OPTIMALITY AND EXISTENCE FOR LIPSCHITZ EQUATIONS

Solutions of certain boundary value problems are shown to exist for the nth order differential equation y(n)=f(t,y,y′,…,y(n−1)), where f is continuous on a slab (a,b)×Rn and f satisfies a Lipschitz condition on the slab. Optimal length subintervals of (a,b) are determined, in terms of the Lipschitz coefficients, on which there exist unique solutions.

We will be concerned with the existence of solutions of boundary value problems for the nth order differential equation y(n) f(t,y,y' (n-l) ,...,y ), ( )   where f is continuous on a slab (a,b) R n and satisfies a Lipschitz condition, on the slab.
A number of papers have appeared in which optimal length sublntervals of (a,b) are determined, in terms of the Lipsch[tz coefficients kl, < i< n, on which solutions of certain oundary value problems for (l.l) are unique; see, for example [I-15].Of motivational importance in this work are the papers by Jackson [ I0-11]   in which he applied methods from control theory in establishing optimal length subintervals, in terms of the Lipschltz coefficients, on which solutions of conjugate boundary value problems and right focal point boundary value problems for (I.I) are unique.It then follows from uniqueness implies exlstence results due to Hartman [16-17] and (laasen [18] in the conjugate case and Henderson [19] in the right focal point case, that unique solutions exist on the optimal intervals given in la [7-8], we adapted-ackso,l's coatr,)l theory arguments, [q c,)nj,1ct[on wth u[qaees-[,pl[e e[.tece e,It, ad determined optimal length teals of (a,b) on h[ch thee et m[que sol.t[os of seerl classes of boundary value pobl.e.,s ..) third an fo,Jth order ordinary differential equat[o sat [sfylag L[pschItz cond[tIons.In a recent ork [9], we followed the pattera of [7-, i0 < _<... <_%_ <b, 0_<k<.<_., . ,,t_< i<_ lq thls work, .atnow ad|res the prob!e,n o. existence of solutions of (1.9) on the optI,,al Intervsls for U,l[.Iseness .!-%[e,l In [9].We ste In Section 2 some o t1e results concerning optimality and ualqeness obtalned [n [9] which are pert[,ent to the argument here.Then l Section , we are able to prove that on subintervals of length less tha the optimal !enth given n gect[o 2 and for certaln values of k and h, sol.at[ons of ([.I), (l.) exist.For It, Is restricted set f k and h, the e.[tence result s so:,e sense analogous to the iqueness lmpl[es existence results In [16-19].
)OF.Let a < < t 2 < b, with t 2 t < y, and Yi ' j J n, be give.We prove the ,xthace of solutions for a much larger family of boundary vale problems than those in the statement of the theorem In fact, we prove the existence of solutions of the two-point problems which bel)g , tha :lass of prohle'., la Corollary 2.
For induction pltl)oq; J, :ctange these problems in a lower triangular array, (I,I) where the bo,.aa'y ,,,:.flueproblem for (I) associated with the (,v)-posltlon, < < u < h, satisfies y(i)(tl) Yi+l, O _< i < n-v-l, y(t)(t2 Yn-+(i+l)' -v <_ t Under this arrangement, the boundary value problems for (1.1) along the principal diagonal (u,u), u h, are e.onJuga_t_e type problems, whereas the boundary value problems In the statement of this theorem are assoelated Ith the entries along the bottom row (h,v), < v < h.
By Corollary 2, solutions of all the problems In th[ array are unique on subintervals of length less than .Moreover, by the constraints on h and k, It follows that solutions of all conjugate type boundary value problems for (I.I) are unique.Then it follows from the uniqueness i,aplIes existence result of Hartman [16-17] and Klaasen [18] that the c,>,ljigate boundary value problems, and in parti- cular those associated with the e,tctes on the maln diagonal, have unique solutions.
(This is the reason for the constraints on h and k.)For existence of solutions of the remaining problems associated with the array, we will use the shooting method coupled wlth an Induction along the subdiagonals on the array.
In that direction, choose any boundary value problem for (I) associated with the first subdlagonal (,u-l), where 2 < < h; that Is, e are concerned with solutions of (I) satisfying Y(i)(t2) Yn-B+(i+l)' <_ I < -I.
We claim that S is also a closed subset of R. Assuming the claim to be false, It follows that there is a limit point r 0 e S \S.Hence, there exists a strictly monotone sequence rj) S of numbers converging to r O.We may assume with- out .loss of generality that rj/rO.For each J > I, let yj(t) denote the solution of (I) given by the definition of S satisfying, y "(t 1) z "(tL) O < i < J(n-) tl J .vj(1)(t2 z(f)(t2 ), 1 !! Fro Corolla a 2, it follows that, oc each I, yj(t) < Yj+l(t) on (tl,t2]. Futeote slice f satLsLes the LlpschLtz codLtLon (2)
It follows that there eElsts a subsequence {yjk(t)} such that, for each k I, y (a-u)(t) [tesects u(-u)(t) at .point Pk e (tl' tl + $) and yj (n-u)(t) , t I) nd Ok+t and Ok+tI.By choos[ng successive subsequences and telabel[ng, we may assume that tl-< k < tl< k < tl + ate the first points where these intersections occur.
t follos that a compactness eodtLo on sequences of solutLons o (L) Is satisfied, (see [I0]); ts compactness coudtLon d the fct that r O , h.e tlat {yj(t)} Ls ot uuformly bou.lT .ach compact subLnterval of (a,b), aud n prtLcular, s not uufoly bounded above o each eo,npact subtrql of [tl,t2].ow let (t) be the soltLo of the proble for (l) assocLated wth the value problems for (I) corresponding to the (u-l, u-2)-pos[t[on coupled with , gu,ent similar to the one use.[n the proof of the first theorem of [13, Thin.I] boundary It aaa be argued that I Is a aonempty subset of R which Is bsth , I ,losed, so that S R. Choosing Yn-SI' the :orespoadlng h, have unique solutions.