2021 UNC Greensboro PDE Conference,

Electronic Journal of Differential Equations, Conference 26 (2022), pp. 33–43.

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

BASISNESS OF FUČÍK EIGENFUNCTIONS FOR THE

DIRICHLET LAPLACIAN

FALKO BAUSTIAN, VLADIMIR BOBKOV

Abstract. We provide improved sufficient assumptions on sequences of Fuč́ık

eigenvalues of the one-dimensional Dirichlet Laplacian which guarantee that

the corresponding Fuč́ık eigenfunctions form a Riesz basis in L2(0, π). For
that purpose, we introduce a criterion for a sequence in a Hilbert space to be

a Riesz basis.

1. Introduction

We study basis properties of sequences of eigenfunctions of the Fuč́ık eigenvalue
problem for the one-dimensional Dirichlet Laplacian

−u′′(x) = αu+(x)− βu−(x), x ∈ (0, π),

u(0) = u(π) = 0,
(1.1)

where u+ = max(u, 0) and u− = max(−u, 0). The Fuč́ık spectrum is the set Σ(0, π)
of pairs (α, β) ∈ R2 for which (1.1) possesses a nontrivial classical solution. Any
(α, β) ∈ Σ(0, π) is called Fuč́ık eigenvalue and any corresponding nontrivial classical
solution of (1.1) is called Fuč́ık eigenfunction. The Fuč́ık eigenvalue problem (1.1)
was introduced in [4] and [6] to study elliptic equations with “jumping” nonlinear-
ities, and it has since been widely investigated in various aspects and for different
operators, see, e.g., the surveys [3], [8, Chapter 9.4], and references therein. To the
best of our knowledge, basisness of sequences of Fuč́ık eigenfunctions was considered
for the first time in [2]. In that article, we provided several sufficient assumptions on
sequences of Fuč́ık eigenvalues to obtain Riesz bases of L2(0, π) consisting of Fuč́ık
eigenfunctions. Let us recall that a sequence is a Riesz basis in a Hilbert space if it
is the image of an orthonormal basis of that space under a linear homeomorphism,
see, e.g., [9]. The aim of the present note is to use more general techniques to
significantly improve the results of [2].

Let us describe the structure of the Fuč́ık spectrum Σ(0, π). It is not hard to
see that the lines {1}×R and R×{1} are subsets of Σ(0, π), since they correspond
to sign-constant solutions of (1.1) which are constant multiples of sinx, the first
eigenfunction of the Dirichlet Laplacian in (0, π). The remaining part of Σ(0, π) is

2020 Mathematics Subject Classification. 34L10, 34B08, 47A70.
Key words and phrases. Fucik spectrum; Fucik eigenfunctions; Riesz basis;

Paley-Wiener stability.
©2022 This work is licensed under a CC BY 4.0 license.

Published August 25, 2022.

33



34 F. BAUSTIAN, V. BOBKOV EJDE-2022/CONF/26

exhausted by the hyperbola-type curves

Γn =
{

(α, β) ∈ R2 :
n

2

π√
α

+
n

2

π√
β

= π
}

for even n ∈ N, and

Γn =
{

(α, β) ∈ R2 :
n+ 1

2

π√
α

+
n− 1

2

π√
β

= π
}
,

Γ̃n =
{

(α, β) ∈ R2 :
n− 1

2

π√
α

+
n+ 1

2

π√
β

= π
}

for odd n ≥ 3, see, e.g., [6, Lemma 2.8]. Evidently, (α, β) ∈ Γn for odd n ≥ 3

implies (β, α) ∈ Γ̃n. If u is a Fuč́ık eigenfunction for some (α, β), then so is tu for
any t > 0, while −tu is a Fuč́ık eigenfunction for (β, α). Hence, we neglect the curve

Γ̃n from our investigation of the basis properties of Fuč́ık eigenfunctions. Each sign-
changing Fuč́ık eigenfunction consists of alternating positive and negative bumps,
where positive bumps are described by C1 sin(

√
α(x − x1)), while negative bumps

are described by C2 sin(
√
β(x− x2)), for proper constants C1, C2, x1, x2 ∈ R.

We want to uniquely specify a Fuč́ık eigenfunction for each point of Σ(0, π). In
slight contrast to [2], we normalize Fuč́ık eigenfunctions in such a way that they
are “close” to the functions

ϕk(x) =

√
2

π
sin(kx), k ∈ N,

which form a complete orthonormal system in L2(0, π). This choice will be helpful
in the proof of our main result, Theorem 1.3, below.

Definition 1.1. Let n ≥ 2 and (α, β) ∈ Γn. The normalized Fuč́ık eigenfunction
gnα,β is the C2-solution of the boundary value problem (1.1) with (gnα,β)′(0) > 0 and
which is normalized by

‖gnα,β‖∞ = sup
x∈[0,π]

|gnα,β(x)| =
√

2

π
.

For n = 1, we set g1
α,β = ϕ1 for every (α, β) ∈ ({1} × R) ∪ (R× {1}).

Piecewise definitions of the Fuč́ık eigenfunctions fnα,β =
√
π/2 gnα,β can be found

in the equations (1.2) and (1.3) in [2]. In accordance to [2], we study the basisness
of sequences of Fuč́ık eigenfunctions described by the following definition.

Definition 1.2. We define the Fuč́ık system Gα,β = {gnα(n),β(n)} as a sequence

of normalized Fuč́ık eigenfunctions with mappings α, β : N → R satisfying α(1) =
β(1) = 1 and (α(n), β(n)) ∈ Γn for every n ≥ 2.

We can now formulate our main result on the basisness of Fuč́ık systems which
presents a non-trivial generalization of [2, Theorems 1.4 and 1.9].

Theorem 1.3. Let Gα,β be a Fuč́ık system. Let N be a subset of the even natural
numbers and N∗ = N \N . Assume that∑

n∈N∗

[
1−
〈gnα,β , ϕn〉2

‖gnα,β‖2
]

+ E2
(

sup
n∈N

{4 max(α(n), β(n))

n2

})
< 1, (1.2)



EJDE-2018/CONF/26 BASISNESS OF FUČÍK EIGENFUNCTIONS 35

with supn∈N
{

4 max(α(n), β(n))/n2
}
∈ [4, 9). Here, E : [4, 9) → R is a strictly

increasing function defined as

E(γ) =
2
√

2

π

γ2

√
γ − 1

(
√
γ − 2) sin

(
π√
γ

)
(γ − 1)(2

√
γ − 1)

+
((3 + π2)γ + (9− 2π2)

√
γ − 6)(

√
γ − 2)

3(
√
γ − 1)(

√
γ + 2)(3

√
γ − 2)

+
4√
3π

γ2

√
γ − 1

(
√
γ − 2) sin

(
− 3π√

γ

)
(9− γ)(2

√
γ − 3)(4

√
γ − 3)

+
2

π

γ2

√
γ − 1

(
√
γ − 2)

(16− γ)(3
√
γ − 4)(5

√
γ − 4)

+

√
6

5

2

π

γ2(
√
γ − 2)

√
γ − 1

∞∑
k=5

1

(k2 − γ)((k − 1)
√
γ − k)((k + 1)

√
γ − k)

.

(1.3)

Then Gα,β is a Riesz basis in L2(0, π).

The proof of this theorem is given in Section 3 and is based on a general basisness
criterion provided in Section 2. We visualize special cases of domains on the (α, β)-
plane described in Theorem 1.3 in Figures 1 and 2 below.

Notice that, thanks to the orthonormality of {ϕn}, the terms in the first sum in
(1.2) satisfy

0 ≤ 1−
〈gnα,β , ϕn〉2

‖gnα,β‖2
= ‖gnα,β−ϕn‖2−

(‖gnα,β‖2 − 〈gnα,β , ϕn〉)2

‖gnα,β‖2
≤ ‖gnα,β−ϕn‖2, (1.4)

and we have the explicit bounds

‖gnα,β − ϕn‖2 ≤


8(3+π2)

9
(max(

√
α,
√
β)−n)2

n2 for even n,
8n2(n2+1)

(n−1)4
(
√
α−n)2

n2 for odd n ≥ 3 with α ≥ n2,
10n2(n2+1)

(n+1)4
(
√
β−n)2

n2 for odd n ≥ 3 with β > n2,

(1.5)

see the estimates (3.2), (3.4), (3.5), (3.6) in [2, Section 3]. In view of (1.4), if we
choose N = ∅, then Theorem 1.3 is an improvement of [2, Theorem 1.4].

Let us summarize a few properties of the function E defined in Theorem 1.3, see
the end of Section 3 for a discussion.

Lemma 1.4. The function E has the following properties:

(i) E is continuous in [4, 9).
(ii) Each summand in the definition (1.3) of E is strictly increasing in [4, 9).
(iii) We have E(4) = 0 and E(6.49278 . . .) = 1.
(iv) The infinite sum in the definition (1.3) of E in (4, 9) can be expressed as

follows:√
6

5

2

π

γ2(
√
γ − 2)

√
γ − 1

∞∑
k=5

1

(k2 − γ)((k − 1)
√
γ − k)((k + 1)

√
γ − k)

=

√
6

5

2

π

√
γ

√
γ − 1

∞∑
k=5

( 1

k2 − γ
− 1

k2 − γ
(
√
γ−1)2

)



36 F. BAUSTIAN, V. BOBKOV EJDE-2022/CONF/26

=

√
6

5

1

π(
√
γ − 1)

(
π(
√
γ − 1) cot

( π
√
γ

√
γ − 1

)
− π cot(π

√
γ)− (

√
γ − 2)

)
−
√

6

5

2

π

γ2(
√
γ − 2)

√
γ − 1

4∑
k=1

1

(k2 − γ)((k − 1)
√
γ − k)((k + 1)

√
γ − k)

.

The interval [4, 9) appears naturally in the proof of Theorem 1.3. In fact, Lemma
1.4 (iii) indicates that the highest possible value of supn∈N

{
4 max(α(n), β(n))/n2

}
to satisfy the assumption (1.2) is even smaller than 9.

We obtain the following practical corollary of Theorem 1.3 by applying the upper
bounds (1.5) for the case that N is the set of all even natural numbers, see Figure 1.

Corollary 1.5. Let Gα,β be a Fuč́ık system, and ε > 0. Assume that

sup
n∈N even

{4 max(α(n), β(n))

n2

}
< 6.49278 . . .

and

max(α(n), β(n)) ≤
(
n+
√
cnn

(1−ε)/2
)2

for all odd n ≥ 3,

where

0 ≤ cn <
1− E2

(
sup

n∈N even

{4 max(α(n), β(n))

n2

})
45
((

1− 1
21+ε

)
ζ(1 + ε)− 1

)
with the Riemann zeta function ζ. Then Gα,β is a Riesz basis in L2(0, π).

(A) (B)

Figure 1. The assumptions of Corollary 1.5 are satisfied for
(α(n), β(n)) belonging to bold parts of curves Γn inside the
shaded regions. We have ε = 0.5 for both panels and

supn∈N even

{ 4 max(α(n),β(n))
n2

}
= 5, 6 in panel (A), (B), respectively.

If we assume that the first sum of (1.2) in Theorem 1.3 is vanishing, which
corresponds to cn = 0 for all odd n ≥ 3 in the previous corollary, we obtain the
following result.



EJDE-2018/CONF/26 BASISNESS OF FUČÍK EIGENFUNCTIONS 37

Corollary 1.6. Let Gα,β be a Fuč́ık system such that gnα,β = ϕn for any odd n.
Assume that

sup
n∈N even

{4 max(α(n), β(n))

n2

}
< 6.49278 . . . (1.6)

Then Gα,β is a Riesz basis in L2(0, π).

Figure 2. The assumption (1.6) is satisfied for (α(n), β(n)) be-
longing to bold parts of curves Γn inside the shaded region.

We remark that Corollaries 1.5 and 1.6 are significant improvements of [2, The-
orem 1.9] since each point (α(n), β(n)) ∈ Γn for even n ≥ 2 is free to belong to the
whole angular sector in between the line

β =

(√
sup

n∈N even

{4 max(α(n), β(n))

n2

}
− 1

)−2

α

and its reflection with respect to the main diagonal α = β, and the angle of that
sector is allowed to be larger than the one provided by [2, Theorem 1.9]. We refer
to Figure 2 for the domain on the (α, β)-plane given by Corollary 1.6. Moreover,
Corollary 1.5 improves [2, Theorem 1.9] in the sense that gnα,β for odd n ≥ 3 might
differ from ϕn, see Figure 1.

2. Basisness criterion

In this section, we formulate a useful generalization of the separation of variables
approach of [5] in a real Hilbert space X. The provided criterion will be applied to
the space L2(0, π) to prove our main result, Theorem 1.3, in the subsequent section.

Theorem 2.1. Let M ∈ N. Let N∗, Nm ⊂ N, 1 ≤ m ≤ M , be pairwise disjoint
sets which form a decomposition of the natural numbers, i.e.,

N∗ ∪
M⋃
m=1

Nm = N.

Let {φn} be a complete orthonormal sequence in X and {fn} ⊂ X be a sequence
that can be represented as

fn = φn +

∞∑
k=1

Cmn,kT
m
k φn for every n ∈ Nm, 1 ≤ m ≤M, (2.1)



38 F. BAUSTIAN, V. BOBKOV EJDE-2022/CONF/26

and satisfies

Λ∗ :=
( ∑
n∈N∗

[
1− 〈fn, φn〉

2

‖fn‖2
])1/2

<∞.

In the representation formula (2.1), {Tmk } is a family of bounded linear mappings
from X to itself with bounds ‖Tmk ‖∗ ≤ tmk on the operator norm and {Cmn,k} is a

family of constants with uniform bounds |Cmn,k| ≤ cmk that satisfy

Λm :=

∞∑
k=1

cmk t
m
k <∞. (2.2)

Then {fn} is a basis in X provided that

Λ2
∗ +

M∑
m=1

Λ2
m < 1. (2.3)

If, in addition, the subsequence {fn}n∈N∗ is bounded, then {fn} is a Riesz basis in
X.

Proof. Denote f̃n = ρnfn, where ρn = 1 for n ∈ N \ N∗, and the values of ρn
for n ∈ N∗ will be specified later. Let {an}n∈Ñ be an arbitrary finite sequence of

constants with a finite index set Ñ ⊂ N. Setting Ñ∗ = N∗ ∩ Ñ and Ñm = Nm ∩ Ñ
for every 1 ≤ m ≤M , we obtain

∥∥ ∑
n∈Ñ

an(f̃n − φn)
∥∥ ≤ M∑

m=1

∥∥ ∑
n∈Ñm

an(fn − φn)
∥∥+

∥∥ ∑
n∈Ñ∗

an(ρnfn − φn)
∥∥. (2.4)

For the first sum on the right-hand side of (2.4), we apply the representation (2.1)
and obtain

M∑
m=1

∥∥ ∑
n∈Ñm

an(fn − φn)
∥∥

=

M∑
m=1

∥∥ ∑
n∈Ñm

an

∞∑
k=1

Cmn,kT
m
k φn

∥∥ =
M∑
m=1

∥∥ ∞∑
k=1

Tmk
∑
n∈Ñm

Cmn,kanφn
∥∥

≤
M∑
m=1

∞∑
k=1

∥∥Tmk ∑
n∈Ñm

Cmn,kanφn
∥∥ ≤ M∑

m=1

∞∑
k=1

tmk
∥∥ ∑
n∈Ñm

Cmn,kanφn
∥∥

≤
M∑
m=1

∞∑
k=1

tmk c
m
k

∥∥ ∑
n∈Ñm

anφn
∥∥ =

M∑
m=1

Λm
∥∥ ∑
n∈Ñm

anφn
∥∥,

while for the second sum we obtain∥∥ ∑
n∈Ñ∗

an(ρnfn − φn)
∥∥ ≤ ( ∑

n∈Ñ∗

‖ρnfn − φn‖2
)1/2( ∑

n∈Ñ∗

|an|2
)1/2

.

Let us choose ρn to be a minimizer of the distance ‖ρfn − φn‖2 with respect to ρ.
Since

‖ρfn − φn‖2 = ρ2‖fn‖2 − 2ρ〈fn, φn〉+ 1,



EJDE-2018/CONF/26 BASISNESS OF FUČÍK EIGENFUNCTIONS 39

we readily see that

‖ρnfn−φn‖2 = min
ρ∈R
‖ρfn−φn‖2 = 1− 〈fn, φn〉

2

‖fn‖2
= ‖fn−φn‖2−

(‖fn‖2 − 〈fn, φn〉)2

‖fn‖2

with ρn = 〈fn, φn〉/‖fn‖2. Evidently, we have |ρn| ≤ 1. We remark that in case
of ρn = 0, we get Λ∗ ≥ 1 which violates the assumption (2.3). Applying now the
Cauchy inequality, we deduce from (2.4) that∥∥ ∑

n∈Ñ

an(f̃n − φn)
∥∥ ≤ M∑

m=1

Λm
∥∥ ∑
n∈Ñm

anφn
∥∥+ Λ∗

( ∑
n∈Ñ∗

|an|2
)1/2

≤
( M∑
m=1

Λ2
m + Λ2

∗

)1/2∥∥ ∑
n∈Ñ

anφn
∥∥.

We conclude from the assumption (2.3) that the sequence {f̃n} is Paley-Wiener
near to the complete orthonormal sequence {φn} and, thus, it is a Riesz basis in

X, see, e.g., [9, Chapter 1, Theorem 10]. Clearly, {fn} = {ρ−1
n f̃n} is a basis in X.

Assume that the subsequence {fn}n∈N∗ is bounded. Then there exists 0 < c < 1

such that |ρn| ≥ c for all n ∈ Ñ∗. This is evident for finite N∗ since ρn 6= 0. In the
case of infinite N∗, if we suppose that ρn goes to zero up to a subsequence, then
the sum

Λ∗ =
( ∑
n∈N∗

[
1− 〈fn, φn〉

2

‖fn‖2
])1/2

=
( ∑
n∈N∗

[
1− ρ2

n‖fn‖2
])1/2

does not converge. Recalling ρn = 1 for every n ∈ N\N∗, we obtain 1 ≤ |ρ−1
n | ≤ c−1

for all n ∈ N which implies that {fn} is a Riesz basis in X, see, e.g., [9, Chapter 1,
Theorem 9]. �

In the case N1 = N, Theorem 2.1 simplifies to Theorem D from [5] and for
N∗ = N we get the result of Theorem V-2.21 and Corollary V-2.22 i) from [7] which
were discussed in [2].

Remark 2.2. It can be seen from the proof of Theorem 2.1 that if we weaken the
definition of Λ∗ to

Λ̃∗ :=
( ∑
n∈N∗

‖fn − φn‖2
)1/2

≤ Λ∗,

then we can formulate the following result under the assumptions of Theorem 2.1:
the sequence {fn} is a Riesz basis in X provided that

Λ̃2
∗ +

M∑
m=1

Λ2
m < 1.

The boundedness of the subsequence {fn}n∈N∗ is not required under this modified
assumption.

3. Proof of Theorem 1.3

We prove Theorem 1.3 by applying the general basisness criterion introduced in
the previous section. To determine the bounds on the family of constants {Cmn,k}
in Theorem 2.1 we will make use of the Fourier coefficients of Fuč́ık eigenfunctions
corresponding to Fuč́ık eigenvalues on the first nontrivial curve Γ2. Namely, we



40 F. BAUSTIAN, V. BOBKOV EJDE-2022/CONF/26

provide estimates for the Fourier coefficients of the odd Fourier expansion of the
function

g2
γ,γ/(

√
γ−1)2 =

∞∑
k=1

Ak(γ)ϕk(x)

for γ > 4 which are given by

Ak(γ) =

∫ π

0

g2
γ,γ/(

√
γ−1)2(x)ϕk(x) dx =

2

π

γ2

√
γ − 1

(2−√γ) sin
(
kπ√
γ

)
(k2 − γ)(k2(

√
γ − 1)2 − γ)

,

and of the function

g2
δ/(
√
δ−1)2,δ

=

∞∑
k=1

Ãk(δ)ϕk(x)

for δ > 4 which are given by

Ãk(δ) =

∫ π

0

g2
δ/(
√
δ−1)2,δ

(x)ϕk(x) dx = (−1)kAk(δ).

In the case γ = δ = 4, we have A2 = 1 and Ak = 0 for any other k ∈ N.
Obviously, we have

|A1(γ)| = B1(γ) :=
2

π

γ2

√
γ − 1

(
√
γ − 2) sin

(
π√
γ

)
(γ − 1)(2

√
γ − 1)

(3.1)

and it was shown in [2, Section 5] that

|A2(γ)− 1| ≤ B2(γ) :=
((3 + π2)γ + (9− 2π2)

√
γ − 6)(

√
γ − 2)

3(
√
γ − 1)(

√
γ + 2)(3

√
γ − 2)

. (3.2)

For γ ∈ [4, 9), we clearly have

|A3(γ)| = B3(γ) :=
2

π

γ2

√
γ − 1

(
√
γ − 2)

(
− sin

(
3π√
γ

))
(9− γ)(2

√
γ − 3)(4

√
γ − 3)

(3.3)

and for k ≥ 4 we use the simple estimate

|Ak(γ)| ≤ Bk(γ) :=
2

π

γ2

√
γ − 1

(
√
γ − 2)

(k2 − γ)((k − 1)
√
γ − k)((k + 1)

√
γ − k)

. (3.4)

Evidently, the same bounds hold for Ãk. Numerical calculations with the exact
coefficients show that the used estimates in (3.2) and (3.4) do not influence the
results in a significant way.

Lemma 3.1. Let γ ∈ [4, 9) and k ∈ N. Then Bk is strictly increasing.

Proof. For simplicity, we introduce the change of variables x =
√
γ ∈ [2, 3). The

first derivative of Bk(x2) with k ∈ N \ {1, 3} is a rational function with a positive
denominator and we can easily check that the numerator is positive, as well. Hence,
Bk(γ) with k ∈ N \ {1, 3} is strictly increasing for γ ∈ [4, 9). The first derivative of
B1(x2) takes the form

2x2(x− 1) cos
(
π
x

) [
x(2x4 − 4x3 − x2 + 15x− 8) tan

(
π
x

)
− π(2x4 − 5x3 + 5x− 2)

]
π(x− 1)2(x2 − 1)2(2x− 1)2

.

Noting that x(2x4 − 4x3 − x2 + 15x − 8) > 0 for x ∈ [2, 3), we can use the simple

lower bound tan
(
π
x

)
≥
√

3 to show that the expression in square brackets is positive.



EJDE-2018/CONF/26 BASISNESS OF FUČÍK EIGENFUNCTIONS 41

Since all other terms in the derivative are also positive, we conclude that B1(γ) is
strictly increasing for γ ∈ [4, 9).

Finally, the numerator of the first derivative of B3(x2) is given by

− 2x2
[
x(10x5 + 90x4 − 765x3 + 1872x2 − 1863x+ 648) sin

(3π

x

)
+ 3π(8x6 − 42x5 + 7x4 + 315x3 − 693x2 + 567x− 162) cos

(3π

x

)]
,

(3.5)

whereas the denominator is a positive polynomial. We have sin
(

3π
x

)
< 0 and

cos
(

3π
x

)
< 0 for x ∈ [2, 3), and taking into account that

x(10x5 + 90x4 − 765x3 + 1872x2 − 1863x+ 648) < 0,

3π(8x6 − 42x5 + 7x4 + 315x3 − 693x2 + 567x− 162) > 0,

we employ the estimates

sin
(3π

x

)
< −

(3π

x
− π

)
+

1

6

(3π

x
− π

)3

and cos
(3π

x

)
> −1.

As a result, the expression (3.5) is estimated from below by a polynomial which is
positive for x ∈ [2, 3). Thus, B3(γ) is strictly increasing for γ ∈ [4, 9). �

Now we are ready to prove our main result.

Proof of Theorem 1.3. We apply Theorem 2.1, where we consider X = L2(0, π),
the sequence {fn} is the Fuč́ık system, which is bounded by definition, and the
complete orthonormal set {φn} is given by {ϕn}. We set M = 1 and N1 = N and
choose N∗ = N \ N as assumed in Theorem 1.3. We define the linear operators
T 1
k : L2(0, π)→ L2(0, π) as

T 1
k g(x) = g∗

(kx
2

)
,

where

g∗(x) = (−1)κg(x− πκ) for πκ ≤ x ≤ π(κ+ 1), κ ∈ N ∪ {0},

is the 2π-antiperiodic extension for arbitrary functions g ∈ L2(0, π). In particular,
we have T 1

k sin(nx) = sin
(
knx

2

)
for every even n. It was proven in [2, Appendix B]

that ‖T 1
k ‖∗ = 1 for even k and ‖T 1

k ‖∗ =
√

1 + 1/k for odd k.
Let n ∈ N be fixed and recall that n is even. To begin with, we assume that

α(n) > n2. The Fuč́ık eigenfunction gnα,β has the dilated structure

gnα,β(x) = g2
γn,γn/(

√
γn−1)2

(nx
2

)
with γn =

4α(n)

n2

and, thus, has the odd Fourier expansion

gnα,β(x) = g2
γn,γn/(

√
γn−1)2

(nx
2

)
=

∞∑
k=1

Ak(γn)ϕk

(nx
2

)
=

∞∑
k=1

Ak(γn)T 1
kϕn(x).

From this, we directly see that the representation (2.1) of gnα,β in terms of {ϕn} holds

with the constants C1
n,k = Ak(γn) for k 6= 2 and C1

n,2 = 1 − A2(γn). The bounds

for the constants |C1
n,k| are given by the functions Bk(γn) defined in (3.1), (3.2),



42 F. BAUSTIAN, V. BOBKOV EJDE-2022/CONF/26

(3.3), and (3.4), which are strictly increasing in the interval [4, 9) by Lemma 3.1.
For the case β(n) > n2, the Fuč́ık eigenfunction has the form

gnα,β(x) = g2
δn/(
√
δn−1)2,δn

(nx
2

)
with δn =

4β(n)

n2
,

and by analogous arguments we get the bounds |C1
n,k| ≤ Bk(δn). If α(n) = n2, and

hence β(n) = n2, then we set C1
n,k = 0 for every k ∈ N.

In view of the monotonicity, we have

|C1
n,k| ≤ Bk

(
sup
n∈N

max(γn, δn)
)
.

Therefore, we can provide the following upper estimate on the constant Λ1 defined
in (2.2):

Λ1 ≤
√

2B1

(
sup
n∈N

max(γn, δn)
)

+B2

(
sup
n∈N

max(γn, δn)
)

+

√
4

3
B3

(
sup
n∈N

max(γn, δn)
)

+B4

(
sup
n∈N

max(γn, δn)
)

+

√
6

5

∞∑
k=5

Bk

(
sup
n∈N

max(γn, δn)
)

= E
(

sup
n∈N

max(γn, δn)
)

= E
(

sup
n∈N

{4 max(α(n), β(n))

n2

})
,

with the function E introduced in Theorem 1.3, and E is strictly increasing in [4, 9).
Noticing that we have

Λ∗ =
( ∑
n∈N∗

[
1−
〈gnα,β , ϕn〉2

‖gnα,β‖2
])1/2

,

the assumption (1.2) yields the assumption Λ2
∗ + Λ2

1 < 1 in Theorem 2.1. This
completes the proof of Theorem 1.3. �

We conclude this note by discussing Lemma 1.4. The monotonicity statement (ii)
directly follows from Lemma 3.1, and to obtain the alternative representation (iv),
we make use of the identity

∞∑
k=1

1

k2 − a2
=

1

2a2
− π cot(πa)

2a
, a 6∈ N,

see, e.g., [1, (6.3.13)]. The representation (iv) shows that the function E is con-
tinuous in [4, 9). The combination of the continuity and monotonicity of E allows
us to compute values of E with an arbitrary precision. In particular, we have
E(6.49278 . . .) = 1.

Acknowledgements. V. Bobkov was supported in the framework of implementa-
tion of the development program of Volga Region Mathematical Center (agreement
no. 075-02-2022-888). This work is supported by the German-Russian Interdisci-
plinary Science Center (G-RISC) funded by the German Federal Foreign Office via
the German Academic Exchange Service (DAAD), Project F-2021b-8 d.



EJDE-2018/CONF/26 BASISNESS OF FUČÍK EIGENFUNCTIONS 43

References

[1] M. Abramowitz, I. A. Stegun; Handbook of mathematical functions with formulas, graphs,
and mathematical tables, U.S. Government Printing Office, 1972.

[2] F. Baustian, V. Bobkov; Basis properties of Fuč́ık eigenfunctions, Anal. Math., 48 (2022),

619–648.
[3] M. Cuesta; On the Fuč́ık spectrum of the Laplacian and p-Laplacian, Proceedings of the

“2000 Seminar in Differential Equations”, Kvilda (Czech Republic), 2000.

[4] E. N. Dancer; On the Dirichlet problem for weakly non-linear elliptic partial differential
equations, P. Roy. Soc. Edinb. A, 76 (1977), 283–300.

[5] R. J. Duffin, J. J. Eachus; Some notes on an expansion theorem of Paley and Wiener, Bull.
Am. Math. Soc., 48 (1942), 850–855.

[6] S. Fuč́ık; Boundary value problems with jumping nonlinearities, Čas. Pěst. Mat., 101 (1976),

69–87.
[7] T. Kato; Perturbation theory for linear operators, Springer, 1980.

[8] D. Motreanu, V. V. Motreanu, N. S. Papageorgiou; Topological and variational methods with
applications to nonlinear boundary value problems, Springer, 2014.

[9] R. M. Young; An introduction to nonharmonic Fourier series, Academic Press, 1980.

Falko Baustian

Institute of Mathematics, University of Rostock, Germany
Email address: falko.baustian@uni-rostock.de

Vladimir Bobkov
Institute of Mathematics, Ufa Federal Research Centre, Russia

Email address: bobkov@matem.anrb.ru


	1. Introduction
	2. Basisness criterion
	3. Proof of Theorem ??
	Acknowledgements

	References