Periodic solutions for prescribed mean curvature Rayleigh equation with a deviating argument

where τ , e ∈ C(R,R) are T-periodic, and f , g ∈ C(R × R,R) are T-periodic in the first argument, T >  is a constant. In recent years, there are many results on the existence of periodic solutions for various types of delay differential equation with deviating arguments, especially for the Liénard equation and Rayleigh equation (see [–]). Now as the prescribed mean curvature ( x ′(t) √ +x′(t) )′ of a function x(t) frequently appears in different geometry and physics (see [– ]), it is interesting to try to consider the existence of periodic solutions of prescribed mean curvature equations. However, to our best knowledge, the studies of delay equations with prescribed mean curvature is relatively infrequent. The main difficulty lies in the nonlinear term ( x ′(t) √ +x′(t) ) ′, the existence of which obstructs the usual method of finding a priori bounds for delay Liénard or Rayleigh equations from working. In [], Feng discussed a delay prescribed mean curvature Liénard equation of the form ( x′ √  + x′ )′ + f ( x(t) ) x′(t) + g ( t,x ( t – τ (t) )) = e(t), (.)