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Forced oscillations with rapidly vanishing nonlinearities. (English) Zbl 0721.34042

This paper considers the existence of \(2\pi\)-periodic solutions of the nonlinear boundary value problem \(x''+f(x)=p(t),\quad x(0)=x(2\pi),\quad x'(0)=x'(2\pi),\) where p(t) is \(2\pi\)-periodic and f(x) assumes the value zero infinitely often. The case when the distance between successive zeros of f approaches infinity for large x, e.g., \(f(x)=\sin (\log x)^{1/2}\), was studied by S. Fučík in 1980 wherein the distances between zeros of f were termed “expansive”. This paper presents a unifield result for the cases when the distances between the zeros of f are expansive, equispaced, or shrinking; the latter two cases are exemplified by \(f(x)=e^ x| \sin x|\) and \(f(x)=e^ x| \sin x^ 2|\), respectively. The main result is an existence theorem based on certain assumptions on the function \(f(x)=g(x)h(x)\) where g(x) increases without bound as \(x\to \infty\), h(x) has an infinite number of zeros, and g(x)h(x) is bounded as \(x\to -\infty\). The proof is based on the Leray-Schauder principle and some a priori estimates.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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