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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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