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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 Carath√©odory conditions; moreover, a sublinear condition of the form |f(t,x)|g(t)|x| α +h(t) is assumed, where g,hL 1 ((0,2π); + ) and α[0,1). The first result deals with the periodic boundary value problem associated to the given equation. Assuming (setting F(t,x)= 0 x f(t,s)ds) that lim inf |x|+ |x| -2α F(t,x)>(1/2)g 1 2 , the existence of at least one 2π-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,,p, where 0=t 0 <t 1 <<t p+1 =2π and the impulse functions I j : are continuous for all j. Besides the same assumptions on f considered in the first result, it is assumed that, for some a,b0 and γ[0,α), one has |I j (x)|a|x| γ +b and | 0 x I j (s)ds|a|x| 2γ +b for all x,j. Under these hypotheses, the existence of at least one 2π-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)