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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Discrete Fučík spectrum - anchoring rather than pasting

Petr Stehlík

Author Affiliations

Department of Mathematics, University of West Bohemia, Univerzitni 22, Pilsen, 30614, Czech Republic

Boundary Value Problems 2013, 2013:67  doi:10.1186/1687-2770-2013-67


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/67


Received:23 November 2012
Accepted:8 March 2013
Published:29 March 2013

© 2013 Stehlík; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M1">View MathML</a>. Based on this equivalence, we state a conjecture that already the second branch has no elementary parametrization, i.e., it cannot be described by a finite number of elementary functions.

MSC: 39A12, 34B15.

Keywords:
Fučík spectrum; nonlinear difference equation; resonance; boundary value problem; elementary functions; second branch

Dedication

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 out-of-touch 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 [1-3]). 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 [10-13]). Those works reveal the complexity of the matrix problem, which prevents to fully describe the spectrum beyond <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M2">View MathML</a> matrices.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M3">View MathML</a>

(1)

Assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M4">View MathML</a>, we seek the Fučík spectrum, i.e., pairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M5">View MathML</a> such that the problem (1) has a nontrivial solution.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M1">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M8">View MathML</a>. Finally, in Section 5 we state and discuss the conjecture that the parametrization of the second-branch of (1) is not elementary, i.e., it cannot be described by a finite number of elementary functions.

2 Pasting in continuous case

First, let us briefly recall the pasting technique for the original continuous problem [4]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M9">View MathML</a>

(2)

Lemma 1The piecewise nonlinear BVP (2) has a nontrivial solution if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M10">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M11">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M12">View MathML</a>

(3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M13">View MathML</a>

(4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M14">View MathML</a>

(5)

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M15">View MathML</a> .

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M16">View MathML</a> so that we could illustrate the pasting technique. On <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M16">View MathML</a> the solution could be split in the positive part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M18">View MathML</a> and negative part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M19">View MathML</a>. Without loss of generality, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M20">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M21">View MathML</a>. Then the negative part must have the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M22">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M23">View MathML</a> is chosen so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25">View MathML</a> meet at m. The constant C is then chosen so that this connection is continuously differentiable (cf. Figure 1). From this we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M27">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M28">View MathML</a>

 □

thumbnail 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M29">View MathML</a>. We realize first that the problem (1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M30">View MathML</a> points can change sign only <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M31">View MathML</a>-times, which implies that the spectrum on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M30">View MathML</a> points will have only N branches <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M33">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M34">View MathML</a>

Naturally, the first Fučík branch <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M35">View MathML</a> of (1) is given by the eigenvalues of the discrete problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M36">View MathML</a>

(6)

As we use the eigenvalues in the sequel, we present a concise proof.

Theorem 2The eigenvalues of (6) are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M37">View MathML</a>

(7)

and the corresponding eigenvectors are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M38">View MathML</a>

(8)

Proof For a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M39">View MathML</a>, direct substitution of (8) into (6) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M40">View MathML</a>

Thus, we have N independent eigenvectors, which finishes the proof. □

The first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M41">View MathML</a> generates the first Fučík branch <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M42">View MathML</a> of (1). Since the corresponding eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M43">View MathML</a> does not change its sign, we obtain that the problem (1) has a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M44">View MathML</a> for an arbitrary couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M45">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M46">View MathML</a>. Similarly, the problem (1) has a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M47">View MathML</a> for an arbitrary couple <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M48">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M49">View MathML</a>.

4 Second branch and anchoring

Let us move on to the second discrete Fučík branch <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M50">View MathML</a>. It corresponds to the solutions of (1) that change the sign exactly once (see Figure 2 for illustration and basic notation). As in the continuous case, we try to paste together two sine functions. Without loss of generality, we assume that the solution is positive first. We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M51">View MathML</a> and consider solutions which are nonnegative on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M52">View MathML</a>,a with values lying on the sine function (cf. Theorem 2 and Figure 2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M54">View MathML</a>

(9)

Since the solutions on the second branch change sign exactly once and lie on the sine function again, we seek constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M55">View MathML</a> (cf. Theorem 2 and Figure 2) in the nonpositive part

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M56">View MathML</a>

(10)

Then the vector (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M58">View MathML</a> denote floor and ceiling functions)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M59">View MathML</a>

is the solution of (1) coupled with the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M60">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M61">View MathML</a>

(11)

thumbnail Figure 2 . Discrete anchoring procedure. Illustration of the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M62">View MathML</a> in general.

Above, we considered <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M63">View MathML</a>. This follows from the fact that if we had chosen <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M64">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M65">View MathML</a>, there would have been no positive/negative part and the solution would have laid on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M42">View MathML</a> instead.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M67">View MathML</a>, and we could easily compute β and C in (10). In other words, we could still talk about pasting in this case.

Lemma 3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M51">View MathML</a>is an integer number, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M69">View MathML</a>

(12)

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25">View MathML</a>are equal in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M72">View MathML</a>, mand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M73">View MathML</a>.

Proof If m is integer, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M74">View MathML</a>. Consequently, the difference equation (1) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M75">View MathML</a> reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M76">View MathML</a>

Exploiting the symmetry of sine functions, both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25">View MathML</a> are equal in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M72">View MathML</a>, m and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M73">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M81">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M82">View MathML</a>

Finally, the equality of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25">View MathML</a>, e.g., at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M73">View MathML</a>, implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M86">View MathML</a>

 □

The analysis gets more complicated once we consider non-integer values of m. In this case, the transition between positive and negative parts occurs between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M58">View MathML</a>. Our first result states that the values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25">View MathML</a> coincide at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M58">View MathML</a> (cf. Figure 2).

Lemma 4 (Necessary condition)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M51">View MathML</a>be non-integer. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M94">View MathML</a>be a nontrivial solution of (1) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25">View MathML</a>given by (9) and (10). Then the following equalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M97">View MathML</a>

(13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M98">View MathML</a>

(14)

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24">View MathML</a> is given by (9), then the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M100">View MathML</a>

holds for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M101">View MathML</a> if and only if (see the proof of Theorem 2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M102">View MathML</a>

If we consider the difference equation (1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M103">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M104">View MathML</a>

This equality holds if and only if (see (1))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M105">View MathML</a>

which verifies (14).

Using the same argument at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M58">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25">View MathML</a>, we obtain that (13) holds as well. □

This result enables us to get both peripheral parts of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M50">View MathML</a>.

Corollary 5

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M109">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M110">View MathML</a>

Proof Let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M111">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M112">View MathML</a>

Then we can rewrite the equation (1) in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M113">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M114">View MathML</a>

This proves the former part of the statement. The latter follows from the mirror argument. □

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M115">View MathML</a> for m sufficiently close to N. This implies that (11) cannot hold for any β, i.e., not all the solutions on the second branch can be obtained as a composition of sine functions!

Since the problem is solved for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M116">View MathML</a>, we could turn our attention to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M117">View MathML</a> in the following. Applying (9) and (10), we can rewrite conditions (13) and (14) in the following way.

Corollary 6 (Necessary condition II)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M118">View MathML</a>be non-integer. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M24">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M25">View MathML</a>have the form (9) and (10), then the following equality is satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M121">View MathML</a>

(15)

Proof One can rewrite equalities (13) and (14) into

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M122">View MathML</a>

Isolating C on the right-hand sides of both equations, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M123">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M124">View MathML</a> does not solve (15) for all m. In other words, if we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M125">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M126">View MathML</a> for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M63">View MathML</a> (see Figure 2).

We have shown that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M128">View MathML</a>

where (see Lemma 3 and (11))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M129">View MathML</a>

and the parametrization of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M130">View MathML</a> is given by (5). Finally, we would like to get the parametrization of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M131">View MathML</a>, i.e., consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M117">View MathML</a>. Corollary 6 implies that one could get the parametrization of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M131">View MathML</a> if and only if one could solve (15). Equation (15) is equivalent to the problem of solving the transcendent equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M134">View MathML</a>

(16)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M8">View MathML</a>. Indeed, if we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M137">View MathML</a>

then the nonlinear equation (15) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M138">View MathML</a>

Considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M139">View MathML</a>, we see that D and C are constant non-zero integers and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M140">View MathML</a>. Consequently, if we substitute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M142">View MathML</a>, we get (16).

In order to get the parametrization of the complete second Fučík branch <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143">View MathML</a>, we need to solve (15) for β, or, equivalently, (16) for x. While this could be done pretty easily numerically, we state, in the final section, a conjecture that it is not possible to use a finite number of elementary functions to get such a parametrization. Meanwhile, we make two straightforward observations.

Remark 8 Considering non-integer values of m, we can solve equation (15) only in the symmetric case, in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M144">View MathML</a>. Either the symmetry of sine functions or the direct computation yields that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M124">View MathML</a>. Similarly, as in the integer case (cf. Lemma 3), this value coincides with the value of the corresponding continuous problem as the positive and negative parts meet at m.

Combining this with the following straightforward corollary, we observe that the second branch <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143">View MathML</a> of the discrete spectrum slides monotonically along its continuous counterpart, coinciding in the ‘symmetric’ values of m, in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M147">View MathML</a> (see Figure 3).

thumbnail Figure 3 . Illustration of the second Fučík branch for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M148">View MathML</a>with the corresponding continuous second branch<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M149">View MathML</a>.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M150">View MathML</a> is not depicted for presentation purposes. In this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M150">View MathML</a> consists only of two points, in which the orange <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M130">View MathML</a> meets the pink <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M131">View MathML</a>, corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M154">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M155">View MathML</a>.

Corollary 9The value ofνis decreasing inμ.

Proof Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M156">View MathML</a> and differentiating (15), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M157">View MathML</a>

Analyzing the signs of individual terms and observing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M158">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M159">View MathML</a>, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M160">View MathML</a>

and (11) yields the result. □

5 Elementariness of the second branch <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143">View MathML</a>

Since the second-branch <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143">View MathML</a> of (1) is the first nontrivial branch of the simplest discrete Fučík spectrum, we believe that its properties can help to explain difficulties with discrete Fučík spectra. Therefore, we discuss possible ways to prove that its part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M131">View MathML</a> has no elementary parametrization.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M164">View MathML</a> is elementary if it consists of elementary functions.

Under this definition, our conjecture becomes as follows.

Conjecture 11The second branch<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M143">View MathML</a>of (1) has no elementary parametrization.

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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M166">View MathML</a>is not elementary.

Proof Denoting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M169">View MathML</a>, we rewrite the integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M170">View MathML</a>

We compute the following resultant (see [[14], Definition 7.3]):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M171">View MathML</a>

Since the roots of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M172">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M173">View MathML</a>, i.e., not constant, the integral is not elementary (see the Rothstein-Trager theorem [[14], Theorem 12.9]). □

Let us return back to Conjecture 12 and study (16) in more general settings, considering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M174">View MathML</a>.

Conjecture 15Equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M175">View MathML</a>cannot be solved for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M176">View MathML</a>using a finite number of elementary functions.

Let us denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M177">View MathML</a> and consider the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M178">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M179">View MathML</a>, the implicit function theorem can be applied in points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M180">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M181">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M182">View MathML</a>. Therefore, we take into account triplets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M183">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M185">View MathML</a>.

The implicit function theorem implies that the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M176">View MathML</a> starting in those points satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M187">View MathML</a>

Multiply the first equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M188">View MathML</a>, the second equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M189">View MathML</a>, add both and divide the result by the non-zero term in the square brackets to obtain the initial problem for the first-order partial differential equation (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M190">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M191">View MathML</a>

(17)

Characteristic equations of (17) are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M192">View MathML</a>

(18)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M193">View MathML</a> is the value of u along characteristic curves. The third equation of (18) cannot be solved using elementary functions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M194">View MathML</a> as the integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M195">View MathML</a> is not elementary (Lemma 14).

Unfortunately, the existence of non-elementary parametrization does not imply yet that the solution surfaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M196">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M178">View MathML</a> (which arise as a union of the characteristic curves) cannot be expressed using elementary functions.

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

  1. Subscript ℤ denotes the discrete interval, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/67/mathml/M53">View MathML</a>.

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