Periodic solution of differential equation with ϕ-Laplacian and state-dependent impulses

Sufficient conditions for the existence of at least one periodic solution of differential equation with ϕ-Laplacian(ϕ(z′(t)))′=f(t,z(t),z′(t)),for a.e. t∈[0,T]⊂R, and state-dependent impulses△ϕ(z′(t))=Mi(t,z(t)),t=τi(t,z(t)),i=1,…,p are given. Here, T>0, ϕ:R→R is an increasing homeomorphism, ϕ(R)=R, ϕ(0)=0, f:[0,T]×R2→R satisfies Carathéodory conditions, Mi:[0,T]×R→R are continuous and τi:[0,T]×R→(0,T) are continuous for i=1,…,p, p∈N, △ϕ(z′(t))=lims→t+⁡ϕ(z′(s))−lims→t−⁡ϕ(z′(s)). The results are obtained via lower and upper functions method and Schauder fixed point theorem.