Abstract
In this paper, we consider a general high-order delay differential equation in the following form: x ( n ) ( t ) = F ( t , x ( t ) , x ( t − τ ( t ) ) , x ′ ( t ) , … , x ( n − 1 ) ( t ) ) , where τ is a continuous periodic function defined on R with period T ; F is a continuous function defined on R n + 2 and periodic to t , F ( t , c , c , 0 , … , 0 ) ≢ 0 for all c ∈ R . By applying Mawhin’s continuation theorem and a new inequality established in this paper, we obtain sufficient conditions for the existence of periodic solutions for the equation. Our results are new and the methods are different from previous works. Moreover, an example to illustrate the results is given.
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