Infinitely many \(T\)-periodic solutions for a problem arising in nonlinear elasticity. (English) Zbl 0781.34032

The authors begin with an example of radially symmetric vibrations of a thick, hollow, elastic sphere made of homogeneous, isotropic and incompressible material. From the incompressibility condition follows the conclusion that knowledge of the time dependence of any point \(R_ 1(t)= R(t,r_ 1)\) determines the position of any point \(R(t,r)\) within the sphere for all times \(t>0\) and for all admissible values of the radial coordinate \(r\): \(r_ 2>r>r_ 1\). Assuming a Mooney-Rivlin potential the authors derive a second order differential equation (1.1) \(d^ 2/dt^ 2(\Psi(u))= \{u^ 2(p(t)- g(u))\}/2 (\Psi'(u))\), where \(u=R_ 1(t,r_ 1)/r_ 1\), while \(\Psi(u)\), \(g(u)\) are functions introduced by Z.- H. Guo and R. Solecki [Arch. Mech. 15, 427–433 (1963; Zbl 0151.39003)]. This is of the form: (1.2) \(d^ 2 x/dt^ 2+F(x,t)=0\), if we denote \(x=\Psi(u)\). The remainder of this paper is devoted to the study of \(T\)-oscillatory solutions of (1.2). This problem has been studied by several authors in the past. In particular, existence of periodic solutions in the presence of singularities has been studied by A. C. Lazer and S. Solimini [Proc. Am. Math. Soc. 99, 109–114 (1987; Zbl 0616.34033)]. Assuming continuity and local Lipschitz property of \(F\) with respect to \(x\), the authors prove the existence of two \(T\)-periodic solutions of (1.2), if \(F(s,t)\) satisfies \(-\infty<\limsup_{s\to 0+} sF(s,t)<0\), \(\lim_{s\to +\infty} F(s,t)/s=+\infty\), uniformly in \(t\). The proof requires several lemmas, including a lemma on existence of at least two fixed points in an area preserving homeomorphism of an annular region, with some rather technical conditions involving periodicity assigned to that homeomorphism. The Poincaré-Birkhoff theorem is crucial to the argument.
Reviewer: V.Komkov (Roswell)


34C25 Periodic solutions to ordinary differential equations
74K15 Membranes
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