This paper surveys an area of nonlinear functional analysis and its applications. The main application is to the existence and multiplicity of periodic solutions of a possible mathematical models of nonlinearly supported bending beams, and their possible application to nonlinear behavior as observed in large-amplitude flexings in suspension bridges. A second area, periodic flexings in a floating beam, also nonlinearly supported, is covered at the end of the paper.

Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis

Infinite Dimensional Morse Theory and Multiple Solution Problems (K. C. Chang)

The effect of harmonic excitation on suspension bridges is examined as a first step towards the understanding of the effect of wind, and possibly certain kinds of earthquake, excitation on such structures. The Lazer-McKenna suspension bridge model is studied completely for the first time by using a methodology that has been successfully applied to models of rocking blocks and other free-standing rigid structures. An unexpectedly rich dynamical structure is revealed in this way. Conditions for the existence of asymptotic periodic responses are established, via a complicated nonlinear transcen- dental equation. A two-part Poincare map is derived to study the orbital stability of such solutions. Numerical results are presented which illustrate the application of the analytical procedure to find and classify stable and unstable solutions, as well as determine bifurcation points accurately. The richness of the possible dynamics is then illustrated by a menagerie of solutions which exhibit fold and flip bifurca...

A piecewise linear suspension bridge model : nonlinear dynamics and orbital continuation

Functional analysis : with applications

After recalling some preliminary notions from differential geometry, this chapter presents the local and global aspects of the theory of nondegenerate critical manifolds. These manifolds are a natural extension of the notion of non-degenerate critical point.

Nondegenerate Critical Manifolds

In this paper we investigate the existence of weak solutions of a nonlinear beam equation under Dirichlet boundary condition on the interval and periodic condition on the variable , . We show that if satisfies condition , then the nonlinear beam equation possesses at least one weak solution.

A weak solution of a nonlinear beam equation

We investigate the bounded weak solutions for the Hamiltonian system with bounded nonlinearity decaying at the origin and periodic condition. We get a theorem which shows the existence of the bounded weak periodic solution for this system. We obtain this result by using variational method, critical point theory for indefinite functional.

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Bounded weak solution for the hamiltonian system

We obtain the multiple solutions for the fourth order elliptic problem with a variable coefficient and a jumping semilinear term. We have a result that there exist at least two solutions if the variable coefficient of the semilinear term crosses some number of the eigenvalues of the biharmonic eigenvalue problem. We obtain this multiplicity result by applying the Leray-Schauder degree theory.

https://journal.kkms.org/index.php/kjm/article/download/86/59

Elliptic problem with a variable coefficient and a jumping semilinear term

We study the uniqueness and the existence of multiple nontrivial solutions u ( x , t ) for a perturbation [ ( u + v + 1 ) + − 1 ] of the hyperbolic system with Dirichlet boundary condition (0.1) u t t − u x x = μ [ ( u + 2 v + 1 ) + − 1 ] in  ( − π 2 , π 2 ) × R , v t t − v x x = ν [ ( u + 2 v + 1 ) + − 1 ] in  ( − π 2 , π 2 ) × R , where u + = max { u , 0 } , μ , ν are nonzero constants. Here the nonlinearity ( μ + 2 ν ) [ ( w + 1 ) + − 1 ] crosses the eigenvalues of the wave operator.

Nonlinearities and nontrivial solutions for the nonlinear hyperbolic system

We prove the existence of multiple solutions for the fourth order nonlinear elliptic problem with fully nonlinear term. Our method is based on the critical point theory; the variation of linking method and category theory.

/pdf/applications-of-topolological-methods-to-the-semilinear-3smw6mxhrw.pdf

Q-Heung Choi

Papers

A weak solution of a nonlinear beam equation

Bounded weak solution for the hamiltonian system

Elliptic problem with a variable coefficient and a jumping semilinear term

Nonlinearities and nontrivial solutions for the nonlinear hyperbolic system

Applications of topolological methods to the semilinear biharmonic problem with different powers