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Periodic solutions of a second order forced sublinear differential equation with delay. (English) Zbl 1097.34050

The authors consider the existence of 2π-periodic solutions to the second-order sublinear differential equation with delay

ax '' (t)+bx(t)+qx ( t - τ )=p(t),t,(*)

where a,b and τ>0 are real constants, the forcing function p: is a 2π-periodic continuous function and g: is a continuous function. By means of a priori estimation and continuation theorems, the authors obtain criteria for the existence of 2π-periodic solutions of equation (*) under a sublinear condition on the function g. The main results of this paper are the following new criteria:

(1) If 0<|b|<|a|/π 2 and if there are constants ρ>0, β>0 and α[0,1) such that |g(t)|β|x| α for |x|>ρ, then (*) has a 2π-periodic solution.

(2) If b=0, a=1 and if there are constants ρ>0 and β(0,1/2π 2 ) such that g(x)=-β|x| for x-ρ, or g(x)β|x| for xρ, and xg(x)>0 for |x|ρ, then (*) has a 2π-periodic solution.

(3) If b=0, a=1 and there are constants ρ>0, β>0 and α[0,1) such that g(x)-β|x| α for x-ρ, or g(x)β|x| for xρ, and xg(x)>0 for |x|ρ, then (*) has a 2π-periodic solution.

MSC:
34K13Periodic solutions of functional differential equations