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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.

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