Electronic Journal of Differential Equations, Vol. 2015 (2015), No. 88, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
LYAPUNOV-TYPE INEQUALITIES FOR FRACTIONAL
BOUNDARY-VALUE PROBLEMS
MOHAMED JLELI, BESSEM SAMET
Abstract. In this article, we establish some Lyapunov-type inequalities for
fractional boundary-value problems under Sturm-Liouville boundary conditions. As applications, we obtain intervals where linear combinations of certain Mittag-Leffler functions have no real zeros. We deduce also nonexistence
results for some fractional boundary-value problems.
1. Introduction
The well-known Lyapunov result [9] states that if a nontrivial solution to the
boundary-value problem
u00 (t) + q(t)u(t) = 0,
a < t < b,
u(a) = u(b) = 0,
exists, where q : [a, b] → R is a continuous function, then
Z b
4
|q(s)| ds >
.
b−a
a
This result found many practical applications in differential and difference equations
(oscillation theory, disconjugacy, eigenvalue problems, etc.); see [1, 2, 11, 13, 14, 15]
and references therein.
The search for Lyapunov-type inequalities in which the starting differential equation is constructed via fractional differential operators has begun very recently. The
first work in this direction is due to Ferreira [4], where he derived a Lyapunov-type
inequality for differential equations depending on the Riemann-Liouville fractional
derivative; that is, for the boundary-value problem
(a Dα u)(t) + q(t)u(t) = 0,
a < t < b, 1 < α ≤ 2,
u(a) = u(b) = 0,
2000 Mathematics Subject Classification. 4A08, 34A40, 26D10, 33E12.
Key words and phrases. Lyapunov’s inequality; Caputo’s fractional derivative;
Sturm-Liouville boundary condition.
c
2015
Texas State University - San Marcos.
Submitted January 2, 2015. Published April 10, 2015.
1
2
M. JLELI, B. SAMET
EJDE-2015/88
where a Dα denotes the Riemann-Liouville fractional derivative of order α. Precisely, the author proved that if the above problem has a nontrivial solution, then
Z b
4 α−1
|q(s)| ds > Γ(α)
.
b−a
a
Clearly, if we let α = 2 in the above inequality, one obtains Lyapunov’s standard
inequality. In [5], a Lyapunov-type inequality was obtained by the same author for
the Caputo fractional boundary-value problem
α
(C
a D u)(t) + q(t)u(t) = 0,
a < t < b, 1 < α ≤ 2,
u(a) = u(b) = 0,
α
where C
denotes the Caputo fractional derivative of order α. In this work,
aD
Ferreira proved that if the above problem has a nontrivial solution, then
Z b
Γ(α)αα
.
|q(s)| ds >
[(α − 1)(b − a)]α−1
a
For other works on Lyapunov-type inequalities for fractional boundary-value problems we refer the reader to [6, 7].
Motivated by the above works, we consider a Caputo fractional differential equation with Sturm-Liouville boundary conditions. More precisely, we consider the
fractional boundary-value problem
α
(C
a D u)(t) + q(t)u(t) = 0,
a < t < b, 1 < α < 2,
(1.1)
with the boundary conditions
pu(a) − ru0 (a) = u(b) = 0,
(1.2)
where p > 0, r ≥ 0 and q : [a, b] → R is a continuous function. We distinguish two
b−a
b−a
cases: the case pr > α−1
and the case 0 ≤ pr ≤ α−1
. For each case, a Lyapunov-type
inequality is derived. The obtained results recover several existing inequalities from
the literature. As applications, we obtain intervals where linear combinations of
certain Mittag-Leffler functions have no real zeros. We deduce also nonexistence
results for some fractional boundary-value problems.
Before presenting our main results, let us start by recalling the concepts of the
Riemann-Liouville fractional integral and the Caputo fractional derivative of order
α ≥ 0. For more details, we refer to [8].
Let α ≥ 0 and let f be a real function defined on a certain interval [a, b]. The
Riemann-Liouville fractional integral of order α is defined by
(a I 0 f )(t) = f (t)
and
Z t
1
(a I f )(t) =
(t − s)α−1 f (s) ds, α > 0, t ∈ [a, b].
Γ(α) a
The Caputo fractional derivative of order α ≥ 0 is defined by
α
0
(C
a D f )(t) = f (t)
and
α
m−α m
(C
D f )(t),
a D f )(t) = (a I
where m is the smallest integer greater or equal to α.
α > 0,
EJDE-2015/88
LYAPUNOV-TYPE INEQUALITIES
3
2. Main results
2.1. Integral representation of the solution. We start by writing (1.1)-(1.2)
in its equivalent integral form.
Lemma 2.1. u ∈ C[a, b] is a solution to (1.1)-(1.2) if and only if u is a solution
to the integral equation
Z b
u(t) =
G(t, s)q(s)u(s) ds, t ∈ [a, b],
a
where G, the Green function associated to (1.1)-(1.2), is given by
 r
( +t−a)(b−s)α−1
− (t − s)α−1 , a ≤ s ≤ t ≤ b,
1  p
γ
G(t, s) =
α−1
r
Γ(α)  ( p +t−a)(b−s)
,
a ≤ t ≤ s ≤ b,
γ
where γ =
r
p
+ b − a.
Proof. The general solution to (1.1) is
1
u(t) = c0 + c1 (t − a) −
Γ(α)
Z
t
(t − s)α−1 q(s)u(s) ds,
a
where c0 and c1 are real constants. Taking the derivative of u(t), we obtain
Z
(α − 1) t
(t − s)α−2 q(s)u(s) ds.
u0 (t) = c1 −
Γ(α) a
Using the boundary condition pu(a) − ru0 (a) = 0, we obtain
pc0 − rc1 = 0.
(2.1)
The boundary condition u(b) = 0 gives us
Z b
1
c0 + c1 (b − a) −
(b − s)α−1 q(s)u(s) ds = 0.
Γ(α) a
(2.2)
Then (2.1) and (2.2) yield
c0 =
r
r
c1 =
p
pγΓ(α)
Z
b
(b − s)α−1 q(s)u(s) ds .
a
Therefore,
u(t) =
r
pγΓ(α)
−
1
Γ(α)
Z
b
a
Z t
(b − s)α−1 q(s)u(s) ds +
(t − a)
γΓ(α)
Z
b
(b − s)α−1 q(s)u(s) ds
a
(t − s)α−1 q(s)u(s) ds,
a
which concludes the proof.
2.2. Green function estimates. Let
( pr + t − a)(b − s)α−1
g1 (t, s) =
− (t − s)α−1 , a ≤ s ≤ t ≤ b,
γ
( pr + t − a)(b − s)α−1
, a ≤ t ≤ s ≤ b.
g2 (t, s) =
γ
We distinguish two cases.
4
M. JLELI, B. SAMET
Case
r
p
>
EJDE-2015/88
b−a
α−1 .
Lemma 2.2. Suppose that
b−a
r
>
.
p
α−1
Then
0 ≤ G(t, s) ≤ G(s, s),
max G(s, s) =
a≤s≤b
(t, s) ∈ [a, b] × [a, b],
r
α−1
1 p (b − a)
.
Γ(α) ( pr + b − a)
Proof. Obviously, the function g2 satisfies the following inequalities:
0 ≤ g2 (t, s) ≤ g2 (s, s),
a ≤ t ≤ s ≤ b.
Now, let us compute the derivative of g2 (s, s) on (a, b). After some simplifications,
we obtain
(b − s)α−2 r
(g2 (s, s))0 =
− αs + (1 − α)( − a) + b .
γ
p
0
Then (g2 (s, s)) has a unique zero, attained at the point
s∗ =
b + (1 − α)( pr − a)
.
α
It is easy to see that (g2 (s, s))0 > 0 on (−∞, s∗ ) and (g2 (s, s))0 < 0 on (s∗ , b). On
b−a
the other hand, from the condition pr > α−1
, we obtain easily that s∗ < a. By
continuity of g2 , we deduce that
max g2 (s, s) = g2 (a, a) =
a≤s≤b
Thus
0 ≤ g2 (t, s) ≤
r
α−1
p (b − a)
( pr + b − a)
r
α−1
p (b − a)
,
( pr + b − a)
.
a ≤ t ≤ s ≤ b.
Now, we turn our attention to the function g1 (t, s). Let s ∈ [a, b) be fixed. Differentiating g1 (t, s) with respect to t, we obtain
∂t g1 (t, s) =
(b − s)α−1
− (α − 1)(t − s)α−2 ,
γ
s < t.
It follows from the above equality that ∂t g1 (t, s) = 0 if and only if
t = t∗ = s +
1
(b − s)α−1 α−2
,
γ(α − 1)
provided t∗ ≤ b, i.e. as long as a ≤ s ≤ b − (α − 1)γ. However, from the condition
r
b−a
p > α−1 , we observe easily that b − (α − 1)γ < a. Then we deduce that s >
b − (α − 1)γ, i.e. t∗ > b. In this case, ∂t g1 (t, s) < 0, i.e. g1 (·, s) is strictly decreasing
and, since g1 (b, s) = 0, we conclude that
0 ≤ g1 (t, s) ≤ g1 (s, s) = g2 (s, s) ≤ g2 (a, a) ≤
which concludes the proof.
r
α−1
p (b − a)
( pr + b − a)
a ≤ s ≤ t ≤ b,
EJDE-2015/88
Case 0 ≤
r
p
LYAPUNOV-TYPE INEQUALITIES
≤
5
b−a
α−1 .
Lemma 2.3. Suppose that
0≤
r
b−a
≤
.
p
α−1
Then
Γ(α)|G(t, s)| ≤ max{A(α, r/p), B(α, r/p)},
(t, s) ∈ [a, b] × [a, b],
where
A(α, r/p) =
1
α−2
(b − a)α−1 r
(b − a)α−1
(2
−
α)
−
,
( pr + b − a)
( pr + b − a)(α − 1)α−1
p
r
(α − 1)α−1
.
B(α, r/p) = ( + b − a)α−1
p
αα
Proof. Following the proof of Lemma 2.2, we have
0 ≤ g2 (t, s) ≤ g2 (s, s),
a≤t≤s≤b
and (g2 (s, s))0 has a unique zero, attained at the point
s∗ =
b + (1 − α)( pr − a)
α
.
b−a
Under the condition 0 ≤ pr ≤ α−1
, it is easy to observe that s∗ ∈ [a, b]. Moreover,
0
∗
(g2 (s, s)) > 0 on (−∞, s ) and (g2 (s, s))0 < 0 on (s∗ , b). Then
max g2 (s, s) = g2 (s∗ , s∗ ) = B(α, r/p).
a≤s≤b
Thus we have
0 ≤ g2 (t, s) ≤ B(α, r/p),
a ≤ t ≤ s ≤ b.
Following the proof of Lemma 2.2, for a fixed s ∈ [a, b), ∂t g1 (t, s) = 0 if and only if
t = t∗ = s +
1
(b − s)α−1 α−2
,
γ(α − 1)
provided t∗ ≤ b, i.e. as long as a ≤ s ≤ b − (α − 1)γ. So, if s > b − (α − 1)γ (i.e.
∂t g1 (t, s) has no zeros), then ∂t g1 (t, s) < 0, i.e. g1 (·, s) is strictly decreasing and,
since g1 (b, s) = 0, we obtain
max g1 (t, s) = g1 (s, s) = g2 (s, s),
s≤t≤b
s ∈ (b − (α − 1)γ, b).
It is easy to check that
s∗ ∈ (b − (α − 1)γ, b).
Thus we have
0 ≤ g1 (t, s) ≤ g2 (s∗ , s∗ ) = B(α, r/p),
b − (α − 1)γ < s ≤ t ≤ b.
Now, we have to check the case when a ≤ s ≤ b − (α − 1)γ; i.e., t∗ ≤ b. It is easy
to see that ∂t g1 (t, s) < 0 for t < t∗ and that ∂t g1 (t, s) ≥ 0 for t ≥ t∗ . This together
with the fact that g1 (b, s) = 0 implies that g1 (t∗ , s) ≤ 0 and, therefore, we only
have to show that
|g1 (t∗ , s)| ≤ max {A(α, r/p), B(α, r/p)} ,
s ∈ [a, b − (α − 1)γ].
6
M. JLELI, B. SAMET
EJDE-2015/88
After some simplifications, we obtain
∗
|g1 (t , s)| =
(b − s)
γ
α−1
α−2
(α−1)2
α−2
(2 − α)
(α − 1)
α−1
α−2
−
r
(b − s)α−1
(s − a + ).
γ
p
Let us define the function
h(s) =
(b − s)
(α−1)2
α−2
(2 − α)
α−1
−
α−1
γ α−2 (α − 1) α−2
(b − s)α−1
r
(s − a + ),
γ
p
s ∈ [a, b − (α − 1)γ].
Now, we differentiate h in the interior of [a, b − (α − 1)γ]. We obtain
0
h (s) =
(b − s)
(α−1)2
α−2
3−α
−1
+
α−1
(α − 1)(s − a + pr )(b − s)α−2
γ
(α − 1) α−2 γ α−2
−
(b − s)α−1
.
γ
0
It is clear that h is an increasing function in [a, b − (α − 1)γ]. Then we have
h0 (s) ≤ h0 (b − (α − 1)γ).
On the other hand, after some simplifications, we obtain
h0 (b − (α − 1)γ) = 0,
which yields h0 (s) ≤ 0. Therefore,
h(s) = h(a) = A(α, r/p),
max
a≤s≤b−(α−1)γ
which concludes the proof.
2.3. Lyapunov-type inequalities. We are ready to state and prove our main
results.
Theorem 2.4. If there exists a nontrivial continuous solution of the fractional
boundary-value problem
α
(C
a D u)(t) + q(t)u(t) = 0,
a < t < b, 1 < α < 2,
0
pu(a) − ru (a) = u(b) = 0,
where p > 0,
r
p
>
b−a
α−1
and q : [a, b] → R is a continuous function, then
Z
b
a
Γ(α)
p
.
|q(s)| ds ≥ 1 + (b − a)
r
(b − a)α−1
(2.3)
Proof. Let X = C[a, b] be the Banach space endowed with the norm
kyk∞ = max{|y(t)| : a ≤ t ≤ b}.
It follows from Lemma 2.1 that
Z b
u(t) =
G(t, s)q(s)u(s) ds,
t ∈ [a, b].
a
We obtain
Z
|u(t)| ≤ kuk∞ max |G(t, s)|a≤t,s≤b
b
|q(s)| ds.
a
Now, Lemma 2.2 yields
kuk∞ ≤ kuk∞
r
α−1 Z b
1 p (b − a)
|q(s)| ds,
Γ(α) ( pr + b − a) a
from which the inequality (2.3) follows.
EJDE-2015/88
LYAPUNOV-TYPE INEQUALITIES
7
Similarly, using Lemma 2.1 and Lemma 2.3, we obtain the following result.
Theorem 2.5. If there exists a nontrivial continuous solution of the fractional
boundary-value problem
α
(C
a D u)(t) + q(t)u(t) = 0,
a < t < b, 1 < α < 2,
0
pu(a) − ru (a) = u(b) = 0,
where p > 0, 0 ≤
r
p
≤
Z
b−a
α−1
and q : [a, b] → R is a continuous function, then
b
|q(s)| ds ≥
a
Γ(α)
.
max{A(α, r/p), B(α, r/p)}
(2.4)
2.4. Particular cases.
Case r = 0. In the case r = 0, from Theorem 2.5, taking r = 0 in (2.4), we obtain
Z b
Γ(α)
|q(s)| ds ≥
.
max{A(α, 0), B(α, 0)}
a
On the other hand, we have
2−α
α−1
,
α−1 (b − a)
(α − 1) α−2
(α − 1)α−1
B(α, 0) =
(b − a)α−1 .
αα
Using the inequality (see [5])
A(α, 0) =
2−α
(α − 1)
α−1
α−2
≤
(α − 1)α−1
,
αα
1 < α < 2,
we deuce that
max{A(α, 0), B(α, 0)} = B(α, 0).
Thus we obtain the following result (see [5, Theorem 1]).
Corollary 2.6. If there exists a nontrivial continuous solution of the fractional
boundary-value problem
α
(C
a D u)(t) + q(t)u(t) = 0,
a < t < b, 1 < α < 2,
u(a) = u(b) = 0,
where q : [a, b] → R is a continuous function, then
Z b
Γ(α)αα
.
|q(s)| ds ≥
[(α − 1)(b − a)]α−1
a
b−a
b−a
with α ' 2. In the case pr = α−1
, from Theorem 2.5, taking
Case pr = α−1
r
b−a
p = α−1 in (2.4), we obtain
Z b
Γ(α)
.
|q(s)| ds ≥
b−a
b−a
max{A α, α−1 , B α, α−1
}
a
An easy computation gives us
A α,
b − a (b − a)α−1 2 − α
=
−1 ,
1
α−1
α
α−2
α
8
M. JLELI, B. SAMET
B α,
EJDE-2015/88
b − a (b − a)α−1
=
.
α−1
α
Thus we have
A α,
b−a
b − a (b − a)α−1 2 − α
−2 .
− B α,
=
1
α−1
α−1
α
α α−2
On the other hand,
lim−
α→2
2−α
1
= +∞.
α α−2
Then there exists δ > 0 such that
2−δ <α<2⇒
2−α
1
> 2.
α α−2
Thus for 2 − δ < α < 2, we have
b − a b−a
b−a
, B α,
= A α,
.
max A α,
α−1
α−1
α−1
Hence we have the following result.
Corollary 2.7. There exists δ > 0 such that if there exists a nontrivial continuous
solution of the fractional boundary-value problem
α
(C
a D u)(t) + q(t)u(t) = 0,
a < t < b, 2 − δ < α < 2,
0
pu(a) − ru (a) = u(b) = 0,
where
r
p
=
b−a
α−1
and q : [a, b] → R is a continuous function, then
α−1
Z b
Γ(α)α α−2
.
|q(s)| ds ≥
1
a
(b − a)α−1 (2 − α − α α−2 )
Case p ' 0. Letting p → 0+ in the inequality (2.3), from Theorem 2.4 we obtain
the following result.
Corollary 2.8. If there exists a nontrivial continuous solution of the fractional
boundary-value problem
α
(C
a D u)(t) + q(t)u(t) = 0,
a < t < b, 1 < α < 2,
0
u (a) = u(b) = 0,
where q : [a, b] → R is a continuous function, then
Z b
Γ(α)
|q(s)| ds ≥
.
(b − a)α−1
a
(2.5)
Taking α = 2 in the inequality (2.5), we obtain the following result.
Corollary 2.9. If there exists a nontrivial continuous solution of the boundaryvalue problem
u00 (t) + q(t)u(t) = 0,
a < t < b,
0
u (a) = u(b) = 0,
where q : [a, b] → R is a continuous function, then
Z b
1
|q(s)| ds ≥
.
b
−
a
a
EJDE-2015/88
LYAPUNOV-TYPE INEQUALITIES
9
3. Applications
In this section, we present some applications of our main results.
3.1. Real zeros of certain Mittag-Leffler functions. Let α, β > 0 be fixed.
The complex function
∞
X
zk
Eα,β (z) =
, α > 0, β > 0, z ∈ C
Γ(kα + β)
k=0
is analytic in the whole complex plane; it will be referred to [10, 12] as the MittagLeffler function with parameters (α, β).
Next, using the above Lyapunov-type inequalities, we give intervals where linear
combinations of some Mittag-Leffler functions have no real zeros.
Theorem 3.1. Let 1 < α < 2. The Mittag-Leffler function Eα,1 (x) has no real
zeros for
x ∈ (−Γ(α), 0].
Proof. Let (a, b) = (0, 1), and consider the fractional Sturm-Liouville eigenvalue
problem
α
(C
0 D u)(t) + λu(t) = 0,
0 < t < 1,
0
u (0) = u(1) = 0.
By [3], we know that the eigenvalues λ ∈ R of the above problem satisfy
λ>0
and Eα,1 (−λ) = 0.
The corresponding eigenfunctions are
u(t) = AEα,1 (−λtα ),
t ∈ [0, 1].
By Corollary 2.8, if a real eigenvalue λ exists; i.e., Eα,1 (−λ) = 0, then λ ≥ Γ(α),
which concludes the proof.
Theorem 3.2. Let 1 < α < 2, p > 0,
Mittag-Leffler functions given by
r
p
>
1
α−1 .
The linear combination of
pEα,2 (x) + qrEα,1 (x)
has no real zeros for
p
x ∈ (−(1 + )Γ(α), 0].
r
Proof. Let (a, b) = (0, 1), and consider the following fractional Sturm-Liouville
eigenvalue problem
α
(C
0 D u)(t) + λu(t) = 0,
0 < t < 1,
0
pu(0) − ru (0) = u(1) = 0.
By [3], we know that the eigenvalues λ ∈ R of the above problem satisfies
λ>0
and pEα,2 (−λ) + qrEα,1 (−λ) = 0.
The corresponding eigenfunctions are
p
u(t) = A Eα,1 (−λtα ) + tEα,2 (−λtα ) , t ∈ [0, 1].
r
By Theorem 2.4, if a real eigenvalue λ exists, then λ ≥ (1+ pr )Γ(α), which concludes
the proof.
10
M. JLELI, B. SAMET
EJDE-2015/88
3.2. Applications to fractional boundary-value problems. In this section, we
apply the results on the Liapunov-type inequalities obtained previoulsy to study the
nonexistence of solutions for certain fractional boundary-value problems. Consider
the fractional boundary-value problem
α
(C
0 D u)(t) + q(t)u(t) = 0,
0 < t < 1, 3/2 < α < 2,
(3.1)
with the boundary conditions
u(0) − 2u0 (0) = u(1) = 0,
(3.2)
where q : [a, b] → R is a continuous function. We have the following result.
Theorem 3.3. Assume that
Z
1
|q(s)| ds <
0
3
Γ(α).
2
(3.3)
Then(3.1)-(3.2) has no nontrivial solution.
Proof. Assume the contrary, i.e. (3.1)-(3.2) has a nontrivial solution u(t). By
Theorem 2.4 with (p, r) = (1, 2), we obtain
Z 1
3
|q(s)| ds ≥ Γ(α),
2
0
which contradicts assumption (3.3).
Consider now the fractional boundary-value problem
α
(C
0 D u)(t) + q(t)u(t) = 0,
0 < t < 1, 1 < α < 2,
(3.4)
with the boundary conditions
2u(0) − u0 (0) = u(1) = 0,
(3.5)
where q : [a, b] → R is a continuous function. We have the following result.
Theorem 3.4. Assume that
Z 1
|q(s)| ds <
0
Γ(α)
.
max{A(α, 1/2), B(α, 1/2)}
(3.6)
Then (3.4)-(3.5) has no nontrivial solution.
Proof. Assume the contrary; i.e., (3.4)-(3.5) has a nontrivial solution u(t). By
Theorem 2.5 with (p, r) = (2, 1), we obtain
Z 1
Γ(α)
|q(s)| ds ≥
,
max{A(α, 1/2), B(α, 1/2)}
0
which contradicts assumption (3.6).
Acknowledgements. The authors would like to extend their sincere appreciation
to the Deanship of Scientific Research at King Saud University for the funding of
this research through the Research Group Project No. RGP-VPP-237.
EJDE-2015/88
LYAPUNOV-TYPE INEQUALITIES
11
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Mohamed Jleli
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455,
Riyadh 11451, Saudi Arabia
E-mail address: jleli@ksu.edu.sa
Bessem Samet
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455,
Riyadh 11451, Saudi Arabia
E-mail address: bsamet@ksu.edu.sa