among the solutions if the problem has more than one solution.If BVP (5.13), (5.14) has two solutions x 1 (t) and x 2 (t), then there exists a c ∈ (a, b) such that I 1−α a + x 1 (c) = I 1−α a + x 2 (c).
As before, we have the following corollary from Corollary 4.2.Then BVP (5.13), (5.14) have a unique solution on (a, b) for any k 1 , k 2 ∈ R.
Finally, we point out that the applications of the results in this paper involve evaluations of fractional derivatives of functions.However, conditions involving fractional derivatives and integrals are hard to check analytically, even with pointwise BCs.So computer programs and numerical algorithms are the main tools for applications.We refer the reader to [13] for numerical algorithms for computing fractional derivatives.Here, we give an example to illustrate the application of Theorem 5.3.A similar example for Theorem 5.7 can be easily elaborated and hence is left to the interested reader.

Corollary 5 . 4 .
With a similar argument, from Corollary 3.2 we obtain the result below.Assume D 2−α b − [G(t, s)q(s)] ≥ 0 in [a, b] and b a q
Then y(t) is continuous on [a, b].Note that x(t) = (D 1−α a + y)(t).As shown in the proof of Theorem 3.1, we have D α a + x (t) = y (t).It follows that BVP (4.1), (4.3) becomes the fractional BVP has at most one solution for any k 1 , k 2 ∈ R. Assume the contrary, i.e., it has two solutions x 1 (t) and x 2 (t) in (a, b).Let x(t) = x 1 (t) − x 2 (t).