Existence of solutions for right focal boundary value problems

Nordinem Analysis, Theory, Methods Printed in Great Britain.

& Applicalions,

Vol. 18, No. 2, pp. 191-197,

1992 0

EXISTENCE OF SOLUTIONS BOUNDARY VALUE JEFFREY EHME Department of Algebra, Combinatorics,

0362-546X/92 $5.00+ .OO 1992 Pergamon Press plc

FOR RIGHT FOCAL PROBLEMS

and DARREL HANKERSON

and Analysis, Auburn University, Auburn, AL 36849, U.S.A.

(Received 25 October 1990; received for publication 13 March 1991) Key words and phrases: Right focal boundary value problem, existence, uniqueness implies existence.

1. INTRODUCTION

WE

WILL

be concerned with solutions of boundary

value problems for an nth order scalar

equation x(“) = f( t, x, x’ , . . . , xcn- 1’))

t E (a, b).

(1)

We assume throughout that: (A) f is continuous on (a, b) x R”; (B) solutions of initial value problems for (1) are unique and extend to (a, b). The boundary value problems with which we are concerned will be described using notation introduced by Muldowney [I] and Peterson [2]. Definition 1. (a) Let a < t, 5 ... ~t, . . . > n,,, = 0, and let m, ,. . . , ml be nonnegative integers such that iiI

mi

=

n,

f:

VljIni,

2sir1.

(2)

j=i

Corresponding to an increasing partition (rr ; . . . ; tl) of n points of (a, b) with I5j ( = mj(where JrjiJ is the cardinality of Zj), we consider boundary conditions where mj values of x”-‘) are specified at rj, in the sense that, at each t E ‘Sj, x “-l’(t) is given, j 5 i < j + m, where m is the multiplicity of t in rj. Boundary conditions of this form will be called right (ml ; . . . ; ml) focaf boundary conditions. (c) We say that (1) is right (ml ; . . .; ml) disfocal on (a, 6) provided there do not exist distinct solutions of (1) satisfying the same right (m, ; . . . ; m,) focal boundary conditions. Example 1. For the purposes of illustration, when n = 4, let I = 3 and consider the sequence (nj)j4_, where n, = 4, n2 = 3, n3 = 1, n4 = 0. One corresponding sequence (mj]j= 1 satisfying (2) is given by m, = 2, m2 = 1, m3 = 1. For such a sequence, a partition (rr ; 7, ; TV) = (tl , tz ; t, ; t4) 191

192

J. EHME and D.

HANKERSON

is increasing, if t, < t2 I t3 5 t4 or if t, = tz < t, 5 t4. Right (2; 1; 1) focal boundary conditions associated with the above increasing partitions of points consist of specifying x(t,), x(tZ), x’(t3), x”(t,), or x(tl), x’(&), x’(t3), x”(t,), respectively. Another sequence rmj]j3=1 satisfying (2) with respect to the above (nj]j4=, is given by m, = 3, m2 = 0, m3 = 1. For this sequence, a partition (zr ; r2; TV) = (tl, t,, t3;; tJ is increasing, if tI < t2 < t3 5 t4, or t, < t2 = t3 5 t4, or t, = t, < t3 2 t4, or tl = t, = t, --c t4. Then (3; 0; 1) focal conditions associated with these increasing partitions of points consist of specifying Wr), .%), x(Q, x”(Q, or NJ, x&), x’(Q, x”(Q, or x(tr), x’(Q, x&), x”(tJ, or x(tl), x’(t,), x”(&), x”(t,), respectively. In the case when I = 1, right (n) focal boundary value problems are commonly referred to as conjugate problems, whereas if 1 = n, right (1; . . . ; 1) focal boundary value problems have been referred to as right focal “point” boundary value problems. Also, if (1) is right disfocal in the sense of right focal point problems, then (1) is right (m, ; . . . ; m,) disfocal for all m, ; . . . ; m, . On the other hand, if (1) is right (ml ; . . .; ml) disfocal for some m, , . . . , m,, then (1) is disconjugate; that is, solutions of conjugate problems, when they exist, are unique. Given integers I and (nj] as described above, we assume: (C) equation (1) is right (ml ; . . . ; m,) disfocal for all integers m, , . . . , ml satisfying (2). In an unpublished result, Schrader [3] established the following compactness condition which we will need in Section 3. THEOREM 1. Assume that (A) and (B) are satisfied with respect to (l), and that (1) is disconjugate on (a, b). If lx,) is a sequence of solutions of (1) and [c, d] is a compact subinterval of (a, b) such that (xk] is uniformly bounded on [c, d], then there exists a subsequence (xkj] such that lx?) converges uniformly on [c, d], 0 5 i 5 n - 1. The goal of this paper is to establish “uniqueness implies existence” results for right (m,; . . . . m,) focal boundary value problems. Using appropriate uniqueness assumptions in place of (C), other authors have established similar results for various types of boundary conditions. If, instead of condition (C), it is assumed that all n-point conjugate problems have at most one solution, then results due to Hartman [4] and Klaasen [5] give existence for all conjugate boundary value problems. In Henderson [6], instead of (C), it is assumed that each right (1; . . . . 1) focal point problem has at most one solution, then it is shown that each right focal point problem has a unique solution. Goecke and Henderson [7] and Henderson [8] established a number of uniqueness and existence results for right (m, ; . . .; m,) focal problems for third order equations, under the assumption that rn; # 0, 1 5 i 5 1. Later, Henderson and McGwier [9] established existence for right (m, ; . . . ; ml) focal problems for n = 4, again under the assumption that mi # 0 for 1 5 i 5 1. Part of the motivation for considering these type of boundary value problems is due to a theorem from Muldowney [l] in which the disfocality condition (C) for linear homogeneous equations is characterized in terms of uniqueness of solutions of related two-point problems. Hankerson and Henderson [lo] used this relationship to establish optimal length intervals (in terms of Lipschitz coefficients) on which solutions of right (ml ; . . .; m,) boundary value problems are unique. This left the existence question open. In Section 2, we make some observations and develop some useful tools concerning right focal boundary value problems. In Section 3, we present the existence result.

Existence for right focal BVPs

193

2. PRELIMINARIES

Throughout the remainder of the paper, we assume that the integers I and (nj)~=l, are given and satisfy the conditions given in definition l(b). We will say that a set of boundary conditions is admissible if they correspond to some right (ml ; . . . ; m,) focal boundary conditions, where m,; ..,; ml satisfy (2). Similarly, we say that an increasing partition r = (ri ; . . . ; zI) of n points of (a, b) is admissible if the numbers m, = Iri I, 1 I i I I, satisfy (2). Although an admissible partition t only indicates which derivatives are specified at the points, if the values at the various points are understood (or if we are only interested in the form of the conditions) we will say that T represents the boundary conditions. We remark that this representation need not be unique; for example, the (3; 0; 1) boundary conditions of example 1 represented by r = (tl, t2, t,;; t3), t1 < t, < t,, can also be considered (2; 1; 1) boundary conditions represented by r = (tl , t2; tz; t3). We will consider the collection of I-tuples (ml ; . . .; WI,) as being ordered lexicographically; that is, (m; ; . . .; mj) > (m, ; . . .; ml) if there exists 1 CCj I I such that rn; = m, for 1 I i < j and rn; > mj. We say that T is in standard form if the right (m, ; . . .; ml) focal boundary conditions represented (ml = 1zi I) have been written with (m, ; . . . ; m,) largest. In the example above, r = (tl , I,, t2;; tJ is in standard form. The following lemma determines the structure of some boundary conditions and shows that certain boundary conditions are admissible. The technical manipulations discussed in the lemma will be used in the proof of the existence result. LEMMA 1. Let r = (r,; . . . . r,) represent an admissible set of boundary conditions with z in standard form, and set mj = (ri (. Assume that tl < ... < tk are the distinct points of 5, and let the numbers {Xj,x)and the indexing sets Aj, 1 5 j 5 k, be such that the boundary conditions are

lljlk. Then: 1. The boundary conditions at t, are conjugate-type; that is, A, = (0, . . .,j) for somej. 2. If )A11 = 1, then the boundary conditions at tz are at most one step from conjugate; that is, A2 = (0, 1, . . . . lA,l]\ljJ for some 0 ~j I (A21. 3. Let r; be obtained from zr by replacing the last occurrence of t, by t2. If we choose v -z 0 to be the first integer such that v $ AZ, then 5’ = (5; ; T, ; . . . ; z,) represents an admissible set of boundary conditions of the form x”‘(t.)

=

x.

J

x%)

AEAjj,

J,h,

=

A

Xl,h,

x’X’(t.) = x. J

1

J,i’

E

A,\(max Ail,

EAj,

2sjsk,

x’“‘(t,) = x,,,. (However, r’ is not necessarily in standard form.) 4. Assume that m, < n and let v 2 0 be the first integer such that v $ AZ. The boundary conditions x’X’(t.) = x. llj
J

J.h

’

x@)(t,) = x2,,, xYt,)

= &,x 9

A E A,\(max

AJ,

194

J. EHMEand D. HANKERSON

are admissible. These boundary conditions can be represented by T' = (7; ; ...; Ti) where 2; is obtained from TV by adding an instance of t, , and, if j is the largest integer such that tj # 0, T; is obtained from Zj by removing an instance of fk , and all other entries match entries in z. In particular, T’ represents right (m;; . ..; m;) focal boundary conditions with (m; ; . . . ; m;) > (m,; . . . . ml).

Proof. First, note that the requirement that t be in standard form implies that if t has multiplicity m in Tj, then t $ 5i for j < i s mint/, j + m). (The definition of an increasing partition implies that t $ ~~for j < i s min(l, j + m - 1). If j + m I I and t E rj+, , then it must be verified that 7’ obtained from T by “moving” a copy of t from TV+,,, to 7j leads to an admissible partition which represents the same boundary conditions, contradicting the fact that T is in standard form.) Also, the fact that (nj) is strictly decreasing and condition (2) imply that TljSTl-j+

1

and

m, + *.a + mj ?j.

Item 1. In particular, the statements above imply that ml 2 1 (and hence, tl E TJ, and that fl $ ‘cifor 1 < i s min(l, m + I J, where m is the multiplicity of tl in fl. If all f, s appear in TV, then the result holds. Suppose, on the contrary, that there exists a first j, m + 1 < j I 1, such that t, E Zj. Necessarily, TV= IZJ for 2 I i
Item 2. Suppose, on the contrary, that there exist smallest integers 0 I i < j < max A2 such that i, j $ AZ. Since t is in standard form, it follows that m, = i + 1, m2 = .a. = mi+l = 0 (if i > 0), mi+2 = j - i - 1, and mi+3 = *a* = mj+l = 0 (if j > i + 1). Hence, we have that m, + *a. + j, which is a contradiction. T?lj+l

=

Item 3. First, note that by item 1, all t, s in T appear in TV. Since the only difference between t and T’ is the replacement of the last tl in 21 by t, , to show that 7’ is increasing, it suffices to check the multiplicity condition on t2 in t; ; i.e. it must be verified that t, $ ti for 2 I i I mint/, m), where m is the multiplicity of t2 in 7;. By the remarks at the beginning of the proof, the requirement that 5 be in standard form (and v $ A,) implies that t2 appears in 51 exactly v times, and, if v > 0, t2 $ Ti for 2 5 i 5 min(l, v + 11. Hence, the multiplicity of t2 in t; is m = v + 1 and t2 $ 7i for 2 5 i 5 mini/, m]. It follows that 7’ is increasing. Finally, it is clear that 7’ is admissible since Ir;l = m,.

Item 4. As in the proof of item 3, we see that t2 appears in ‘51exactly v times, and t2 $ 'sifor 2 I i 5 min(l, v + 1). It follows that z’ is increasing. Since m, < n (i.e. the boundary conditions are not conjugate), it follows that j > 1, where j is the largest integer such that Tj # 0. We have m; = m, + l,mj= mj- l,andm:= m,fori$(l,j).Itiseasilyverifiedthatm~,...,m~satisfy condition (2).

Existence 3. THE

for right focal BVPs

EXISTENCE

195

RESULT

THEOREM 2. Let the integers 1 5 I I n and n = n, > . . . > n,,, = 0 be given, and assume that, with respect to (I), conditions (A)-(C) are satisfied. Then every right (ml ; . . . ; mr) focal boundary value problem for (1) has a unique solution.

Proof. Firstly, note that our assumptions guarantee that every conjugate problem has at most one solution. As mentioned earlier, this implies that every right (n) focal (i.e. conjugate) problem has a solution. The proof will have three levels of induction: (outer level) (middle level) (inner level)

number of distinct points, number of conditions at tl, the “order” of the problem in terms of (m, ; . . .; ml).

Assume that we have an admissible set of boundary conditions represented by 5 = (Ti ; . . . ; T,), and set mi = (ri (, 1 5 i I I. Let t, < . .. < tk be the distinct points in ‘5. If k = 1, then we have an initial value problem and the result holds. Assume, then, that k > 1 and that we can solve all admissible problems with less than k distinct points. Let the numbers {x~,~] and the indexing sets Aj be such that the boundary conditions are given by x@)(t) = x. J

?LEAjj,

J,h*

lsjsk.

If iA11 = 1, then by “fewer conditions at tl” we will mean a (k - I)-point problem. With this understanding, we assume by way of induction that we can solve every admissible k-point problem with fewer than IAil conditions at f1 (this is the “middle” induction hypothesis). To begin the inner induction, note that we can solve every right (n) focal (i.e. conjugate) boundary value problem. Hence, we assume that m, < n and that, among k-point problems with (Ail conditions at t,, we can solve every right (m; ; . . . ; m;) focal problem with (m; ; . . . ; m;) > (m,; . . . . ml) (this is the “inner” induction hypothesis). If T is not in standard form, then, by definition, if r’ = (5; ; . . .; 5;) represents the boundary conditions in standard form with m; = IT: I, it follows that (m; ; . . . ; m;) > (m, ; . . , ; m,). Hence, without loss of generality, we can assume that r is in standard form. Let p = max A,, let v L 0 be the first integer such that v $ A,, and let S be the set of numbers of the form xcp)(tI) where x is a solution of (1) satisfying x’%,)

=

X”‘(tj)

= Xj,X 9

x1,x,

A

E Al\t~l),

A

EAjT

2sj
Note that S is nonempty since these boundary conditions together with x’“‘(t,) = 0 are admissible by lemma 1 and have fewer conditions at t, . A standard argument using the Brouwer invariance of domain theorem shows that S is open (see, for example, [l 1I). We claim that S is also closed. Suppose, on the contrary, that there exists to E S\S. We consider the case that there exists a strictly increasing sequence (ri ] G S converging to r, . Let xi correspond to ri; i.e. Xi is the solution of (1) satisfying x94)

= x1,x,

X(‘)(tl)

=

x@‘(b)

J

A o Ai\l~l,

ri,

= x.

J.1'

A

EAjv

2sjsk.

196

J. EHMEand D. HANKERSON

It is clear that xi(t) < xi+i (t) on (ti , tl + E) for some E > 0. If there exists s E (ti , f2) such that Xi(S) = Xi+i(S), then Xi and Xi+l would be distinct solutions of an admissible problem, contradicting the disfocality assumption (C). Hence, Xi(t) < Xi+1 (t) for t E (ti , f2). Similarly, (-l)‘Xi(t) < (-l)“Xi+i(t) on (a, tt). By theorem 1, it follows that (Xi) is not bounded above on compact subintervals of (ti , f2), and ((-1)“Xi) is not bounded above on compact subintervals of (a, t1). Consider

the boundary

conditions Y%)

=

x1,x>

P)(h)

=

ro,

y’“‘&)

=

0,

Y”‘Ctj)

=

xj,X,

Y%J

= -%,A,

A

E

A,\bI,

A

E

Ajj,

2sjck,

I E A,\(max

A,].

conditions with From lemma 1, these are admissible right (rn; ; . . .; mj) focal boundary (m;; . . . . m;) > (rni ; . . . ; m,). Hence, there exists a solution y of (1) satisfying these boundary conditions. Since (Xi) is not bounded above on compact subintervals of (tl , t2), and l(-l)“xi] is not bounded above on compact subintervals of (a, tl), it follows that there exists i, and points Q < s1 < tl < s2 < tz such that Xio(Si) = Y(s~) for i = 1, 2. But then xi, and y both satisfy the boundary conditions i = 1,2, x(si) = YCsi)7 X%)

=

X”‘(tj)

x’Yt,)

x1,x,

A

E A,\b4,

= Xj,X,

~

EAj,

= Xk,X,

A E A,\(max

2sj
which contradicts the disfocality assumptions. Hence, S = R and the inner induction is complete. Note that this shows that we can solve all k-point problems with iA1 1 conditions at t, , and hence completes the “middle” induction step, showing that all admissible k-point problems have solutions. In turn, this completes the induction step on the number of distinct points, and the proof is complete. As mentioned in the introduction, boundary value problems of the type discussed in this paper are considered in [lo], where optimal length intervals (in terms of Lipschitz coefficients) are determined on which solutions of right (ml ; . . . ; 02,) boundary value problems are unique. We can apply theorem 2 to show that solutions actually exist. Specifically, we have the following theorem. THEOREM 3.

Assume

that

f satisfies the Lipschitz

If(t,Xlr

-.-,Xn)

-f(t,Y~v

condition ..*tYn)I

5

n C kiIxi - YiI i=l

on (a, b) x R”. Let

(3)

Existence

where yk is the smallest problem

positive

number

197

for right focal BVPs

such that there is a solution

x(t) of the boundary

value

9 1 X (‘-“(0) X (i-l&)

=

0,

liil?Z-k,

=

0,

hSirh+k-1,

with x(t) > 0 on (0, yk), or yk = +oo if no such solution exists. Then each of the right value problems for (1) has a unique solution, provided (m,; . . . . m,) focal boundary t, - c, < y. Moreover, this result is best possible for the class of all nth order differential equations satisfying the Lipschitz condition (3). REFERENCES 1. MULDOWNEY J., On invertibility

12, 368-384 (1981). 2. PETERSON A., A disfocality 741-752 (1982). 3. SCHRADER K., Uniqueness

of linear ordinary

function

for a nonlinear

implies existence

differential

boundary

ordinary

differential

for solutions

of nonlinear

value problems, equation, boundary

math. Analysis

SIAMJ.

.I. Math. 12,

Rocky Mountain value problems,

Abstracts Am.

math. Sot. 6, 235 (1985). families and interpolation 4. HARTMAN P., On n-parameter Trans. Am. math. Sot. 154, 201-226 (1971). for boundary value 5. KLAASEN G., Existence theorems

problems problems

for nonlinear for nth

order

ordinary

differential

differential

equations,

equations,

Rocky

Mountain J. Math. 3, 451-472 (1973). value problems for ordinary differential 6. HENDERSON J., Existence of solutions of right focal point boundary equations, Nonlinear Analysis 5, 989-1002 (1981). D. M. & HENDERSON J., Uniqueness of solutions of right focal problems for third order differential 7. GOECKE equations, Nonlinear Analysis 8, 253-259 (1984). equations, J. Math. 8. HENDERSON J ., Right (m, ; . . ; m,) focal boundary value problems for third order differential phys. Sci. 18, 405-413 (1984). Lipschitz equations, J. 9. HENDERSON J. & MCGWIER R. JR, Uniqueness, existence, and optimality for fourth-order

d$f. Eqns 67, 414-440 (1987). for boundary value problems for Lipschitz equations, J. diff. Eqns 10. HANKERSON D. & HENDERSON J., Optimality 77, 392-404 (1989). of solutions of boundary value problems with respect to boundary 11. EHME J. & HENDERSON J., Differentiation conditions, Applicable Analysis (to appear).