ASYMPTOTIC BEHAVIOUR OF NTH ORDER DIFFERENTIAL EQUATIONS WITH MIDDLE TERMS AND FORCINGS

Equations of the following types are studied:

(1) x('(n)) + P(t)x('(n-1)) + H(t,x) = Q(t), n (GREATERTHEQ) 4, even,

(2) x('(n)) + P(t)x('(n-2)) + H(t,x) = Q(t), n (GREATERTHEQ) 4, even,

(3) {p(t)x('(n-k))}('(k)) + H(t,x) = Q(t), n, k even, 2 (LESSTHEQ) k (LESSTHEQ) n - 2,

(4) x('(n)) + p(t)x('(n-1)) + q(t)x('(n-2)) + H(t,x) = 0, n (GREATERTHEQ) 4, even or odd.

Several results are obtained concerning the oscillation and other asymptotic properties of the solutions of such equations. The methods involved include the recently developed comparison method of Kartsatos as well as the method of functionals that has been used for third order equations by Lazer, Erbe, and for nth order equations by Kartsatos.

The results obtained utilizing these nonlinear functionals are actually weaker than the corresponding results for third order equations. This is due to the fact that certain terms in the analysis of an nth order equation, containing derivatives of order n - 3, do not enter the corresponding research of third order equations.

Numerous results of Erbe, Foster and Grimmer, Kartsatos, and Kartsatos and Toro are extended to equations of the above types. These results are actually dealing with unforced equations of order 3 and Type (4) (Erbe) or with nth order equations without middle terms and/or forcings.

Since very little research has been done concerning such nonlinear functionals, this work actually constitutes a first attempt towards a general theory of oscillation of equations with middle terms via the study of such functionals. It should also be mentioned that several results in this dissertation are new even in the linear case (H(t,x) (TBOND) r(t)x).