QUASI-ADJOINT THIRD ORDER DIFFERENCE EQUATIONS : OSCILLATORY AND ASYMPTOTIC BEHAVIOR

In this paper, asymptotic properties of solutions of Δ 3 V n + P n − 1 V n + 1 = 0 ( E + ) are investigated via the quasi-adjoint equation Δ 3 U n + P n U n + 2 = 0. ( E − ) A necessary and sufficient condition for the existence of oscillatory solutions of ( E + ) is given. An example showing that it is possible for ( E + ) to have only nonoscillatory solutions is also given.

By the graph of a solution V {V we mean n-i n the polygonal path connecting the points (n,V) n > I.A point of contact of the n graph of V with the real axis is a node.A solution V of (E+) is said to be oscil- latory if it has arbitrarily large nodes; otherwise it is said to be nonoscillatory.
Hereafter the term "solution" shall mean "nontrivial solution".
The oscillation criterion established by Lazer [I, Theorem 1.2], for the differen- tial equation y"' + p(x)y' + q(x)y 0, where p(x) !O, q(x) > 0 is proved for the difference equation (E+).We also include an example which demonstrates that is pos- sible for (E+) to have only nonoscillatory solutions.
In studying (E+) we will make use of its quasi-adjoint equation (E-).For general properties and definitions concerning (E+) and (E-) we refer to Fort's book [2], and to the paper by this author.
The posltivlty of the coefficient function p places severe restrictions on the behavior of the nonoscillatory solutions of (E+).
PROOF.Assume that V is a nonoscillatory solution of (E+), where without loss of generality Vn > 0 for each n _> N. Note that A3Vn A(A2Vn) -Pn-IVn+l < 0 for al.! n > N hence A2V is decreasing and is eventually one sign.It follows that M n exists, M > N for which AV and A2V are sign definite, for all n > M. Hence n n V AV A2V # 0, for every n > M. The following cases must be considered: The cases (c) and (d) are clearly impossible since JV AJ+Iv > 0 for all n n n sufficiently large implies that sgnAJ-lv sgnAJV eventually.To complete the proof n n apply Lemma 2.1 to the case (a).
Denote by (S-) the solution space of (E-) and (S+) the solution space of (E+).For (U,V) e (S-) x (S+) define It is easy to verify.The function defined by (2.3) is a constant determined by the initial values of U and V, hence, F represents the discrete LaGrange billnear n concomitant for solutions of (E-) and (E+).Using (2.3), proofs of the following two results can be modeled after the analogous results appearing in [3].THEOREM 2.3.Suppose that U is a nonoscillatory solution of (E-).If (E+) has an oscillatory solution, then there are two independent oscillatory solutions of (E+) that satisfy AV A2U A() + (U n-i Vn+l 0. (2.4) n n Un+l COROLLARY 2.4.If (E+) has an oscillatory solution, there exists a basis for the solution space (S+) consisting of k nonoscillatory solutions and 3-k oscillatory solutions for k=0, i.
REMARK.Since the nodes of linearly independent solutions of (2.4) separate each other, and those of linearly dependent solutions coincide, it follows that solutions of (2.4) are oscillatory.See for example, Fort [3, p.221].
We conclude this section with the following easily verifiable result.
LEMMA 2.5.The graph of a sequence X {X Let Y be a sequence.Then, (1) G(r, (a Xn + b Yn )) a G(r, X n) + b G(r, Yn n r n+l, n I, where a and b are constants. (ll) If AX > 0 for some integral m, then m Xm < G(r, X m) < Xm+ (X m G(r, X m) > Xm+l).(ill) If Xn > 0 < 0) for each n, then G(r, X n) > 0 < 0) for each r.

MAIN RESULT.
In case (E+) has oscillatory solutions, our main result shows that more strin- gent requirements are imposed on the nonoscillatory solutions of (E+), than those posed by Theorem 2.2.We will show that solutions satisfying relations (2.2) cannot be 'ntroduced" into the solution space (S+) without "forcing" out all the oscillatory solutions.
THEOREM.A necessary and sufficient condition for (E+) to have oscillatory solu- tions is that for any nonoscillatory solution V the relations (2.1) are satisfied.PROOF.The sufficiency is clear.If every nonoscillatory solution V satisfies the relations (2.1) by Lemma 2 5 G(r, V for such a solution is of one sign for n each n I.It follows that any solution with a node is oscillatory; thus we see that initial values can be used to construct oscillatory solutions of (E+).To prove the necessity, suppose that (E+) has an oscillatory solution, and that (E+) has a solution Z satisfying the relations (2.2).By Corollary 2.3 and the above remark, there exists a basis for (S+) consisting of one nonoscillatory solution R satisfying conditions (2.1) and two oscillatory solutions S and T with every linear combination of S and T oscillatory.Now Zn CIRn + C2Sn + C3Tn where CI C 2 C 3 are scalars, B. SMITH C + C + C # 0. Conditions (2.2) imply lira IZnl Let {r i} be an increasing n+ of nodes of {C2S n + C3T n} Then at each sequence r i G(ri,Zn) CIG(ri,R . (2.5) The left member of (2.5) is bounded as i This contradiction completes the proof of the theorem.
In conclusion we present an example showing that it is possible for every solution of (E+) to be nonoscillatory.The relations (2.2) are satisfied by the funciton V -n+l dfined by V n + 2 It is easily seen that V is a solution of n A3v + 0 n 4[ (n+l) 2n+l) vn+l Consequently every solution of (E) is nonoscillatory.ACKNOWLEDGEMENT.This research was partially supported by Texas Southern University Faculty Research #16512.