Multiple Periodic Solutions to a Suspension Bridge Wave Equation with Damping

In 1 , also see 2–6 , the author considered a horizontal cross-section of the center span of a suspension bridge and proposed a partial differential equationmodel for the torsional motion of the cross-section and treated the center span of the bridge as a beam of length L and width 2l suspended by cables. Consider the horizontal cross-section of mass m located at position x along the length of the span. She treated this cross-section as a rod of length 2l and mass m suspended by cables. Let y x, t denote the downward distance of the center of gravity of the rod from the unloaded state and let θ x, t denote the angle of the rod from horizontal at time t. Assume that the cables do not resist compression, but resist elongation according to Hooke’s law with spring constant K. Then the torsional and vertical motion of the span satisfy


Introduction
In 1 , also see 2-6 , the author considered a horizontal cross-section of the center span of a suspension bridge and proposed a partial differential equation model for the torsional motion of the cross-section and treated the center span of the bridge as a beam of length L and width 2l suspended by cables.Consider the horizontal cross-section of mass m located at position x along the length of the span.She treated this cross-section as a rod of length 2l and mass m suspended by cables.Let y x, t denote the downward distance of the center of gravity of the rod from the unloaded state and let θ x, t denote the angle of the rod from horizontal at time t.Assume that the cables do not resist compression, but resist elongation according to Hooke's law with spring constant K. Then the torsional and vertical motion of the span satisfy where u max{u, 0}, ε 1 , ε 2 are physical constants related to the flexibility of the beam, δ is the damping constant, h 1 and h 2 are external forcing terms, and g is the acceleration due to gravity.The spatial derivatives describe the restoring force that the beam exerts, and the time derivatives θ t and y t represent the force due to friction.The boundary conditions reflect the fact that the ends of the span are hinged.
Throughout the paper 1 the author assumes that the cables never lose tension; that is, it is assumed that y ± sin θ ≥ 0. In this case, we see that 1.1 becomes uncoupled, and the torsional and vertical motions satisfy, respectively,

1.3
In paper 1 , removing the damping term; that is, let δ 0, changing variables, and imposing boundary and periodicity conditions, the author rewrites 1.2 as 1.4 And it proves that 1.4 has at least two solutions in the subspace H of L 2 .Where H is defined as Notice that 1.4 is particular in no damping and the selection of H. Hence, in 1 the author left a problem which is relevant to this case.Problem 1. "Under appropriate hypotheses on the forcing term, does a similar result hold for the damped equation?" Motivated by this problem, in this paper, we suppose that the damping is present, that is, δ / 0, and study the following problem: 1.6

Preliminaries
Let N {0, 1, . ..} and Z be the set of integers, Λ N × N. Let Ω 0, π × 0, π and L 2 Ω be usual space of square integrable functions with usual inner product •, • and corresponding norm • .For the Sobolev space H 1 Ω , we denote the standard inner product by u, v 1 u, v u x , v x u t , v t and norm by u 1 .Define the operator L δ u u tt − u xx δu t : H → H by

2.1
We know that the eigenvalues and corresponding eigenfunctions of L δ are

2.2
In order to seek the solutions of 1.6 , we first investigate the properties of operator L δ .We have the following Lemma.Proof.Because we are restricted to the subspace H of L 2 , and λ mn / 0, we easily know L −1 δ exists.
We prove L −1 δ : H → H is compact below.We find that

ISRN Mathematical Analysis
Hence, On the other hand,

2.8
Hence, By 2.6 and 2.9 , we can find that the operator Finally, we prove L −1 δ 1.By 2.2 and

2.11
Hence, we complete the proof of this lemma.
Definition 2.2.One says that u ∈ H is a solution to 1.6 if To establish the existence of multiple periodic solutions to 1.6 , we use Leray-Schauder degree theory to prove the existence of multiple zeros of a related operator T 1 .To compute the degree of T 1 , we continuously deform it to a linear operator T 0 , the Gâteaux derivative of T 1 , and compute its degree via a direct calculation.
It is not difficult to show that the homotopy property of Leray-Schauder degree ensures that the degree of an operator T 1 is preserved as T 1 is continuously deformed to its Fréchet derivative under appropriate hypotheses.However, the nonlinear term in 1.6 , f u sin u, is not Fréchet differentiable in L 2 at u 0.
There is a theorem in paper 1 , in which, the author shows that, under certain conditions on the nonlinear term f and the differential operator L, Leray-Schauder degree is indeed preserved under homotopy from the operator T 1 to its Gâteaux derivative T 0 .This result can be used to establish multiplicity of solutions to equations of the form 1.6 .The result follows.where L, f, and h satisfy the following: H6 the Gâteaux derivative df 0, u exists and satisfies df 0, u ρu, where ρ > 0 and −ρ is not an eigenvalue of L.

2.15
Then for ε sufficiently small, there exists γ > 0 such that

Result and Proof
The main result of this paper is as follows.
Proof.Let L L δ and f u b sin u, it is easy to know that L and f satisfy the conditions H1-H5 in Lemma 2.3.
Reply Lemma 2.3, we define T 0 : H → H by and And note that zeros of T 1 correspond to solutions of 1.6 .To prove the theorem, we will show the following: Then, since deg T 1 , B γ 0 , 0 / 0, there exists a zero of T 1 i.e., a solution of 1.6 in B γ 0 .Moreover, by the additivity property of degree, deg T 1 , B R 0 \ B γ 0 , 0 / 0 and hence 1.6 has a second solution in the annulus B R 0 \ B γ 0 .
To establish C1 , define or β ∈ 0, 1 , and note that this definition of T 1 is consistent with our previous definition.Note also that T 0 is simply the identity map; hence, for any R > 0 we have deg T 0 , B R 0 , 0 1.The homotopy property of degree ensures that deg T β , B R 0 , 0 is constant provided that 0∈T β ∂B R 0 for all β ∈ 0, 1 .
Fix β ∈ 0, 1 and suppose u ∈ H solves T β u 0. We will show that u is bounded above by some R 0 > 0 and that this bound is independent of β.
Since T β u 0, we have and C1 above holds.

ISRN Mathematical Analysis 7
To establish C2 , let ε < ε 0 ; we will determine the value of ε 0 later.For μ ∈ 0, 1 define and note again that this definition of T 1 is consistent with our previous definitions.We will again apply the homotopy property of degree via Lemma 2.3 and a standard degree calculation to show that for some γ > 0 deg T 1 , B γ 0 , 0 deg T 0 , B γ 0 , 0 −1.

3.7
Observe that for L L δ and f u : b sin u, hypotheses H1 -H5 of Lemma 2.3 are satisfied.To verify hypothesis H6 , we need to show that df 0, u bu.

3.8
By definition of the Gâteaux derivative, b sin hu h .

3.9
We will show that the limit above in H is bu.

3.20
The proof of the theorem is complete.

Lemma 2 . 3 .
Let I 1 , I 2 be open, bounded intervals in R, and define Q : I 1 × I 2 .Let B be a subspace of L p Q , p ≥ 1, and define u : u L p .Consider the problem Lu f u εh x, t , 2.13
L 2 , | sin hu /h − u| 2 is dominated in L 1 ; thus by the dominated convergence theorem, Consider the finite dimensional subspace M N span{Φ mn } N 0 of H and recall that, by compactness, bL −1 δ can be approximated in operator norm by the operators B N : M N → M N given by