We shall be concerned at first with some properties of the solutions of the matrix differential equation1.1whereis an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).

In this paper we prove a theorem on the existence and asymptotic behaviour of nonoscillatory solutions of the equation x″+p(t)x=0, where p(t) = (−λ2 + h(t)) t−2α with λ > 0 and 0 < α < 1. The coefficient p need not be onesigned. Examples show that the same asymptotic formula can hold either when p(t) is eventually negative, or when it oscillates. Moreover, the result can be applied to cases where p is negative, but is such that the classical Liouville‐Green approximation formula cannot be used.

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In this paper we show that under certain conditions a unique {complex valued) solution exists for the non-linear elliptic equation (1.1) AU = [(x, y, u, u,~, uu) , which satisfies a generalized boundary condition on each of the analytic curves I'~=-{(x,y) ly=f~(x), o~x~_a} and F2=-{(x,y) lx=f~(y}, o~y~'b} namely u~(~, y) = ~o(X) u(x, y) + ~(~ uAx, y) + ~(x~) o, r~ (1.2) u~(x, y)= ~o(Y)u(x, y) -~ ~:(y)u,~(x, y) -~ h(y) on F2(1.3) u(0, 0) : y (0, 0)= F, A F2.The coefficients ~a, ~a, g, h and y are assumed to be complex valued. By separating real and imaginary parts in equations {1.1)-(1.3) our results may be seen to apply to a system of real partial differential equations.