ON THE SOLVABILITY OF A SYSTEM OF WAVE AND BEAM EQUATIONS

where h = (h1,h2) is a given function in L2( ;R2) with =]0,2π [×]0,π [, β > 0, the function g(t,x,s) = (g1(t,x,s),g2(t,x,s)) from ×R2 to R2 is 2π periodic in t , measurable in (t,x) for each s ∈ R2 and continuous in s for almost all (t,x) ∈ . Moreover, we assume that g(t,x, ·) has at most linear growth. With suitable nonlinearity g (1.1) provides a reasonable model for a suspension bridge, where the main cable is described as a vibrating string and the road bed as a vibrating beam.


Introduction
We consider a system of wave and beam operators with linear coupling and damping having the form where h = (h 1 , h 2 ) is a given function in L 2 ( ; R 2 ) with =]0, 2π[×]0, π[, β > 0, the function g(t, x, s) = (g 1 (t, x, s), g 2 (t, x, s)) from ×R 2 to R 2 is 2πperiodic in t, measurable in (t, x) for each s ∈ R 2 and continuous in s for almost all (t, x) ∈ .Moreover, we assume that g(t, x, •) has at most linear growth.With suitable nonlinearity g (1.1) provides a reasonable model for a suspension bridge, where the main cable is described as a vibrating string and the road bed as a vibrating beam.
We will study the existence of weak solutions and therefore it is relevant to transform system (1.1) into the operator equation 2 , where ᏺ is the Nemytskii operator generated by g, Ꮽ is the constant multiplication operator induced by the coupling matrix A := (a lk ) and ᏸ : D(ᏸ) ⊂ Ᏼ → Ᏼ is the abstract realization of the linear differential operator.We define the matrix spectrum of ᏸ as the set 3) The set σ M (ᏸ) is closed in R 2×2 and hence the "resolving set" of ᏸ is divided into open components.We will apply the extension of the Leray-Schauder degree introduced by Berkovits and Mustonen [4] for a class of mappings related to our model problem.If A / ∈ σ M (ᏸ), we can use the homotopy argument to obtain nonresonance results for (1.2) (see [8]).In this note we deal with the resonance case A ∈ σ M (ᏸ).Using suitable reduction to invariant subspaces we can find solution for (1.2), provided ᏺ and h satisfy some auxiliary symmetry conditions.Indeed, if the coupled wave-beam operator ᏸ − Ꮽ is completely reduced by a closed linear subspace V and N(V ) ⊂ V , any solution of the reduced equation is also a weak solution for the original equation (1.2).It is possible that A ∈ σ M (ᏸ) but ᏸ| V − Ꮽ| V is injective in the subspace V and hence we can apply the known nonresonance results for the reduced equation (1.4).The method of invariant subspaces is widely used in the study of ordinary differential equations and for a single wave equation by Coron, Vejvoda, among others.(See [5,9,11,15].) Problem (1.1) has been studied mainly in the case of gradient type nonlinearities ᏺ with no damping, that is, with β = 0. We refer the reader to the papers of Brezis and Nirenberg [10], Amann [1], Mawhin [13,14], Fonda and Mawhin [12], where also a survey on the recent results and relevant references can be found.For previous results on the existence of the periodic solutions of the systems of wave equations we also refer to [2,6,7].

Prerequisites
Let H be a real separable Hilbert space with inner product •, • and corresponding norm • .We recall some basic definitions.A mapping F : H → H is • bounded, if it takes any bounded set into a bounded set, • demicontinuous, if u j → u (norm convergence) implies F (u j ) F (u) (weak convergence), • of class (S + ), if for any sequence with u j u, lim sup F (u j ), u j − u ≤ 0, it follows that u j → u, • quasimonotone, if for any sequence u j u, lim sup F (u j ), u j −u ≥ 0.
The class of mappings considered in this paper is given in the following definitions.
Definition 2.1.A linear densely defined operator L : Denote by P and Q = I −P the orthogonal projections to Ker L and Im L = (Ker L) ⊥ , respectively.Definition 2.2.A bounded, demicontinuous map N : H → H is an admissible perturbation in H , if there exists a bounded demicontinuous map S : H → H of class (S + ) such that P N = P S.
We are interested in the case where L is not selfadjoint and therefore we include the complex spectrum of L into consideration.We recall that the complexification H C = H + iH of H has the usual linear structure and inner product •, • C induced by H .For each w = u + iv ∈ H C it is natural to denote w = u − iv.We define the complex linear operator (2.1) Note that for any complex eigenvalue µ j = α j +iβ j also its complex conjugate μj = α j − iβ j is an eigenvalue with corresponding eigenvector φj .For each u ∈ D(L) we have the spectral representation For any map N : H → H the equation can be written equivalently as The equivalence of (2.3) and (2.4) is due to the fact that KQ − P is the right inverse of L−P .If N is bounded, demicontinuous and of class (S + ), then there exists a topological degree for mappings of the form where C is compact (see [4]).In fact, it is sufficient that N is admissible, that is, there exists an auxiliary bounded demicontinuous map S : H → H of class (S + ) such that P N = P S. This observation is quite obvious, since only the P -component of N appears in F .However, it has some interesting implications as we will see later on (see also [3]).The degree theory given in [4] is a unique extension of the classical Leray-Schauder degree.It is single-valued and has the usual properties of degree, such as additivity and invariance under homotopies.
Let the corresponding degree function be d H .In order to simplify our notations we define a further degree function "deg H " by setting . By the term reference map we refer to any linear injection L−N 0 with L and N 0 admissible.For a reference map we have (see [4]) (2.6)

On systems
Let H be a real separable Hilbert space and denote Ᏼ = H n with n ≥ 2. We assume that L k : We shall further assume that the inverse K k of each L k is compact and hence each L k is admissible.We define the diagonal operator ᏸ : where ) n and ᏸ as well as ᏸ C inherits the properties of the component operators.We shall use the notations •, • and • for the inner product and norm in any real Hilbert space and the subscript "C" whenever the norm, inner product or spectrum is complex.For simplicity we shall frequently use the same symbol for an operator and its complexification.The inverse We denote by ᏼ and ᏽ the orthogonal projections onto Ker ᏸ and Im ᏸ, respectively.Let ᏺ : Ᏼ → Ᏼ be a (possibly nonlinear) bounded demicontinuous map.As described in Section 2, a topological degree is available for maps of the form ᏸ − ᏺ, where ᏸ is admissible and ᏺ is any admissible perturbation.
Consider first linear maps of the following type.Let A = (a lk ) be a real n × n-matrix and Ꮽ : Ᏼ → Ᏼ the constant multiplication operator induced by A, that is, for any with w l = n k=1 a lk u k , l = 1, 2, . . ., n. Clearly σ (Ꮽ) = σ (A), a real point spectrum which may be empty if n is even.Similarly σ C (Ꮽ) = σ C (A) for the complex spectra.If the matrix A is strictly positive, it is not hard to prove that where α = min{(Ax, x) R n ; |x| R n = 1} is positive.Hence the operator Ꮽ is of class (S + ).In order to tackle more specific situations we assume that dim Ker L k = ∞ for k = 1, 2, . . ., p and dim Ker L k < ∞ for k = p + 1, . . ., n, where 0 ≤ p ≤ n.If p = n we assume that A > 0 and if p = 0, no positivity is needed.
For the general case, we formulate the condition: (PC) The matrix (a lk ) p l,k=1 is strictly positive.We have the following useful result adopted from [3].
Lemma 3.1.Assume that 1 ≤ p ≤ n and the positivity condition (PC) holds.Then there exists a bounded linear operator S A : Ᏼ → Ᏼ of class (S + ) such that ᏼS A = ᏼᏭ, that is, Ꮽ is an admissible linear perturbation.
It is important to realize the meaning of S A ; it is only needed to guarantee the existence of the topological degree.All concrete calculations will be done with Ꮽ, not with S A .We shall impose a further "common eigenbasis"-assumption: (CE) The operators L k , k = 1, 2, . . ., n, have a common complex eigenbasis {ψ j } j ∈ .
Here we can assume that the index set ⊂ Z.We denote the corresponding complex eigenvalues by {µ

. , n. Although assumption (CE) is very restrictive from the general point of view it can be verified in many applications. It trivially holds in case
It is easy to see that for any u ∈ D(ᏸ) we have a quasidiagonal representation where M j = diag(µ . By (3.5) we get the following result (cf.[8]).

Lemma 3.2. Assume (CE). The operator ᏸ − Ꮽ is injective if and only if det(M
Note that in case L k = L for all k = 1, 2, . . ., n, the injectivity condition in Lemma 3.2 can be written as σ C (L)∩σ C (A) = ∅.By condition (PC) we get the following result (cf.[8]).
Lemma 3.3.Assume (PC), that is, the matrix (a lk ) p l,k=1 is strictly positive.If the operator ᏸ − Ꮽ is injective, then it is bijective.
We get a simple formula for the norm of (ᏸ − Ꮽ) −1 .Lemma 3.4.Assume that (CE) and (PC) hold.Then det(M j − A) = 0 for all j ∈ Z, if and only if the operator ᏸ − Ꮽ is bijective.Moreover, (ᏸ − Ꮽ) −1 is bounded and In general, the operator ᏸ − Ꮽ can be injective without being surjective (cf.[3]).However, in the special case injectivity implies bijectivity without any positivity condition (see [7]).We recall now the concept of the matrix spectrum introduced in [8].Given a linear operator -with domain and range in Ᏼ = H n , we define the matrix resolvent of the operatoras the set The matrix spectrum of the operator -is then the set It is not hard to prove that the matrix spectrum σ M (-) is a closed set in R n×n (see [8]).In the particular case where ᏸ has the properties given at the beginning of the section, that is, ᏸ is an admissible diagonal operator, it is not hard to prove that Im(ᏸ−Ꮽ) is closed whenever the positivity condition (PC) holds.Then also the degree theory is available and any invertible operator ᏸ − Ꮽ is a reference map.Recall that a linear operator -is completely reduced by a closed linear subspace V ⊂ Ᏼ if where P V is the orthogonal projection from Ᏼ onto V .The use of degree theory in an invariant subspace is justified by the following result.

Juha Berkovits 221
Lemma 3.5.Assume that L and N are admissible in H , L is completely reduced by a closed linear subspace V ⊂ H and N(V ) ⊂ V .Then L| V and N| V are admissible in V .
Proof.Denote L = L| V and Ñ = N| V .It is easy to verify that L : D(L)∩V → V is admissible in V and P = P P V = P V P is the orthogonal projection onto Ker L. Since N is admissible, there exists a bounded demicontinuous map S : H → H of class (S + ) with P S = P N. Denote S = P V S| V .Then S : V → V is a bounded demicontinuous map of class (S+).Moreover, P S(u) = P V P S(u) = P V P N(u) = P Ñ (u) for all u ∈ V .Hence Ñ is admissible in V .
In the sequel we will need a generalization of the well-known result σ (- The concept of the matrix spectrum is defined in the space H n and hence it is too narrow to deal with invariant subspaces.Therefore we define more general "spectrum-like" sets.Indeed, let H be any real separable Hilbert space and T a linear operator with domain and range in H . Denote (3.10)Here "BL" stands for "bounded linear" and L(H ) is the space of bounded linear operators in H.The set σ BL (T ) is closed in L(H ).We have in analogy with Lemma 3.3 the following result.Lemma 3.6.Assume that L : D(L) ⊂ H → H is admissible and S ∈ L(H ) is some admissible linear perturbation.Then S ∈ σ BL (L) if and only if L − S is not injective.
Clearly A ∈ σ M (ᏸ) if and only if Ꮽ ∈ σ BL (ᏸ).Moreover, if ᏸ is completely reduced by a closed subspace V ⊂ Ᏼ, then the sets σ BL (ᏸ| V ) and σ BL (ᏸ| V ⊥ ) are well defined.We obtain the following result.Lemma 3.7.Assume that ᏸ and Ꮽ are admissible and Ꮽ is a constant multiplication operator induced by the matrix A ∈ R n×n .Assume that both ᏸ and Ꮽ are completely reduced by a closed subspace V ⊂ H n .Then The straightforward proof of Lemma 3.7 is omitted here.If the maps ᏸ and Ꮽ are admissible, then it is easy to prove that the "geometric multiplicity" of any "eigenmatrix" A ∈ σ M (ᏸ) is finite, that is, dim Ker(ᏸ − Ꮽ) < ∞.Assuming (CE), that is, the existence of a common complex eigenbasis, we can write σ M (ᏸ) = (∪ j ∈ σ j ) ∪ σ ∞ , where ∈ ∪ j ∈ σ j , sup j (M j − A) −1 = ∞}.Now ∪ j ∈ σ j corresponds to the usual point spectrum and σ ∞ to the continuous spectrum.As noted above, Lemma 3.3 implies A / ∈ σ ∞ whenever the positivity condition (PC) holds for A and ᏸ is admissible.By homotopy argument, we get the following basic existence result.Using the results in [7,8] together with Lemmas 3.5 and 3.7 we obtain the following result.
Theorem 3.8.Assume that ᏸ is admissible, the condition (PC) holds for A ∈ σ M (ᏸ) and the operators ᏸ and Ꮽ are completely reduced by a closed linear subspace V ⊂ Ᏼ. Assume that ᏺ : then the equation admits at least one solution for any h ∈ V .

Wave-beam system
We consider first a linear system of wave and beam equations with linear coupling and damping having the form where Clearly L 1 is selfadjoint, Ker L 1 is infinite-dimensional and L 1 has a pure point spectrum σ (L 1 ) = {λ Note that the spectrum is unbounded from below and from above.The beam operator (with damping) ∂ tt +∂ xxxx +β∂ t has an analogous realization where λ (2) In case β = 0 the operator L 2 is selfadjoint with infinite-dimensional kernel.We will always assume that β > 0. The diagonal operator ᏸ = diag(L 1 , L 2 ) is defined on D(ᏸ) = D(L 1 ) × D(L 2 ) ⊂ Ᏼ = H 2 .Then ᏸ is normal with compact (partial) inverse from Im ᏸ into Im ᏸ.A vector w = (u, v) T ∈ [L 2 ( )] 2 is a weak solution of the wave-beam system (4.1) if and only if it is a solution of the operator equation where h = (h 1 , h 2 ) T ∈ Ᏼ. Denote M jk = diag(λ for all j ∈ Z + , k ∈ Z.Moreover, assuming the positivity condition (PC), which in this case means that a 11 > 0, the operator Ꮽ is admissible and by Lemma 3.3 the injectivity of ᏸ − Ꮽ implies its surjectivity and hence ᏸ − Ꮽ is a reference map.We now study more closely the case A ∈ σ M (ᏸ).If a 11 > 0, then dim Ker(ᏸ−Ꮽ) < ∞.In fact, for a wave-beam system (with β > 0) it is easy to see that dim Ker(ᏸ − Ꮽ) = ∞ if and only if a 11 = 0 and a 12 a 21 = 0. Note that the condition det(M j 0 −A) = 0 is equivalent to j 6 −a 11 j 4 −a 22 j 2 +det A = 0.Moreover, det(M jk − A) = 0, k = 0, only in the special case, where a 11 = λ (1) jk and a 12 a 21 = 0.

Existence results
Consider now the linearly coupled system (1.1) of wave and beam equations with some nonlinear perturbation.Indeed, let the given function g(t, x, s) = (g 1 (t, x, s), g 2 (t, x, s)) from × R 2 to R 2 be 2π-periodic in t, measurable in a linear densely defined closed, normal operator with Im L k = (Ker L k ) ⊥ for each k = 1, 2, . . ., n.The inverse K k of the restriction of each L k to Im L k ∩ D(L k ) is a bounded linear operator on Im L k .
+ iβk).The operator ᏸ − Ꮽ is injective if and only if det M jk − A = det λ Z + , k ∈ Z, the set {φ jk } forms an orthonormal basis in H C .The wave operator ∂ tt − ∂ xx with periodic Dirichlet boundary conditions has in L 2 ( ) the abstract realization