Abstract
In this short note we study a simple discrete Fučík spectrum. Trying to imitate standard
continuous pasting procedures, we derive a more complicated discrete analogue  anchoring.
Using this technique, we show that the problem of finding the parametrization of the
second discrete Fučík branch is equivalent to solving a transcendent equation
MSC: 39A12, 34B15.
Keywords:
Fučík spectrum; nonlinear difference equation; resonance; boundary value problem; elementary functions; second branchDedication
In 1979, deep in the dark days of the Soviet occupation of Central and Eastern Europe, a car with a Belgian couple is crossing the heavily guarded border between West Germany and Czechoslovakia. In their luggage they hide a package with a big amount of cash. If revealed, Jean Mahwin and his wife would end up in a serious trouble. Custom officers wouldn’t believe the true story about a donation collection of West European mathematicians for the widow of the recently deceased young mathematician Svatopluk Fučík. The organizer of the collection is none other than Jean Mawhin himself. The driver not only brings dollars, but also brightens faces of hundreds of decent people who learn about this story. Even today. Mathematicians are commonly depicted as outoftouch and asocial beings. Few people would connect them to acts of courage and compassion. Jean Mawhin defies this stereotype more than anyone else. I am very happy that I can offer my wishes to his 70th birthday. Happy birthday!
1 Introduction
Comparison of related continuous and discrete nonlinear problems reveals a very interesting relationship between these two worlds. In some cases, the finite dimension of discrete function spaces could significantly simplify analysis and provide general results (see [13]). In other situations, the broken discrete topology, in which sequences or vectors appear instead of continuous curves, causes difficulties without analogies in the continuous world. The goal of this paper is to show that the Fučík spectrum is one of the most astonishing examples of the latter type.
The nonlinear generalization of the eigenvalue problem for ODEs by Svatopluk Fučík
[4] was quickly applied in the theory of semilinear boundary problems (e.g., [5,6]). Since then this concept has been extended to more complicated differential operators
(e.g., [7,8]) and studied in general settings of Banach spaces (e.g., [9]). Attempts to analyze the Fučík spectrum for matrices and difference operators have
been less successful (see [1013]). Those works reveal the complexity of the matrix problem, which prevents to fully
describe the spectrum beyond
One possible answer to these complications could be to concentrate on a special class of matrices corresponding to specific difference operators. In this brief note, we follow this avenue and try to apply Fučík’s pasting technique [4] to the simplest discrete problem, a direct counterpart of the original continuous problem
Assuming that
In Section 2, there is a short summary of continuous pasting technique. In Section 3,
we deal with the trivial first branch of (1). In Section 4, we show that (i) one should
rather talk about anchoring than pasting in the case of the second branch, and that
(ii) the problem of finding its parametrization is equivalent to the problem of solving
2 Pasting in continuous case
First, let us briefly recall the pasting technique for the original continuous problem [4]
Lemma 1The piecewise nonlinear BVP (2) has a nontrivial solution if and only if
for
Proof The detailed proof could be found, e.g., in [[6], Chapter 42]. We only provide a seemingly clumsy proof of the construction of the
second branch
□
Figure 1 . Continuous pasting procedure. See Lemma 1 and its proof.
3 Trivial first branch
Let us return back to the discrete problem (1). Following the continuous notation,
we denote the discrete spectrum by
Naturally, the first Fučík branch
As we use the eigenvalues in the sequel, we present a concise proof.
Theorem 2The eigenvalues of (6) are
and the corresponding eigenvectors are
Proof For a fixed
Thus, we have N independent eigenvectors, which finishes the proof. □
The first eigenvalue
4 Second branch and anchoring
Let us move on to the second discrete Fučík branch
Since the solutions on the second branch change sign exactly once and lie on the sine
function again, we seek constants
Then the vector (
is the solution of (1) coupled with the pair
Figure 2
. Discrete anchoring procedure. Illustration of the fact that
Above, we considered
Our first observation is trivial and considers integer values of m. In this case, the transition between the positive and negative parts occurs exactly
at
Lemma 3If
Moreover,
Proof If m is integer, then
Exploiting the symmetry of sine functions, both
Finally, the equality of functions
□
The analysis gets more complicated once we consider noninteger values of m. In this case, the transition between positive and negative parts occurs between
Lemma 4 (Necessary condition)
Let
Proof If
holds for
If we consider the difference equation (1) in
This equality holds if and only if (see (1))
which verifies (14).
Using the same argument at
This result enables us to get both peripheral parts of
Corollary 5
and
Proof Let us consider
Then we can rewrite the equation (1) in
This proves the former part of the statement. The latter follows from the mirror argument. □
Obviously,
Since the problem is solved for
Corollary 6 (Necessary condition II)
Let
Proof One can rewrite equalities (13) and (14) into
Isolating C on the righthand sides of both equations, we get
Now, it suffices to multiply this equality by both denominators to get (15). □
Remark 7 (Anchoring)
Corollary 6 implies that the (continuous extensions of) sine functions (9) and (10)
do not intersect at m in general. Indeed, we could see that
We have shown that
where (see Lemma 3 and (11))
and the parametrization of
with
then the nonlinear equation (15) can be rewritten as
Considering
In order to get the parametrization of the complete second Fučík branch
Remark 8 Considering noninteger values of m, we can solve equation (15) only in the symmetric case, in which
Combining this with the following straightforward corollary, we observe that the second
branch
Figure 3
. Illustration of the second Fučík branch for
Corollary 9The value ofνis decreasing inμ.
Proof Putting
Analyzing the signs of individual terms and observing that
and (11) yields the result. □
5 Elementariness of the second branch
C
2
d
Since the secondbranch
Definition 10 We say that a function is elementary if it is a finite composition of rational, algebraic, exponential, logarithmic, trigonometric,
inverse trigonometric, hyperbolic and inverse hyperbolic functions. We say that a
parametrization of a curve in
Under this definition, our conjecture becomes as follows.
Conjecture 11The second branch
Our analysis in the previous section implies that one could rephrase this conjecture in the following way.
Conjecture 12The solution of equation (16) cannot be solved in elementary functions.
Since there is a developed theory of elementary integration (see [14]) and (to our knowledge) there is no suitable tool dealing with elementary parametrizations of transcendent equations like (16), we try to use the theory of elementary integration to attack Conjectures 11 and 12.
Definition 13 We say that the integral is elementary if it can be expressed in terms of elementary functions.
One could use the Risch algorithm [[14], Chapter 12] to determine whether an integral is elementary or not. We use this procedure to analyze an integral directly connected to (15).
Lemma 14The integral
Proof Denoting
We compute the following resultant (see [[14], Definition 7.3]):
Since the roots of
Let us return back to Conjecture 12 and study (16) in more general settings, considering
Conjecture 15Equation
Let us denote
The implicit function theorem implies that the solution
Multiply the first equation by
Characteristic equations of (17) are given by
where
Unfortunately, the existence of nonelementary parametrization does not imply yet
that the solution surfaces
6 Conclusion
The goal of this paper was to shed some light on the problems which have arisen in the study of the discrete Fučík spectrum or related resonance problems. Although we were unable to fully prove the nonexistence of elementary parametrization of the second branch of the simplest discrete Fučík spectrum, we believe that the anchoring technique and the relationship to the transcendent equation (16) help to understand better the troubles which occur in this area.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
This research brought the author to the corners of mathematics he had not been familiar with. Therefore, he is very thankful to Jochen Merker and Petr Nečesal for their guidance. His thanks are also directed to Pavel Drábek, Gabriela Holubová and Komil Kuliev. This research has been supported by the grants of Ministry of Education, Youth and Sports of the Czech Republic ME09109 and MSM 4977751301.
End notes
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