Multiple Solutions for Lie´nard Type Generalized Equations
ANITA KIRICHUKA
Daugavpils University,
Vienibas Street 13, LV-5401 Daugavpils,
LATVIA
FELIX SADYRBAEV
University of Latvia,
Institute of Mathematics and Computer Science,
Raina bulvaris 29, LV-1459, Riga,
LATVIA
Abstract: - Two-point boundary value problems for second-order ordinary differential equations of Lie´nard
type are studied. A comparison is made between equations x´´ + f (x) x´2 + g(x) = 0 and x´´ + f (x) x´ + g(x) =
0. In our approach, the Dirichlet boundary conditions are considered. The estimates of the number of solutions
in both cases are obtained. These estimates are based on considering the equation of variations around the
trivial solution and some additional assumptions. Examples and visualizations are supplied.
Key-Words: - Lie´nard equation, ordinary differential equation, phase plane, boundary value problems,
multiplicity of solutions, number of solutions.
Received: September 11, 2022. Revised: April 24, 2023. Accepted: May 11, 2023. Published: June 2, 2023.
1 Introduction
Two-point boundary value problems (BVP) often
arise in various types of mathematical models of
real-world phenomena. Consequently, the literature
on BVP for ordinary differential equations (ODE) is
im- mense. We refer to the books and chapters in
the books, [1], [3], [10], [11], where BVPs for ODE
were studied. Numerous articles are devoted to this
subject.
The first results on BVP for ODE were focused on
the existence of solutions and uniqueness. The
Dirichlet and the Neumann problems shared the
main attention. Then the following issues were
studied: dependence of solutions on parameters;
different boundary conditions; periodic boundary
conditions; multipoint boundary conditions;
functional boundary conditions; BVP for higher
order ODE; BVP for systems of ODE; BVP for
fractional differential equations; BVP for time
scales; the multiplicity of solutions.
The last issue is of great importance. It is a
remarkable feature of nonlinear ODE to have
multiple solutions. Estimates of the number of
solutions were the focus of study in many papers.
In this article, we wish to contribute to the special
approach that is applied in the studies of multiple
solutions to BVP. In the early papers of [10], the
following idea was used.
Imagine that the boundary value problem has a
solution ξ(t). Suppose the variational equation
around ξ(t) is oscillatory. If some other solution,
satisfying the first of boundary conditions exists,
and it is not oscillatory, then between those two
solutions may exist other solutions of the same
BVP.
More precisely, suppose that the problem
x´´+ g(x) = 0, g
C1 (1)
x(a) = 0, x(b) = 0 (2)
is to be studied. Let g (0) ≡ 0. Then there is the
trivial solution ξ(t) ≡ 0 for the problem. Suppose
that:
(A1) There exist solutions x+(t) (resp.: x−(t)) to
the Cauchy problem x´´+g(x) = 0, x(a) = 0, x´(0)
> 0 (resp.: x´´(0) < 0) such that x±(t) do not vanish
in the interval (a, b].
Does this problem have a solution other
than the trivial one?
It has, if the equation of variations
y´´ + gx (0) y = 0 (3)
is oscillatory.
To be definite, the following is true: if
solutions
y+(t) and y−(t) of the Cauchy problems
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y´´ + gx (0) y = 0, y(a) = 0, y´(a) = ±1 (4)
have i zeros in the interval (a, b), and the condition
(A1) holds, then the BVP (1), (2) has at least 2i
more solutions.
The conditions hold for the problem
x´´ + ax − bx3 = 0, x(a) = 0, x(b) = 0, (5)
for instance.
This approach was used in the study of
BVPs in the articles, [5], [6], [7], [8], [9].
In the recent work, [4], the two-point BVPs
were considered for the equation
x´´ + f (x)x´2 + g(x) = 0, (6)
which can be classified as the Liénard equation with
the quadratic dependence on x´. The function g(x) =
ax − bx3 was specified. Several choices of f (x) were
tested. The conclusion was that for the problem (6),
(2) the number of solutions is not less than that for
the problem (5).
The method of investigation was based on
the variable transformation proposed by M. Sabatini
in [12]. After applying this transformation the
equation (6) reduces to the conservative equation of
the form u´´ + h(u) = 0. Sabatini’s approach is
briefly described below.
It seems natural to consider the formally
similar equation
x´´ + f (x)x´ + g(x) = 0 (7)
and to compare the results, concerning the
multiplicity of solutions to BVPs (6), (2) and (7),
(2). This is the main thrust of this paper. In our
considerations functions f(x) and g(x) are
continuously differentiable in [a, b].
After the introduction brief description of
Sabatini’s transformation and the results for
equation (6) follow.
Then the BVP (7), (2) is considered and the
multiplicity results are formulated.
In the Examples section, several BVPs are
studied for different choices of f(x). The function
g(x) is cubic, that is, g(x) = ax − bx3, where a and b
are constants.
2 Equation x´´ + f(x)x´2 + g(x) = 0
2.1 Reduction to Shorter Equation
It is known that this equation by the variable
change
(8)
is reduced to the conservative equation
u´´ + h(u) = 0, h(u) = g(x(u)) eF (x(u)). (9)
The differential equation (9) is of the same form as
equation (1). The boundary conditions (2) in the
new variable are the same
u(a) = 0, u(b) = 0. (10)
The BVP (9), (10) can be qualitatively studied by
the same method, making use of the fact that zeros
of x(t) and the respective u(t) coincide.
The result of this study is the main
conclusion: BVP
x´´ + f (x)x´2 + g(x) = 0, x(a) = 0, x(b) = 0, (11)
generally, has at least the same number of solutions,
as the BVP
x´´ + g(x) = 0, x(a) = 0, x(b) = 0 (12)
had.
2.2 Alternative Approach
The problem (11) could be studied directly. It has
the trivial solution ξ(t) ≡ 0. The variational equation
around the trivial solution is
y´´ + gx (0) y = 0. (13)
Suppose that the solution y(t) with the initial
conditions y(a) = 0, y´(a) = 1 has exactly i zeros in
the interval (a, b) and y(b) ≠ 0. Assume also that
there exist
solutions x+(t) (resp.: x−(t)) to the Cauchy problem
x´´ + f (x)x´2 + g(x) = 0, x(a) = 0, x´(a)>0 (resp.:
x´(a)< 0) such that x±(t) do not vanish in the interval
(a, b] (refer to this assumption as A1quadr.) Then
there exist at least 2i nontrivial solutions of the BVP
(11). The condition A1quadr. fulfils for g(x) = ax −
bx3.
3 Dissipative Equation
Equation (9) is conservative, so equation (6) also is.
Our intent now is to consider the dissipative
equation
x´´ + f (x)x´ + g(x) = 0, (14)
together with the boundary conditions (2).
Suppose that g (0) = 0, so the problem has the
trivial solution ξ(t) ≡ 0. The variational equation at
ξ(t) is
y´´ + (fxx´ + gx) |x≡0y + f |x≡0y´ = 0 (15)
or, the same,
y´´ + f (0) y´ + gx (0) y = 0. (16)
Formulate the condition:
(B1) There exist solutions x+(t) (resp.: x−(t)) to the
Cauchy problem x´´ + f (x)x´ + g(x) = 0, x(a) = 0,
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x´(a) > 0 (resp.: x´(a) < 0) such that x(t) is positive
(resp.: negative) in the interval (a, b].
The following result can be proved.
Theorem 1 Let the condition (B1) hold. Suppose
that the solution y(t) of the Cauchy problem
y´´ + f (0) y´ + gx (0)y = 0, y(a) = 0, y´(a) = 1 (17)
has n zeros in the interval (a, b). Then the BVP (14),
(2) has at least 2n nontrivial solutions.
Scheme of the proof. To avoid many technicalities
and discussion about types of non-extendibility of
solutions of the second order equations, we make
one additional assumption: solutions of (14), x(a) =
0, x´(a) = α, α
[0, αmax] extend to the interval [a,
b], where αmax is the initial value for the first
derivative of a solution x(t) from the condition (B1).
Consider the set S of solutions of the problem
(14), x(a) = 0, x´(a) = α, α
[0, αmax]. Due to the
extendibility of solutions, this set is bounded in
C1[a, b] (Theorem 15.1 in [10]). Then there exists a
number δ > 0 such that for any solution x(t) of S the
arbitrary consecutive t1 and t2 are separated by this
δ, that is, |t1 − t2| > δ. The number δ is dependent on
S only, not on the choice of a solution. This was
proved in [2], with reference to Valle`e Poussin
Theorem, [13]. All solutions in S cross the zero
solution transversally. From these facts and the
continuous dependence on the initial data follows
that zeros ti(α) of solutions in S are continuous
functions of α. Any zero escapes the interval (a, b]
when α goes from zero value to αmax. On any
occasion, ti(αi) = b, a new solution to the boundary
value problem (14), (2) emerges. This process, when
performed up and down (for α positive and
negative) yields at least 2n nontrivial solutions.
4 Conclusion
The estimations of the number of solutions to the
BVP for equations
x´´ + f (x)x´2 + g(x) = 0 (18)
and
x´´ + f (x)x´ + g(x) = 0 (19)
can be produced using the equations of variations (at
the trivial solution ξ(t) ≡ 0)
y´´ + gx (0) y = 0, (20)
for equation (18), and
y´´ + f (0) y´+ gx (0) y = 0, (21)
for equation (19).
Since equations of variations are different, the
estimates of the number of solutions to both
BVPs are also different.
5 Examples
In this section, examples are constructed for the
specific choice of g(x) = ax − bx3.
5.1 Conservative Equation
Consider the problem
x´´ + µx´2 + (ax − bx3) = 0, (22)
x (0) = 0, x (1) = 0. (23)
The phase portrait of equation (22), a = 50, b = 25,
µ = 1 is depicted in Fig. 1. It corresponds to f (x) =µ
and g(x) = ax – bx3 in (18).
Fig. 1: The phase portrait of x´´+µx´
2
+ (ax −
bx
3
) =0, a
=
50, b
=
25, µ
=
1.
The equation of variations around the trivial
solution ξ(t) ≡ 0 is
y´´ + ay = 0. (24)
Let a = 50, b = 25. The equation of variations is
y´´ + 50y = 0. The solution of y (0) = 0, y´ (0) = 1 is
. It has exactly two zeros in the
interval (0, 1) and it is not zero at t = 1. The
condition (A1) is fulfilled also ([4]) and BVP (18),
(2) has at least four nontrivial solutions: three
nontrivial solutions for the initial conditions (see
Fig.2, α ≈ −3.11, α ≈ −3.6605, α ≈ 3.1) in the region
bounded by a homoclinic trajectory and one solution
outside this region (see Fig.2, α ≈ 16.4802).
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Fig. 2:
Graphs x(t) for solutions of the problem
x´´+x´
2
+ (50x − 25x
3
) = 0, x (0) = 0, x (1) = 0, α
≈ 16.4802, α ≈ 3.1, α ≈ −3.11, α ≈ −3.6605.
Consider the equation
x´´ + µ (x
2
− 1) x′
2
+ g(x) = 0, (25)
with the conditions (23), where µ = 1 and g(x)
=50x − 25x
3
. The phase portrait is depicted in
Fig.3.
For equation (25) the number of solutions
for the Dirichlet problem is the same as for
equation
x´´ + µx´2 + (ax − bx3) = 0
. The Dirichlet
problem (25),
(23) has three solutions in the
region bounded by a homoclinic trajectory (α ≈
3.367, α ≈ 4.214, α ≈ −3.378) and one solution
outside this region (α ≈ −12.3317, see in Fig. 4).
Fig. 3: The phase portrait of x´´ + (x
2
– 1) x´
2
+50x−25x
3
= 0.
Fig. 4: Graphs x(t) for solutions of the x´´ + (x
2
–
1) x´
2
+50x−25x
3
, x (0) = 0, x (1) = 0, α ≈ 3.367, α
≈ 4.214, α ≈ −3.378, α ≈ −12.3317.
5.2
Dissipative Equation
Consider the problem
x´´ + µx´+ (ax − bx3) = 0
, x (0) = 0, x (1) = 0. (26)
It corresponds to f (x) = µ and g(x) = ax − bx
3
in
equation (19). The equation of variations around
the trivial solution ξ(t) ≡ 0 is
y´´ − µy´ + ay = 0. (27)
Let again a = 50, b = 25, and µ be an arbitrary
positive number. The solution y(t) of (27) is
y(t) =C
1
+ C
2
, (28)
where λ´ s are solutions of the characteristic
equation
λ
2
− µ λ + a = 0. (29)
The roots of (29) are
λ =
µ ±
. (30)
The complex roots (then
µ
2
− a < 0) correspond
to the oscillatory solutions
y(t) =C
1
t+
C
2
(31)
The solution of (31) with the initial conditions
y (0) = 0, y´ (0) = 1 is
y(t) =C
1
t, (32)
where C1 is the appropriate positive constant.
The estimate for the number of solutions to
the BVP (25) depends on the number of zeros of the
function (32) in the interval (0, 1).
Lemma 2 The number of zeros of the function (32)
in the interval (0, 1) is not greater than the number
of zeros of the function .
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This means that the estimate of the number
of solutions to BVP for the dissipative equation is
not better (better=more solutions) than the estimate
for the conservative equation.
On the other hand, it may be worst. Since µ in (32)
can be chosen so that
is arbitrarily
small, it is possible that
. (33)
Then the function (32) does not vanish in (0, 1].
If additionally, the condition
(B1)
holds, the BVP
may have no nontrivial solutions.
Theorem 3
Suppose that
(34)
for some positive integer i. Then problem (22), and
(23) has at least 2i nontrivial solutions.
Suppose that
(35)
Then problem (26) may have less nontrivial
solutions than problems (22), (23) with the same a,
µ.
If a = 50, then to satisfy the inequality (33), choose
12.67129 < µ < 14.14215. Suppose µ = 14. It
appears (look at the phase portrait in Fig. 5) that the
condition (B1) for the dissipative equation holds.
Fig. 6 shows that there are no nontrivial solutions
for the BVP.
Fig. 5: The phase portrait of x´´ + µx´ + 50x −
25x3 = 0,
µ
=
14.
Fig. 6: Graphs x(t) for solutions of the problem x´´
+ µx´ + 50x − 25x3 = 0, µ = 14, α ≈ 3.367, α ≈
4.214, 0.1 < α <6.1, step 0.5.
Consider the example where in problem (26) a = 50,
b = 25, µ = 1. The phase portrait is depicted in Fig.
7. The variational equation
y´´ − y´ + ay = 0 (36)
along with the initial conditions y (0) = 0, y′ (0) = 1
has a solution y (t) with exactly two zeros in the
interval (0,1) and y (1) ≠ 0. Therefore, by Theorem
3, the corresponding BVP (22), (23) must have at
least 4 nontrivial solutions. It has four, as may be
concluded, looking at Fig. 7.
All four nontrivial solutions are depicted in Fig. 8.
Fig. 7: The phase portrait of x´´ + x´ + 50x − 25x
3
= 0
Fig. 8: Graphs x(t) for solutions of the problem x´´
+ x´ + 50x − 25x
3
=
0, α
≈
5.951, α
≈
8.0183, α
≈
−5.951, α
≈−8.0183
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Contribution of Individual Authors to the
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Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and conclusions.
Sources of Funding:
No funding was received for conducting this study.
Conflict of Interest
The authors have no conflicts of interest to declare
that are relevant to the content of this article.
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