Abstract
We establish the existence of positive periodic solutions of the second-order differential equation via Schauder’s fixed point theorem, where , , f is a Caratheodory function and it is singular at . Our main results generalize some recent results by Torres (J. Differ. Equ. 232:277-284, 2007). MSC: 34B10, 34B18.
Highlights
In this paper, we are concerned with the existence of positive periodic solutions of the second-order differential equation x + a(t)x = f (t, x) + c(t) ( . )with a ∈ L (R/TZ; R+), c ∈ L (R/TZ; R), f ∈ Car(R/TZ × (, ∞); R) is a L Caratheodory function, and f is singular at x = .The interest on this type of equations began with the paper of Lazer and Solimini [ ].They dealt with the case that a ≡ and f (x) =xλ reduces to the special equation x = xλ + c(t), which was initially studied by Lazer and Solimini [ ]
Which was initially studied by Lazer and Solimini [ ]. They proved that for λ ≥, a necessary and sufficient condition for the existence of a positive periodic solution of ( . ) is that the mean value of c is negative, c =
By a direct application of Schauder’s fixed point theorem, the proof is finished if we prove that F maps the closed convex set defined as
Summary
They proved that for λ ≥ (called a strong force condition in the terminology first introduced by Gordon [ , ]), a necessary and sufficient condition for the existence of a positive periodic solution of If < λ < (weak force condition) they found examples of functions c with negative mean values and such that periodic solutions do not exist. If compared with the literature available for strong singularities, see [ – ] and the references therein, the study of the existence of periodic solutions in the presence of a weak singularity is much more recent and the number of references is considerably smaller.
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