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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 $g,h \in L^1((0,2\pi); {\Bbb R}^+)$ 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 $I_j: {\Bbb R} \to {\Bbb R}$ 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:
34C25Periodic solutions of ODE
34B37Boundary value problems for ODE with impulses
47J30Variational methods (nonlinear operator equations)
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References:
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