Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 62, pp. 1–14.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

SMALLEST EIGENVALUES FOR BOUNDARY VALUE

PROBLEMS OF TWO TERM FRACTIONAL DIFFERENTIAL

OPERATORS DEPENDING ON FRACTIONAL

BOUNDARY CONDITIONS

PAUL W. ELOE, JEFFREY T. NEUGEBAUER

Abstract. Let n ≥ 2 be an integer, and let n − 1 < α ≤ n. We consider

eigenvalue problems for two point n− 1, 1 boundary value problems

Dα0+u+ a(t)u+ λp(t)u = 0, 0 < t < 1,

u(i)(0) = 0, i = 0, 1, . . . , n− 2, Dβ
0+
u(1) = 0,

where 0 ≤ β ≤ n − 1 and Dα0+ and Dβ0+ denote standard Riemann-Liouville
differential operators. We prove the existence of smallest positive eigenvalues

and then obtain comparisons of these smallest eigenvalues as functions of both

p and β.

1. Introduction

Let n ∈ N, n ≥ 2, and n − 1 < α ≤ n. Assume a ∈ C[0, 1]. In this paper, we
consider the folowing boundary value problems:

Dα
0+u+ a(t)u+ λ1p(t)u = 0, 0 < t < 1, (1.1)

satisfying the boundary conditions

u(i)(0) = 0, i = 0, 1, . . . , n− 2, Dβ1

0+u(1) = 0, (1.2)

or the problem
Dα

0+u+ a(t)u+ λ2q(t)u = 0, 0 < t < 1, (1.3)

satisfying the boundary conditions

u(i)(0) = 0, i = 0, 1, . . . , n− 2, Dβ2

0+u(1) = 0, (1.4)

where 0 < β1 ≤ β2 ≤ n− 1, or the problem

Dα
0+u+ a(t)u+ λ3r(t)u = 0, 0 < t < 1, (1.5)

satisfying the boundary conditions

u(i)(0) = 0, i = 0, 1, . . . , n− 2, u(1) = 0, (1.6)

where Dα
0+ , and Dβi

0+ , i = 1, 2, are the standard Riemann-Liouville fractional deriva-
tives. Here p, q, and r are continuous nonnegative functions on [0, 1] that do not

2010 Mathematics Subject Classification. 26A33, 34A08, 34A40, 26D20.
Key words and phrases. Riemann-Liouville fractional differential equation;

boundary value problem; principal eigenvalue; fractional boundary conditions.
c©2021. This work is licensed under a CC BY 4.0 license.

Submitted October 14, 2020. Published July 7, 2021.

1



2 P. W. ELOE, J. T. NEUGEBAUER EJDE-2021/62

vanish identically on any nondegenerate compact subinterval of [0, 1] and through-
out this paper, we assume a(t) ≥ 0, 0 ≤ t ≤ 1.

The purpose of this work is to apply Krein-Rutman theory [16] to first, show
the existence of smallest eigenvalues of each of the boundary value problems (1.1),
(1.2), or (1.3), (1.4), or (1.5), (1.6) and second, to compare these eigenvalues when
0 ≤ r(t) ≤ p(t) ≤ q(t) and 0 < β1 ≤ β2 ≤ n− 1.

There is a long tradition to apply Krein-Rutman theory to obtain smallest or
principal eigenvalues for boundary value problems for ordinary differential equations
and we cite for example, [7, 13, 17, 20, 21]. These methods have been applied to and
similar results have been developed for boundary value problems for finite difference
equations and dynamic equations on time scales; see, for example, [1, 9, 12].

With the recent rapid advancements in the study of fractional calculus and frac-
tional differential equations, these methods have applied to boundary value prob-
lems for fractional differential equations (both of Riemann-Liouville and of Caputo
type) and analogous result have been obtained; see [5, 6, 10, 11, 14, 18].

Concerning the first purpose of this work, comparison theorems of Green’s func-
tions of families of boundary value problems have played a key role in the develop-
ment of comparison of principal eigenvalues. For example, in [3], a partial order was
defined on the type of boundary conditions that were specified at the right, and then
comparison theorems for Green’s functions, obtained by Peterson and Ridenhour
[19], were employed to compare principal eigenvalues as a function of the partial
order on the boundary conditions. For the purpose of this article, this is analogous
to comparing principal eigenvalues of (1.1), (1.2) in the case 0 < β1 ≤ β2 ≤ n− 1.
Comparison theorems for Green’s functions of two-point boundary value problems
related to (1.1), (1.2), as a function of β have been obtained [4]; the application to
the comparison of principal eigenvalues is made for the first time in this paper.

Concerning the second purpose of this work, to date, comparisons of principal
eigenvalues for fractional equations have been restricted to the fractional operator
Dα

0+. The comparison of principle eigenvalues for a fractional operator (Dα
0+ + aI)

is new. On the surface, it appears that the analogous comparison theory for ordi-
nary differential equations applies to a general nth order linear ordinary differential
operator. But in the references cited above, the operators are assumed to be dis-
conjugate or right disfocal on the given domains and so, with the Frobenius factor-
ization of disconjugate operators [2], the operator behaves as a one–term operator.
Following the lead provided in [8], we obtain a Neumann series representation for
a Green’s function for the boundary value problem associated with a two–term op-
erator, (Dα

0+ + aI), with boundary conditions (1.2) or (1.6). With this approach,
we obtain the necessary comparison theorems for the associated Green’s functions
and then obtain the comparisons of the eigenvalues.

In what follows, we provide preliminary definitions and results related to the
application of Krein-Rutman theory in Section 2. In Section 3, we construct the
Green’s function for the fractional operator (Dα

0+ + aI) with the boundary condi-
tions (1.2), for β1 = β and 0 < β ≤ n − 1 and for the boundary conditions (1.6)
(with β = 0). We obtain the comparisons of the Green’s function as a function of
β, analogous to the comparison theorems obtained in [4]. In Section 4, we define
the appropriate linear operators associated with each of the boundary value prob-
lems (1.1), (1.2) or (1.3), (1.4) or (1.5), (1.6). We first show the compactness of



EJDE-2021/62 TWO TERM FRACTIONAL EQUATIONS 3

the operators. Then we apply the methods outlined in Section 2 and obtain and
compare smallest eigenvalues.

2. Preliminary definitions and theorems

We first give the definitions of the Riemann-Liouville fractional integral and
fractional derivative.

Definition 2.1. Let ν > 0. The Riemann-Liouville fractional integral of a function
u of order ν, denoted Iν0+u, is defined as

Iν0+u(t) =
1

Γ(ν)

∫ t

0

(t− s)ν−1u(s)ds,

provided the right-hand side exists. Moreover, let n denote a positive integer and
assume n − 1 < α ≤ n. The Riemann-Liouville fractional derivative of order α of
the function u : [0, 1]→ R, denoted Dα

0+u, is defined as

Dα
0+u(t) =

1

Γ(n− α)

dn

dtn

∫ t

0

(t− s)n−α−1u(s)ds = DnIn−α0+ u(t),

provided the right-hand side exists.

Definition 2.2. Let B be a Banach space over R. A closed nonempty subset P of
B is said to be a cone provided

(i) αu+ βv ∈ P, for all u, v ∈ P and all α, β ≥ 0, and
(ii) u ∈ P and −u ∈ P implies u = 0.

Definition 2.3. A cone P is solid if the interior, P◦, of P is nonempty. A cone
P is reproducing if B = P − P; i.e., given w ∈ B, there exist u, v ∈ P such that
w = u− v.

Krasnosel’skĭi [15] showed that every solid cone is reproducing.

Definition 2.4. Let P be a cone in a real Banach space B. If u, v ∈ B, u ≤ v with
respect to P if v − u ∈ P. If both M,N : B → B are bounded linear operators,
M ≤ N with respect to P if Mu ≤ Nu for all u ∈ P.

Definition 2.5. A bounded linear operator M : B → B is u0-positive with respect
to P if there exists u0 ∈ P, u0 6= 0 such that for each u ∈ P, u 6= 0, there exist
k1(u) > 0 and k2(u) > 0 such that k1u0 ≤Mu ≤ k2u0 with respect to P.

The following three results are fundamental to our comparison results and are
attributed to Krasnosel’skĭi [15]. The proof of Theorem 2.7 can be found in Kras-

nosel’skĭi’s book [15]. Theorem 2.8 is provided by Keener and Travis [13] as an ex-

tension of Krasonel’skĭi’s results; a slightly more general result was recently proved
by Webb [22].

Lemma 2.6. Let B be a Banach space over the reals, and let P ⊂ B be a solid
cone. If M : B → B is a linear operator such that M : P\{0} → P◦, then M is
u0-positive with respect to P.

Theorem 2.7. Let B be a real Banach space and let P ⊂ B be a reproducing cone.
Let L : B → B be a compact, u0-positive, linear operator. Then L has an essentially
unique eigenvector in P, and the corresponding eigenvalue is simple, positive, and
larger than the absolute value of any other eigenvalue.



4 P. W. ELOE, J. T. NEUGEBAUER EJDE-2021/62

Theorem 2.8. Let B be a real Banach space and P ⊂ B be a cone. Let both
M,N : B → B be bounded, linear operators and assume that at least one of the
operators is u0-positive. If M ≤ N , Mu1 ≥ λ1u1 for some u1 ∈ P and some
λ1 > 0, and Nu2 ≤ λ2u2 for some u2 ∈ P and some λ2 > 0, then λ1 ≤ λ2.
Furthermore, λ1 = λ2 implies u1 is a scalar multiple of u2.

3. Two term differential operator

To develop the appropriate compact operators, we introduce the appropriate
Banach spaces. Define the Banach Space

B = {u : u = tα−1v, v ∈ C[0, 1]},

with the norm

‖u‖ = |v|0,
where |v|0 = sup

t∈[0,1]

|v(t)| denotes the usual supremum norm. Notice that for u ∈ B,

|u|0 = |tα−1v|0 ≤ tα−1‖u‖,

implying |u|0 ≤ ‖u‖. We also define the Banach space

B1 = {u : u = tα−1v, v ∈ C1[0, 1], v(1) = 0},

with the norm given by ‖u‖1 = |v′|0.
Note that for v ∈ C1[0, 1] and since v(1) = 0, then for 0 ≤ t ≤ 1,

|v(t)| = |v(t)− v(1)| =
∣∣∣ ∫ t

1

v′(s)ds
∣∣∣ ≤ (1− t)|v′|0 ≤ ‖u‖1.

Therefore, |v|0 ≤ ‖u‖1 = |v′|0 and

|u|0 = |tα−1v|0 ≤ tα−1‖u‖1,

implies

|u|0 ≤ ‖u‖1. (3.1)

Let n ∈ N, n ≥ 2, and n − 1 < α ≤ n. Assume a ∈ C[0, 1] a(t) ≥ 0, 0 ≤ t ≤ 1,
and consider a boundary value problem for a nonhomogeneous two–term fractional
differential equation

Dα
0+u+ a(t)u(t) + h(t) = 0, 0 < t < 1, (3.2)

u(i)(0) = 0, i = 0, 1, . . . , n− 2, Dβ
0+u(1) = 0, (3.3)

where 0 ≤ β ≤ n − 1, and Dα
0+ and Dβ

0+ are the standard Riemann-Liouville
derivatives.

The following construction of a Neumann series representation of a Green’s func-
tion can be found in [8]. We provide some details because of our choice of Banach
spaces.

Let 0 ≤ β ≤ n− 1. Let G0(β; t, s) denote the Green’s function for −Dα
0+u = 0,

satisfying the boundary conditions u(i)(0) = 0, i = 0, 1, . . . , n − 2, Dβ
0+u(1) = 0,

which is given by

G0(β; t, s) =


tα−1(1−s)α−1−β

Γ(α) , 0 ≤ t ≤ s ≤ 1,

tα−1(1−s)α−1−β−(t−s)α−1

Γ(α) , 0 ≤ s < t ≤ 1.
(3.4)



EJDE-2021/62 TWO TERM FRACTIONAL EQUATIONS 5

We define

v0(β; t, s) =


(1−s)α−1−β

Γ(α) , 0 ≤ t ≤ s ≤ 1,

(1−s)α−1−β

Γ(α) − (1− st )α−1

Γ(α) , 0 ≤ s < t ≤ 1.
(3.5)

Note that G0(β; t, s) = tα−1v0(β; t, s).
If 0 < β ≤ n− 1, let h ∈ B. It has been shown in [11] that u ∈ B is a solution of

(3.2), (3.3) if, and only if, u ∈ B and u satisfies

u(t) =

∫ 1

0

G0(β; t, s)(a(s)u(s) + h(s)u(s))ds

=

∫ 1

0

G0(β; t, s)a(s)u(s)ds+

∫ 1

0

G0(β; t, s)h(s)ds

= A1u(t) +Ah(t),

(3.6)

where A1 and A have now been respectively defined as

A1u(t) =

∫ 1

0

G0(β; t, s)a(s)u(s)ds, Au(t) =

∫ 1

0

G0(β; t, s)u(s)ds, 0 ≤ t ≤ 1.

(3.7)
If 0 = β, let h ∈ B1. It has been shown in [5] that u ∈ B1 is a solution of (3.2),
(1.6) if, and only if, u ∈ B1 and u satisfies

u(t) =

∫ 1

0

G0(0; t, s)(a(s)u(s) + h(s)u(s))ds

=

∫ 1

0

G0(0; t, s)a(s)u(s)ds+

∫ 1

0

G0(0; t, s)h(s)ds

= A1u(t) +Ah(t).

(3.8)

Remark 3.1. We will suppress dependence on β in the operators A1 and A with
the understanding that if 0 < β ≤ n − 1, the supporting Banach space is B and if
0 = β, the Banach space is B1.

Solving (3.6) for u to obtain (I − A1)u = Ah, or, formally

u =
( ∞∑
n=0

An1
)
Ah.

Before stating and outlining a proof of Theorem 3.3, we state a lemma (see [23,
p. 795]).

Lemma 3.2. Let B denote a Banach space, and assume A : B → B is a linear
operator with operator norm ‖A‖. Let r(A) denote the spectral radius of A. Then

(i) r(A) ≤ ‖A‖;
(ii) if r(A) < 1, then (I − A)−1 =

∑∞
n=0An, where I denotes the identity

operator.

Theorem 3.3. Assume a ∈ C[0, 1], and assume |a|0 < Γ(α). If 0 < β ≤ n − 1,
then a function u ∈ B is a solution of the boundary value problem (3.2), (3.3) if,
and only if, u ∈ B and

u(t) =

∫ 1

0

G(β; t, s)h(s)ds,



6 P. W. ELOE, J. T. NEUGEBAUER EJDE-2021/62

where

G(β; t, s) =

∞∑
n=0

Gn(β; t, s), (3.9)

and for n ≥ 1, n an integer,

Gn(β; t, s) =

∫ 1

0

a(τ)G0(β; t, τ)Gn−1(β; τ, s)dτ. (3.10)

If 0 = β, then a function u ∈ B1 is a solution of the boundary value problem (3.2),
(1.6) if, and only if, u ∈ B1 and

u(t) =

∫ 1

0

G(0; t, s)h(s)ds,

where

G(0; t, s) =

∞∑
n=0

Gn(0; t, s), (3.11)

and for n ≥ 1, n an integer,

Gn(0; t, s) =

∫ 1

0

a(τ)G0(0; t, τ)Gn−1(0; τ, s)dτ. (3.12)

Proof. To obtain (3.9) inductively from (3.10) (or respectively (3.11) from (3.12)),
compute each An1Ah inductively. If

An1Ah =

∫ 1

0

Gn(β; t, s)h(s)ds

then

An+1
1 Ah = A1An1Ah

=

∫ 1

0

G0(β; t, s)a(s)

∫ 1

0

Gn(β; s, r)h(r)drds

=

∫ 1

0

(∫ 1

0

a(τ)G0(β; t, τ)Gn(β; τ, s)dτ

)
h(s)ds

=

∫ 1

0

Gn+1(β; t, s)h(s)ds.

To address the convergence in (3.9), it is shown in [4] that for 0 < β ≤ n− 1,

G0(β;t, s) ≥ 0, (t, s) ∈ [0, 1]× [0, 1],

and so it follows from (3.4) that

0 ≤ G0(β; t, s) ≤ tα−1(1− s)α−1−β

Γ(α)

∣∣∣
(t=1,s=0)

=
1

Γ(α)
.

To see this, it is clear that for s ∈ [0, 1],

G0(β; t, s) ≤ tα−1(1− s)α−1−β

Γ(α)

∣∣∣
t=1

=
(1− s)α−1−β

Γ(α)
.

So now maximize the function of s at s = 0. Assume inductively that for n ≥ 1,

|Gn(β1; t, s)| ≤ |a|n0
Γn+1(α)

. (3.13)



EJDE-2021/62 TWO TERM FRACTIONAL EQUATIONS 7

Then

|Gn+1(β1; t, s)| ≤
∫ 1

0

|a(τ)||G0(β1; t, τ)||Gn(β1; τ, s)|dτ ≤ |a|n+1
0

Γn+2(α)
.

So, (3.13) is valid for each n ≥ 1. Straightforward applications of the Weierstrass
M -test and the ratio test imply the uniform and absolute convergence of (3.9) on
[0, 1]× [0, 1].

For β = 0, to address convergence in (3.11), first define, for n ≥ 1,

vn(0; t, s) =

∫ 1

0

a(τ)v0(0; t, τ)Gn−1(0; τ, s)dτ, (3.14)

where v0(β; t, s) has been defined in (3.5). Then Gn(0; t, s) = tα−1vn(0; t, s) and

G(0; t, s) = tα−1
∞∑
n=0

vn(0; t, s) = tα−1V (0; t, s), (3.15)

where V (0; t, s) =
∑∞
n=0 vn(0; t, s).

It is shown in [5] that
∫ 1

0
v0(0; t, s)ds ∈ C1[0, 1]. Moreover, if h ∈ B1 and

‖h‖1 = 1, which implies by (3.1) that |h|0 ≤ 1, then∣∣∣ d
dt

∫ t

0

(1− s
t )
α−1

Γ(α)
a(s)h(s)ds

∣∣∣ ≤ ∫ t

0

(α− 1)(1− s
t )
α−2

Γ(α)

s

t2
ds|a|0 =

|a|0
Γ(α)

. (3.16)

Thus,

‖vn(0; t, s)‖1 ≤
|a|n0

Γn+1(α)
,

the analogue of (3.13).
To apply Lemma 3.2 and complete the proof, for β = 0 or 0 < β ≤ n − 1,

calculate

‖A1‖ = sup
h∈B,‖h‖=1

‖A1h‖

= sup
h∈B,‖h‖=1

∥∥∥∫ 1

0

G0(β; t, s)a(s)h(s)ds
∥∥∥ ≤ |a|0

Γ(α)
< 1. �

The following inequalities are known for G0 and v0.

Lemma 3.4. The following hold.

(1) G0(β; t, s) ≥ 0 for (t, s) ∈ [0, 1]× [0, 1), 0 ≤ β ≤ n− 1;
(2) G0(β; t, s) > 0 for (t, s) ∈ (0, 1] × [0, 1) for β > 0 and G0(0; t, s) > 0 for

(t, s) ∈ (0, 1)× (0, 1);
(3) v0(β; 0, s) > 0 for s ∈ (0, 1), 0 ≤ β ≤ n− 1;
(4) If 0 ≤ β1 < β2 ≤ n − 1, then G(β1; t, s) < G(β2; t, s) for (t, s) ∈ (0, 1) ×

(0, 1);
(5) If 0 ≤ β1 < β2 ≤ n− 1, then v0(β1; 0, s) < v0(β2; 0, s) for s ∈ (0, 1);
(6) v0(0; 1, s) = 0 for s ∈ (0, 1);
(7) v′0(0; 1, s) < 0 for s ∈ (0, 1).

Proof. The proofs of (1), (2), and (4) can be found in [4]. The proof of (3) can be
found in [18]. For (5), notice that

v0(β2; 0, s)− v0(β1; 0, s) =
1

Γ(α)

[
(1− s)α−1−β2 − (1− s)α−1−β1

]



8 P. W. ELOE, J. T. NEUGEBAUER EJDE-2021/62

=
1

Γ(α)
(1− s)α−1−β2

[
1− (1− s)β2−β1

]
> 0.

So (5) holds. Property (6) can be verified directly. For property (7), notice

v0(0; 1, s) = − (α− 1)s(1− s)α−2

Γ(α)
< 0. �

Because of the construction of G(β; t, s) through (3.10) and (3.9) for β > 0 or
through (3.11) and (3.12), the following extension to Lemma 3.4 is valid. So we
need the following inequalities.

Lemma 3.5. The following hold.

(1) G(β; t, s) ≥ 0 for (t, s) ∈ [0, 1]× [0, 1), 0 ≤ β ≤ n− 1;
(2) G(β; t, s) > 0 for (t, s) ∈ (0, 1] × [0, 1) for β > 0 and G(0; t, s) > 0 for

(t, s) ∈ (0, 1)× (0, 1);
(3) V (β; 0, s) > 0 for s ∈ (0, 1), 0 ≤ β ≤ n− 1;
(4) If 0 ≤ β1 < β2 ≤ n − 1, then G(β1; t, s) < G(β2; t, s) for (t, s) ∈ (0, 1) ×

(0, 1);
(5) If 0 ≤ β1 < β2 ≤ n− 1, then V (β1; 0, s) < V (β2; 0, s) for s ∈ (0, 1);
(6) V (0; 1, s) = 0 for s ∈ (0, 1);
(7) V ′(0; 1, s) < 0 for s ∈ (0, 1).

Proof. If a ≡ 0, G(β; t, s) = G0(β; t, s) and so (1)-(7) hold. Suppose a 6≡ 0. Let
0 ≤ β1 < β2 ≤ n − 1. For (2) and (4), notice for (t, s) ∈ (0, 1) × (0, 1), 0 <
G0(β1; t, s) < G0(β2; t, s). Now assume for k ∈ N, 0 < Gk(β1; t, s) < Gk(β2; t, s).
Then for (t, s) ∈ (0, 1)× (0, 1),

Gk+1(β2; t, s) =

∫ 1

0

a(τ)G0(β2; t, τ)Gk(β2; τ, s)dτ

>

∫ 1

0

a(τ)G0(β1; t, τ)Gk(β1; τ, s)dτ

= Gk+1(β1; t, s) > 0.

So for each n ∈ N, 0 < Gn(β1; t, s) < Gn(β2; t, s) for (t, s) ∈ (0, 1)× (0, 1). Then

G(β2; t, s) =
∞∑
n=0

Gn(β2; t, s) >
∞∑
n=0

Gn(β1; t, s) = G(β1; t, s) > 0.

The proof of (1) is similar. For (3) and (5), similarly notice for s ∈ (0, 1), 0 <
v0(β1; 0, s) < v0(β2; 0, s). Assume for k ∈ N, 0 < vk(β2; 0, s) < vk(β1; 0, s). For
s ∈ (0, 1),

vk+1(β2; 0, s) =

∫ 1

0

a(τ)v0(β2; 0, τ)vk(β2; τ, s)dτ

>

∫ 1

0

a(τ)v0(β1; 0, τ)vk(β1; τ, s)dτ

= vk+1(β1; t, s) > 0.

Thus, for each n ∈ N, 0 < vn(β1; 0, s) < vn(β2; 0, s) for s ∈ (0, 1). This implies

V (β2; 0, s) =

∞∑
n=0

vn(β2; 0, s) >

∞∑
n=0

Gn(β1; 0, s) = V (β1; 0, s) > 0.



EJDE-2021/62 TWO TERM FRACTIONAL EQUATIONS 9

The proofs of (6) and (7) are similar. �

4. Comparison of smallest eigenvalues

We derive existence and comparison results. To do this, we will define integral
operators whose kernels are the Green’s function for −Dα

0+u−a(t)u = 0, satisfying

the boundary conditions u(i)(0) = 0, i = 0, 1, . . . , n − 2, Dβ
0+u(1) = 0, which are

given by (3.9). So u solves (1.1), (1.2) if, and only if,

u(t) = λ1

∫ 1

0

G(β1; t, s)p(s)u(s)ds.

Similarly, u solves (1.3), (1.4) if, and only if,

u(t) = λ2

∫ 1

0

G(β2; t, s)q(s)u(s)ds,

and u solves (1.5), (1.6) if, and only if,

u(t) = λ3

∫ 1

0

G(0; t, s)r(s)u(s)ds.

We define the linear operators

Mu(t) =

∫ 1

0

G(β1; t, s)p(s)u(s)ds, (4.1)

Nu(t) =

∫ 1

0

G(β2; t, s)q(s)u(s)ds, Lu(t) =

∫ 1

0

G(0; t, s)r(s)u(s)ds. (4.2)

Theorem 4.1. The operators M,N,L : B → B are compact. Also, L : B1 → B1 is
compact.

Proof. Let 0 ≤ β ≤ n−1. It is proved in [14] that if 0 < β ≤ n−1, then A : B → B
is compact, where A has been defined in (3.7). For β = 0, it is proved in [5] that
A : B → B is compact.

For the sake of completeness, we remind the reader the technique of proof. Let
h ∈ B so h = tα−1v. If β > 0, v ∈ C[0, 1]; if β = 0, v ∈ C1[0, 1]. Write

Ah(t) = tα−1

∫ 1

0

v0(β; t, s)sα−1v(s)ds = tα−1K0(β)v(t),

where

K0(β)v(t) =

∫ 1

0

v0(β; t, s)sα−1v(s)ds,

and v0 has been defined in (3.5). For β > 0, A : B → B is compact if, and only if,
K0(β) : C[0, 1]→ C[0, 1] is compact; for β = 0, A : B → B is compact if, and only
if, K0(β) : C1[0, 1]→ C1[0, 1] is compact. For β > 0, a standard application of the
Arzela-Ascoli theorem then gives the compactness of K0(β). For β = 0, (3.16) is
employed.

It is clear that the operators A and A1 commute and if h ∈ B, then( ∞∑
n=0

(A1)nA
)
h = A

∞∑
n=0

(A1)nh.



10 P. W. ELOE, J. T. NEUGEBAUER EJDE-2021/62

Thus, if u ∈ B, then pu ∈ B and

Mu(t) =

∫ 1

0

G(β1; t, s)p(s)u(s)ds

=

∞∑
n=0

(A1)nApu

= A
( ∞∑
n=0

An1
)
pu.

Once we argue that u ∈ B implies
(∑∞

n=0An1
)
pu ∈ B; then the compactness of M

is proved by the compactness of A.
The analysis to show the uniform and absolute convergence of (3.9) on [0, 1] ×

[0, 1] can be applied to (
∑∞
n=0An1 )h. Now Ḡ0(β; t, s) = a(s)G0(β; t, s),

Ḡn(β; t, s) =

∫ 1

0

Ḡ0(β; t, τ)Ḡn−1(β; τ, s)dτ,

for each n ≥ 1, and

Ḡ(β; t, s) =

∞∑
n=0

Ḡn(β; t, s) = tα−1
∞∑
n=0

v̄n(β; t, s).

The assumption |a|0 < Γ(α) implies that Ḡ(β; t, s) converges uniformly and abso-
lutely on [0, 1]× [0, 1]. Thus, (

∑∞
n=0An1 ) pu ∈ B.

In a similar way, N : B → B is compact. In [5], it was shown that K0(0) :
C1[0, 1]→ C1[0, 1] is compact and K0(0)u(1) = 0 for any u ∈ B1. Then L : B1 → B1

is compact, which implies L : B → B is also compact. �

We define the cone

P = {u ∈ B : u(t) ≥ 0 for t ∈ [0, 1]},

and the set Ω := {u = tα−1v ∈ B : u(t) > 0 for t ∈ (0, 1], v(0) > 0}. We also define
the cone

P1 = {u ∈ B1 : u(t) ≥ 0 for t ∈ [0, 1]},
and the set Ω1 := {u = tα−1v ∈ B : u(t) > 0 for t ∈ (0, 1), v(0) > 0, v′(1) < 0}.
The proof of the following Lemma 4.2 can be found in [6].

Lemma 4.2. The set Ω ⊂ P◦. Hence the cone P is solid in B and therefore
reproducing.

The proof of the following Lemma 4.3 can be found in [5].

Lemma 4.3. The set Ω1 ⊂ P◦1 . Hence the cone P1 is solid in B1 and therefore
reproducing.

Lemma 4.4. The operators M,N are u0-positive with respect to P.

Proof. We first show M : P → P. Let u ∈ P. Then

Mu(t) =

∫ 1

0

G(β; t, s)p(s)u(s)ds ≥ 0.



EJDE-2021/62 TWO TERM FRACTIONAL EQUATIONS 11

So Mu ∈ P and M : P → P. Next, let u ∈ P \ {0}. Now, there exists a compact
subinterval [a, b] ⊂ [0, 1] such that p(t) > 0 and u(t) > 0 for t ∈ [a, b]. So for
t ∈ (0, 1],

Mu(t) =

∫ 1

0

G(β; t, s)p(s)u(s)ds ≥
∫ b

a

G(β; t, s)p(s)u(s)ds > 0.

Let Mu(t) = tα−1v(t). Then

v(0) =

∫ 1

0

V (β; 0, s)p(s)u(s)ds > 0.

So M : P \ {0} → Ω ⊂ P◦. By Lemma 2.6, M is u0-positive with respect to P.
Similarly, N is u0-positive with respect to P. �

Lemma 4.5. The operators L is u0-positive with respect to P1.

Proof. Following the proof of the previous theorem, L : P1 → P1 and if u ∈ P1\{0},
then Lu(t) > 0 for t ∈ (0, 1). Let Lu(t) = tα−1v(t). Again, similar to above,
v(0) > 0. Finally,

v′(1) =

∫ 1

0

V ′(0; 1, s)r(s)u(s)ds < 0.

So L : P1 \{0} → Ω1 ⊂ P◦1 . By Lemma 2.6, L is u0-positive with respect to P1. �

The following result is a direct consequence of Theorem 2.7.

Theorem 4.6. Let B, B1, P, P1 M , N , and L be defined as earlier. Then M (and
N) has an eigenvalue that is simple, positive, and larger than the absolute value of
any other eigenvalue, with an essentially unique eigenvector that can be chosen to
be in P◦. Similarly, L has an eigenvalue that is simple, positive, and larger than
the absolute value of any other eigenvalue, with an essentially unique eigenvector
that can be chosen to be in P◦1 .

Theorem 4.7. Let B, B1, P, P1 M , N , and L be defined as earlier. Let r(t) ≤
p(t) ≤ q(t) on [0, 1]. Let Λ1, Λ2, and Λ3 be the eigenvalues defined in Theorem 4.6
associated with M , N , and L, respectively, with the essentially unique eigenvectors
u1, u2 ∈ P◦, u3 ∈ P◦1 . Then Λ3 < Λ1 ≤ Λ2, and Λ1 = Λ2 if and only if p(t) = q(t)
on [0, 1] and β1 = β2.

Proof. Let p(t) ≤ q(t) on [0, 1]. So for any u ∈ P and t ∈ [0, 1],

(N −M)u(t) =

∫ 1

0

G(β2; t, s)q(s)u(s)ds−
∫ 1

0

G(β1; t, s)p(s)u(s)ds

≥
∫ 1

0

G(β1; t, s)p(s)u(s)ds−
∫ 1

0

G(β1; t, s)p(s)u(s)ds = 0.

So (N −M)(u) ∈ P for all u ∈ P, or M ≤ N with respect to P. Then, by Theorem
2.8, Λ1 ≤ Λ2.

If p(t) = q(t) on [0, 1] and β1 = β2, then Λ1 = Λ2. Next, suppose p(t) 6= q(t) or
β1 6= β2. If p(t) 6= q(t), then p(t) < q(t) on some subinterval [a, b] ⊂ [0, 1], which
implies (N −M)u1(t) > 0 for t ∈ (0, 1]. Let (N −M)u1(t) = tα−1v(t). So

v(0) =

∫ 1

0

V (β2; 0, s)q(s)u(s)ds−
∫ 1

0

V (β1; 0, s)p(s)u(s)ds



12 P. W. ELOE, J. T. NEUGEBAUER EJDE-2021/62

≥
∫ 1

0

V (β1; 0, s)q(s)u(s)ds−
∫ 1

0

V (β1; 0, s)p(s)u(s)ds

=

∫ 1

0

V (β1; 0, s)(q(s)− p(s))u(s)ds.

Since p(t) < q(t) on [a, b] ⊂ [0, 1], then v(0) > 0. So, (N−M)u1 ∈ Ω ⊂ P◦. So there
exists ε > 0 such that (N −M)u1 − εu1 ∈ P. So Λ1u1 + εu1 = Mu1 + εu1 ≤ Nu1,
implying Nu1 ≥ (Λ1 + ε)u1. Since M ≤ N and Nu2 = Λ2u2, Theorem 2.8 implies
Λ1 + ε ≤ Λ2, or Λ1 < Λ2. Next, suppose β1 6= β2 and p(t) = q(t) on [0, 1].
Then β1 < β2, and by Lemma 3.4 (4), (N − M)u1(t) > 0 for t ∈ (0, 1]. Let
(N −M)u1(t) = tα−1v(t). Then

v(0) =

∫ 1

0

V (β2; 0, s)q(s)u(s)ds−
∫ 1

0

V (β1; 0, s)p(s)u(s)ds

>

∫ 1

0

V (β1; 0, s)q(s)u(s)ds−
∫ 1

0

V (β1; 0, s)p(s)u(s)ds = 0.

So, (N −M)u1 ∈ Ω ⊂ P◦. A similar argument gives that Λ1 < Λ2.
Finally, let p(t) ≥ r(t) on [0, 1]. For u ∈ P and t ∈ [0, 1],

(M − L)u(t) =

∫ 1

0

G(β1; t, s)p(s)u(s)ds−
∫ 1

0

G(0; t, s)r(s)u(s)ds

≥
∫ 1

0

G(0; t, s)r(s)u(s)ds−
∫ 1

0

G(0; t, s)r(s)u(s)ds = 0.

So (M −L)(u) ∈ P for all u ∈ P, or L ≤M with respect to P. Notice Theorem 2.8
only requires M be u0-positive with respect to P. Consequently, by Theorem 2.8,
Λ3 ≤ Λ1. Since u3 ∈ P1, u3 ∈ P. By Lemma 3.4, (M − L)u3(t) > 0 for t ∈ (0, 1].
Let (M − L)u3(t) = tα−1v(t). Then

v(0) =

∫ 1

0

V (β1; 0, s)q(s)u(s)ds−
∫ 1

0

V (0; 0, s)r(s)u(s)ds

>

∫ 1

0

V (0; 0, s)q(s)u(s)ds−
∫ 1

0

V (0; 0, s)r(s)u(s)ds = 0.

So, (M −L)u3 ∈ Ω ⊂ P◦. So there exists ε > 0 such that (M −L)u3− εu3 ∈ P. So
Λ3u3 + εu3 = Lu3 + εu3 ≤ Mu3, implying Mu3 ≥ (Λ3 + ε)u3. Since M ≤ M and
Mu1 = Λ1u1, by Theorem 2.8, Λ3 + ε ≤ Λ1, or Λ3 < Λ1. �

Lemma 4.8. The eigenvalues of (1.1), (1.2) are reciprocals of eigenvalues of M ,
and conversely. Similarly, eigenvalues of (1.3), (1.4) are reciprocals of eigenvalues
of N , and conversely, and eigenvalues of (1.5), (1.6) are reciprocals of eigenvalues
of N , and conversely.

The main result is a direct consequence of Theorem 4.7 and Lemma 4.8.

Theorem 4.9. Assume the hypotheses of Theorem 4.6. Then there exists smallest
positive eigenvalues λ1 and λ2 of (1.1), (1.2) and (1.3), (1.4), and λ3 of (1.5), (1.6),
respectively, each of which is simple, positive, and less than the absolute value of any
other eigenvalue of the corresponding problems. Also, eigenfunctions corresponding
to λ1 and λ2 may be chosen to belong to P◦, and eigenfunctions corresponding to
λ3 can be chosen to belong to P◦1 . Finally, λ3 > λ1 ≥ λ2, and λ1 = λ2 if and only
if p(t) = q(t) for all t ∈ [0, 1] and β1 = β2.



EJDE-2021/62 TWO TERM FRACTIONAL EQUATIONS 13

Acknowledgments. The authors thank Lingju Kong, whose question motivated
this work.

References

[1] C. Chyan, J. Davis, J. Henderson, W. K. C. Yin; Eigenvalue comparisons for differential
equations on a measure chain, Electron. J. Differential Eqns., 1998, (1998), No. 35, 7 pp.

[2] W. Coppel; Disconjugacy, Lecture Notes in Mathematics, 220, Springer-Verlag, New

York/Berlin, 1971.
[3] P. Eloe, J. Henderson; Focal points and comparison theorems for a class of two point boundary

value problems, J. Differential Equations, 103 (1993), No. 2, 375–386.

[4] P. Eloe, J. Lyons, J. Neugebauer; An ordering on Green’s functions for a family of two-
point boundary value problems for fractional differential equations, Commun. Appl. Anal.,

19 (2015), 453–462.

[5] P. Eloe, J. Neugebauer; Existence and comparison of smallest eigenvalues for a fractional
boundary value problem, Electron. J. Differential Equations, 2014, (2014), No. 43, 10 pp.

[6] P. Eloe, J. Neugebauer; Smallest eigenvalues for a right focal boundary value problem, Fract.
Calc. Appl. Anal., 19, (2016), No. 1 11—18.

[7] R. Gentry, C. Travis; Comparison of eigenvalues associated with linear differential equations

of arbitrary order, Trans. Amer. Math. Soc., 223 (1967), 167–179.
[8] J. R. Graef, L. Kong, Q. Kong, M. Wang; Existence and uniqueness of solutions for a fractional

boundary value problem with Dirichlet boundary condition. Electron. J. Qual. Theory Differ.

Equ., 2013 (2013), No. 55, 11 pp.
[9] D. Hankerson, A. Peterson; Comparison of eigenvalues for focal point problems for nth order

difference equations, Differential Integral Egns., 3 (1990), No. 2, 363–380.

[10] J. Henderson, N. Kosmatov; Eigenvalue comparison for fractional boundary value problems
with the Caputo derivative, Fract. Calc. Appl. Anal., 17, (2014), No. 3, 872–880.

[11] J. Henderson, J. Neugebauer; Comparison of smallest eigenvalues for fractional-order nonlocal

boundary value problems, Adv. Dyn. Syst. Appl., 14 (2019), No. 2, 189–199.
[12] J. Hoffacker; Green’s functions and eigenvalue comparisons for a focal problem on time scales,

Comput. Math. Appl., 45 (2003), No. 6–9, 1339–1368.
[13] M. Keener, C. Travis; Positive cones and focal points for a class of nth order differential

equations. Trans. Amer. Math. Soc., 237 (1978), 331–351.

[14] A. M. Koester, J. T. Neugebauer; Smallest Eigenvalues for a fractional boundary value prob-
lem with a fractional boundary condition. J. Nonlinear Funct. Anal., 2017 (2017) Article ID

1, 1–16.

[15] M. Krasnosel’skĭi; Positive Solutions of Operator Equations. Fizmatgiz, Moscow, 1962; Engl.

Transl. P. Noordhoff Ltd., Gronigen, 1964.
[16] M. G. Krein, M. A. Rutman; Linear operators leaving a cone invariant in a Banach space.

Translations Amer. Math. Soc., Series 1, Volume 10, 199-325, American Mathematical Soci-

ety, Providence, RI, 1962.

[17] J. Neugebauer; Methods of extending lower order problems to higher order problems in the
context of smallest eigenvalue comparisons, Electron. J. Qual. Theory Differ. Equ., 2011

(2011), No. 99, 16 pp.
[18] J. Neugebauer; Classifying first extremal points for a fractional boundary value problem with

a fractional boundary condition, Mediterr. J. Math., 14 (2017), No. 4, Paper No. 171, 11 pp.

[19] A. Peterson, J. Ridenhour; Comparison theorems for Green’s functions for focal boundary
value problems, In: Recent Trends in Ordinary Differential Equations, World Sci. Ser. Appl.

Anal., 1, World Sci. Publ., River Edge, NJ, 1992, 493–506.

[20] K. Schmitt, H. Smith; Positive solutions and conjugate points for systems of differential
equations, Nonlinear Anal., 2 (1978), No. 1, 93–105.

[21] E. C. Tomastik; Comparison theorem for conjugate points of systems of nth order nonselfad-

joint differential equations, Proc. Amer. Math. Soc., 96 (1986), No. 3, 437–442.
[22] Jeffrey R. L. Webb; A class of positive linear operators and applications to nonlinear boundary

value problems. Topol. Methods Nonlinear Anal. 39 (2012), No. 2, 221-242.

[23] E. Zeidler; Nonlinear Functional Analysis and its Applications I: Fixed Point Theorems,
Springer-Verlag, New York, 1986.



14 P. W. ELOE, J. T. NEUGEBAUER EJDE-2021/62

Paul W. Eloe

Department of Mathematics, University of Dayton, Dayton, OH 45469, USA

Email address: peloe1@udayton.edu

Jeffrey T. Neugebauer

Department of Mathematics and Statistics, Eastern Kentucky University, Richmond,
KY 40475, USA

Email address: Jeffrey.Neugebauer@eku.edu


	1. Introduction
	2. Preliminary definitions and theorems
	3. Two term differential operator
	4. Comparison of smallest eigenvalues
	Acknowledgments

	References