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 which is symmetric with respect to the diagonal. c1999 Elsevier Science B.V. All rights reserved.
Keywords:Fucik spectrum; Discrete Laplacian; M-matrices
1. Introduction
Let a be bounded domain contained in the plane R2 and u be dened on and then dene
u+= Max (u;0); u−
= Min (u;0):
Then the Fucik spectrum of the Laplacian with a Dirichlet boundary condition is dened as the set of those (; ) such that
−u=u+−u−
on ;
u= 0 on @ ; (1)
has a nontrivial weak solution.
In the discrete case, assume that x∈Rn and then dene
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
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 dened as in [4]. It is known that M-matrices are real, symmetric, irreducible, diagonally dominant and positive denite. Hence their eigenvalues are all positive and the rst eigen-value 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 x6= 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
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 denitions:
and then dene the sets Nr and Mr by
Nr={x∈Rn: 1(x) = 0}; Mr={x∈Rn: 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.
Denition 3. Dene the function by
(x) =hAx; xi −1kxk2 :
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 ! ∈ M 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 constantsa,b,c and a vectoruthat belongs to the subgradient (see [2]) of 2(x) at ! and such that
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 Denition 1 then
1. The function (x) given in Denition 3 attains its minimum on Mr∩Nr at the point !. Thus
there exists a positive function
(r) = min{ (x): x∈Mr∩Nr}= (!)
2. ! is a solution of the equation
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) andv∈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
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
(n) =
(2 +1
r2 +r): r ¿0
:
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 Fu cik 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.
