Uniqueness of Nonlinear Inverse Problem for Sturm–Liouville Operator with Multiple Delays

Abstract

The inverse problem concerns how to reconstruct the operator from given spectral data. The main goal of this paper is to address nonlinear inverse Sturm–Liouville problem with multiple delays. By using a new technique and method: zero function extension, we establish the uniqueness result and practical method for recovering the nonlinear inverse problem from two spectra.

Full text

Vol.:(0123456789)
Journal of Nonlinear Mathematical Physics (2024) 31:13
https://doi.org/10.1007/s44198-024-00176-2
1 3
RESEARCH ARTICLE
Uniqueness ofNonlinear Inverse Problem
forSturm–Liouville Operator withMultiple Delays
ChenghuaGao1· GaofengDu1
Received: 30 October 2023 / Accepted: 30 January 2024
© The Author(s) 2024
Abstract
The inverse problem concerns how to reconstruct the operator from given spectral
data. The main goal of this paper is to address nonlinear inverse Sturm–Liouville
problem with multiple delays. By using a new technique and method: zero function
extension, we establish the uniqueness result and practical method for recovering the
nonlinear inverse problem from two spectra.
Keywords Nonlinear inverse spectral problem· Multiple delays· Uniqueness
Mathematics Subject Classification 34B24· 34A55· 34L05
1 Introduction
With the emergence of various applications in natural sciences, there appeared a
growing interest in inverse problems for differential operator with constant delay.
Regarding the inverse problems of these delay differential operator models, the main
focus is on how to reconstruct the operator from given spectral data. Specifically, for
i=0, 1
, denote by
{𝜆n
,
i}∞
n=1
the spectrum of the boundary value problem of the form
with the delay
𝜏∈(0, 𝜋)
and complex-value potential
q(x)∈L2(𝜏,𝜋)
. It is signifi-
cant in some specific mathematical models, such as, long-term economic forecasting
and automatic control theory. These mathematical models with the delay
𝜏>0
can
describe the real processes more comprehensively than the case of
𝜏=0
. We refer
the reader to monographs [1–3] and the references therein.
(1.1)
−y��(x)+q(x)y(x−𝜏)=𝜆y(x)
,
x∈(
0,
𝜋),
(1.2)
y(
0
)=y(i)(
𝜋
)=
0,
i=0, 1
* Chenghua Gao
gaokuguo@163.com
1 College ofMathematics andStatistics, Northwest Normal University, Lanzhou730070, China

Journal of Nonlinear Mathematical Physics (2024) 31:13
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13 Page 2 of 15
For different delay value
𝜏
, the inverse problems of delay differential operator can
be divided into linear inverse problems and nonlinear inverse problems. In details,
for
𝜏
⩾
𝜋∕2
, the linear inverse problems mean that the characteristic functions
depend linearly on the potential and the various aspects of linear inverse problem
(1.1)–(1.2) were widely studied in [4–6] and other works.
If
0<𝜏<𝜋∕2
, then the inverse problem (1.1)–(1.2) is nonlinear. As well for suf-
ficiently small constant delay
𝜏→0
, the inverse problem (1.1)–(1.2) is closer to the
classical Sturm–Liouville inverse problem [7] and many scholars expect to obtain
unique solvability and non-unique solvability of this kind of problems, see [8–10].
Therefore, it is much more complicated to give unique results for this kind of prob-
lems to some extent than linear ones. Recently, an affirmative answer has been given
in [11] to the solvability of inverse problem (1.1)–(1.2) in nonlinear case. It is worth
noting whether it is linear case or nonlinear case, the classical methods (transforma-
tion operator method, method of spectral mappings) of the inverse spectral problems
do not give reliable results for various classes of nonlocal operators (see [12, 13]).
While there are a number of results about both direct and inverse problems for
various difference or differential operators, see [14–22], there are just a few results
related to differential operators with two or more delays, see [23, 24]. Inspired by
the above mentioned literatures, the purpose of this paper is to study the nonlinear
inverse problem for Sturm–Liouville operator with multiple delays. We set out to
investigate the boundary value problems
S−Li(qj),(i=0, 1, j=1, 2)
of the form
where
𝜆
is the spectral parameter,
2
𝜋
∕5
⩽𝜏
1
<𝜏
2
<𝜋
∕2
,
qj(x)
(
j=1, 2
) are com-
plex-valued functions such that
qj(x)∈L2[
𝜏
j,
𝜋
]
and
qj(x)≡0, x∈[0,
𝜏
j)
. More pre-
cisely, the main goal is to recover Sturm–Liouville operator from given spectra. Let
{
𝜆
ni}∞
n=1,i=0, 1
be the eigenvalues of
S−Li(qj),(i=0, 1, j=1, 2)
and the inverse
problem is formulated as follows:
Inverse problem Given the spectra
{
𝜆
n0}∞
n=1
and
{
𝜆
n1}∞
n=1
, determine the poten-
tial
qj(x)
,
j=1, 2
.
We try to develop a global program for the inverse problem of Sturm–Liouville
operator with multiple delays in nonlinear case and establish a uniqueness theo-
rem. During the whole process of algorithm construction, in order to overcome
the obstacles caused by multiple delays and complex integral terms, we propose
a new technique and method (see Algorithm3.2, Step 7): zeroing the potential
terms in the proper integrand interval, so as to remove some undesirable non-
linear integral terms. Moreover, we generalize the result of [25] and develop the
concept of [11] for inverse Srurm–Liouville problem with constant delay. For
this purpose, we study the relationship between given spectra and characteris-
tic function of
S−Li(qj),(i=0, 1, j=1, 2)
and present some auxiliary assertions
about eigenvalues and characteristic functions in Sect. 2. Compared with the
results of [23] and [24], we obtain the uniqueness theorem and reconstruction
algorithm for
S−Li(qj),(i=0, 1, j=1, 2)
in nonlinear case for a smaller range of
(1.3)
−y��(x)+q1(x)y(x−𝜏1)+q2(x)y(x−𝜏2)=𝜆y(x)
,
x∈(
0,
𝜋),
(1.4)
y(
0
)=y(i)(𝜋)=
0,
i=0, 1,

1 3
Journal of Nonlinear Mathematical Physics (2024) 31:13 Page 3 of 15 13
delay
𝜏j,j=1, 2
. The main result Theorem3.1 and constructive Algorithm3.2 are
given in Sect.3.
2 Preliminary
Let
S(x,𝜆)
be the solution of equation (1.1) satisfying the initial conditions
Then the function
S(x,𝜆)
satisfies the integral equation
We solve the integral equation (2.1) by the method of successive approximations
(see [25]) and following lemma holds.
Lemma 2.1 The Volterra-type integral equation (2.1) has a unique solution
where
Denote the
Δi
(𝜆) ∶= S
(i)
(𝜋,𝜆),i=
0, 1
. Then from (2.2) we have
Lemma 2.2 The eigenvalues
{𝜆ni}∞
n=1
of the boundary value problems
S−Li(qj),(i=0, 1, j=1, 2)
coincide with the zeros of characteristic function
Δi(𝜆),i=0, 1
.
S(0, 𝜆)=0, S�(0, 𝜆)=1.
(2.1)
S
(x,𝜆)= sin
√
𝜆x
√
𝜆
+1
√
𝜆
2
�
k=1∫x
𝜏k
qk(t)sin
√
𝜆(x−t)S(t−𝜏k,𝜆)dt
.
(2.2)
S(x,𝜆)=S0(x,𝜆)+S1(x,𝜆)+S2(x,𝜆),x
⩾
2𝜏2,
S
0(x,𝜆)= sin
√
𝜆x
√𝜆
,
S
1(x,𝜆)= 1
𝜆
2
�
k=1∫x
𝜏k
qk(t)sin √𝜆(x−t)sin √𝜆(t−𝜏k)dt
,
S
2(x,𝜆)= 1
√𝜆
2
�
k=
1∫x
2𝜏k
qk(t)sin
√
𝜆(x−t)S1(t−𝜏k,𝜆)dt.
(2.3)
Δ
0(𝜆)= sin
√
𝜆𝜋
√𝜆
+1
𝜆
2
�
k=1∫𝜋
𝜏k
qk(t)sin
√
𝜆(𝜋−t)sin
√
𝜆(t−𝜏k)dt +S2(𝜋,𝜆),
Δ
1(𝜆)=cos
√
𝜆𝜋 +1
√𝜆
2
�
k=
1∫𝜋
𝜏k
qk(t)cos
√
𝜆(𝜋−t)sin
√
𝜆(t−𝜏k)dt +S�
2(𝜋,𝜆)
.

Journal of Nonlinear Mathematical Physics (2024) 31:13
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13 Page 4 of 15
Proof From Lemma 2.1, it can be known that
S(x,𝜆)
is a solution of initial value
problem. Let
𝜆
be an eigenvalue of boundary value problem (1.1)–(1.2), and
S(x,𝜆)
is the corresponding eigenfunction satisfying the boundary condition (1.2). In view
of classical Sturm–Liouville theory, the eigenvalues of the boundary value problem
S−Li(qj),(i=0, 1, j=1, 2)
coincide with characteristic function
Δi(𝜆),(i=0, 1)
and
it is entire function of
𝜆
.
◻
Note that the characteristic functions
Δi(𝜆),(i=0, 1)
depend nonlinearly on the
potential
qj(x),(j=1, 2)
. By the method mentioned in reference [26] (Ch.1), we
obtain the asymptotical formulae for the eigenvalues
{𝜆ni}∞
n=1,(i=0, 1)
for
n→∞
given by following lemma:
Lemma 2.3 The eigenvalues
{𝜆ni}∞
n=1
, (
i=0, 1
) of the boundary value problem
S−Li(qj),(i=0, 1, j=1, 2)
satisfy asymptotic formula:
In view of Hadamard’s factorization theorem (see [1]), characteristic function
is uniquely determined up to a multiplicative constant by its zeros. Here, the fol-
lowing lemma holds.
Lemma 2.4 (see [25]) The specification of the spectrum
{𝜆ni}∞
n=1
, (
i=0, 1
) uniquely
determines the characteristic function
Δi(𝜆)
by the formulas
Now, from the spectra of
S−Li(qj),(i=0, 1, j=1, 2)
, combined with Lem-
mas2.1–2.4, we construct the connection between the potential
qj(x),j=1, 2
and
the characteristic functions
Δi(𝜆),i=0, 1
. Firstly, we denote
Then
(2.4)

𝜆n0=n+
2

k=1
∫𝜋
𝜏kqk(t)dt cos n𝜏k
2𝜋n+o

1
n
,
(2.5)

𝜆n1=

n−1
2

+
2

k=1
∫𝜋
𝜏kqk(t)dt cos(n−
1
2)𝜏k
2𝜋n+o

1
n
.
(2.6)
Δ
0(𝜆)=𝜋
∞
∏
n=1
𝜆n0−𝜆
n2,Δ1(𝜆)=
∞
∏
n=1
𝜆n1−𝜆
(n−1∕2)2
.
(2.7)

Δ
0(𝜆) ∶=𝜆

Δ0(𝜆)− sin

𝜆𝜋

𝜆
,

Δ1
(𝜆) ∶=

𝜆(Δ
1
(𝜆)−cos

𝜆𝜋)
.

1 3
Journal of Nonlinear Mathematical Physics (2024) 31:13 Page 5 of 15 13
Next, in view of Lemmas2.1–2.4 and relation (2.7)–(2.8), we need to separate the
constant term from the potential function integral term. In order to achieve this goal,
we calculate
(2.8)

Δ
0(𝜆)=
2
�
k=1∫𝜋
𝜏k
qk(t)sin
√
𝜆(𝜋−t)sin
√
𝜆(t−𝜏k)dt +𝜆S2(𝜋,𝜆),

Δ
1(𝜆)=
2
�
k=1
∫𝜋
𝜏k
qk(t)cos
√
𝜆(𝜋−t)sin
√
𝜆(t−𝜏k)dt +
√
𝜆S�
2(𝜋,𝜆)
.
(2.9)
𝜆S
2
(𝜋,𝜆)
=A1
sin √𝜆(𝜋−2𝜏1)
4+A2
sin √𝜆(𝜋−𝜏1−𝜏2)
4
+A3
sin √𝜆(𝜋−𝜏1−𝜏2)
4+A4
sin √𝜆(𝜋−2𝜏2)
4
+∫𝜋−2𝜏1
2𝜏1−𝜋
(B1(x1)+C1(y1)+D1(z1)) sin √𝜆𝜉
8d𝜉
+∫𝜋−2𝜏2
2𝜏2−𝜋
(B4(x4)+C4(y4)+D4(z4)) sin √𝜆𝜉
8d𝜉
+∫𝜋−𝜏1−𝜏2
3𝜏1−𝜏2−𝜋
(B2(x2)+C22(y2)) sin √𝜆𝜉
8d𝜉
+∫𝜋−𝜏1−𝜏2
3𝜏2−𝜏1−𝜋
(B3(x3)+C32(y3)) sin √𝜆𝜉
8d𝜉
+∫3𝜏1−𝜏2−𝜋
𝜏1+𝜏2−𝜋
(C21(y2)) sin √𝜆𝜉
8d𝜉+∫3𝜏2−𝜏1−𝜋
𝜏1+𝜏2−𝜋
(C31(y3)) sin √𝜆𝜉
8d
𝜉
+∫𝜋−𝜏1−𝜏2
𝜋−3𝜏1+𝜏2
D21(z2)sin √𝜆𝜉
8d𝜉+∫𝜋−𝜏1−𝜏2
𝜏1+𝜏2−𝜋
D22(z2)sin √𝜆𝜉
8d𝜉
+∫𝜋−𝜏1−𝜏2
𝜋
−3
𝜏2
+
𝜏1
D31(z3)sin
√
𝜆𝜉
8d𝜉+∫𝜋−3𝜏2+𝜏1
𝜏1
+
𝜏2
−
𝜋
D32(z3)sin
√
𝜆𝜉
8d𝜉,

Journal of Nonlinear Mathematical Physics (2024) 31:13
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13 Page 6 of 15
At this point, we have completed the separation of integral terms of potential. In
(2.9) and (2.10), for detailed data on the integrand, see the Appendix.
We assume that
qj(x)∈AC[𝜏j,𝜋],j=1, 2
. Generally, some minor technical mod-
ifications are needed. Denote
p1
(x)=q
�
1
(x
)
and p
2
(x)=q
�
2
(x
)
. Taking (2.8)–(2.10)
into account and applying integration by part, we arrive at
where
N1=2(q1(𝜋)+q1(𝜏1))
, p01 (𝜉)=p1(
𝜋+𝜏
1
−𝜉
2)
,
L
1=
∫𝜋
𝜏
1
q1(t)
dt
.
(2.10)
√
𝜆S
′
2(𝜋
,
𝜆)
=A1
cos √𝜆(𝜋−2𝜏1)
4+A2
cos √𝜆(𝜋−𝜏1−𝜏2)
4
+A3
cos √𝜆(𝜋−𝜏1−𝜏2)
4+A4
cos √𝜆(𝜋−2𝜏2)
4
−∫𝜋−2𝜏1
2𝜏1−𝜋
(B1(x1)+C1(y1)−D1(z1)) cos √𝜆𝜉
8d𝜉
−∫𝜋−2𝜏2
2𝜏2−𝜋
(B4(x4)+C4(y4)−D4(z4)) cos √𝜆𝜉
8d𝜉
+∫𝜋−𝜏1−𝜏2
3𝜏1−𝜏2−𝜋
(−B2(x2)−C22(y2)) cos √𝜆𝜉
8d𝜉
+∫𝜋−𝜏1−𝜏2
3𝜏2−𝜏1−𝜋
(−B3(x3)−C32(y3)) cos √𝜆𝜉
8d𝜉
+∫3𝜏1−𝜏2−𝜋
𝜏1+𝜏2−𝜋
(−C21(y2)) cos √𝜆𝜉
8d𝜉+∫3𝜏2−𝜏1−𝜋
𝜏1+𝜏2−𝜋
(−C31(y3)) cos √𝜆𝜉
8d𝜉
+∫𝜋−𝜏1−𝜏2
𝜋−3𝜏1+𝜏2
D21(z2)cos √𝜆𝜉
8d𝜉+∫𝜋−𝜏1−𝜏2
𝜏1+𝜏2−𝜋
D22(z2)cos √𝜆𝜉
8d𝜉
+∫𝜋−𝜏1−𝜏2
𝜋−3𝜏2+𝜏1
D31(z3)cos
√
𝜆𝜉
8
d𝜉+∫𝜋−3𝜏2+𝜏1
𝜏1+𝜏2−𝜋
D32(z3)cos
√
𝜆𝜉
8
d𝜉.
(2.11)
4
𝜆
Δ0(𝜆)
=𝜆∫3𝜏1−𝜏2−𝜋
𝜏1+𝜏2−𝜋
(C21(y2)) sin 𝜆𝜉d𝜉
+∫𝜋−𝜏1−𝜏2
𝜋−3𝜏1+𝜏2
D21(z2)sin 𝜆𝜉d𝜉+∫𝜋−𝜏1−𝜏2
𝜏1+𝜏2−𝜋
D22(z2)sin 𝜆𝜉d𝜉
+𝜆∫𝜋−2𝜏1
2𝜏1−𝜋
(B1(x1)+C1(y1)+D1(z1)) sin 𝜆𝜉d𝜉
+∫𝜋−𝜏1−𝜏2
3𝜏1−𝜏2−𝜋
(B2(x2)+C22(y2)) sin 𝜆𝜉d𝜉
−4𝜆L1cos 𝜆(𝜋−𝜏1)+N1sin 𝜆(𝜋−𝜏1)−∫𝜋−𝜏1
𝜏1−𝜋
p01(𝜉)sin 𝜆𝜉d
𝜉
+2

𝜆

A1sin

𝜆(𝜋−2𝜏1)+A2sin

𝜆(𝜋−𝜏1−𝜏2)

,

1 3
Journal of Nonlinear Mathematical Physics (2024) 31:13 Page 7 of 15 13
In relation (2.11), we do zero function extension for partial integrand, that is, it
is always zero outside its corresponding interval (for example
D21(z2)≡0
outside
the interval
(𝜋−𝜏1−𝜏2,𝜋−3𝜏1+𝜏2)
). Therefore, we have
In a similar way, we get
where
N2=2(q2(𝜋)+q2(𝜏2))
, p02 (𝜉)=p2(
𝜋+𝜏
2
−𝜉
2)
,
L
2=
∫𝜋
𝜏
2
q2(t)
dt
.
Similar to the processing techniques in (2.11) and (2.12), we do further calcu-
lations for

Δ1(
𝜆
)
. Therefore, we have
where
M1=−2(q1(𝜋)−q1(𝜏1))
, p01 (𝜉)=p1(
𝜋+𝜏
1
−𝜉
2)
, L1=
∫𝜋
𝜏
1
q1(t)
dt
.
4
√
𝜆
Δ0(𝜆)=
√
𝜆
∫𝜋−3𝜏1+𝜏2
3𝜏1−𝜏2−𝜋
(
−C21(y2)−D21 (z2)+D22 (z2)+B2(x2)+C22 (y2)
)
sin
√
𝜆𝜉d
𝜉
−4√𝜆L1cos √𝜆(𝜋−𝜏1)+N1sin √𝜆(𝜋−𝜏1)−∫𝜋−𝜏1
𝜏1−𝜋
p01(𝜉)sin √𝜆𝜉 d𝜉
+√𝜆∫𝜋−2𝜏1
2𝜏1−𝜋(B1(x1)+C1(y1)+D1(z1))sin √𝜆𝜉d𝜉
+2
√
𝜆
(
A1sin
√
𝜆(𝜋−2𝜏1)+A2sin
√
𝜆(𝜋−𝜏1−𝜏2)
)
.
(2.12)
4√
𝜆
Δ0(𝜆)=
√
𝜆
∫𝜋−𝜏1−𝜏2
𝜏1+𝜏2−𝜋
(
C31(y3)+D31 (z3)+D32(z3)+B3(x3)+C32 (y3)
)sin √
𝜆𝜉d
𝜉
−4√𝜆L1cos √𝜆(𝜋−𝜏1)+N1sin √𝜆(𝜋−𝜏1)−∫𝜋−𝜏2
𝜏2−𝜋
p02(𝜉)sin √𝜆𝜉 d𝜉
+√𝜆∫𝜋−2𝜏2
2𝜏2−𝜋(B4(x4)+C4(y4)+D4(z4))sin √𝜆𝜉d𝜉
+2
√
𝜆
(
A3sin
√
𝜆(𝜋−𝜏1−𝜏2)+A4sin
√
𝜆(𝜋−2𝜏1)
)
,
(2.13)
4
𝜆
Δ1(𝜆)=𝜆∫
𝜋−3𝜏
1
+𝜏
2
3𝜏1−𝜏2−𝜋C21(y2)+D21 (z2)+D22(z2)−B2(x2)−C22 (y2)cos 𝜆𝜉d
𝜉
−4𝜆L1cos 𝜆(𝜋−𝜏1)+N1sin 𝜆(𝜋−𝜏1)−∫𝜋−𝜏1
𝜏1−𝜋
p01(𝜉)sin 𝜆𝜉d𝜉
−𝜆∫𝜋−2𝜏1
2𝜏1−𝜋
(B1(x1)+C1(y1)−D1(z1)) cos 𝜆𝜉d𝜉
+2

𝜆

A1cos

𝜆(𝜋−2𝜏1)+A2cos

𝜆(𝜋−𝜏1−𝜏2)

,

Journal of Nonlinear Mathematical Physics (2024) 31:13
1 3
13 Page 8 of 15
where
M2=−2(q2(𝜋)−q2(𝜏2))
, p02 (𝜉)=p2(
𝜋+𝜏
2
−𝜉
2)
, L2=
∫𝜋
𝜏2
q2(t)
dt
.
Next, we separate the constant term. Denote
Let
Then
(2.14)
4√
𝜆
Δ1(𝜆)=
√
𝜆
∫𝜋−𝜏1−𝜏2
𝜏1+𝜏2−𝜋
(
−C31(y2)+D31 (z3)+D32(z3)−B3(x3)−C32 (y2)
)cos √
𝜆𝜉d
𝜉
+4√𝜆L2sin √𝜆(𝜋−𝜏1)+M2cos √𝜆(𝜋−𝜏1)−∫𝜋−𝜏2
𝜏2−𝜋
p02(𝜉)cos √𝜆𝜉 d𝜉
−√𝜆(∫𝜋−2𝜏2
2𝜏2−𝜋
(B4(x4)+C4(y4)−D4(z4)) cos √𝜆𝜉d𝜉)
+2
√
𝜆
(
A3cos
√
𝜆(𝜋−𝜏1−𝜏2)+A4cos
√
𝜆(𝜋−2𝜏1)
)
,
(2.15)
b
0(𝜆) ∶= 4

𝜆
Δ0(𝜆)+4

𝜆L1cos

𝜆(𝜋−𝜏1)−N1sin

𝜆(𝜋−𝜏1
)
−2

𝜆

A1sin

𝜆(𝜋−2𝜏1)+A2sin

𝜆(𝜋−𝜏1−𝜏2)

,
(2.16)
b
1(𝜆) ∶= 4

𝜆
Δ0(𝜆)+4

𝜆L2cos

𝜆(𝜋−𝜏2)−N2sin

𝜆(𝜋−𝜏2
)
−2

𝜆

A3sin

𝜆(𝜋−𝜏1−𝜏2)+A4sin

𝜆(𝜋−2𝜏2)

;
(2.17)
c
0(𝜆) ∶= 4

𝜆
Δ1(𝜆)−4

𝜆L1sin

𝜆(𝜋−𝜏1)−M1cos

𝜆(𝜋−𝜏1
)
−2

𝜆

A1cos

𝜆(𝜋−2𝜏1)+A2cos

𝜆(𝜋−𝜏1−𝜏2)

,
(2.18)
c
1(𝜆) ∶= 4

𝜆
Δ1(𝜆)−4

𝜆L2sin

𝜆(𝜋−𝜏1)−M2cos

𝜆(𝜋−𝜏1
)
−2

𝜆

A3cos

𝜆(𝜋−𝜏1−𝜏2)+A4cos

𝜆(𝜋−2𝜏1)

.
M
1(𝜉)=
√
𝜆(B1(x1)+C1(y1)+D1(z1)),
M2(𝜉)=
√𝜆(−C21(y2)−D21 (z2)+D22(z2)+B2(x2)+C22 (y2)).

1 3
Journal of Nonlinear Mathematical Physics (2024) 31:13 Page 9 of 15 13
where
M(𝜉)=p01(𝜉)+M1(𝜉)+M2(𝜉)
,
M1(𝜉)≡0
outside the interval
(2𝜏1−𝜋,𝜋−2𝜏1)
and
M2(𝜉)≡0
outside the interval
(3𝜏1−𝜏2−𝜋,𝜋−3𝜏1+𝜏2)
.
Then we get
Let
Then
where
R(𝜉)=−p02 (𝜉)+R1(𝜉)+R2(𝜉)
,
R1(𝜉)≡0
outside the interval
(2𝜏2−𝜋,𝜋−2𝜏2)
and
R2(𝜉)≡0
outside the interval
(𝜏1+𝜏2−𝜋,𝜋−𝜏1−𝜏2)
.
Therefore, we have
Let
Then
(2.19)
b
0(𝜆)=−
∫𝜋−𝜏
1
𝜏1−𝜋
p01(𝜉)sin 𝜆𝜉d𝜉
+∫𝜋−2𝜏1
2𝜏1−𝜋
M1(𝜉)sin 𝜆𝜉d𝜉+∫𝜋−3𝜏1+𝜏2
3𝜏1−𝜏2−𝜋
M2(𝜉)sin 𝜆𝜉d𝜉
=∫𝜋−𝜏1
𝜏1−𝜋−p01(𝜉)+M1(𝜉)+M2(𝜉)sin 𝜆𝜉d𝜉
=∫𝜋−𝜏1
𝜏
1
−𝜋
M(𝜉)sin 𝜆𝜉d𝜉,
(2.20)
p01(𝜉)=M1(𝜉)+M2(𝜉)−M(𝜉).
R
1(𝜉)=
√
𝜆(B4(x4)+C4(y4)+D4(z4)),
R2(𝜉)=
√𝜆(C31(y3)+D31 (z3)+D32(z3)+B3(x3)+C32 (y3)).
(2.21)
b
1(𝜆)=−
∫
𝜋−𝜏
2
𝜏2−𝜋
p02(𝜉)sin 𝜆𝜉d𝜉+∫
𝜋−2𝜏
2
2𝜏2−𝜋
R1(𝜉)sin 𝜆𝜉d
𝜉
+∫𝜋−𝜏1−𝜏2
𝜋−𝜏1−𝜏2
R2(𝜉)sin 𝜆𝜉d𝜉
=∫𝜋−𝜏2
𝜏2−𝜋−p02(𝜉)+R1(𝜉)+R2(𝜉)sin 𝜆𝜉d𝜉
=∫𝜋−𝜏2
𝜏2−𝜋
R(𝜉)sin 𝜆𝜉d𝜉,
(2.22)
p02(𝜉)=R1(𝜉)+R2(𝜉)−R(𝜉).
O
1(𝜉)=
√
𝜆(−B1(x1)−C1(y1)+D1(z1)),
O2
(𝜉)=
√
𝜆(C
21
(y
2
)+D
21
(z
2
)+D
22
(z
2
)−B
2
(x
2
)−C
22
(y
2
))
.

Journal of Nonlinear Mathematical Physics (2024) 31:13
1 3
13 Page 10 of 15
where
O(𝜉)=−p01 (𝜉)+O1(𝜉)+O2(𝜉)
,
O1(𝜉)≡0
outside the interval
(2𝜏1−𝜋,𝜋−2𝜏1)
and
O2(𝜉)≡0
outside the interval
(3𝜏1−𝜏2−𝜋,𝜋−3𝜏1+𝜏2)
.
Then we get
Let
Then
where
Q(𝜉)=−p02 (𝜉)+Q1(𝜉)+Q2(𝜉)
,
Q1(𝜉)≡0
outside the interval
(2𝜏2−𝜋,𝜋−2𝜏2)
and
Q2(𝜉)≡0
outside the interval
(𝜏1+𝜏2−𝜋,𝜋−𝜏1−𝜏2)
.
Then we get
Therefore, we obtain a linear system
Moreover,
(2.23)
c
0(𝜆)=−
∫𝜋−𝜏
1
𝜏1−𝜋
p01(𝜉)cos 𝜆𝜉d𝜉
+∫𝜋−2𝜏1
2𝜏1−𝜋
O1(𝜉)cos 𝜆𝜉d𝜉+∫𝜋−3𝜏1+𝜏2
3𝜏1−𝜏2−𝜋
O2(𝜉)cos 𝜆𝜉d
𝜉
=∫𝜋−𝜏1
𝜏1−𝜋−p01(𝜉)+O1(𝜉)+O2(𝜉)cos 𝜆𝜉d𝜉
=∫𝜋−𝜏1
𝜏
1
−𝜋
O(𝜉)cos 𝜆𝜉d𝜉,
(2.24)
p01(𝜉)=−O(𝜉)+O1(𝜉)+O2(𝜉).
Q
1(𝜉)=
√
𝜆(−B4(x4)−C4(y4)+D4(z4)),
Q2(𝜉)=
√𝜆(−C31(y2)+D31 (z3)+D32(z3)−B3(x3)−C32 (y2)).
(2.25)
c
1(𝜆)=−∫
𝜋−𝜏
2
𝜏2−𝜋
p02(𝜉)cos

𝜆𝜉d𝜉+∫
𝜋−2𝜏
2
2𝜏2−𝜋
Q1(𝜉)cos

𝜆𝜉d𝜉+∫
𝜋−𝜏
1
−𝜏
2
𝜏1+𝜏2−𝜋
Q2(𝜉)cos

𝜆𝜉d
𝜉
=∫𝜋−𝜏2
𝜏2−𝜋−p02(𝜉)+Q1(𝜉)+Q2(𝜉)cos 𝜆𝜉d𝜉
=∫𝜋−𝜏2
𝜏2−𝜋
Q(𝜉)cos

𝜆𝜉d𝜉,
(2.26)
p02(𝜉)=−Q(𝜉)+Q1(𝜉)+Q2(𝜉).
(2.27)
⎧
⎪
⎨
⎪
⎩
p01(𝜉)=M1(𝜉)+M2(𝜉)−M(𝜉)
,
p02(𝜉)=R1(𝜉)+R2(𝜉)−R(𝜉),
p01(𝜉)=O1(𝜉)+O2(𝜉)−O(𝜉),
p
02
(𝜉)=Q
1
(𝜉)+Q
2
(𝜉)−Q(𝜉).

1 3
Journal of Nonlinear Mathematical Physics (2024) 31:13 Page 11 of 15 13
In particular, this yields
3 Inverse Problem
In this section, we present main results: Theorem3.1 and constructive Algorithm3.2.
Theorem3.1 The two spectra
{𝜆ni}∞
n=1
of
S−Li(qj),(i=0, 1, j=1, 2)
can uniquely
determine the potential
q1(x)
and
q2(x)
.
Through the analysis of Sect. 2, the proof of Theorem 3.1 comes from the
Algorithm3.2:
Algorithm 3.2 Solve Inverse problem. Require: Let the spectra
{𝜆ni}∞
n=1ofS−Li(qj),(i=0, 1, j=1, 2)
be given.
(2.28)
p01(𝜉)=
−M(𝜉)−O(𝜉)+M1(𝜉)+O1(𝜉)+M2(𝜉)+O2(𝜉)
2
,
𝜉∈(
2
𝜏1−𝜋
,
𝜋−
2
𝜏1)
,
p02(𝜉)= −R(𝜉)−Q(𝜉)+R1(𝜉)+Q1(𝜉)+R2(𝜉)+Q2(𝜉)
2
,𝜉∈(2𝜏2−𝜋,𝜋−2𝜏2).
(2.29)
{
p1(x)= M(𝜋+𝜏1−2x)−O(𝜋+𝜏1−2x)−M1(𝜋+𝜏1−2x)+O1(𝜋+𝜏1−2x)−M2(𝜋+𝜏1−2x)+O2(𝜋+𝜏1−2x)
2,x∈(
3𝜏1
2,𝜋−𝜏1
2)
,
p2(x)= R(𝜋+𝜏2−2x)−Q(𝜋+𝜏2−2x)−R1(𝜋+𝜏2−2x)+Q1(𝜋+𝜏2−2x)−R2(𝜋+𝜏2−2x)+Q2(𝜋+𝜏2−2x)
2
,x∈(
3𝜏2
2
,𝜋−𝜏2
2
).

Journal of Nonlinear Mathematical Physics (2024) 31:13
1 3
13 Page 12 of 15
Remark 3.2 In the whole construction of the algorithm, we solve the uniqueness
of the inverse problem on some intervals by means of zero function extension. In
details, as in Step 7 and Step 8, we reconstruct
p1(x)
for
x
∈(𝜏1,
3𝜏
1
2
)∪(𝜋−
𝜏
1
2
,𝜋
)
,
p2(x)
for
x
∈(𝜏2,
3𝜏
2
2
)∪(𝜋−
𝜏
2
2
,𝜋
)
. Besides, by linear system (2.27), we determine
the uniqueness of the potential function on the residual interval. It says that the
uniqueness result for nonlinear inverse problem for Sturm–Liouville operator with
multiple delays from two spectra can be established on the whole interval.

1 3
Journal of Nonlinear Mathematical Physics (2024) 31:13 Page 13 of 15 13
Appendix
Acknowledgements The authors are grateful to the referees for his/her careful reading and very helpful
suggestions which improved and strengthened the presentation of this manuscript.
Author Contributions GC and GD study conceptualization and writing the manuscript, while CG gives
the main idea of the paper and supervises GD. All authors completed the paper together and contribute
equally.
Funding This research is supported by National Natural Science Foundation of China [Grant
No.11961060].
Data Availability No data was used for the research described in the article.
A1=− 1
√𝜆∫
𝜋
2𝜏1
q1(t)∫
t−𝜏
1
𝜏1
q1(s)dsdt,A2=− 1
√𝜆∫
𝜋
2𝜏1
q1(t)∫
t−𝜏
1
𝜏2
q1(s)dsdt,
A3=− 1
√𝜆∫𝜋
2𝜏2
q1(t)∫t−𝜏2
𝜏1
q1(s)dsdt,A4=− 1
√𝜆∫𝜋
2𝜏2
q2(t)∫t−𝜏2
𝜏2
q2(s)dsdt.
B1(x)= 1
√𝜆∫x
𝜏1
q1(s)q1(x+𝜏1)ds,B2(x)= 1
√𝜆∫x
𝜏2
q2(s)q1(x+𝜏1)ds,
B3(x)= 1
√𝜆∫x
𝜏1
q1(s)q2(x+𝜏2)ds,B4(x)= 1
√𝜆∫x
𝜏2
q2(s)q2(x+𝜏2)ds.
C1(x)= 1
√𝜆∫𝜋
x
q1(s)q1(s−x+𝜏1)ds,C21(x)= 1
√𝜆∫x
2𝜏1
q1(s)q2(s−x+𝜏2)ds
,
C
22(x)= 1
√𝜆∫𝜋
x
q1(s)q2(s−x+𝜏2)ds,C31(x)= 1
√𝜆∫𝜋
2𝜏2
q2(s)q1(s−x+𝜏1)ds
,
C
32(x)= 1
√𝜆∫𝜋
x
q2(s)q1(s−x+𝜏1)ds,C4(x)= 1
√𝜆∫𝜋
x
q2(s)q2(s−x+𝜏2)ds.
D1(x)=− 1
√𝜆∫𝜋
x
q1(s)q1(x−𝜏1)ds,D21(x)=− 1
√𝜆∫𝜋
2𝜏1
q1(s)q2(x−𝜏1)ds,
D
22(x)=− 1
√𝜆∫𝜋
x
q1(s)q2(x−𝜏1)ds,D31(x)=− 1
√𝜆∫𝜋
2𝜏2
q2(s)q1(x−𝜏2)ds,
D
32(x)=− 1
√𝜆∫𝜋
x
q2(s)q1(x−𝜏2)ds,D4(x)=− 1
√𝜆∫𝜋
x
q2(s)q2(x−𝜏2)ds.
x1=𝜉
2+𝜋
2,y1=𝜉
2+𝜋
2+𝜏1z1=𝜋
2−𝜉
2+𝜏1.
x2=𝜉
2+𝜋
2−𝜏1
2+𝜏2
2,y2=𝜉
2+𝜋
2+𝜏1
2+𝜏2
2,z2=𝜋
2−𝜉
2+𝜏1
2+𝜏2
2.
x3=𝜉
2+𝜋
2+𝜏1
2−𝜏2
2,y3=𝜉
2+𝜋
2+𝜏1
2+𝜏2
2,z3=𝜋
2−𝜉
2+𝜏1
2+𝜏2
2.
x4=𝜉
2
+𝜋
2
,y4=𝜉
2
+𝜋
2
+𝜏2,z4=x4=𝜋
2
−𝜉
2
+𝜏2.

Journal of Nonlinear Mathematical Physics (2024) 31:13
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13 Page 14 of 15
Declarations
Conflict of interest All authors declare no conflicts of interest in this paper.
Ethical approval Not applicable.
Consent to participate Not applicable.
Consent for publication All authors approved the final manuscript and the submission to this journal.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-
mons licence, and indicate if changes were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons licence and your intended use is
not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission
directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen
ses/ by/4. 0/.
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