Research Article | Open Access
Shanshan Wang, Yukun An, "Multiple Periodic Solutions to a Suspension Bridge Wave Equation with Damping", International Scholarly Research Notices, vol. 2011, Article ID 154806, 9 pages, 2011. https://doi.org/10.5402/2011/154806
Multiple Periodic Solutions to a Suspension Bridge Wave Equation with Damping
This paper is concerned with the existence of multiple periodic solutions for a suspension bridge wave equation with damping. By using Leray-Schauder degree theory, the authors prove that the damped wave equation has multiple periodic solutions.
In , 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 and width suspended by cables. Consider the horizontal cross-section of mass located at position along the length of the span. She treated this cross-section as a rod of length and mass suspended by cables. Let denote the downward distance of the center of gravity of the rod from the unloaded state and let denote the angle of the rod from horizontal at time . Assume that the cables do not resist compression, but resist elongation according to Hooke’s law with spring constant . Then the torsional and vertical motion of the span satisfy where , , are physical constants related to the flexibility of the beam, is the damping constant, and are external forcing terms, and is the acceleration due to gravity. The spatial derivatives describe the restoring force that the beam exerts, and the time derivatives and represent the force due to friction. The boundary conditions reflect the fact that the ends of the span are hinged.
Throughout the paper  the author assumes that the cables never lose tension; that is, it is assumed that . In this case, we see that (1.1) becomes uncoupled, and the torsional and vertical motions satisfy, respectively,
In paper , removing the damping term; that is, let , changing variables, and imposing boundary and periodicity conditions, the author rewrites (1.2) as And it proves that (1.4) has at least two solutions in the subspace of . Where is defined as
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, , and study the following problem:
Let and be the set of integers, . Let and be usual space of square integrable functions with usual inner product and corresponding norm . For the Sobolev space , we denote the standard inner product by and norm by .
Define the operator by
We know that the eigenvalues and corresponding eigenfunctions of are
In order to seek the solutions of (1.6), we first investigate the properties of operator . We have the following Lemma.
Lemma 2.1. exists, is compact, and .
Proof. Because we are restricted to the subspace of , and 0, we easily know exists.
We prove is compact below. We find that for all . For any , we have then Hence, On the other hand, while Hence, By (2.6) and (2.9), we can find that the operator is compact since the embedding is compact.
Finally, we prove 1. By (2.2) and Set , such that, . Therefore, Hence, we complete the proof of this lemma.
Definition 2.2. One says that 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 . To compute the degree of , we continuously deform it to a linear operator , the Gâteaux derivative of , 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 is preserved as is continuously deformed to its Fréchet derivative under appropriate hypotheses. However, the nonlinear term in (1.6), , is not Fréchet differentiable in at .
There is a theorem in paper , in which, the author shows that, under certain conditions on the nonlinear term and the differential operator , Leray-Schauder degree is indeed preserved under homotopy from the operator to its Gâteaux derivative . This result can be used to establish multiplicity of solutions to equations of the form (1.6). The result follows.
Lemma 2.3. Let , be open, bounded intervals in , and define . Let be a subspace of , , and define . Consider the problem where , , and satisfy the following:(H1) is compact;(H2)≤ 1;(H3);(H4) is Lipschitz with Lipschitz constant ;(H5) and ;(H6) the Gâteaux derivative exists and satisfies , where and is not an eigenvalue of .Define : by and by Then for sufficiently small, there exists such that
3. Result and Proof
The main result of this paper is as follows.
Theorem 3.1. Let with , and let , . Then there exists such that if , (1.6) has at least two solutions in .
Proof. Let and , it is easy to know that and satisfy the conditions (H1–H5) in Lemma 2.3.
Reply Lemma 2.3, we define : by and : by And note that zeros of correspond to solutions of (1.6). To prove the theorem, we will show the following:(C1) there exists such that for , ;(C2) there exists such that .Then, since , there exists a zero of (i.e., a solution of (1.6)) in . Moreover, by the additivity property of degree, and hence (1.6) has a second solution in the annulus .
To establish (C1), define or , and note that this definition of is consistent with our previous definition. Note also that is simply the identity map; hence, for any we have . The homotopy property of degree ensures that is constant provided that for all .
Fix and suppose solves . We will show that u is bounded above by some and that this bound is independent of .
Since , we have if we choose .
Thus, for , we have and (C1) above holds.
To establish (C2), let ; we will determine the value of later. For define and note again that this definition of 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
Observe that for and , hypotheses (H1)–(H5) of Lemma 2.3 are satisfied. To verify hypothesis (H6), we need to show that By definition of the Gâteaux derivative, We will show that the limit above (in ) is .
Note first that in we have and hence as . Invoking the convexity of , we have
Since , is dominated in ; thus by the dominated convergence theorem, as ; therefore (3.8) holds. Moreover, by the form of eigenvalue of and our choice of , is not an eigenvalue of ; therefore hypothesis (H6) of Lemma 2.3 holds. Thus, by Lemma 2.3, for sufficiently small , we have
Finally, we will show that
Consider the finite dimensional subspace of and recall that, by compactness, can be approximated in operator norm by the operators given by By definition of Leray-Schauder degree, for sufficientlylarge, where is the Jacobian determinant of at .
Since can be identified with an diagonal matrix whose entries are , we have Now we consider the following two cases.(D1)If contains imaginary part, suppose a pair of conjugate complex numbers are , then, (D2)If is real, then , here .
Since , and , the only negative value of occurs at . Therefore, The proof of the theorem is complete.
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Copyright © 2011 Shanshan Wang and Yukun An. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.