USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 55–64 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Asymptotic behaviour of the solvability set for pendulum-type equations with linear damping

and homogeneous Dirichlet conditions ^{∗}

A. Cañada & A. J. Ureña

Abstract

We show some results on the asymptotic behavior of the solvability set for a nonlinear resonance boundary-value problem, with linear damping, periodic nonlinearity and homogeneous Dirichlet boundary conditions. Our treatment of the problem depends on a multi-dimensional generalization of the Riemann-Lebesgue lemma.

1 Introduction

Solvability of the nonlinear boundary-value problem

\begin{matrix}−u′′(x) − αu′(x) − λ_{1}(α)u(x) + g(u(x)) = h(x), x ∈ [0, π], \\ u(0) = u(π) = 0 , \end{matrix}  (1.1)

has been studied by several authors under the following set of hypotheses.

[H] α is a given real number, λ_{1}(α) = 1 + α^{2}/4 is the first eigenvalue of the eigenvalue problem

\begin{matrix}−u′′(x) − αu′(x) = λu(x), x ∈ [0, π] \\ u(0) = u(π) = 0 , \end{matrix}  (1.2)

g : ℝ → ℝ is a continuous and T -periodic function with zero mean value, and h ∈ L^{1}[0, π] .

The case α = 0 can be found in [1, 4, 9, 10], while the case α ≠ 0 has been recently treated in [2]. These type of problems, with periodic nonlinearity, are important in applications and (1.1) models, for example, the motion of a pendulum clock ([6, 8]). If g is not identically zero and ψ(x) = exp(αx/2) sin(x) is the principal positive eigenfunction of the adjoint problem to (1.2) for λ = λ_{1}(α), it was proven in ([2]) that for a given h̃ ∈ L^{1}[0, π], with ∫_{0}^{π} h̃(x)ψ(x) dx = 0, there exist real numbers a_{1}(h̃) < 0 < a_{2}(h̃), such that (1.1), with h given by h(x) = aψ(x) + h̃(x), (a ∈ ℝ), has solution if, and only if, a ∈ [a_{1}(h̃), a_{2}(h̃)]. However, very little is known on the behavior of the functionals a_{1} and a_{2}. In this paper we deal with their asymptotic behavior. More precisely, we shall show that if

∗Mathematics Subject Classifications: 34B15, 70K30.

Key words: Pendulum-type equations, linear damping, Dirichlet boundary conditions,

solvability set, asymptotic results, Riemann-Lebesgue lemma, Baire category.

\begin{matrix}c \\ ◯ \end{matrix}2001 Southwest Texas State University.

Published January 8, 2001.

Supported by DGES, by grant PB98-1343 from the Spanish Ministry of Education and

Culture, and by grant FQM116 from Junta de Andalucía.

L̃^{1}[0, π] = {h ∈ L^{1}[0, π] : ∫_{0}^{π} h(x)ψ(x) dx = 0},

then there exist a subset F ⊂ L̃^{1}[0, π], (which will be explicitly described) of first category in L̃^{1}[0, π] in the sense of Baire, such that for each h̃ ∈ L̃^{1}[0, π]\F,

lim_{|λ|→∞} a_{1}(λh̃) = lim_{|λ|→∞} a_{2}(λh̃) = 0 .  (1.3)

As a trivial consequence, the set of functions h̃ ∈ L̃^{1}[0, π] for which (1.3) is true, is a dense and second category subset of L̃^{1}[0, π]. In the final remarks we briefly comment why this result cannot be strengthened very much, since, under hypotheses [H], it may happens that (1.3) does not occur also for a dense subset of L̃^{1}[0, π] (see [3]).

Let us point out that related results for the case of periodic boundary conditions and α = 0 can be found in [7]. However, to the best of our knowledge, properties like (1.3) for the problem (1.1) and periodic nonlinearity g, have not been previously treated in the literature, even for the case α = 0. In the proofs we use the Liapunov-Schmidt reduction, The Baire’s category theorem, some notions on measure theory and the multi-dimensional version of the Riemann-Lebesgue lemma developed in Lemma 3.1 (see [4, 7, 11] for the classical one-dimensional version).

Through this paper, ⟨·, ·⟩ will stand for the Euclidean inner product in ℝ^{N}, while for any x ∈ ℝ^{N}, ∥x∥ := \sqrt{⟨x, x⟩} will denote its associated norm and x_{1}, …, x_{N} its components. We will write as ∥ · ∥_{1} and ∥ · ∥_{∞} the usual norms in L^{1}[0, π] and L_{∞}[0, π] respectively. A function h ∈ L^{1}[0, π] will be called a step function if there exists a partition 0 = x_{0} < x_{1} < … < x_{m−1} < x_{m} = π of the interval [0, π], and constants c_{i}, 1 ≤ i ≤ m such that h|_{(x_{i−1},x_{i})} ≡ c_{i}, 1 ≤ i ≤ m. If, furthermore, all constants c_{i}, i : 1, …, m, are not zero, h will be called a non-vanishing step function. Finally, for every measurable set I ⊂ [0, π], we will denote by χ_{I} its characteristic function, and by meas I its one-dimensional Lebesgue measure.

2 Liapunov-Schmidt reduction

Let any h ∈ L^{1}[0, π] be written in the form h(x) = aψ(x) + h̃(x), a ∈ ℝ, ∫_{0}^{π} h̃(x)ψ(x) dx = 0. Let W_{0}^{2,1}[0, π] = {u ∈ W^{2,1}[0, π], u(0) = u(π) = 0} be the usual Sobolev space with the usual W_{0}^{2,1}[0, π] norm, and define the operators

L : W_{0}^{2,1}[0, π] → L^{1}[0, π], Lu = −u′′ − αu′ − λ_{1}(α)u,

N : W_{0}^{2,1}[0, π] → L^{1}[0, π], (Nu)(x) = aψ(x) + h̃(x) − g(u(x)) .

Then (1.1) is equivalent to the operator equation

Lu = Nu .  (2.1)

Let φ(x) = exp(\frac{−α}{2}x) sin(x) be the principal eigenfunction associated with λ = λ_{1}(α) of the eigenvalue problem (1.2). Each u ∈ W_{0}^{2,1}[0, π] can be written in the form u(x) = cφ(x) + ũ(x), c ∈ ℝ, ∫_{0}^{π} ũ(x)φ(x) dx = 0. Consider the linear, continuous projections

P : W_{0}^{2,1}[0, π] → W_{0}^{2,1}[0, π], cφ + ũ ↦ cφ,
Q : L^{1}[0, π] → L^{1}[0, π], aψ + h̃ ↦ aψ .

(so that im P = ker L, im L = ker Q = L̃^{1}[0, π]), and let K : ker Q → ker P be the inverse of the mapping L : ker P → ker Q. With this notation, (2.1) is equivalent to the system

ũ = K(I − Q)N(cφ + ũ)  (2.2)
a = \frac{1}{∫_{0}^{π}(ψ(x))^{2} dx} ∫_{0}^{π} g(cφ(x) + ũ(x))ψ(x) dx  (2.3)

(auxiliary and bifurcation equation, respectively). Since the natural embedding of W_{0}^{2,1}[0, π] into C[0, π] is compact, we get that for any fixed c ∈ ℝ, there exists at least one solution ũ ∈ ker P of (2.2) ([4], [5]). Denote by Σ the solution set of equation (2.2), i.e.,

Σ = {(c, ũ) ∈ ℝ × ker P : ũ = K(I − Q)N(cφ + ũ)}

and let Γ : Σ → ℝ, be defined by

Γ(c, ũ) = \frac{1}{∫_{0}^{π}(ψ(x))^{2} dx} ∫_{0}^{π} g(cφ(x) + ũ(x))ψ(x) dx  (2.4)

Hence, for a given h̃, BVP (1.1), with h(x) = aψ(x) + h̃(x) has solution, if and only if, a belongs to the set Γ(Σ). The next Theorem, which describes the solvability of (1.1) may be seen in [2].

Theorem 2.1 Let us assume the hypotheses [H] with g not identically zero. Then for each h̃ ∈ L̃^{1}[0, π], there exist real numbers a_{1}(h̃) < 0 < a_{2}(h̃) such that (1.1) with h(x) = aψ(x) + h̃(x) has a solution if, and only if, a ∈ [a_{1}(h̃), a_{2}(h̃)].

In the next section we deal with the asymptotic behavior of the functionals a_{1} and a_{2} as h̃ becomes ‘large’.

3 Asymptotic behavior of the solvability set

In what follows, choose one of the functionals a_{1}, a_{2}, and denote it simply by a. Let h̃ ∈ L̃^{1}[0, π] be given. Taking into account the results of the previous section, we obtain that, for each λ ∈ ℝ, there is (c_{λ}, u_{λ}) ∈ ℝ × ker P such that

u_{λ} = K(I − Q)N_{λ}(c_{λ}φ + u_{λ})  (3.1)

a(λh̃) = \frac{1}{∫_{0}^{π}(ψ(x))^{2} dx} ∫_{0}^{π} g(c_{λ}φ(x) + u_{λ}(x))ψ(x) dx  (3.2)

where N_{λ}u(x) = aψ(x)+λh̃(x)−g(u(x)) = λh̃(x)+N_{0}u(x). Therefore, equation (3.2) becomes

\begin{matrix}a(λh̃) ∫_{0}^{π}(ψ(x))^{2} dx = \\ ∫_{0}^{π} g(c_{λ}φ(x) + λKh̃ + K(I − Q)N_{0}(c_{λ}φ + u_{λ}))ψ(x) dx \end{matrix}  (3.3)

Since K(I −Q)N_{0}(c_{λ}φ+u_{λ}) = K(I −Q)(−g(c_{λ}φ(.)+u_{λ}(.)), there is a constant M > 0 independent of λ ∈ ℝ, such that

\begin{matrix}|K(I − Q)N_{0}(c_{λ}φ + u_{λ})(x)| ≤ M, ∀ x ∈ [0, π], \\ |(K(I − Q)N_{0}(c_{λ}φ + u_{λ}))′(x)| ≤ M, ∀ x ∈ [0, π] \end{matrix}  (3.4)

Previous discussion motivates the next multidimensional generalization of the Riemann-Lebesgue lemma.

Lemma 3.1 Let g : ℝ → ℝ be a continuous and T -periodic function with zero mean value and let u_{1}, …, u_{N} ∈ C^{1}[0, π] be given functions satisfying the following property:

[P] If ρ_{1}, …, ρ_{N} are real numbers such that

meas {x ∈ [0, π] : ∑_{i=1}^{N} ρ_{i}u′_{i}(x) = 0} > 0 ,

then ρ_{1} = … = ρ_{N} = 0.

Let B ⊂ C^{1}[0, π] be such that the set {b′, b ∈ B} is uniformly bounded in C[0, π]. Then, for any given function r ∈ L^{1}[0, π], we have

lim_{∥ρ∥→∞} ∫_{0}^{π} g (∑_{i=1}^{N} ρ_{i}u_{i}(x) + b(x)) r(x) dx = 0  (3.5)

uniformly with respect to b ∈ B.

Proof. Let r ∈ L^{1}[0, π] be a given function and let {ρ^{n}, n ∈ ℕ} ⊂ ℝ^{N} and {b^{n}, n ∈ ℕ} ⊂ B be given sequences with ∥ρ^{n}∥ → ∞. If we define μ^{n} = ρ^{n}/∥ρ^{n}∥, we have, at least for a subsequence, that μ^{n} → μ for some μ ∈ ℝ^{N} with μ_{1}^{2} + … + μ_{N}^{2} = 1. If u = (u_{1}, …, u_{N}), then by hypothesis, meas(Z) = 0, where Z = {x ∈ [0, π] : ⟨μ, u′(x)⟩ = 0}. This implies that the linear span of the set

S = {⟨μ, u′⟩χ_{I} : I is any compact subinterval of [0, π], I ∩ Z = ∅} (3.6)

is a dense set in L^{1}[0, π]. To see this, let us define

S_{1} = {χ_{I} : I is any compact subinterval of [0, π], I ∩ Z = ∅}  (3.7)

Then, for any open subset A ⊂ [0, π] (in particular, for any open subinterval of [0, π]), meas(A \ Z) = meas(A). Since A \ Z is also open, there exists an at most countable collection {I_{i}, i ∈ ℕ} of pairwise disjoint open intervals such that A \ Z = ∪_{i∈ℕ} I_{i} and meas(A \ Z) = ∑_{i∈ℕ} meas(I_{i}). Consequently, the linear span of the set S_{1} is a dense set in the set of step functions and therefore in L^{1}[0, π].

Now, let χ_{I} be a given element of S_{1}. Write w = ⟨μ, u′⟩ and m = inf_{I} |w| (m > 0). Finally, fix ϵ > 0. Choose a partition of I = [a, b], a = a_{0} < a_{1} < … < a_{m−1} < a_{m} = b such that if x, y ∈ J_{i} = [a_{i−1}, a_{i}], 1 ≤ i ≤ m, then |w(x) − w(y)| ≤ ϵ. Then, for any x ∈ I, there is some i, 1 ≤ i ≤ m, such that x ∈ J_{i} and

|χ_{I}(x) − ∑_{i=1}^{m} \frac{wχ_{J_{i}}(x)}{w(a_{i})}| = |\frac{w(a_{i}) − w(x)}{w(a_{i})}| ≤ ϵ/m,

so that

∥χ_{I} − ∑_{i=1}^{m} \frac{wχ_{J_{i}}}{w(a_{i})}∥_{1} ≤ ϵπ/m .

Consequently, we deduce that the linear span of S is dense in S_{1} and therefore in L^{1}[0, π].

On the other hand, if l^{∞} denotes the space of bounded sequences of real numbers with the usual norm, the linear operator T : L^{1}[0, π] → l^{∞}, s → {(T s)^{n}, n ∈ ℕ}, defined by

(T s)^{n} = ∫_{0}^{π} g(⟨ρ^{n}, u(x)⟩ + b^{n}(x))s(x) dx, ∀ s ∈ L^{1}[0, π], ∀ n ∈ ℕ,

is trivially continuous. Recall that our purpose is to prove that T (L^{1}[0, π]) ⊂ l_{0}, the closed subspace of l^{∞} of all sequences which converge to zero. Since T is continuous and l_{0} is closed, to prove the lemma it is sufficient to demonstrate that T (S) ⊂ l_{0}, i.e.,

lim_{n→∞} ∫_{I} g(⟨ρ^{n}, u(x)⟩ + b^{n}(x))(⟨μ, u′(x)⟩) dx = 0,  (3.8)

for any compact subinterval I of [0, π] such that I ⋂ Z = ∅. But, if v^{n}, v : I → ℝ are defined as

\begin{matrix}v^{n}(x) = ⟨μ^{n}, u(x)⟩ + b^{n}(x)/∥ρ^{n}∥, \\ v(x) = ⟨μ, u(x)⟩, ∀ x ∈ [0, π], \end{matrix}

we trivially have

lim_{n→∞} ∫_{I} g(∥ρ^{n}∥v^{n}(x))(v′(x) − (v^{n})′(x)) dx = 0  (3.9)

and

lim_{n→∞} ∫_{I} g(∥ρ^{n}∥v^{n}(x))(v^{n})′(x)) dx  (3.10)
= lim_{n→∞} \frac{G(∥ρ^{n}∥v^{n}(max I)) − G(∥ρ^{n}∥v^{n}(min I))}{∥ρ^{n}∥} = 0

where G is any primitive function of function g. Now, (3.9) and (3.10) imply (3.8).

Remark. It is clear that both the conclusion and the proof of the previous lemma are still true under more general hypotheses on the function g. It is sufficient that g be a continuous and bounded function with bounded primitive G.

Next, we apply the previous lemma to the specific problem of the asymptotic behavior of the functional a whose expression was given in (3.3).

Corollary 3.2 Let h̃ ∈ L̃^{1}[0, π] be a given function and suppose that the functions Kh̃ and φ satisfy the following property

[P1] If ρ_{1}, ρ_{2} are real numbers such that

meas{x ∈ [0, π] : ρ_{1}(Kh̃)′(x) + ρ_{2}φ′(x) = 0} > 0,

then ρ_{1} = ρ_{2} = 0.

Let B ⊂ L̃^{1}[0, π] be any bounded subset. Then

lim_{|λ|→∞} a(λh̃ + b) = 0,  (3.11)

uniformly with respect to b ∈ B.

Proof. For each λ ∈ ℝ and each b ∈ B, there is (c_{λ,b}, ũ_{λ,b}) ∈ Σ_{λ,b} such that

a(λh̃ + b) ∫_{0}^{π}(ψ(x))^{2} dx = ∫_{0}^{π} g(c_{λ,b}φ(x) + λKh̃(x) + Kb(x) (3.12)
+K(I − Q)N_{0}(c_{λ,b}φ + ũ_{λ,b})(x))ψ(x) dx

where Σ_{λ,b} is the corresponding solution set of the auxiliary equation for λh̃+b. Since the set

{Kb + K(I − Q)N_{0}(c_{λ,b}φ + ũ_{λ,b}), λ ∈ ℝ, b ∈ B}

is bounded in C^{1}[0, π] (see (3.4)), the conclusion follows from the previous lemma.

The following equivalent version of previous corollary will be very useful for our purposes.

Corollary 3.3 Let h̃ ∈ L̃^{1}[0, π] be a given function and suppose that, for every ρ ∈ ℝ,

meas{x ∈ [0, π] : (Kh̃)′(x) = ρφ′(x)} = 0.

Let B ⊂ L̃^{1}[0, π] be any bounded subset. Then

lim_{|λ|→∞} a(λh̃ + b) = 0,  (3.13)

uniformly with respect to b ∈ B.

Now, we state and prove our main result.

Theorem 3.4 There exists a subset F ⊂ L̃^{1}[0, π], of first category in L̃^{1}[0, π], such that for any given h̃ ∈ L̃^{1}[0, π] \ F, and each given bounded subset B ⊂ L̃^{1}[0, π], one obtains

lim_{|λ|→∞} a(λh̃ + b) = 0  (3.14)

uniformly with respect to b ∈ B.

Proof. Let

F = {h̃ ∈ L̃^{1}[0, π] : ∃ ρ ∈ ℝ s.t. meas {x ∈ [0, π] : (Kh̃)′(x) = ρφ′(x)} > 0}

Then F = ∪_{n∈ℕ} F_{n}, where

F_{n} = {h̃ ∈ L̃^{1}[0, π] : ∃ ρ ∈ ℝ s.t. meas{x ∈ [0, π] : (Kh̃)′(x) = ρφ′(x)} ≥ 1/n}

Let us prove that each subset F_{n} is closed and has an empty interior. To see this, let us fix F_{n}. Then, since K : ker Q → ker P is a topological isomorphism, F_{n} is a closed subset of ker Q if and only if K(F_{n}) ≡ G_{n} is a closed subset of ker P. Now, it is clear that G_{n} is the set of functions

{u ∈ ker P : ∃ρ ∈ ℝ s.t. meas{x ∈ [0, π] : u′(x) = ρφ′(x)} ≥ 1/n}

Let {u_{m}, m ∈ ℕ} ⊂ G_{n} be a sequence such that {u_{m}} → u in ker P. Then, for any m ∈ ℕ, we can find ρ_{m} ∈ ℝ such that

meas{x ∈ [0, π] : u′_{m}(x) = ρ_{m}φ′(x)} ≥ 1/n

Since

meas{x ∈ [0, π] : φ′(x) = 0} = 0,

the sequence {ρ_{m}} must be bounded and, after possibly passing to a subsequence, we can suppose, without loss of generality, that {ρ_{m}} → ρ. Moreover, if we define

M_{m} = {x ∈ [0, π] : u′_{m}(x) = ρ_{m}φ′(x)}

then meas M_{m} ≥ 1/n, ∀ m ∈ ℕ and meas (⋂_{m=1}^{∞} [⋃_{s=m}^{∞} M_{s}]) ≥ 1/n. Finally, let us observe that if x ∈ ⋂_{m=1}^{∞} [⋃_{s=m}^{∞} M_{s}], then u′(x) = ρφ′(x), so that meas{x ∈ [0, π] : u′(x) = ρφ′(x)} ≥ 1/n and consequently u ∈ G_{n}.

Next, we prove that F (and therefore each F_{n}) has an empty interior. To see this, let us define the function φ_{1} as the primitive of φ with zero mean value and φ_{2} as the primitive of φ_{1} satisfying φ_{2}(0) = φ_{2}(π) = 0. Then, φ_{2} ∈ W_{0}^{2,1}[0, π], φ′′_{2} = φ and for any u ∈ W_{0}^{2,1}[0, π], we have

∫_{0}^{π} uφ = − ∫_{0}^{π} u′φ_{1} = ∫_{0}^{π} u′′φ_{2}.

As a consequence, the mapping Φ : ker P → L̃_{φ_{2}}^{1}[0, π], u → u′′ is a topological isomorphism, where

L̃_{φ_{2}}^{1}[0, π] = {h ∈ L^{1}[0, π] : ∫_{0}^{π} h(x)φ_{2}(x) dx = 0}

Therefore, F has an empty interior in L̃^{1}[0, π] provided Φ(K(F)) has an empty interior in L̃_{φ_{2}}^{1}[0, π]. This last result is an easy consequence of the following lemma.

Lemma 3.5 Let us denote by A the subset of L^{1}[0, π] given by all the step functions and by B the subset of L^{1}[0, π] given by all the non-vanishing step functions. Then,

1. The set A ∩ L̃_{φ_{2}}^{1}[0, π] is dense in L̃_{φ_{2}}^{1}[0, π].

2. The set B ∩ L̃_{φ_{2}}^{1}[0, π] is dense in L̃_{φ_{2}}^{1}[0, π].

3. B ∩ Φ(K(F)) = ∅

Proof.

1. Let us choose any h ∈ L̃_{φ_{2}}^{1}[0, π] and ϵ > 0. Then, there exists s ∈ A such

that ∥h − s∥_{1} < min \{ϵ/2π, \frac{∥φ_{2}∥_{1}}{∥φ_{2}∥_{∞}} \}. Now, the function s̃ = s + ^{\frac{∫_{0}^{π} sφ_{2}}{∥φ_{2}∥_{1}}} is again a step function which belongs to L̃_{φ_{2}}^{1}[0, π] and such that ∥h−s̃∥_{1} < ϵ.

2. Let us demonstrate that B ∩ L̃_{φ_{2}}^{1}[0, π] is dense in A ∩ L̃_{φ_{2}}^{1}[0, π]. To see this, let us take u ∈ A ∩ L̃_{φ_{2}}^{1}[0, π]. If a, b ∈ ℝ, define the function u_{a,b} = u + aχ_{[0,π/2]} + bχ_{[π/2,π]}. The condition for u_{a,b} to belong to L̃_{φ_{2}}^{1}[0, π] is

a ∫_{0}^{π/2} φ_{2} + b ∫_{π/2}^{π} φ_{2} = 0

Since ∫_{0}^{π/2} φ_{2} < 0 and ∫_{π/2}^{π} φ_{2} < 0, (think that, by the maximum principle, φ_{2} < 0 in (0, π)), it is clear that we may choose a and b both different from zero but as small as we want in absolute value such that u_{a,b} ∈ B ∩ L̃_{φ_{2}}^{1}[0, π].

3. If s ∈ B ∩ Φ(K(F)), then there is h̃ ∈ F such that K(h̃) = u, Φ(u) = u′′ = s. Since h̃ ∈ F, there exists ρ ∈ ℝ such that meas {x ∈ [0, π] : u′(x) = ρφ′(x)} > 0. Choose some nontrivial compact interval I ⊂ [0, π] satisfying s|_{I} ≡ c ≠ 0 and such that meas {x ∈ I : u′(x) = ρφ′(x)} > 0. This implies that meas {x ∈ I : c = u′′(x) = ρφ′′(x)} > 0, which is a contradiction with the form of the function φ.

Final Remark. Under the hypotheses [H], it is possible to show that, in many cases, the set of functions h̃ ∈ L̃^{1}[0, π] for which lim_{|λ|→∞} a(λh̃) is not zero, is also dense in L̃^{1}[0, π]. For example, this is true for the oscillating function δg in the place of g provided that |δ| is small enough. In this case, it may be proved that the previous limit is not zero if the function u = K(h̃) belongs to the set of functions in ker P for which there exists a partition 0 = x_{0} < x_{1} < … < x_{p−1} < x_{p} = π and 1 ≤ i_{0} ≤ p and constants μ ≠ 0, c ≠ 0, such that

i) u′′_{[x_{i−1},x_{i}]} is a constant function, for any 1 ≤ i ≤ p, i ≠ i_{0}.

ii) u(x) = μφ(x) + c, ∀ x ∈ [x_{i_{0}−1}, x_{i_{0}}].

After this, it may be proved that this set is dense in ker P. The detailed proof may be found in [3].

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A. Cañada Dep. Análisis Matemático, Univ. de Granada, 18071-Granada, Spain e-mail: acanada@ugr.es A. J. Ureña Dep. Análisis Matemático, Univ. de Granada, 18071-Granada, Spain e-mail: ajurena@ugr.es