A necessary and sufficient condition for uniqueness of solutions of singular differential inequalities

The author proves that the abstract differential inequality [[u’ (t) A(t)u(t),,2[[ 7 (t) + ()d in which the linear operator A(t) M(t) + 0 N(t), M symmetric and N antisymmetric, is in general unbounded, w(t) t-2(t)[[u(t)[[ 2 + [[M(t)u(t)[[ [[u(t)[[ and 7 is a positive constant has a nontrivial solution near t-0 i which vanishes at t-0 if and only if ft-l(t)dt . The author also shows that the 0 t second order differential inequality [[u"(t) A(t)u(t)[[ 2 7[(t) + f()d] in which 0 (t) t-40(t)[[u(t)[[ 2 + t’2l(t)[[u’(t)[[ 2 has a nontrivial solution near t-0 such i I that u(0)-u’(0)-O if and only if either ft-l0(t)dt or ft-ll(t)dt . Some 0 0 mild restrictions are placed on the operators M and N. These results extend earlier uniqueness theorems of Hile and Protter.


INTRODUCTION.
Let H be a complex Hilbert space with the usual inner product and norm notation and let A be an linear, in general unbounded, operator defined on a non-trivial domain D in H. Assuming  (1.5) 0 Our results extend those of Hile and Protter [i] who prove that the only solution of (I.i) and likewise for (1.2) with homogenous initial conditions is the trivial one provided the functions t'2(t), t-40(t and t'2l(t) are bounded. Thus which is the well known abstract Euler-Polsson-Darboux (EPD) equation if (t) k/t, k constant, and prove uniqueness for the initial value problem provided (t) z -i/t. These results of Donald and Goldstein [3] have been extended by Goldstein [5] as well as Arrate and Garcia [6]. Ames [4] also considers (1.6) with =(t) 4(t)/t (where 4 has properties somewhat similar to ours) but requires only that the operator A be symmetric (and independent of t). Furthermore it is known that the solution to the EPD equation (A the Laplacian) is not unique if k < 0 (See e.g., [4]. (2.1) Consequently the function 4(t)/t is nonincreasing and hence t4'(t) 4(t). (2.2) We now give assumptions on the linear operator A which, except for (iii) and (iv), match those of [i] while (iil) and (iv) are more general than the similar conditions given in [I]. It should be noted that not all of these will be needed in the proof of sufficiency.

I1()112
Sufficiency. Although the proof of necessity will require that the operator A satsify condition (I), sufficiency will not require properties (ili) and (iv).  Then the monotoniclty of @ gives lim @(t) 0. Also, without loss of generality, we t$O may assume lim @(t)/t-. Indeed lim @(t)/t exists (possibly infinite) since @(t)/t t&O t0 is nonlncreasing; and furthermore, if lim @(t)/t < , inequality (I.I) is still valid on (0,T] if @(t) is replaced with Ct I/2 for a sufficiently large constant C (depending only on T) and hence lim @(t)/t-. Additionally, as a consequence of (2.6) and t$0 the monotoncity of @(t)/t, we have (2.7) t Before proving necessity (Theorem 2), we need some preliminary lemmas.  (7))2d. The last integral in (2.9) admits the estimate 1/2 1/2 -17 "I p()d for t < T. Since t 0 s -R(7)9( 7-i(7)d7 t [P(7)] 0 and application of L'Hospital's rule gives (2.10) where the last equality holds because r is bounded near zero and @(t)/t , we get llm R(W)() 0. Using this result, we integrate by parts in the last integral in (2.10) and obtain Since A(t) and (t) are nonnegative while R(t) is nonpositive, we may discard the first expression on the right side of (2.11). Also (2.7) with k 2 gives exactly -I s 1 so that -R()'() S r2(). Substitution of this into (2.11) and the resulting inequality into (2.9) yields (2.8). This completes the proof. .1,pe2r + C..,r/ge2r -/9 + 2 Re I(Nv,Mv) + Illv'-Nvll 2 I 1 + + 16. Using estimates for I I through 13 identical to those in [i, proof of Lemma I] and estimates virtually identical to those of 14 and 15 in the same lemma (the only difference is the iin [I] is replaced with 1 here) and using (2.12) above to estimate 16 gives (2.14) and the proof is complete.
We may now prove necessity. It should be noted that Theorem

Re(M(t)u(t),N(t)u(t)) >_ -F(t).
Sufficiency. Not all of Condition (II) will be needed to prove sufficiency, and as in the the first order case, we show that our solution actually satisfies a much sharper estimate than (1.2). (See inequalities (3.4) and (3.10).) However, before proving sufficiency, we need a preliminary result. If this were not the case, it must be that 4 @0 < C near zero or 4 C < @0 near 0 and in either case the result would hold trivially. Choose a subsequence (anj} of {a n such that anla I, and 2anj+IS an. for all j. Since 4(t)/t is nonlncreasing and 4(an)/a n C/a n, we  . We define the function (suppressing its dependence on e since e will be chosen to be 1/2 later (in the proof of Lemma i0)) by /@(t)/t dt < (as a result of (3.11)).
We now apply this result along with (3.22) and (3.36) to (3.51) to obtain I.