Ezeilo pioneered the use of Leray-Schauder degree type arguments to obtain existence results for periodic solutions of
ordinary differential equations.
Third-order ODEs In 1959, Ezeilo studied the stability of the third-order ODE \overset{...}{x} + f(x, \dot{x})\ddot{x} + g(x)\dot{x} + h(x) = 0 near the trivial solution x = 0. Stability holds if f(x, y)\geq\delta_0 for some positive constant \delta_0, g(y)/y\geq\delta_1for some positive constant \delta_1, h(x)/x\geq\delta_2 for some positive constant \delta_2, and h'(x)\leq c for some constant
c where \delta_0\delta_1 - c\geq 0, and y\frac{\partial f}{\partial y} is always nonpositive. In 1966, he, together with
H. O. Tejumola, generalized this work even further to study vector equations of the form \overset{...}{X} + A\ddot{X} + B\dot{X} + H(X) = P(t, X, \dot{X}, \ddot{X}).
Fourth and fifth order ODEs In 1962, Ezeilo studied the stability of the fourth-order equation x^{(4)} + f(\ddot{x})\overset{...}{x} + \alpha_2\ddot{x} + g(\dot{x}) + h(x) + \alpha_4 x = p(t). In 1978 he studied the fifth-order equations x^{(5)} + a_1 x^{(4)} + a_2\overset{...}{x} + a_3\ddot{x} + a_4\dot{x} + f(x) = 0 and the two generalizations x^{(5)} + a_1x^{(4)} + a_2\overset{...}{x} + h(\dot{x})\ddot{x} + g(x)\dot{x} + f(x) = 0 and x^{(5)} + \psi(\ddot{x})\overset{...}{x} + \phi(\ddot{x}) + \theta(\dot{x}) + f(x) = 0 . == Honour ==