ON THE CARDINALITY OF SOLUTIONS OF MULTILINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS

We study the existnece and cardinality of solutions of multilinear differential equations giving upper bounds on the number of solutions.

n(i) are never vanishing real and con- tinuous on I.
We give necessary and sufficient conditions for a (m-l)-tuple (GI,@ 2 ,m_l) to be a multiple ordinary branching of a solution to (i.i) where  I, e e 1,2,... ,m-l.
We also study the existence and cardinality of solutions to the initial value problem Dn(1)u(z) zi, i 1,2 n(1)-i (1.3) where z,z i I, giving upper bounds on the number of solutions with n multiple branchings.
Finally we study the problem when it assumes the form (1.1) for some function %.
of the form We will seek solutions u (2.2) e ,x e+l am_l x for aeI, e 1,2 m-i and aeN(Le) U N(Le+I).A function of the form (2.1)   in N(M) will be said to have a single ordinar branchin5 at x on e [ae_l,ae+l].A function of the form (2.2) will be said to have a multiple ordinary branching at (i,2 m_l on I [a,b]  We(Uli,U2i,...,Un(i)i, Ul(i+l)(xo) by W __(x 0), e 1,2 m-l.e The following theorem shows when N(M) will contain functions having a multiple ordinary branching.THEOREM i. Assume that n(e) (e) + n(e+l) n(1) + i, e 2 ,m-i (2.3) and if () E has Just one function __ulj(x)' J 2,...,m, then there exists u N(M) having a multiple ordinary branching at (l m-1 if and only if We(e o, e i, m-i <=> (LiUl(i+l)  It is enough to find numbers, Cll ,Cln(1), C21 C2n(2) Cm/, so that u C n(l )(I).Therefore we must have .CeU c j=l e (ae Uln(e+l)(e) 0, e=1,2 m-l, k 0,1,...,n(1) (2.6) where Cln(e+l # 0 (we take Cln(e+l i).The homogeneous equation (2.6) has a nontrivial solution if and only if (2.4) holds.
Note that it is easy to verify that W e (%) W e(ule,u2e ,Un(e)e,Ul(e+l)(De) -i (e)We(Ule e(e) )(e e=l,2 ,m-i en e

U2e
Un e LeUl e+l CASE (ii).If (LeU s (e+l))(e) 0 e 1,2 m-1 we let c 1 and s (e+l) e e the rest coefficients zero.We then work as in Case (i).Otherwise we write (2.5)Note now, that the rank of the coefficients matrix on the left hand side is (n(1)+l) and thus we have a unique solution for the coefficients on the left hand side for any choice of the coefficients on the right hand side and for any I, e e 1,2,...,m-l.
The next theorem characterizes the conditions with the coefficients in (2.2) must satisfy in order that multiple branching can occur at (l,a2  (Le+l[j= CejUej])(%) 0, ke 0,i, n(e+l) n(e) (2.8) (2.9) In particular, (2.8) with Ce+lj # 0 for at least one Ue+lj Ej and (2.9) with Cej # 0 for at least on Uej E Be-Ee+l are both necessary and sufficient conditions for U N(M) to have a multiple branching at (,{z 2 am_ 1 on [c,d]. PROOF.If B E # 0, e 1,2 ..,m-2 the result is trivially true.Other- e e+l wise as in Theorem i, we have that u N(M) if and only if n(e) (k) n(e+l) . CejUej (e) Z c + u + e lJ e lJ J =i j =i (k (e)' k 0,i n( 1), e 1,2,... ,m-l.
The above can be written in the form n(e Un( e )i )(Ce )LeUe (n (e)+i (Ce)' but Ce+ljUe+lj (e) 0, k 0,1,2 n(1), e 1 m-i or in matrix form, where A is the coefficient matrix in (2.12) and the unknown e e vector.There will exist a nontrivial solution d # On the other hand, u has a nontrivial branching at (l m-1 if and only if (2.10) has a nontrivial solution for the coefficients on the right hand side.As of A is n(e+l).Now elementary row operations on A show that this is equivalent e e to (2.9).We now show that N(M) may contain infinitely many linearly independent functions.THEOREM 3. Assume that either Case (i) holds in theorem for infinitely many (li'a2 m-li )' i 1,2 or Case (ii) holds.In either case, there is a sequence u i=l N(M) such that u has a multiple i'"%-%'"%-.branching at (li,2i...m_li) with ei < e+li' e i m-2, i 1,2 and the set [Uli2i" "m-li ]n,i=l is linearly independent on I for every n.
DEFINITION 2. Define the set S. by setting 1 S.
(2.1h) if N(Le # N(Le+I), e 1,2,...,m-l, then we have at least m solutions, the unique solutions belonging to N(L ), e 1,2,...,m-1.In addition according to e Theorems 1 and 2 we may have solutions with one or many multiple ordinary branchings.e 1,2, ,m have constant coefficients we proceed as In the event that Le, follows:-let Sje, J 1,2,...,n(1), e 1,2, (2.17) Note that each one of the equations in (2.15) can have at most n(1)-i real 's and t j's are all real [7].
solutions if the dje e th Denote by apl,p2 pn(1)-i the solutions obtained in the equation in (2.15), p 1,2,...,m-i and assume that (the other cases can be dealt analogously) Inequality (2.18) shows that we can have at most (n(1)-l) m-I ordinary multiple branchings, e.g.(i'21 am-ll is one of them.We have thus proved.
THEOREM 5.If Li, i 1,2,...,m have constant coefficients, then there exists a solution u E N(M) (u as in (2.2)) to the intial value problem (2.1h) having a multiple ordinary branching (al, 2 m_l with e E I, e 1,2 m-1 if and only if ge is a root of the exponential polynomial (2.15), where the tj's are all real and they are given by (2.16) and (2.17) e Moreover if (2.18) holds there are at most (n(1)-l) m-1 solutions (u as in (2.2)).
(2.20) J=0 PROOF.Without loss of generality we can assume that l denote the first point at which a branching occurs and u N(L l) on some subinterval I(z) [z,all].
Then u N(L 2) on [ii,//+ ], for some > 0. There are at most m-i possi- ble values for I" Suppose w > ii is the next point at which a multiple branching of u occurs.Then u N(L 2) on [ll,W].Hence there exist uniquely determined cJ2(ll )' j 1,2 ,n(1) such that u(x) n( ).dj2 (ail)Uj2(x) J=l ]n(1) on [l,W] where uj2oj=l span N(L2).By Theorem 2, n() ILl( E dj(ll)Uj2)](v) 0 j=l at v GII and v w.Hence there are m-2 possible w's with w > Ii" This argument applies again for the next branching.Since this argument can be applied in any of the m-1 rows in (2.18), this proves (2.19).
Finally (2.20) can easily be proved if we use (2.19) for J 0,1,2 n and add the results.REMARK i. (a) We can assume in Theorem 6 that any h points h E [1,2 m-lS are fixed from (a l,a 2 am_I) then proceeding as in Theorem 6 we can prove that the corresponding relations for (2.19) and (2.20) are respectively (m-].-(h-1) )(m(1 )-l )h(n(i)-2 )h-1 2.21 and (b) above by assuming that (2.18) is true and u as in (2.2).But (2.2) can be written in (m-l)'.different ways by interchanging the role of the L.'s, i 1,2,...,m.
Therefore in general all the cardinality results obtained up till now can be multi- plied by (m-i).' (c) If the L i, i 1,2, m are nonconstant but continuous (as in the Introduction) we can restate Theorem 5 and (2.2.However the conclusions and the proofs are going to be exactly analogous. We now provide examples for Theorems and 6 and (1.4).Ul(X) e f(x) up(x) e 2f(x) LlU u' f'(x)u, L2u u' 2f'(x)u u N(M) can be written as f(x) That is, 0 is a limit point of branching points of u.
"e-l'ae+l" c [ae_ I' ,ae+ I' ].Then by Case (i) in Theorem i there exist con- Cje are uniquely determined by (2.1h).By (2.9) we must have Le+l(U(e)) O, e 1 l-(h-i) (n(l)-l hnl(n(1)-2 )h 2.22J=0Up till now we obtained the cardinality results in Theorems 5, 6 and in(a)    and consider the function f defined by CARDINALITY OF SOLUTIONS OF MULTILINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS 759