A symmetric solution of a multipoint boundary value problem with one-dimensional p-Laplacian at resonance

We apply an extension of Mawhin’s continuation theorem due to Ge to show the existence of at least one symmetric solution of the multipoint boundary value problem for the one-dimensional pp-Laplacian at resonance (ϕp(x′(t)))′=f(t,x(t),x′(t)),t∈(0,1), subject to the boundary conditions x(0)=∑i=1nμix(ξi),x(1)=∑i=1nμix(ηi), where ϕp(s)=|s|p−2s,p>1,0<ξ1<ξ2<⋯<ξn<1/2,ξi+ηi=1,i=1,2,…,n,∑i=1nμi=1,f:[0,1]×R2→R with f(t,u,v)=f(1−t,u,−v)f(t,u,v)=f(1−t,u,−v) for (t,u,v)∈[0,1]×R2(t,u,v)∈[0,1]×R2, satisfying the Carathéodory conditions.