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 8>>>>>< >>>>>: .􀀀1/mu.2m/ D f1.t; u;􀀀u00; : : : ; .􀀀1/m􀀀1u.2m􀀀2/; v;􀀀v00; : : : ; .􀀀1/n􀀀1v.2n􀀀2//; .􀀀1/nv.2n/ D f2.t; u;􀀀u00; : : : ; .􀀀1/m􀀀1u.2m􀀀2/; v;􀀀v00; : : : ; .􀀀1/n􀀀1v.2n􀀀2//; 0u.2i/.0/ 􀀀 0u.2iC1/.0/ D 1u.2i/.1/ C 1u.2iC1/.1/ D 0 .i D 0; 1; : : : ;m 􀀀 1/; 0v.2j/.0/ 􀀀 0v.2jC1/.0/ D 1v.2j/.1/ C 1v.2jC1/.1/ D 0 .j D 0; 1; : : : ; n 􀀀 1/: Here f1; f2 2 C.T0; 1U RmCn C ; RC/.RC VD T0;C1//. Our hypotheses imposed on the nonlinearities f1 and f2 are formulated in terms of nonnegative matrices.Weuse fixed point index theory to establish our main results based on a priori estimates of positive solutions.