Positive solutions of singular sub-linear boundary value problems for fourth-order and second-order differential equation systems

This paper investigates the existence of positive solutions of singular sub-linear boundary value problems for fourth-order and second-order differential equation systems. First of all, we establish some important Lemmas. Then, we define a partial ordering in C2[a, b] ∩ C4(a, b) × C[a, b] ∩ C2(a, b) and construct lower and upper solutions to give a necessary and sufficient condition for the existence of C2[0, 1] × C[0, 1] positive solutions as well as C3[0, 1] × C1[0, 1] positive solutions. Our nonlinearity f(t, x1, x2, x3), g(t, x1, x2) may be singular at x1 = 0, x2 = 0, x3 = 0, t = 0 and or t = 1.