Periodic solutions for a kind of prescribed mean curvature Liénard equation with a singularity and a deviating argument

摘要

In this paper, we study the existence of periodic solutions to the following prescribed mean curvature Liénard equation with a singularity and a deviating argument: ( u ′ ( t ) 1 + ( u ′ ( t ) ) 2 ) ′ + f ( u ( t ) ) u ′ ( t ) + g ( u ( t − σ ) ) = e ( t ) , $$iggl(rac{u'(t)}{sqrt{1+(u'(t))^{2}}}iggr)'+figl(u(t)igr)u'(t)+g igl( u(t-sigma)igr)=e(t), $$ where g has a strong singularity at x = 0 $x=0$ and satisfies a small force condition at x = ∞ $x=infty$ . By applying Mawhin’s continuation theorem, we prove that the given equation has at least one positive T-periodic solution. We will also give an example to illustrate the application of our main results.