Symmetric solutions of a multi-point boundary value problem

We apply the fixed point theorem of Avery and Peterson to the nonlinear second-order multi-point boundary value problem −u″(t)=a(t)f(t,u(t),|u′(t)|),t∈(0,1),u(0)=∑i=1nμiu(ξi),u(1−t)=u(t),t∈[0,1], where 0<ξ1<ξ2<⋯<ξn⩽12, μi>0 for i=1,…,n with ∑i=1nμi<1, n⩾2. We show that under the appropriate growth conditions on the inhomogeneous term symmetric about t=12 the problem has triple symmetric solutions.