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Periodic solutions for sublinear systems via variational approach. (English) Zbl 1209.34046

This paper deals with a scalar second order equation of the form
\[ x''+f(t,x)=0, \]
where \(f\) satisfies a variant of the Carathéodory conditions; moreover, a sublinear condition of the form \(|f(t,x)| \leq g(t) |x|^{\alpha} +h(t)\) is assumed, where \(g,h \in L^1((0,2\pi); {\mathbb R}^+)\) and \(\alpha \in [0,1)\). The first result deals with the periodic boundary value problem associated to the given equation. Assuming (setting \(F(t,x)=\int_0^x f(t,s)\,ds\)) that \(\liminf _{|x| \to +\infty}|x|^{-2\alpha}F(t,x) > (1/2) \|g\|^2_1\), the existence of at least one \(2\pi\)-periodic solution is proved.
In the second result, the periodic boundary condition is replaced by the impulsive condition \(x'(t_j^+)-x'(t_j^-)=I_j(x(t_j))\), \(j=1, \dots, p,\) where \(0=t_0<t_1< \dots < t_{p+1}=2\pi\) and the impulse functions \(I_j: {\mathbb R} \to {\mathbb R}\) are continuous for all \(j\). Besides the same assumptions on \(f\) considered in the first result, it is assumed that, for some \(a,b \geq 0\) and \(\gamma \in [0,\alpha)\), one has \(|I_j(x)| \leq a|x|^{\gamma}+b\) and \(|\int_0^x I_j(s) ds| \leq a|x|^{2\gamma}+b\) for all \(x,j\). Under these hypotheses, the existence of at least one \(2\pi\)-periodic solution is proved. The proofs are performed by applying the saddle point theorem.

MSC:

34C25 Periodic solutions to ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations
47J30 Variational methods involving nonlinear operators
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