Two-Parameter Nonlinear Oscillations: The Neumann Problem

Boundary value problems of the form x = −λf(x) + μg(x) (i), x(a) = 0 = x(b) (ii) are considered, where λ, μ > 0. In our considerations functions f and g are generally nonlinear. We give a description of a solution set of the problem (i), (ii). It consist of all triples (λ, μ, α) such that (λ,μ, x(t)) nontrivially solves the problem (i), (ii) and |x(z)| = α at zero points z of the function x(t) (iii). We show that this solution set is a union of solution surfaces which are centro-affine equivalent. Each solution surface is associated with nontrivial solutions with definite nodal type. Properties of solution surfaces are studied. It is shown, in particular, that solution surface associated with solutions with exactly i zeroes in the interval (a, b) is centro-affine equivalent to a solution surface of the Dirichlet problem (i), x(a) = 0 = x(b), (iii) corresponding to solutions with odd number of zeros 2j − 1 (i 6= 2j) in the interval (a, b).


Introduction
Two-parameter nonlinear boundary value problems have been extensively studied in the literature, see for example [1,2,3,5,8,11,14,16]. Some of the mentioned references deal with the so called asymmetric oscillators. The simplest asymmetric oscillator is described by the Fučík differential equation x ′′ = −λx + + µx − , where x + = max{x, 0}, x − = max{−x, 0}. This equation possesses the property of positive homogeneity and any function cx(t) solves it for c ≥ 0 if x(t) is a solution. The sum of two solutions x 1 (t) and x 2 (t) need not to be a solution.
A nonlinear asymmetric oscillator (in the meaning that restoring forces on the left and right sides nonlinearly depend on replacements x(t)) may be associated with equation x ′′ = −λf (x + ) + µg(x − ), where f and g are nonlinear nonnegative functions. This equation with the Dirichlet boundary conditions was considered in [4,5,7]. In [4] the Dirichlet boundary value problem was investigated together with a normalization condition which is not needed in case of the Fučík equation since it fulfils automatically. The analytical description of the spectrum was obtained. The spectrum is a set of all pairs (λ, µ) such that the problem has a nontrivial solution. In [7] the Dirichlet problem was studied provided that one of the functions f and g is linear. In [5] some properties of the spectrum were analyzed. It was pointed out that branches of the spectrum may have separate components connected at infinity and even bounded separate components.
This article is continuation and expansion of [8], where the Neumann problem x ′′ = −λf (x + ) + µg(x − ), x ′ (a) = 0 = x ′ (b), (1.1) a < b, was considered. The main result in [8] is an analytical description of the spectrum provided in Theorem 1. In Definition 1 of the present article we substantially expand notion of a spectrum (a solution set is introduced) for the problem (1.1) and give full description of a spectrum in Theorem 4, thereby generalizing the main result in [8].
Sometimes we need the additional technical requirements on f function. These conditions are: (A3) for some k ∈ N: x κ is bounded for x large for some κ ≥ 1.

Remark 1. A solution of the equation
Therefore x ′ (t) is continuous. If z 1 and z 2 are two consecutive zeros of x(t) then it is known that x ′ (z 1 ) = −x ′ (z 2 ) due to the fact that the graph of x(t) between two consecutive zeros is symmetric with respect to the middle point (the condition (A1) is only needed). Thus the relation |x ′ (z)| = α holds at any zero of a solution x(t) for given α > 0.
Remark 2. The conditions (A1) and (A2) together are more general than those imposed on f in [8]. The conditions (A3) and (A4) together ensure that the relations (2.3) are valid. One could require that (2.3) holds instead of (A3) and (A4) in statements below.
Theorem 4 (see Section 4) claims that the solution set is a union F = ∞ i=1 F ± i of solution surfaces. The solution surface F + i (resp.: F − i ) describes all C 2 -solutions of the problem (1.1) with exactly i zeros in the interval (a, b) and positive (resp.: negative) x(a). Notion of a solution surface was introduced in [14] for the case of the Dirichlet boundary value problem.
The problem (1.1) with a normalization condition will be referred to as α-normalized problem. We will call α-spectrum F (α) of α-normalized problem a set of all pairs (λ, µ) such that the α-normalized problem is nontrivially solvable. Description of α-spectrum is similar to that given in [8], where a normalization condition |x ′ (z)| = 1 was used: the αspectrum is a union ) is a set of all (λ, µ) such that there exists a C 2 -solution of the α-normalized problem which 1) has exactly i zeros in the interval (a, b); 2) x(a) is positive (resp.: negative); 3) the condition (1.5) fulfils for some fixed α > 0 (so x(t) is a nontrivial solution).
Remark 3. Actually α-branches are projections of α-level sets of solution surfaces to the (λ, µ)-plane, henceforth we will identify these subsets if it would not lead to confusion.
The special cases of the problem (1.1) are: 1) one-parameter linear problem where f = g = x and λ = µ; 2) two-parameter piece-wise linear problem for the Fučík equation where f = g = x; 3) one-parameter nonlinear problem for all x ≥ 0.

Time Maps
Let T f (α, λ) be the first zero function (the time map) for the Cauchy problem If a function f satisfies the conditions (A1) and (A2) then: 1. The time map T f (α, λ) is a continuous function and If additionally T f (α, λ) is differentiable, then it is a solution of the first order nonhomogeneous partial differential equation for any α, λ > 0 and as a consequence the function T f cannot have (positive) extrema; 3. For any α, β, λ > 0 the rescaling formula is valid 4. If additionally (A3) and (A4) hold then for a fixed α > 0: Proof. The assertions 1 and 2 can be proved as the respective assertions in [6]. 3. For α, β, λ > 0 we have h(x) ≤ Ω < +∞. Therefore by Opial's comparison theorem [12,Theorem 11] one has lim inf u→+∞ τ f (u) |x(t)| and defined for all u > 0; τ h (u) has analogous meaning and calculations show that τ h (u) = cu −(κ−1)/2 , c > 0 for all u > 0. From (A1) and (A2), it follows that the function F is strictly increasing in the interval (0; +∞), besides F (x) → +∞ as x → +∞ since the time map t f (α) exists and is finite for any α > 0. One has for a fixed For the nonlinearity f both relations (2.3) are valid, so the conditions (A3) and (A4) fix rather generic limit behavior of time maps.

α-Spectrum of a Two-Parameter Nonlinear Oscillator
The next theorem gives description of an α-spectrum with arbitrary normalization α > 0.
Proof. The proof is similar to that of Theorem 1 in [8]. ⊓ ⊔ Now let us consider properties of an α-spectrum.

For given
8. Suppose f = g and the function γt f (γ) has a point of strict maximum (there is also a point of strict minimum, in view of (2.3)). Then there exists a normalization β j (j ∈ N) such that the branch F ± j (β j ) has a separate bounded component.
Proof. Statements 1, 4 and 5 are valid due to the assertions 1 and 4 of Theorem 1. Statements 2 and 3 follow from (3.1).
Applying the rescaling formula (2.2) to the previous equation one gets α) and the branches F ± i (α) and F ± j (α) are centro-affine equivalent under the mapping Ω i,j .
The case when the function γt f (γ) has a point of strict minimum can be considered analogously. We have proved that there exists a normalization β 1 > 0 such that the branch F ± 1 (β 1 ) has a separate bounded component. Then the branch F ± j (β j ) with the normalization β j = jβ 1 (j ∈ N) is centro-affine equivalent to the branch F ± 1 (β 1 ), see assertion 7 of the theorem, and therefore has a separate bounded component too.
2) An i-th α-branch of the α-spectrum of the α-normalized two-parameter piece-wised linear problem (1.7), (1.5) is the same set for all normalizations α > 0 and coincide with the i-th branch of the classical Fučík spectrum [3,8]: i . For continuation of this remark see Remark 14.
Example 1. In Fig. 1, branches F ± 1 (1) and F ± 2 (2) of the problem (1.1), (1.5) are depicted, where f = g = x 1 3 + x 32 and b − a = 2.5, and these branches are centro-affine equivalent. Proposition 1. For given i = j branches F ± i (α) and F ± j (β) of spectra with different normalizations • may coincide (in this case at least one of the functions f and g must be • may have points of intersection being distinct ; see Fig. 1. Proof. Suppose i = j and two different normalizations α, β > 0 are given. To determine common points of the branches F ± i (α) and F ± j (β) we need to solve the system with respect to (λ, µ). If f = g = x r , r > 0, then using Remark 4 the system

(3.4)
If r = 1 and α β = ( j i ) − r+1 r−1 the equations of the system (3.4) are equivalent, hence the branches F ± i (α) and F ± j (β) coincide. If r = 1 and α β = ( j i ) − r+1 r−1 or r = 1 (linear case) the system (3.4) has no solutions and the branches F ± i (α) and F ± j (β) do not intersect. Thereby, if the branches F ± i (α) and F ± j (β) with α = β and i = j coincide then at least one of the functions f and g must be nonlinear.
Branches F ± i (α) and F ± j (β) of spectra with i = j and different normalizations α, β > 0 may be different with some points of intersection, see Fig. 1. ⊓ ⊔

Solution Set of a Two-Parameter Nonlinear Oscillator
The next theorem gives description of a solution set of the problem (1.1). N). The solution surfaces have the following properties.

Theorem 4. Let functions f and g satisfy the conditions (A1)-(A4). Then solution set F of the problem (1.1) is a union of solution surfaces
1. Solution surfaces F ± i are nonempty sets for any i ∈ N.

Solution surfaces F ±
i and F ± j do not intersect unless i = j.
4. For given i = j the solution surfaces F ± i and F ± j are centro-affine equivalent under the mapping Φ i,j :
Example 2. If f = g = x r , r > 0, r = 1 then the solution surface F ± i is the graph of a two argument function α = (2

Proof. Consider
If α = 2j i α then applying the rescaling formula (2.2) to the previous equation one has 2j−1 and the solution surfaces F ± i and F ± 2j−1 are centro-affine equivalent under the mapping Φ i,2j . ⊓ ⊔ Corollary 1. If i = j the solution surface F ± 2i−1 is centro-affine equivalent to the solution surface F ± 2j−1 under the mapping Φ i,j .
One can prove the following Theorem in a similar manner having in mind assertion 7 of Theorem 3 and Theorem 5.
under the mapping Ω i,2j : Remark 10. The centro-affine mapping Ω i,2j (i = 2j) preserves areas with the coefficient J(Ω i,2j ) = 16(j/i) 4 and has the inverse mapping N) is a cross point of the ( 2j i α)-branch F ± 2j−1 ( 2j i α) and the line µ = µ1 λ1 λ.  N); the solution curve G + i (G − i ) describes all C 2 -solutions of the problem (1.8) which have exactly i zeros in the interval (a, b) and negative (positive) derivative at the first after t = a zero, −α and α respectively; 2) solution curves G + i and G − i coincide for any i, but solution curves G ± i and G ± j do not intersect if i = j. Using results on centro-affine mappings described in Theorems 4, 5 and Remarks 8, 9 we can supplement the theory of similarity of solution branches (solution curves) outlined in [9,10,15].

For given i = j the solution curves G ±
i and G ± j are centro-affine equivalent under the mapping Γ i,j : Corollary 3. If i = j the solution curve G ± 2i−1 is centro-affine equivalent to the solution curve G ± 2j−1 under the mapping Γ i,j .
Remark 11. The analogue of the condition (1.4) in the Dirichlet problem is |x ′ (a)| = α. Solutions associated with the positive branch satisfy the condition x ′ (a) = α > 0. Respectively, solutions associated with the negative branch satisfy the condition x ′ (a) = −α < 0.

Conclusions
A two-parameter nonlinear oscillator with the Neumann boundary conditions exhibits the following features.
• α-spectrum is similar to the classical Fučík spectrum if the functions T f (α, λ) and T g (α, µ) are monotone in λ and µ respectively.
• Otherwise it is possible that α-branches of an α-spectrum have separate bounded components.
• Different branches F ± i (α) and F ± j (β) of spectra with different normalizations may intersect (even coincide). This means that oscillations near cross-points (λ, µ) may switch instantly from one nodal type to another one. This behavior was recognized for large asymmetrical oscillators such as suspension bridges [11, p. 540]. The branches may not intersect as well.
• A solution set is a union of solution surfaces F ± i , besides any two solution surfaces F ± i and F ± j (i = j) are centro-affine equivalent. For given i = 2j the solution surface F ± i is centro-affine equivalent to the solution surface F ± 2j−1 for the Dirichlet problem. In other words these solution surfaces have the same shape.
• The technique of affine mappings can be applied to the special cases of a two-parameter nonlinear oscillator also -α-normalized problem and oneparameter nonlinear problem. In Table 1, the respective centro-affine equivalence formulae are collected.