Existence of positive solutions for a system of generalized Lidstone problems

In this paper we study the existence of positive solutions for the system of generalized Lidstone problems for ordinary differential equations {(−1)mu(2m)=f1(t,u,−u″,…,(−1)m−1u(2m−2),v,−v″,…,(−1)n−1v(2n−2)),(−1)nv(2n)=f2(t,u,−u″,…,(−1)m−1u(2m−2),v,−v″,…,(−1)n−1v(2n−2)),α0u(2i)(0)−β0u(2i+1)(0)=α1u(2i)(1)+β1u(2i+1)(1)=0(i=0,1,…,m−1),α0v(2j)(0)−β0v(2j+1)(0)=α1v(2j)(1)+β1v(2j+1)(1)=0(j=0,1,…,n−1). Here f1,f2∈C([0,1]×R+m+n,R+)(R+≔[0,+∞)). Our hypotheses imposed on the nonlinearities f1f1 and f2f2 are formulated in terms of nonnegative matrices. We use fixed point index theory to establish our main results based on a priori estimates of positive solutions.