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In this paper, we study the singular fourth-order differential equation with a deviating argument:By using Mawhin's continuation theorem and some analytic techniques, we establish some criteria to guarantee the existence of positive periodic solutions. The significance of this paper is that g has a strong singularity at x = 0 and satisfies a small force condition at x = ∞, which is different from the known ones in the literature.

In this paper, we study the existence of at least two geometrically distinct periodic solutions for a differential equation which models the planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter J 2. And at least one of such two periodic solutions is unstable. The proof is based on the version of the Poincaré-Birkhoff theorem due to Franks. Moreover, we also study the existence and multiplicity of periodic solutions and subharmonic solutions with winding number.

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