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A multiplicity theorem for locally Lipschitz periodic functionals. (English) Zbl 0863.58014

The authors consider the periodic multivalued problem of the forced pendulum \[ \begin{cases} x''(t) + f(t)\in \biggl[\underline g\bigl(x(t) \bigr), \overline g \bigl(x(t) \bigr)\biggr], \quad \text{a.e. } t\in(0,1) \\ x(0) = x(1),\;x'(0) = x'(1) \end{cases} \] where \(f\in L^p(0,1)\), \(g\in L^\infty (\mathbb{R})\) is periodic of period \(T\), the conditions \(\int^T_0 g(s)ds = \int^1_0 f(t)dt =0\) hold and \(\underline g(s) = \lim_{\varepsilon \to 0^+} \text{ess inf}\{g (\sigma): |\sigma - s|< \varepsilon\}\), \(\overline g(s) = \lim_{\varepsilon \to 0^+} \text{ess sup} \{g(\sigma): |\sigma - s |< \varepsilon\}\). They show that the given problem admits at least two geometrically distinct solutions. The proof is based on a result of Lyusternik-Schnirelman type for locally Lipschitz functionals. Results in this direction were already obtained by K. C. Chang [J. Math. Anal. Appl. 80, 102-129 (1981; Zbl 0487.49027)] through an extension of the deformation lemma. In contrast, the authors use the Ekeland variational principle.

MSC:

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34C25 Periodic solutions to ordinary differential equations
49J52 Nonsmooth analysis
58E30 Variational principles in infinite-dimensional spaces

Citations:

Zbl 0487.49027
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