Electronic Journal of Qualitative Theory of Differential Equations Proc. 8th Coll. QTDE, 2008, No. 10 1-14; http://www.math.u-szeged.hu/ejqtde/

Two-parametric nonlinear eigenvalue problems

Armands Gritsans, Felix Sadyrbaev ^{∗}

Abstract

Eigenvalue problems of the form x′′ = −λf(x^{+}) + μg(x^{−}), (i), x(0) = 0, x(1) = 0, (ii) are considered, where x^{+} and x^{−} are the positive and negative parts of x respectively. We are looking for (λ, μ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fučík problem for piece-wise linear equations. In our considerations functions f and g may be nonlinear functions of super-, suband quasi-linear growth in various combinations. The spectra obtained under the normalization condition |x′(0)| = 1 are sometimes similar to usual Fučík spectrum for the Dirichlet problem and sometimes they are quite different. This depends on monotonicity properties of the functions ξt_{1}(ξ) and ητ_{1}(η), where t_{1}(ξ) and τ_{1}(η) are the first zero functions of the Cauchy problems x′′ = −f(x), x(0) = 0, x′(0) = ξ > 0, y′′ = g(y), y(0) = 0, y′(0) = −η, (η > 0) respectively.

1 Introduction

Our goal is to study boundary value problems for two-parameter second order equations of the form

x′′ = −λf(x^{+}) + μg(x^{−}), x(0) = 0, x(1) = 0,  (1)

where f, g : [0, +∞) → [0, +∞) are C^{1}-functions such that f(0) = g(0) = 0, x^{+} = max{x, 0}, x^{−} = max{−x, 0}.

The same equation in extended form

x′′ = \{ \begin{matrix}−λf(x), & if & x ≥ 0 \\ μg(−x), & if & x < 0. \end{matrix}  (2)

We are motivated by the Fučík equation:

x′′ = −λx^{+} + μx^{−}.  (3)

∗1991 Mathematics Subject Classification. Primary: 34B15. Key words and phrases: nonlinear spectra, jumping nonlinearity, asymptoti-

cally asymmetric nonlinearities, Fučík spectrum.

This paper is in final form and no version of it will be submitted for publication elsewhere.

In extended form:

x′′ = \{ \begin{matrix}−λx, & if & x ≥ 0 \\ −μx, & if & x < 0, \end{matrix} x(0) = x(1) = 0.  (4)

The Fučík spectrum is well known and it is depicted in Fig. 1 and Fig. 2 It consists of a set of branches F_{i}^{±}, where the number i = 0, 1, … refers to the number of zeros of the respective nontrivial solution in the interval (0, 1) and an upper index, which is either + or −, shows either x′(0) is positive or negative.

Fig. 1. The classical (λ, μ) Fučík spectrum.

Fig. 2. The classical Fučík spectrum in inverted coordinates (γ = \frac{1}{\sqrt{λ}}, δ = \frac{1}{\sqrt{μ}}).

2 One-parametric problems

Consider first the one-parametric eigenvalue problem of the type

x′′ = −λf(x), x(0) = 0, x(1) = 0,  (5)

where f satisfies our assumptions.

It easily can be seen that this problem may have a continuous spectrum. For example, the problem

x′′ = −λx^{3}, x(0) = 0, x(1) = 0.

has a positive valued in (0, 1) solution x(t) for any λ > 0. The value max_{[0,1]} x(t) := ∥x∥ and λ relate as

∥x∥ · λ = 2\sqrt{2} · ∫_{0}^{1} \frac{dx}{\sqrt{1 − x^{4}}} .

In order to make the problem reasonable one should impose additional conditions. Let us require that

|x′(0)| = 1.

Let us mention that problems of the type (5) were intensively studied in various settings. For the recent review one may consider the paper [2].

3 Two-parametric problems

3.1 Assumptions

We assume that functions f and g satisfy the following conditions: (A1) the first zero t_{1}(α) of a solution to the Cauchy problem

u′′ = −f(u), u(0) = 0, u′(0) = α  (6)

is finite for any α > 0.

Similar property can be assigned to a function g.

We assume that g satisfies the condition: (A2) the first zero τ_{1}(β) of a solution to the Cauchy problem

v′′ = g(−v), v(0) = 0, v′(0) = −β  (7)

is finite for any β > 0.

Functions t_{1} and τ_{1} are the so called time maps ([5]).

3.2 Formulas for nonlinear Fučík type spectra

Consider

x′′ = \{ \begin{matrix}−λf(x), & if & x ≥ 0 \\ μg(−x), & if & x < 0, \end{matrix} x(0) = x(1) = 0, |x′(0)| = 1.  (8)

Let us recall the main result in [4].

Theorem 3.1 Let the conditions (A1) and (A2) hold. The Fučík type spectrum for the problem (8) is given by the relations:

F_{0}^{+} = \{(λ, μ) : λ is a solution of \frac{1}{\sqrt{λ}} t_{1}(\frac{1}{\sqrt{λ}}) = 1, μ ≥ 0\}, (9)
F_{0}^{−} = \{(λ, μ) : λ ≥ 0, μ is a solution of \frac{1}{\sqrt{μ}} τ_{1}(\frac{1}{\sqrt{μ}}) = 1\},  (10)
F_{2i−1}^{+} = \{(λ; μ) : i\frac{1}{\sqrt{λ}} t_{1}(\frac{1}{\sqrt{λ}}) + i\frac{1}{\sqrt{μ}} τ_{1}(\frac{1}{\sqrt{μ}}) = 1\} ,  (11)
F_{2i−1}^{−} = \{(λ; μ) : i\frac{1}{\sqrt{μ}} τ_{1}(\frac{1}{\sqrt{μ}}) + i\frac{1}{\sqrt{λ}} t_{1}(\frac{1}{\sqrt{λ}}) = 1\} ,  (12)
F_{2i}^{+} = \{(λ; μ) : (i + 1)\frac{1}{\sqrt{λ}} t_{1}(\frac{1}{\sqrt{λ}}) + i\frac{1}{\sqrt{μ}} τ_{1}(\frac{1}{\sqrt{μ}}) = 1\} ,  (13)
F_{2i}^{−} = \{(λ; μ) : (i + 1)\frac{1}{\sqrt{μ}} τ_{1}(\frac{1}{\sqrt{μ}}) + i\frac{1}{\sqrt{λ}} t_{1}(\frac{1}{\sqrt{λ}}) = 1\} .  (14)

The same formulas in inverted coordinates γ = \frac{1}{\sqrt{λ}}, δ = \frac{1}{\sqrt{μ}} are:

F_{0}^{+} = {(γ, δ) : γ is a solution of γ t_{1}(γ) = 1, δ > 0}∪  (15)
∪ {(γ, ∞) : γ is a solution of γ t_{1}(γ) = 1},
F_{0}^{−} = {(γ, δ) : γ > 0, δ is a solution of δ τ_{1}(δ) = 1}∪  (16)
∪ {(∞, δ) : δ is a solution of δ τ_{1}(δ) = 1},
F_{2i−1}^{+} = {(γ; δ) : iγ t_{1}(γ) + iδ τ_{1}(δ) = 1, γ > 0, δ > 0},  (17)
F_{2i−1}^{−} = {(γ; δ) : iδ τ_{1}(δ) + iγ t_{1}(γ) = 1, γ > 0, δ > 0},  (18)
F_{2i}^{+} = {(γ; δ) : (i + 1)γ t_{1}(γ) + iδ τ_{1}(δ) = 1, γ > 0, δ > 0},  (19)
F_{2i}^{−} = {(γ; δ) : (i + 1)δ τ_{1}(δ) + iγ t_{1}(γ) = 1, γ > 0, δ > 0}.  (20)

Corollary 3.1 The sets F_{2i−1}^{+} and F_{2i−1}^{−} (respectively F_{2i−1}^{+} and F_{2i−1}^{−}) coincide.

Remark 3.1 Each subset F_{i}^{±} is associated with nontrivial solutions with definite nodal structure. For example, the set

F_{4}^{+} = \{(λ; μ) : 3\frac{1}{\sqrt{λ}} t_{1}(\frac{1}{\sqrt{λ}}) + 2\frac{1}{\sqrt{μ}} τ_{1}(\frac{1}{\sqrt{μ}}) = 1\}

is associated with nontrivial solutions that have three positive humps and two negative ones. The total number of interior zeros is exactly four. Similarly, the set

F_{4}^{−} = \{(λ; μ) : 2\frac{1}{\sqrt{λ}} t_{1}(\frac{1}{\sqrt{λ}}) + 3\frac{1}{\sqrt{μ}} τ_{1}(\frac{1}{\sqrt{μ}}) = 1\}

is associated with nontrivial solutions that have two positive humps and three negative ones.

Remark 3.2 The additional condition |x′(0)| = 1 is not needed if f and g are linear functions (the classical Fučík equation). Then t_{1} and τ_{1} are constants and do not depend on the initial values of the derivatives.

3.3 Samples of time maps

Consider equations

x′′ = −(r + 1)x^{r}, r > 0,  (21)

which may be integrated explicitly. One has that

t_{1}(\frac{1}{\sqrt{λ}}) = 2A λ^{\frac{r−1}{2(r+1)}} , where A = ∫_{0}^{1} \frac{1}{\sqrt{1 − ξ^{r+1}}} dξ,  (22)

so t_{1} is decreasing in λ for r ∈ (0, 1), t_{1} is constant for r = 1, t_{1} is increasing in λ for r > 1.

The function

u(λ) = \frac{1}{\sqrt{λ}}t_{1}(\frac{1}{\sqrt{λ}}) = 2A λ^{− \frac{1}{r+1}}

is decreasing for r > 0.

4 Some properties of spectra

Introduce the functions

u(λ) := \frac{1}{\sqrt{λ}}t_{1}(\frac{1}{\sqrt{λ}}) v(μ) := \frac{1}{\sqrt{μ}}τ_{1}(\frac{1}{\sqrt{μ}}),  (23)

where t_{1} and τ_{1} are the time maps associated with f and g respectively. Due to Theorem 3.1 the spectrum of the problem (8) is a union of pairs (λ, μ) such that one of the relations

\begin{aligned}u(λ) & + & v(μ) & = 1, & F_{1}^{±} \\ 2u(λ) & + & v(μ) & = 1, & F_{2}^{+} \\ u(λ) & + & 2v(μ) & = 1, & F_{2}^{−} \\ 2u(λ) & + & 2v(μ) & = 1, & F_{3}^{±} \\ 3u(λ) & + & 2v(μ) & = 1, & F_{4}^{+} \\ 2u(λ) & + & 3v(μ) & = 1, & F_{4}^{−} \\ … \end{aligned}  (24)

holds. The coefficients at u(λ) and v(μ) indicate the numbers of “positive” and “negative” humps of the respective eigenfunctions.

4.1 Monotone functions u and v

Suppose that both functions u and v are monotonically decreasing. Then the same do the multiples iu and iv, i is a positive integer.

Theorem 4.1 Suppose that the functions u and v monotonically decrease from +∞ to zero. Then the spectrum of the problem (8) is essentially the classical Fučík spectrum, that is, it is a union of branches F_{i}^{±}, which are the straight lines for i = 0, the curves which look like hyperbolas and have both vertical and horizontal asymptotes, for i > 0.

Proof. First of all notice that the value u(λ) = \frac{1}{\sqrt{λ}} t_{1}(\frac{1}{\sqrt{λ}}) is exactly the distance between two consecutive zeros of a solution to the problem x′′ = −λf(x), x(0) = 0, x′(0) = 1. Similarly the value v(μ) = \frac{1}{\sqrt{μ}} τ_{1}(\frac{1}{\sqrt{μ}}) is the distance between two consecutive zeros of a solution to the problem y′′ = μg(−y), y(0) = 0, y′(0) = −1.

Let λ_{1}, λ_{2}, λ_{3} and so on be the points of intersection of u(λ), 2u(λ), 3u(λ), … with the horizontal line u = 1. Respectively μ_{1}, μ_{2}, μ_{3} and so on for the function v(μ) (see the Fig. 3 and Fig. 4).

Fig. 3. The graphs of u(λ), 2u(λ), 3u(λ) (schematically).

Fig. 4. The graphs of v(μ), 2v(μ), 3v(μ) (schematically).

Positive solutions to the problem with no zeros in the interval (0, 1) appear for λ = λ_{1}. Thus F_{0}^{+} is a straight line {(λ_{1}, μ) : μ ≥ 0}. Similarly F_{0}^{−} is a straight line {(λ, μ_{1}) : λ ≥ 0}.

The branches F_{1}^{±} which are defined by the first equation of (24) coincide and look like hyperbola with the vertical asymptote at λ = λ_{1} and horizontal asymptote at μ = μ_{1}.

The branch F_{2}^{+} has the vertical asymptote at λ = λ_{2} and horizontal asymptote at μ = μ_{1}. This can be seen from the second equation of (24).

The branch F_{2}^{−} has the vertical asymptote at λ = λ_{1} and horizontal asymptote at μ = μ_{2}. This is a consequence of the third equation of (24). Notice that the branches F_{2}^{+} and F_{2}^{−} need not to cross at the bisectrix λ = μ unless g ≡ f (in contrast with the case of the classical Fučík spectrum).

The branches F_{3}^{±} coincide and have the vertical asymptote at λ = λ_{2} and horizontal asymptote at μ = μ_{2}.

The branch F_{4}^{+} has the vertical asymptote at λ = λ_{3} and horizontal asymptote at μ = μ_{2}.

The branch F_{4}^{−} has the vertical asymptote at λ = λ_{2} and horizontal asymptote at μ = μ_{3}. The branches F_{4}^{+} and F_{4}^{−} need not to cross at the bisectrix.

In a similar manner any of the remaining branches can be considered. □

Proposition 4.1 The function u(λ) = \frac{1}{\sqrt{λ}}t_{1}(\frac{1}{\sqrt{λ}}), where t_{1} is defined in (6), is monotonically decreasing if

1 − \frac{F (x)F′′(x)}{f^{2}(x)} > 0, F (x) = ∫_{0}^{x} f(s) ds.  (25)

Proof. Let us show that the function α t_{1}(α) is monotonically increasing for α > 0. Consider the Cauchy problem x′′ + f(x) = 0, x(0) = 0, x′(0) = α. A solution x satisfies the relation \frac{1}{2}x^{′2}(t) + F (x(t)) = h, where h = \frac{1}{2}α^{2} = F (x_{+}), x_{+} is a maximal value of x(t).

It was shown in [3, Lemma 2.1] that the function T (h) = 2 ∫_{0}^{x_{+}} \frac{ds}{\sqrt{2(h−F (s))}}

has the derivative

\frac{dT}{dh} = \frac{2}{h} ∫_{0}^{x_{+}} (\frac{1}{2} − \frac{F (x)F′′(x)}{f^{2}(x)})\frac{dx}{\sqrt{2(h − F (x))}}.  (26)

Notice that t_{1}(α) = T (\frac{1}{2}α^{2}). One has that

\begin{aligned}[α t_{1}(α)]′_{α} & = t_{1}(α) + α t′_{1}(α) \\ & = 2 ∫_{0}^{x_{+}} \frac{dx}{\sqrt{α^{2}−2F (x))}} + 4 ∫_{0}^{x_{+}} (\frac{1}{2} − \frac{F (x)F′′(x)}{f^{2}(x)})\frac{dx}{\sqrt{α^{2}−2F (x))}} \\ & = 4 ∫_{0}^{x_{+}} (1 − \frac{F (x)F′′(x)}{f^{2}(x)})\frac{dx}{\sqrt{α^{2}−2F (x))}} \end{aligned} (27)

□

For instance, if x′′ + x = 0, then f = x, F = \frac{1}{2}x^{2}, ω(α) := αt_{1}(α) = πα, ω′ = π. Taking into account that x_{+} = α one obtains from (27)

ω′(α) = 4 ∫_{0}^{α}(1 − \frac{1}{2})\frac{dx}{\sqrt{α^{2} − x^{2}}} = 2 arcsin \frac{x}{α} |_{0}^{α} = π.

4.2 Non-monotone functions u and v

It is possible that the functions u(λ) = \frac{1}{\sqrt{λ}}t_{1}(\frac{1}{\sqrt{λ}}) and v(μ) = \frac{1}{\sqrt{μ}}τ_{1}(\frac{1}{\sqrt{μ}}) are not monotone.

Then spectra may differ essentially from those in the monotone case.

Proposition 4.2 Suppose that u(λ) and v(μ) are not zeros at λ = 0 and μ = 0 respectively and monotonically decrease to zero starting from some values λ_{⋆} and μ_{⋆}. Then the subsets F_{i}^{±} of the spectrum behave like the respective branches of the classical Fučík spectrum for large numbers i, that is, they form curves looking like hyperbolas which have vertical and horizontal asymptotes.

Indeed, notice that for large enough values of i the functions iu(λ) and iv(μ) monotonically decrease to zero in the regions {λ ≥ λ_{Δ}, 0 < u < 1}, {μ ≥ μ_{Δ}, 0 < v < 1} respectively (for some λ_{Δ} and μ_{Δ}) and are greater than unity for 0 < λ < λ_{Δ} and 0 < μ < μ_{Δ} respectively. Therefore one may complete the proof by analyzing the respective relations in (24).

If one (or both) of the functions u and v is non-monotone then the spectrum may differ essentially from the classical Fučík spectrum. Consider the case depicted in Fig. 5.

Fig. 5. Functions u (solid line) and v (dashed line).

Proposition 4.3 Let the functions u and v behave like depicted in Fig. 5, that is, v monotonically decreases from +∞ to zero and u has three segments of monotonicity, u tends to zero as λ goes to +∞. Then the subset F_{1}^{±} consists of two components.

Indeed, let λ_{1}, λ_{2} and λ_{3} be successive points of intersection of the graph of u with the line u = 1. Denote λ_{∗} the point of minimum of u(λ) in the interval (λ_{1}, λ_{2}). Let μ_{∗} be such that u(λ_{∗}) + v(μ_{∗}) = 1. It is clear that there exists a U-shaped curve with vertical asymptotes at λ = λ_{1} and λ = λ_{2} and with a minimal value μ_{∗} at λ_{∗} which belongs to F_{1}^{+}. There also exists a hyperbola looking curve with the vertical asymptote at λ = λ_{3} and horizontal asymptote at μ = μ_{1}, where μ_{1} is the (unique) point of intersection of the graph of v with the line v = 1.

There are no more points belonging to F_{1}^{+}.

5 Examples

Let

0 < a_{1} < a_{2} < a_{3}, b_{1} > b_{2} > 0, b_{3} > b_{2}.

Consider a piece-wise linear function:

f(x) = \{ \begin{matrix}f_{1}(x), & 0 ≤ x ≤ a_{1}, \\ f_{2}(x), & a_{1} ≤ x ≤ a_{2}, \\ f_{3}(x), & x ≥ a_{3}, \end{matrix}  (28)
f_{1}(x) = p_{1}x + q_{1}, f_{2}(x) = p_{2}x + q_{2}, f_{3}(x) = p_{3}x + q_{3},
f_{1}(0) = 0, f_{1}(a_{1}) = f_{2}(a_{1}), f_{2}(a_{2}) = f_{3}(a_{2}), f_{3}(a_{3}) = b_{3}.

Fig. 6. Function f(x).

Notice that

p_{1} = \frac{b_{1}}{a_{1}} ,  q_{1} = 0,
p_{2} = \frac{b_{2} − b_{1}}{a_{2} − a_{1}} ,  q_{2} = \frac{b_{1}a_{2} − a_{1}b_{2}}{a_{2} − a_{1}} ,
p_{3} = \frac{b_{3} − b_{2}}{a_{3} − a_{2}} ,  q_{3} = \frac{b_{2}a_{3} − a_{2}b_{3}}{a_{3} − a_{2}} .

Let t_{1}(α) be the first positive zero of a solution to the initial value problem

x′′ = −f(x), x(0) = 0, x′(0) = α > 0.  (29)

Denote F (x) = ∫_{0}^{x} f(s) ds. Direct calculations ([1]) show that

1. if 0 ≤ α ≤ \sqrt{2F (a_{1})}, then t_{1}(α) = π \sqrt{\frac{a_{1}}{b_{1}}} ;

2. if \sqrt{2F (a_{1})} ≤ α ≤ \sqrt{2F (a_{2})}, then

t_{1}(α) = 2\sqrt{\frac{a_{1}}{b_{1}}} arcsin \frac{\sqrt{a_{1}b_{1}}}{α} +

+ \sqrt{\frac{a_{2} − a_{1}}{b_{1} − b_{2}}} ln \frac{D_{2}(α)}{(−2b_{1} + 2\sqrt{\frac{b_{1}−b_{2}}{a_{2}−a_{1}}} \sqrt{α^{2} − a_{1}b_{1}})^{2}} ,

3. if α ≥ \sqrt{2F_{2}(a_{2})}, then

t_{1}(α) = 2\sqrt{\frac{a_{1}}{b_{1}}} arcsin \frac{\sqrt{a_{1}b_{1}}}{α} + \sqrt{\frac{a_{3} − a_{2}}{b_{3} − b_{2}}} [π − 2 arcsin \frac{2b_{2}}{\sqrt{D_{3}(α)}}] +

+ 2\sqrt{\frac{a_{2} − a_{1}}{b_{1} − b_{2}}} ln |\frac{−b_{2} + \sqrt{\frac{b_{1}−b_{2}}{a_{2}−a_{1}}} \sqrt{α^{2} − a_{1}b_{1} − (a_{2} − a_{1})(b_{1} + b_{2})}}{−b_{1} + \sqrt{\frac{b_{1}−b_{2}}{a_{2}−a_{1}}} \sqrt{α^{2} − a_{1}b_{1}}} | ,

where

D_{2}(α) = 4 \frac{b_{1} − b_{2}}{a_{1} − a_{2}} α^{2} + 4b_{1} \frac{a_{1}b_{2} − a_{2}b_{1}}{a_{1} − a_{2}} , D_{3}(α) = 4 \frac{b_{2} − b_{3}}{a_{2} − a_{3}} α^{2}+

+ 4 \frac{−a_{2}b_{1}b_{2} + a_{1}b_{2}^{2} + a_{3}b_{2}^{2} + a_{2}b_{1}b_{3} − a_{1}b_{2}b_{3} + a_{2}b_{2}b_{3}}{a_{2} − a_{3}} .

The first zero function is asymptotically linear:

lim_{α→+∞} t_{1}(α) = \sqrt{\frac{a_{3} − a_{2}}{b_{3} − b_{2}}} π.

Consider equation

x′′ = −λf(x^{+}) + μf(x^{−}),

where f(x) is a piece-wise linear function depicted in Fig. 6. Let parameters of the piece-wise linear function f(x) be

a_{1} = 0.1,  a_{2} = 0.3,  a_{3} = 0.31,
b_{1} = 9,  b_{2} = 0.5,  b_{3} = 150.

Fig. 7. The graph of y = f(x).

Fig. 8. The graphs of y = γt_{1}(γ)
(γ = \frac{1}{\sqrt{λ}}) and y = 1.

Fig. 9. The subset F_{0}^{+} in the (γ, δ)-plane.

Fig. 10. The subset F_{0}^{+} in the
(λ, μ)-plane.

The subset F_{0}^{+} consists of three vertical lines which correspond to three solutions of the equation \frac{1}{\sqrt{λ}}t_{1}(\frac{1}{\sqrt{λ}}) = 1.

Fig. 11.The subset F_{0}^{−} in the (γ, δ)-plane.

Fig. 12.The subset F_{0}^{−} in the
(λ, μ)-plane.

The subset F_{0}^{−} consists of horizontal lines which correspond to solutions of the equation \frac{1}{\sqrt{μ}}τ_{1}(\frac{1}{\sqrt{μ}}) = 1.

Fig. 13. The subset F_{1}^{+} = F_{1}^{−} in the

(γ, δ)-plane.

Fig. 14. The subset F_{1}^{+} = F_{1}^{−} in the

(λ, μ)-plane.

Properties of the subsets F_{1}^{±} depend on solutions of the equation

u(λ) + v(μ) = 1.

A set of solutions of this equation consists of exactly three components due to non-monotonicity of the functions u(λ) and v(μ). Respectively, properties of the subsets F_{1}^{±} depend on solutions of the equation

γt_{1}(γ) + δτ_{1}(δ) = 1.

Fig. 15. The subset F_{2}^{+} in the
(γ, δ)-plane.

Fig. 16. The subset F_{2}^{+} in the

(λ, μ)-plane.

Fig. 17. The subset F_{2}^{−} in the (γ, δ)-plane.

Fig. 18. The subset F_{2}^{−} in the
(λ, μ)-plane.

The subsets F_{2}^{±} look a little bit different since now their properties depend on a set of solutions of equations

2\frac{1}{\sqrt{λ}}t_{1}(\frac{1}{\sqrt{λ}}) + \frac{1}{\sqrt{μ}}τ_{1}(\frac{1}{\sqrt{μ}}) = 1

and

\frac{1}{\sqrt{λ}}t_{1}(\frac{1}{\sqrt{λ}}) + 2\frac{1}{\sqrt{μ}}τ_{1}(\frac{1}{\sqrt{μ}}) = 1.

Respectively, properties of the subsets F_{2}^{±} depend on solutions of equations

2γt_{1}(γ) + δτ_{1}(δ) = 1

and

γt_{1}(γ) + 2δτ_{1}(δ) = 1.

References

[1] A. Gritsans and F. Sadyrbaev. On nonlinear eigenvalue problems. Proceedings Inst. Math. Comp. Sci. of the University of Latvia “Mathematics. Differential Equations”, vol. 6 (2006), 76 - 86. http://www.lumii.lv/sbornik2006/Sbornik-2006-english.htm

[2] P. Korman. Global solution branches and exact multiplicity of solutions for two-point boundary value problems. In: Handbook of Diff. Equations, ODE, Vol. III. Elsevier - North Holland, Amsterdam, 2006, 548 - 606.

[3] Bin Liu. On Littlewood’s boundedness problem for sublinear Duffing equations. Trans. Amer. Math. Soc., 2000, 353, No. 4, 1567 - 1585.

[4] F. Sadyrbaev and A. Gritsans. Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations. Nonlinear Analysis: Modelling and Control, 2007, 12, No.2, 253 - 267.

[5] R. Schaaf. Global solution branches of two-point boundary value problems. Lect. Notes Math. 1458. Springer-Verlag, Berlin - Heidelberg - New York, 1990.

(Received August 31, 2007)

Faculty of Natural Sciences and Mathematics (DMF), Daugavpils University, Parades str. 1, 5400 Daugavpils, Latvia E-mail addresses: felix@latnet.lv (the address for communication), arminge@inbox.lv