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Journal of Computational and Applied Mathematics 103 (1999) 93–97

Discrete analogue of Fucik spectrum of the Laplacian
Pedro C. Espinoza
Universidad de Lima, Lima, Peru
Received 16 May 1997; received in revised form 26 June 1997

Abstract
The discrete analog of the Fucik spectrum for elliptic equations, namely M-matrices, is shown to have properties
analogous to the continuum. In particular, the Fucik spectrum of a M-matrix contains a continuous and decreasing curve
c 1999 Elsevier Science B.V. All rights reserved.
which is symmetric with respect to the diagonal.
Keywords: Fucik spectrum; Discrete Laplacian; M-matrices

1. Introduction
Let
a be bounded domain contained in the plane R2 and u be dened on
and then dene
u+ = Max (u; 0) ;

u− = Min (u; 0) :

Then the Fucik spectrum of the Laplacian with a Dirichlet boundary condition is dened as the set
 of those (; ) such that
−
u = u+ − u−
u = 0 on @
;

on
;

(1)

has a nontrivial weak solution.
In the discrete case, assume that x ∈ Rn and then dene
x+ = max(x; 0) ;

x− = min(x; 0) :

If A is a real n × n matrix then the discrete analog of the Fucik spectrum for A is the set (n) of
those (; ) such that

A x = x+ − x− ;

x ∈ Rn ;

(2)

has a nontrivial solution. It is well-known that discretizations of (1) are commonly M-matrices,
particularly when nite-dierence methods are used for the discretization (see [4]). Actually, this
c 1999 Elsevier Science B.V. All rights reserved.
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P.C. Espinoza / Journal of Computational and Applied Mathematics 103 (1999) 93–97

is true, more generally, for elliptic operators in divergence form. So we restrict our attention to
M-matrices.
The discrete analogue (n) appears already in Espinoza [3, Proposition 2.3], where the discrete
analogue of semilinear boundary value problems using variational methods are studied. De Figueiredo

and Gossez [1] have obtained a variational characterization of the rst nontrivial curve contained
in the Fucik Spectrum of a general dierential operator in divergence form, with Dirichlet boundary
condition. They have proved that is a continuous and strictly decreasing curve, which is symmetric
with respect to the diagonal, unbounded and asymptotic to the lines 1 × [1 ; ∞i and [1 ; ∞i × 1 ,
where 1 is the rst eigenvalue of the dierential operator.

2. The discrete analogue
Using the ideas described in [1], we prove that the Fucik spectrum for a n × n M-matrix (n)
contains a curve (n) with the same properties, except for one, as the curve
in . Example 2.8
below shows that the remaining property is not true for the discrete case.
2.1. Some properties of (n)
M-matrices are dened as in [4]. It is known that M-matrices are real, symmetric, irreducible,
diagonally dominant and positive denite. Hence their eigenvalues are all positive and the rst eigenvalue 1 has multiplicity one, with a unique associated norm one eigenvector ’1 whose components
are all positive. From this it is easy to see that:
1. The lines 1 × [1 ; ∞i and [1 ; ∞i × 1 are contained in (n).
2. Because (−x)+ = x− and (−x)− = x+ , (n) is symmetric with respect to the diagonal.
Now suppose that (; ) ∈(n). Then there exists a vector x 6= 0 satisfying (2). Multiplying both
sides of (2) by ’1 , and taking into account that ’1 is the eigenvector with eigenvalue 1 , gives
h(1 − ) x+ − (1 − ) x− ; ’1 i = 0 ;

where h·; ·i is the inner product on Rn and k:k is the norm given by this inner product. If (1 −) 6= 0,
then
hx+ − r x− ; ’1 i = 0

where

r=

1 −  −
x ;
1 − 

which says that there exists a orthogonality relationship between the solution x of (2) and the rst
eigenvalue ’1 of A. This relationship motivates the following denitions:
Denition 1. For each r ¿ 0 dene the functions
1 (x) = 1−
2 (x) = hx

+

kx+ k2 −r kx− k2 ;
− r x− ; ’1 i;

1

and

2

on Rn by

P.C. Espinoza / Journal of Computational and Applied Mathematics 103 (1999) 93–97

95

and then dene the sets Nr and Mr by
Nr = {x ∈ Rn :
n

Mr = {x ∈ R :

1 (x) = 0};
2 (x) = 0}:

Remark 2. It is easy to prove the following:
1. 1 (x) is a dierentiable function and 2 (x) is a Lipschitzian and increasing function.
2. Mr is a closed set and Nr is a compact set; hence Mr ∩ Nr is a compact set.
3. If x ∈ Mr ∩ Nr then x+ 6= 0 and x− 6= 0. Moreover hx− ; ’1 i =
6 0 and hx+ ; ’1 i =
6 0.
4. The subspace h’1 i spanned by ’1 is disjoint from Mr ∩ Nr .
Denition 3. Dene the function

by

(x) = hAx; xi − 1 kx k2 :
The function
plays a fundamental role in the study of discrete analogue (n) of the Fucik
spectrum. Since (x) is a function of class C ∞ on Rn , there exists ! ∈ Mr ∩ Nr , not necessarily
unique, such that

(!) = min { (x): x ∈ Mr ∩ Nr } :
From Remark 2, !+ 6= 0 and !− 6= 0; and thus ! 6∈ h’1 i. Furthermore, as the rst eigenvalue 1
of A is positive and strictly less than the other eigenvalues, (!) ¿ 0.
As (x) and 1 (x) are dierentiable functions and 2 (x) is a Lipschitzian function, by the Lagrange
Multiplier Rule (see [2, p. 347]), there exists nonnegative constants a, b, c and a vector u that belongs
to the subgradient (see [2]) of 2 (x) at ! and such that
2a (A w − 1 !) − 2b (!+ − !− ) + c u = 0;

(3)

and a + b + c = 1. Multiply both sides of the Eq. (3) by ’1 to get c = 0, since hu; ’1 i =
6 0.
If a = 0 then b = 0, which contradicts the Lagrange Multiplier Rule, so a 6= 0. Letting (r) = b=a,
we have
Aw = (1 + (r)) !+ − (1 + r (r)) !− :
Multiplying this equality by ! gives the explicit form of the function :
(r) = (!) = min{ (x): x ∈ Mr ∩ Nr } :
Thus we have proved the following proposition.
Proposition 4. If Mr and Nr are as in Denition 1 then
1. The function (x) given in Denition 3 attains its minimum on Mr ∩ Nr at the point !. Thus

there exists a positive function
(r) = min{ (x): x ∈ Mr ∩ Nr } = (!)
dened for every r ¿ 0;

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P.C. Espinoza / Journal of Computational and Applied Mathematics 103 (1999) 93–97

2. ! is a solution of the equation
A x = x+ − x−
with
(r) = 1 + (r) ;

(r) = 1 + r(r) :

Consequently, the curve
(n) = {((r); (r)): r ¿ 0}
is contained in the discrete analogue of the Fucik spectrum: (n).
Now we will prove the continuity of (r) for r ¿ 0. Recall that (r) and (r) are continuous
functions for every r ¿ 0. So let rm be a sequence of real positive numbers that converge to r ¿ 0.

Also let !m ∈ Mrm ∩ Nrm and ! ∈ Mr ∩ Nr be points where (rm ) = (!m ) and (r) = (!). Since !n is
bounded, there exists a subsequence (denoted in the same way) and v ∈ Rn such that !m → v. Because
of the continuity of 1 (x) , 2 (x) and (x) it follows that v ∈ Mr ∩ Nr and (rm ) = (!m ) → (v).
On the other hand, it is quite easy to prove that there exists m and m , positive real numbers,
such that
zm = m !+ − m !+ ∈ Mrm ∩ Nrm :
Hence (!m )6 (zm ). Moreover, since (m ; m ) → (1; 1), and (v) 6 (!), the continuity of (r)
then follows. It is easy to check that x ∈ Mr ∩ Nr if only if
√
(− r x) ∈ M1=r ∩ N1=r ;
√
from which it follows r (x) = (− r x) and consequently r(r) = (1=r):
Finally, we conclude this section by showing that (n) is a decreasing curve. The functions g
and h are dened as
Denition 5.
g(”) = k(x + ”’1 )+ k2 +r k(x + ”’1 )− k2 ;
h(”) = k(x + ”’1 )+ k2 +(r + t)k(x + ”’1 )− k2 :
It is easy to check that g(”) ¿ 1 for all ” ¿ 0 and thus h(”) ¿ 1. There exists ” ¿ 0 such that
z = ! + ”’1 ∈ Mr+t ∩ Nr+t ;
and then also

z
√
∈ Mr+t ∩ Nr+t :
h(”)
Hence
(r) 6



√

z
h(”)



=

1
1

(z) =
(!) ¡ (!) = (r):
h(”)
h(”)

P.C. Espinoza / Journal of Computational and Applied Mathematics 103 (1999) 93–97

97

Thus we have proved the proposition:
Proposition 6. Let (r), (r) and (n) be as in Proposition 2.6. Then for every r ¿ 0
(a) (r) = (1=r);
(b) (r) and (r) are continuous functions;
(c) (r) is a decreasing function.
Example 7. Consider the matrix
A=



2 −1
:
−1 2


In this case 1 = 1 and
1
(n) = (2 + 2 + r): r ¿ 0
r




:

This is a curve which is not asymptotic to the lines 1 × [1 ; ∞i and [1 ; ∞i × 1 .
Consequently, the asymptotic property in the continuum cannot hold in the discrete case.
References
[1] D.G. De Figueiredo, J.P. Gossez, On the rst curve of the Fucik
 spectrum of an elliptic operator, J. Dierential
Integral Equation 7 (1994) 1285–1302.
[2] K.Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
[3] P.C. Espinoza, Positive-ordered solutions of a discrete analogue of a nonlinear elliptic eigenvalue problems, SIAM J.
Numer. Anal. 31 (1994) 760–767.
[4] R.S. Varga, Matrix Iterative Analysis, Prentice-Hall, Englewood Clis, NJ, 1962.