ON SYSTEMS OF TELEGRAPH EQUATIONS

Juha Berkovits and Vesa Mustonen

1. Introduction

In this note we shall consider the existence of 2π-periodic solutions for systems of telegraph equations of the form

(1.1)  \{ \begin{matrix}u_{tt} − u_{xx} − βu_{t} − g(t, x, u) = h(t, x), (t, x) ∈ Ω \\ u(t, 0) = u(t, π) = 0, t ∈]0, 2π[, \end{matrix}

where h is a given function in L^{2}(Ω; ℝ^{n}) with Ω =]0, 2π[×]0, π[ and n is a positive integer, the function g(t, x, s) from Ω × ℝ^{n} to ℝ^{n} is 2π-periodic in t, measurable in (t, x) for each s ∈ ℝ^{n}, and continuous in s for almost all (t, x) ∈ Ω and β is a real diagonal n × n-matrix diag(β_{1}, β_{2}, …, β_{n}). We shall assume that g satisfies the usual growth condition

(1.2)  |g(t, x, s)| ≤ c_{0}|s| + k_{0}(t, x)

for all s ∈ ℝ^{n} and a.a. (t, x) ∈ Ω with some constant c_{0} > 0 and k_{0} ∈ L^{2}(Ω). We are interested in conditions on g such that a solution u ∈ L^{2}(Ω; ℝ^{n}) exists for any h ∈ L^{2}(Ω; ℝ^{n}) (nonresonance) and in conditions on g and h such that a solution exists for a given h (resonance). In fact, we shall study the existence of weak solutions and therefore it is relevant to transform the system (1.1) into the operator equation

(1.3)  Lu − N (u) = h  u ∈ D(L)

in H = [L^{2}(Ω)]^{n}, where N is the Nemytskii operator generated by g and L: D(L) ⊂ H → H is the abstract realization of the telegraph operator ∂_{tt} − ∂_{xx} − β∂_{t} with periodic Dirichlet boundary conditions in H. It is not hard to see that the linear operator L is densely defined, closed, and normal. Moreover, if det β = β_{1}β_{2} ⋯ β_{n} ≠ 0, then L is invertible but otherwise Ker L is infinite dimensional. It is essential that the (partial) inverse of L is compact.

The problem (1.3) has been studied mainly in the case of gradient type nonlinearities N with no damping, i.e., with β = 0. We refer the reader to the papers by Brézis and Nirenberg [BN], Bates and Castro [BC], Amann [Am] and Mawhin [Ma] and to the recent contribution by Fonda and Mawhin [FM], where also a survey of the recent results and relevant references can be found. In this note we shall use a different approach based on the degree theory introduced in our earlier paper [BM1]. In [BM3] we treated the case of nonresonance for systems of wave equations without damping terms. The goal of this note is to deal with more general systems with β ≠ 0 and also to study the resonance case.

Our paper is organized as follows. In Section 2 we first recall the concept of complexification of a real Hilbert space and a linear operator. Essential to our study is the structure of the complex spectrum of normal operator. We shall introduce some classes of nonlinear mappings of monotone type and give a useful continuation theorem based on topological degree argument. This abstract result will be used frequently in the sequel. Section 3 is devoted to the linear case, that is, a system of equations with linear coupling. We obtain some injectivity results and estimates of the norm of inverse operator. In Section 4 we consider the nonresonance for systems of normal operators with linear coupling and with nonlinear perturbations. The abstract result is then applied to a system of telegraph and wave equations with a nonlinearity of special type. In our last section the resonance for systems is briefly studied. We derive abstract results which are then applied to the model problem (1.1).

2. Prerequisites

Let H be a real separable Hilbert space with inner product ⟨·,·⟩ and corresponding norm ∥ · ∥. We recall the standard procedure of complexification. Indeed, the complexification of H, H_{ℂ} = H + iH, has the usual linear structure and inner product induced by H, that is,

⟨u + iv, x + iy⟩_{ℂ} = ⟨u, x⟩ + ⟨v, y⟩ + i(⟨x, v⟩ − ⟨u, y⟩)

for all u + iv, x + iy ∈ H_{ℂ}. Let L: D(L) ⊂ H → H be a densely defined closed, normal linear operator with Im L = (Ker L)^{⊥}. The inverse K of the restriction of L to Im L∩D(L) is a bounded linear operator on Im L. We shall assume that K is compact. Note that Ker L may be infinite dimensional. We are interested in the case when L is not self adjoint and therefore we include the complex spectrum of L into consideration (cf. [BN], [He]). Indeed, we define the complex linear operator L_{ℂ}: D(L_{ℂ}) ⊂ H_{ℂ} → H_{ℂ} by setting D(L_{ℂ}) = D(L) + iD(L) and L_{ℂ}(u + iv) = Lu + iLv for all u + iv ∈ D(L_{ℂ}). It is clear that Im L_{ℂ} = (Ker L_{ℂ})^{⊥}, L_{ℂ} is normal and its partial inverse K_{ℂ} is compact. Consequently, L_{ℂ} has a pure point spectrum σ_{ℂ}(L) = {μ_{j}}_{j∈ℤ} with corresponding orthonormal basis {ϕ_{j}}_{j∈ℤ} of H_{ℂ} such that

L_{ℂ}ϕ_{j} = μ_{j}ϕ_{j} for all j ∈ ℤ.

Note that for any complex eigenvalue μ = α + iβ also its complex conjugate \overline{μ} = α − iβ is an eigenvalue. It is not hard to see that the spectrum of L_{ℂ} can be rewritten in the form

σ_{ℂ}(L) = {μ_{j}: μ_{j} = α_{j} ∈ ℝ} ∪ {μ_{j}^{±}: μ_{j}^{±} = α_{j} ± iβ_{j}, β_{j} > 0},

and there exists an orthonormal basis {ψ_{j}} ∪ {ω_{j}} of the original space H such that

\{ϕ_{j}\} = \{ψ_{j}: μ_{j} ∈ ℝ\} ∪ \{\frac{1}{\sqrt{2}}(ψ_{j} ± iω_{j}): μ_{j}^{±} ∉ ℝ\}.

For each z = u + iv ∈ D(L_{ℂ}) we have the spectral representation

L_{ℂ}z = ∑_{j∈ℤ} μ_{j}⟨z, ϕ_{j}⟩_{ℂ}ϕ_{j}.

Note that L_{ℂ}u = Lu for all u ∈ D(L) and hence

⟨Lu, u⟩ = ∑_{j∈ℤ} Re μ_{j}|⟨u, ϕ_{j}⟩_{ℂ}|^{2}.

Using the real basis we can also write

Lu = ∑_{j∈ℤ} α_{j}⟨u, ψ_{j}⟩ψ_{j} + ∑_{μ_{j}^{±}∉ℝ}{α_{j}⟨u, ω_{j}⟩ω_{j} + β_{j}[⟨u, ω_{j}⟩ψ_{j} − ⟨u, ψ_{j}⟩ω_{j}]},

where one can easily see the representations of the symmetric and antisymmetric parts of the operator L.

Let H = H^{n}, n ≥ 2, and L: D(L) ⊂ H → H be a diagonal operator defined by

Lu = (L_{1}u_{1}, L_{2}u_{2}, …, L_{n}u_{n}), u = (u_{1}, u_{2}, …, u_{n}) ∈ D(L),

where the linear operators L_{k}, k ∈ {1, 2, …, n} satisfy the properties imposed on the operator L above. Note that the complexification H_{ℂ} := H + iH = (H_{ℂ})^{n}. Obviously L and L_{ℂ} inherit the properties of the component operators. We shall use the notations ⟨·,·⟩ and ∥ · ∥ for the inner product and norm in a real Hilbert space and the subscript ’ℂ’ whenever the space is complex. For simplicity we shall frequently use the same symbol for an operator and its complexification.

Let N : H → H be a possibly nonlinear map. We assume that N is bounded in the sense that it takes any bounded set into a bounded set, and demicontinuous, i.e., u_{k} → u (norm convergence) implies N (u_{k}) ⇀ N (u) (weak convergence). We recall that N is monotone if

⟨N (u) − N (v), u − v⟩ ≥ 0 for all u, v ∈ H.

If there exists a constant μ > 0 such that for all u, v ∈ H

⟨N (u) − N (v), u − v⟩ ≥ μ∥u − v∥^{2},

we say that N is strongly monotone. Such mappings belong to a wider class (S_{+}), defined by the property: u_{k} ⇀ u and lim sup⟨N (u_{k}), u_{k} − u⟩ ≤ 0 imply that u_{k} → u. Note that any demicontinuous monotone map is pseudomonotone, i.e., the conditions u_{k} ⇀ u, lim sup⟨N (u_{k}), u_{k} − u⟩ ≤ 0 imply N (u_{k}) ⇀ N (u) and ⟨N (u_{k}), u_{k}−u⟩ → 0. In our previous paper [BM1] we introduced an extension of the classical Leray-Schauder degree suitable for our present purposes. Indeed, if N is demicontinuous, bounded, and of class (S_{+}), G ⊂ H is an open bounded set, and h ∉ (L − N)(∂G ∩ D(L)), then the degree deg(L − N , G, h) is well defined and satisfies the properties of the classical topological degree. Let P: H → Ker L be an orthogonal projection. The injection L − P is important because it is the normalizing map, that is,

deg(L − P, G, h) = \{ \begin{matrix}1, & if h ∈ (L − P)(G ∩ D(L)) \\ 0, & if h ∉ (L − P)(G ∩ D(L). \end{matrix}

By a reference map we mean in the sequel any injection L − N for which deg(L−N , G, h) ≠ 0 whenever h ∈ (L−N)(G∩D(L)). For a given nonlinear problem it is crucial to find an appropriate reference map which reflects the structure of the original equation and enables to apply the continuation principle. We shall deal with reference maps of the form L − A, where A is linear. The following existence theorem is a direct consequence of the homotopy invariance of topological degree (see [BM1]).

Continuation Theorem. Let G be an open bounded convex subset in H with 0 ∈ G, N : H → H a bounded demicontinuous pseudomonotone map, h ∈ H, and let A: H → H be a linear operator such that L−A is a reference map. If

Lu − t(N (u) + h) − (1 − t)Au ≠ 0

for all t ∈ [0, 1] and u ∈ ∂G ∩ D(L), then the equation

Lu − N (u) = h, u ∈ G ∩ D(L),

admits at least one solution.

Note that the Continuation Theorem remains valid also in the following cases. First, the set G need not be convex if the map N is of class (S_{+}). Secondly, if Ker L is finite dimensional, then the classical Leray-Schauder degree can be applied, and we can remove the convexity of G and also any monotonicity condition on N . Naturally, this observation holds for the reference map, too. Hence we shall in the sequel assume that A is strongly monotone in the case dim Ker L = ∞.

3. Linear case

Let H be a real separable Hilbert space and H = H^{n}. We consider a real n×n-matrix A = (a_{lk}) and the constant multiplication operator A: H → H induced by A, that is, for any u = (u_{1}, u_{2}, …, u_{n}) ∈ H

Au = w = (w_{1}, w_{2}, …, w_{n}),

with w_{l} = ∑_{k=1}^{n} a_{lk}u_{k}, l = 1, 2, …, n. In the sequel A also denotes the linear operator in ℝ^{n} and ℂ^{n} corresponding to the matrix (a_{lk}). Clearly σ(A) = σ(A), a point spectrum which may be empty if n is even. Similarly σ_{ℂ}(A) = σ_{ℂ}(A). If the matrix A is strictly positive, it is not hard to prove that

⟨Au, u⟩ ≥ α∥u∥^{2} for all u ∈ H,

where α = min{(Ax, x)_{ℝ^{n}}: |x|_{ℝ^{n}} = 1} is positive. Let L be the diagonal operator L = diag(L_{1}, L_{2}, …, L_{n}), defined on a dense subset D(L) = D(L_{1})×D(L_{2})×⋯×D(L_{n}) of H. We assume that each L_{k} is a closed normal linear operator with Im L_{k} = (Ker L_{k})^{⊥} and that K_{k}, the partial inverse of L_{k}, is compact. Futhermore we shall impose a strong assumption that each L_{k} has the same complex eigenbasis {ϕ_{j}}_{j∈ℤ} with corresponding complex eigenvalues {μ_{j}^{(k)}}_{j∈ℤ}. This condition holds in particular for systems of telegraph equations with different damping terms. From now on we always assume that the matrix A is strictly positive whenever Ker L is infinite dimensional. This is needed in order to employ the Continuation Theorem. For any z = (z_{1}, z_{2}, …, z_{n}) ∈ H_{ℂ} we denote z⃗_{j} = ∑_{k=1}^{n}⟨z_{k}, ϕ_{j}⟩_{ℂ}e_{k}, where {e_{k}} is the standard basis of ℝ^{n}. Hence we can write

z = ∑_{j∈ℤ} z⃗_{j}ϕ_{j} and ∥z∥^{2} = ∑_{j∈ℤ} |z⃗_{j}|_{ℂ^{n}}^{2}.

It is easy to see that for any z = u + iv ∈ D(L) + iD(L) we have

(3.1)  (L − A)(z) = ∑_{j∈ℤ}[(M_{j} − A)z⃗_{j}]ϕ_{j},

where M_{j} = diag(μ_{j}^{(1)}, μ_{j}^{(2)}, …, μ_{j}^{(n)}). By the above formula we get

Lemma 3.1. The operator L−A is injective if and only if det(M_{j} −A) ≠ 0 for all j ∈ ℤ.

Note that in the case L_{k} = L for all k = 1, 2, …, n, the injectivity condition in Lemma 3.1 can be written as σ_{ℂ}(L) ∩ σ_{ℂ}(A) = ∅. We also get a simple formula for the norm of (L − A)^{−1}.

Lemma 3.2. If det(M_{j} − A) ≠ 0 for all j ∈ ℤ, then

∥(L − A)^{−1}∥ = sup_{j} ∥(M_{j} − A)^{−1}∥ < ∞.

Proof. The image of L − A is closed and hence the inverse operator is bounded. By (3.1) we see that

z = (L − A)^{−1}w = ∑_{j∈ℤ}[(M_{j} − A)^{−1}w⃗_{j}]ϕ_{j}

is the unique solution of the equation Lz − Az = w. Hence

∥(L − A)^{−1}∥ ≤ sup_{j} ∥(M_{j} − A)^{−1}∥.

On the other hand, if x⃗ ∈ ℂ^{n} and |x⃗|_{ℂ^{n}} = 1, then clearly (L − A)^{−1}(x⃗ϕ_{j}) = [(M_{j} − A)^{−1}x⃗]ϕ_{j} for any j ∈ ℤ and thus

∥(L − A)^{−1}∥ ≥ sup_{j} ∥(M_{j} − A)^{−1}∥

completing the proof. □

In what follows we shall consider two special cases. If A = τI, where τ > 0 and τ ∉ σ(L), then σ_{ℂ}(L − τI) = {μ_{j}^{(k)} − τ: j ∈ ℤ, k = 1, 2, …, n} and

(3.2)  ∥(L − τI)^{−1}∥ = \frac{1}{min_{j,k} |τ − μ_{j}^{(k)}|}.

If L_{1} = L_{2} = ⋯ = L_{n} = L and A is normal, then L−A can be diagonalized and σ_{ℂ}(L − A) = {μ_{j} − q_{k}: j ∈ ℤ, k = 1, 2, …, n}, where the complex eigenvalues of A are denoted by {q_{k}} and μ_{j}^{(k)} ≡ μ_{j}. In the case L − A is invertible we have

(3.3)  ∥(L − A)^{−1}∥ = \frac{1}{min_{j,k} |q_{k} − μ_{j}|}.

Now any invertible L − A is a reference map. As a direct consequence of the results in [BM1] we get

Lemma 3.3. Assume that A = (a_{lk}) is a real n × n−matrix such that L − A is invertible. For any open bounded set G in H and any h ∈ H such that h ∈ (L − A)(G ∩ D(L)) we have

deg(L − A, G, h) ≠ 0.

In the study of resonance we shall need further estimates of the type

(3.4) −\frac{1}{θ_{−}} ∥Lu − Au∥^{2} ≤ ⟨Lu − Au, u⟩ ≤ \frac{1}{θ_{+}} ∥Lu − Au∥^{2} for all u ∈ D(L),

where L − A may not be injective. In the special case A = τI, τ ∈ ℝ, and possibly τ ∈ σ(L), we have in an analogy with [BM2]

(3.5) θ_{+} = inf_{Re μ_{j}^{(k)}>τ} \frac{|μ_{j}^{(k)} − τ|^{2}}{Re(μ_{j}^{(k)} − τ)},  θ_{−} = inf_{Re μ_{j}^{(k)}<τ} \frac{|μ_{j}^{(k)} − τ|^{2}}{Re(τ − μ_{j}^{(k)})}.

If L_{k} = L for all k = 1, 2, …, n and the matrix A is normal, then

(3.6) θ_{+} = inf_{Re(μ_{j}−q_{k})>0} \frac{|μ_{j} − q_{k}|^{2}}{Re(μ_{j} − q_{k})},  θ_{−} = inf_{Re(μ_{j}−q_{k})<0} \frac{|μ_{j} − q_{k}|^{2}}{Re(μ_{j} − q_{k})}.

In [BM2] the values of θ_{+} and θ_{−} are calculated in the case of a single telegraph equation.

4. Nonresonance for systems

Let L = diag(L_{1}, L_{2}, …, L_{n}) be as defined in section 2 and let N : H → H be a demicontinuous bounded operator. We assume that N is monotone or, more generally, pseudomonotone. Note that this assumption is not needed if Ker L is finite dimensional. By nonresonance we mean that the equation

(4.1)  Lu − N (u) = h,  u ∈ D(L),

admits a solution for any given h in H. From the Continuation Theorem we derive the following abstract nonresonance result.

Theorem 4.1. Assume that L − A is a reference map and N is pseudomonotone if Ker L is infinite dimensional. If there exist constants η > 0 and C ≥ 0 such that

η < ∥(L − A)^{−1}∥^{−1}, ∥N (u) − Au∥ ≤ η∥u∥ + C for all u ∈ H,

then the equation (4.1) admits at least one solution for any h ∈ H.

Proof. It is sufficient to show that there exists R > 0 such that

Lu − t(N (u) + h) − (1 − t)Au ≠ 0

for all t ∈ [0, 1] and u ∈ D(L) with ∥u∥ = R. Indeed, if

Lu − Au = t[N (u) − Au + h],

then the fact that L−A is a reference map together with the other assumptions implies

∥u∥ ≤ ∥(L − A)^{−1}∥[η∥u∥ + C + ∥h∥].

Since ∥(L − A)^{−1}∥η < 1 it suffices to take any

R > (1 − η∥(L − A)^{−1}∥)^{−1}∥(L − A)^{−1}∥(C + ∥h∥)

to get the desired result. □

Instead of (4.1) it is often more natural and more convenient to consider the equation

(4.2)  Lu − Au − Ñ(u) = h,  u ∈ D(L),

where Ñ = N −A and Ñ satisfies the growth condition ∥Ñ(u)∥ ≤ η∥u∥+C in the previous theorem.

Example. In order to illustrate the application of Theorem 4.1 we shall consider a system of a telegraph equation and a wave equation with linear coupling (4.3)

\frac{∂^{2}u_{1}}{∂t^{2}} − \frac{∂^{2}u_{1}}{∂x^{2}} − β_{1} \frac{∂u_{1}}{∂t} − a_{11}u_{1} − a_{12}u_{2} − q_{1}(t, x, u_{1}) = h_{1}(t, x) in Ω

\frac{∂^{2}u_{2}}{∂t^{2}} − \frac{∂^{2}u_{2}}{∂x^{2}} − a_{21}u_{1} − a_{22}u_{2} − q_{2}(t, x, u_{2}) = h_{2}(t, x) in Ω

u_{i}(t, 0) = u_{i}(t, π) = 0, t ∈]0, 2π[, i = 1, 2
u_{i}(·, x) is 2π-periodic, x ∈]0, π[, i = 1, 2,

where Ω =]0, 2π[×]0, π[. We assume that the coupling matrix

A = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} )

is strictly positive, i.e., a_{11} > 0, a_{22} > 0, |a_{12} + a_{21}| < 2\sqrt{a_{11}a_{22}}, and that q_{1}, q_{2} : Ω × ℝ → ℝ are Carathéodory functions satisfying the usual growth condition of the form (1.2) and are 2π-periodic in t.

Denoting H = L^{2}(Ω) and ϕ_{jk}(t, x) = \frac{1}{π} sin(jx) exp(ikt), (t, x) ∈ Ω, j ∈ ℤ_{+}, k ∈ ℤ, the set {ϕ_{jk}} forms an orthonormal basis in H_{ℂ}. Observe the natural indexing used here. The telegraph operator L_{1} = ∂_{tt} − ∂_{xx} − β_{1}∂_{t} of (4.3) has in H_{ℂ} the representation

L_{1}v = ∑_{j,k}(l_{jk} − iβ_{1}k)⟨v, ϕ_{jk}⟩_{ℂ}ϕ_{jk}

for all v ∈ D(L_{1}) with l_{jk} = j^{2} − k^{2} and

D(L_{1}) = {v ∈ H_{ℂ}: ∑_{j,k} |l_{jk} − iβ_{1}k|^{2}|⟨v, ϕ_{jk}⟩_{ℂ}|^{2} < ∞}.

Obviously, Ker L_{1} = {0} and σ_{ℂ}(L_{1}) = {μ_{jk}^{(1)} = l_{jk} − iβ_{1}k: j ∈ ℤ_{+}, k ∈ ℤ}.

Note that σ(L_{1}) = σ_{ℂ}(L_{1})∩ℝ = {μ_{j0}^{(1)} = j^{2} : j ∈ ℤ_{+}}. For the wave operator L_{2} = ∂_{tt} − ∂_{xx} we respectively have an analogous representation, but now Ker L_{2} is infinite dimensional and σ_{ℂ}(L_{2}) = σ(L_{2}) = {l_{jk} = j^{2} − k^{2} : j ∈ ℤ_{+}, k ∈ ℤ}.

The diagonal operator L = diag(L_{1}, L_{2}) is defined on D(L) = D(L_{1}) × D(L_{2}) ⊂ H_{ℂ} = H_{ℂ}^{2}. Since L_{1} is normal and L_{2} is self adjoint both having compact (partial) inverses, also L is normal with compact partial inverse. The system (4.3) corresponds to the equation

Lu − Au − Ñ(u) = h, u ∈ D(L),

where h = (h_{1}, h_{2}) ∈ H, Ñ(u_{1}, u_{2}) = (q_{1}(·, ·, u_{1}), q_{2}(·, ·, u_{2})) and A is the constant multiplication operator induced by the matrix A. In view of Lemma 3.1 the linear part L − A is injective if and only if

det(M_{jk} − A) ≡ det ( \begin{matrix}λ_{jk} + iβ_{1}k − a_{11} & −a_{12} \\ −a_{21} & λ_{jk} − a_{22} \end{matrix} ) ≠ 0

for all j ∈ ℤ_{+}, k ∈ ℤ. Denote A_{jk} = M_{jk} − A, D_{jk} = det A_{jk} and let ∥ · ∥_{E} stand for the Frobenius norm of a matrix, that is, the Euclidean norm in ℝ^{4}. Then it is not hard to see that

∥A_{jk}^{−1}∥ = \frac{1}{2|D_{jk}|^{2}} (∥A_{jk}∥_{E}^{2} + \sqrt{∥A_{jk}∥_{E}^{4} − 4|D_{jk}|^{2}}).

Note that for any matrix B we have ∥B∥ ≤ ∥B∥_{E}, and the Frobenius norm is very easy to calculate. In the two dimensional case as above also ∥B∥ ≥ \frac{1}{\sqrt{2}}∥B∥_{E}. We have

D_{jk} = det(M_{jk} − A) = det(λ_{jk}I − A) + iβ_{1}k(λ_{jk} − a_{22}).

Hence, if a_{12}a_{21} ≠ 0, the operator L − A is injective if and only if j^{2} ∉ σ(A) for all j ∈ ℤ_{+}, that is, σ(L_{1})∩σ(A) = ∅. Similarly, if a_{12}a_{21} = 0, then L−A is injective if and only if σ(L_{2}) ∩ σ(A) = ∅.

Before introducing further conditions for the nonlinearity Ñ required in Theorem 4.1 we need an estimate of the norm ∥(L − A)^{−1}∥ = sup_{jk} ∥A_{jk}^{−1}∥. It is not hard to see that

∥A_{jk}^{−1}∥^{2} ≤ \frac{2d_{jk} + β_{1}^{2}k^{2} + R^{2}}{d_{jk}^{2} + β_{1}^{2}k^{2}(l_{jk} − a_{22})^{2}} ,

where d_{jk} = det(l_{jk}I − A) and R^{2} = (a_{11} − a_{22})^{2} + (a_{12} + a_{21})^{2}. Denoting

κ^{2} = sup_{j,k} \frac{2d_{jk} + β_{1}^{2}k^{2} + R^{2}}{d_{jk}^{2} + β_{1}^{2}k^{2}(l_{jk} − a_{22})^{2}}

we obtain an upper bound for ∥(L−A)^{−1}∥ which is easy to calculate in each specific case when L − A is injective. Since Ker L is infinite dimensional, we shall require that the map A + Ñ is monotone. Indeed, this will be true if we assume that

\frac{q_{i}(t, x, s) − q_{i}(t, x, ŝ)}{s − ŝ} ≥ −α, i = 1, 2

for a.a. (t, x) ∈ Ω and all real s ≠ ŝ, where α = min{(Ax, x)_{ℝ^{2}} : |x| = 1}. Therefore, assuming further that there exists η > 0 such that

lim sup_{|s|→∞}|\frac{q_{i}(t, x, s)}{s}| ≤ η < \frac{1}{κ}, i = 1, 2,

uniformly in (t, x) ∈ Ω, we can conclude by Theorem 4.1 that the system (4.3) admits a weak solution u = (u_{1}, u_{2}) ∈ D(L) for any given h = (h_{1}, h_{2}) ∈ [L^{2}(Ω)]^{2}.

5. Resonance for systems

Let L = diag(L_{1}, L_{2}, …, L_{n}) be a linear operator in H = H^{n} satisfying the properties given in Section 2. Let N : H → H be a bounded demicontinuous map. Assume that there exist a constant γ > 0 and a real valued function C(·) such that

(5.1)  ⟨N (u) − N (z), u − z⟩ ≥ \frac{1}{γ} ∥N (u)∥^{2} − C(z) for all u, z ∈ H.

We shall consider the solvability of the abstract operator equation

(5.2)  Lu − lu − N (u) = h,  u ∈ D(L),

where l ∈ σ(L) and h ∈ H. We assume that lI + N is monotone if Ker L is infinite dimensional. We denote by conv S the set of convex combinations of elements of any set S and by R(N) the range of N . In the sequel we shall use the constant θ_{+} given in (3.5). We start with

Theorem 5.1. Assume that the condition (5.1) holds with 0 < γ < θ_{+} and l ≥ 0. Then

\overline{Im(L − lI) − conv R(N)} = \overline{R(L − lI − N)}

if l ∈ σ(L) and R(L − lI − N) = H if l ∉ σ(L).

Proof (cf. [BM2] for a more detailed proof). It is sufficient to show that Im L_{l} − conv R(N) is contained in the closure of R(L_{l} − N), where L_{l} = L − lI. Each h ∈ Im(L_{l}) − conv R(N) has a representation h = L_{l}v_{0} − p, where p = ∑_{i=1}^{m} α_{i}N (v_{i}) with α_{i} ≥ 0, ∑_{i=1}^{m} α_{i} = 1. Let P_{l} stand for the orthogonal projection to Ker L_{l} and Q_{l} = I − P_{l}. Note that P_{l} = 0 if l ∉ σ(L). By [BM1] the mapping L_{l} − P_{l} is a reference map. If the equation (5.3) now has a solution, the proof is complete. Hence we assume that the equation (5.3) has no solution. In view of the Continuation Theorem this implies the existence of sequences {u_{j}} and {t_{j}} such that ∥u_{j}∥ → ∞, 0 < t_{j} < 1, and

(5.3)  L_{l}u_{j} − (1 − t_{j})P_{l}u_{j} − t_{j}N (u_{j}) = L_{l}v_{0} + (1 − t_{j})P_{l}v_{0} − t_{j}p

for all j ∈ ℤ_{+}. Taking Q_{l}-projection we get

(5.4)  ∥L_{l}(u_{j} − v_{0})∥ ≤ t_{j}(∥N (u_{j})∥ + ∥p∥).

Using the representation of p we have

⟨L(u_{j} − v_{0}), u_{j} − v_{0}⟩ = t_{j} ∑_{i=1}^{m}{α_{i}⟨N (u_{j}) − N (v_{i}), u_{j} − v_{0}⟩}
+ (1 − t_{j})∥P(u_{j} − v_{0})∥^{2}

and by (5.1), (5.4) and (3.5) together with the fact that 0 < t_{j} < t_{j}^{2} < 1 we get further

\frac{t_{j}^{2}}{θ_{+}} (∥N (u_{j})∥ + ∥p∥)^{2} ≥ t_{j}^{2}(\frac{1}{γ} ∥N (u_{j})∥^{2} − C(h)) + (1 − t_{j})∥P_{l}(u_{j} − v_{0})∥^{2},

where C(h) is a constant depending on h only. Since 0 < γ < θ_{+} we conclude that the sequence {t_{j}N (u_{j})} remains bounded which also implies that {L_{l}u_{j}}, {Q_{l}u_{j}} and {(1 − t_{j})∥P_{l}u_{j}∥^{2}} remain bounded. Note that in the case l ∉ σ(L) we have Q_{l}u_{j} = u_{j}, which gives a contradiction showing that h ∈ R(L_{l} − N). Moreover, in this case Im L_{l} = H, proving our second assertion R(L_{l} − N) = H. If l ∈ σ(L), then ∥P_{l}u_{j}∥ → ∞ and hence (1 − t_{j})P_{l}u_{j} → 0 and t_{j} → 1. Thus we get from (5.3) that

L_{l}u_{j} − N (u_{j}) → L_{l}v_{0} − p = h

which shows that h ∈ \overline{R(L_{l} − N)}. □

Note that in the above proof we obtained a sequence {u_{j}} such that ∥u_{j}∥ → ∞ but {N (u_{j})} is bounded as a consequence of the assumption that the equation (5.2) has no solution. Excluding this possibility we obtain the following surjectivity result.

Corollary 1. Assume that N is monotone, (5.1) holds with 0 < γ < θ_{+} and l ∈ σ(L) with l ≥ 0. If ∥N (u)∥ → ∞ whenever ∥u∥ → ∞, then the equation (5.2) admits a solution for any h ∈ H.

Proof. Let h ∈ H be arbitrary. Since R(N) = H, we have h ∈ Im L_{l}−R(N) and the assertion follows from the proof of Theorem 5.1. □

To tackle the equation (5.2) in the resonance case we recall some notation and useful facts from [BN]. The support function of R(N) is defined by

I_{R(N)}^{∗}(v) = sup_{u∈H}⟨N (u), v⟩, v ∈ H,

and the recession function for N by

J_{N} (v) = lim inf_{t→∞,u→v}⟨N (tu), u⟩, v ∈ H.

Clearly we have

J_{N} (v) ≤ I_{R(N)}^{∗}(v) for all v ∈ H.

The following result is a direct consequence of Propositions II.5 and II.6 of [BN] and the fact that Im L_{l} = (Ker L_{l})^{⊥}.

Proposition A. Let S: H → H be any nonlinear map and h ∈ H.

(a) If l ∈ σ(L), l > 0, then the following conditions are equivalent:

(5.5)  I_{R(S)}^{∗}(v) > −⟨h, v⟩ for all v ∈ Ker L_{l}, v ≠ 0;

(5.6)  h ∈ int(Im L_{l} − conv R(S)).

(b) If l = 0, then the following conditions are equivalent:

(5.7)  I_{R(S)}^{∗}(v) ≥ −⟨h, v⟩ for all v ∈ Ker L;

(5.8)  h ∈ \overline{Im L − conv R(S)}.

The difference between the cases l > 0, l = 0 above is due to the fact that Ker L_{l} is always finite-dimensional if l ≠ 0. From Theorem 5.1 and Proposition A we derive Theorem 5.2. Assume that l ∈ σ(L), l ≥ 0, and the condition (5.1) holds with 0 < γ < θ_{+}.

(a) If l > 0 and J_{N} (v) > −⟨h, v⟩ for all v ∈ Ker L_{l}, v ≠ 0, then the equation (5.2) has at least one solution.

(b) If l = 0 and J_{N} (v) ≥ −⟨h, v⟩ for all v ∈ Ker L, then the equation (5.2) is almost solvable in the sense that h ∈ \overline{R(L − N)}.

Proof. The case (b) follows immediately from Theorem 5.1 and Proposition A. To prove (a) we observe first by Proposition A that h ∈ Im L_{l} − conv R(N). Hence we can argue exactly in the same way as in the proof of Theorem 5.1. Thus we assume that the equation (5.2) does not have a solution. But this implies the existence of a sequence {u_{j}} ⊂ D(L) such that

(5.9)  L_{l}u_{j} − N (u_{j}) → h.

By the proof of Theorem 5.1 we also know that {L_{l}(u_{j})}, {N (u_{j})} and \{Q_{l}u_{j}\} remain bounded while ∥P_{l}u_{j}∥ → ∞. Denoting v_{j} = \frac{u_{j}}{∥u_{j}∥} and s_{j} = ∥u_{j}∥ we have v_{j} ⇀ v and L_{l}v_{j} ⇀ w ∈ Im L_{l} for a subsequence. Since Q_{l}v_{j} → 0 and Ker L_{l} is finite-dimensional, we have v_{j} → v = P_{l}v. Therefore ⟨L_{l}v_{j}, v_{j}⟩ → ⟨w, v⟩ = 0 and by (5.9) ⟨N (u_{j}), v_{j}⟩ → ⟨−h, v⟩. Hence we get

−⟨h, v⟩ = lim_{j→∞}⟨N (s_{j}v_{j}), v_{j}⟩ ≥ J_{N} (v),

giving a contradiction. □

Note that the results presented in this section so far are purely abstract. In particular, they are independent of the fact that we consider systems. For this reason we assume that H = [L^{2}(Ω)]^{n} and L = diag(L_{1}, L_{2}, …, L_{n}) is the telegraph operator, i.e., Lu = u_{tt} − u_{xx} − βu_{t} as given in our model problem (1.1). Let g: Ω × ℝ^{n} → ℝ^{n} be a Carathéodory function satisfying the usual growth condition of the form (1.2). Moreover, we assume that l ≥ 0 is an eigenvalue of L and the map lI + g is monotone if det β = 0. We notice that in the one-dimensional case the condition (5.1) follows from the growth and monotonicity conditions (see [BN]) and the recession function can be estimated by

J_{N} (v) ≥ ∫_{v>0} g_{+}v + ∫_{v<0} g_{−}v, v ∈ L^{2}(Ω),

where g_{+}(t, x) = lim inf_{s→+∞} g(t, x, s) and g_{−}(t, x) = lim sup_{s→−∞} g(t, x, s). We as-

sume that the function g: Ω × ℝ^{n} → ℝ^{n} satisfies condition

(5.10)  ⟨g(t, x, s) − g(t, x, s̃), s − s̃⟩_{ℝ^{n}} ≥ \frac{1}{γ} |g(t, x, s)|^{2} − C(s̃)

for a.a. (t, x) ∈ Ω and all s, s̃ ∈ ℝ^{n}, where γ > 0 and C(·) a real-valued function. Now it is easy to see that (5.1) holds. For a discussion on various types of functions satisfying (5.10) we refer to [BN]. In place of g_{+} and g_{−} we consider now the recession function J_{g} of g defined by

J_{g}(t, x, s) = lim inf_{r→∞,z→s}⟨g(t, x, rz), z⟩.

By (5.10) and Fatou’s lemma we obtain the estimate

J_{N} (v) ≥ ∫_{v≠0} J_{g}(t, x, v) for all v ∈ H.

Hence we conclude from Theorem 5.2 the following Landesman-Lazer type result (cf. [LL]).

Corollary 2. Assume that l = j^{2} for some j ∈ ℤ_{+} and that the function g satisfies the condition (5.10). If h ∈ H and

∫_{v≠0} J_{g}(t, x, v) > −⟨h, v⟩ for all v ∈ Ker(L − j^{2}I), v ≠ 0

holds, then the system

u_{tt} − u_{xx} − βu_{t} − j^{2}u − g(t, x, u) = h(t, x), (t, x) ∈ Ω u(t, 0) = u(t, π) = 0, t ∈ [0, 2π] u(·, x) is 2π-periodic in t

admits at least one weak solution.

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Address of J. Berkovits: Department of Mathematical Sciences University of Oulu Linnanmaa, P.O. Box 333 90570 Oulu Finland