geometric existence proofs for nonlinear boundary value problems

Geometric Existence Proofs for Nonlinear Boundary Value ProblemsAuthor(s): Herbert W. HethcoteSource: SIAM Review, Vol. 14, No. 1 (Jan., 1972), pp. 121-128Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2028913 .

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SIAM REVIEW

Vol. 14, No. 1, January 1972

GEOMETRIC EXISTENCE PROOFS FOR NONLINEAR BOUNDARY VALUE PROBLEMS*

HERBERT W. HETHCOTEt

Abstract. Nonlinear ordinary differential equations in the forms x(n)(t) = f(t, x, x', * * , x(n-1)),

where n ? 2 and x'(t) = f(t, x) and x(t) is a vector, are considered when If I is bounded by a linear function in the space variables. Existence of solutions of boundary value problems is proved by analysis of the funnel of solutions. For linear boundary conditions, our proof is a more elementary and geometrically more intuitive proof of a known result due to Z. Opial [J. Differential Equations, 3 (1967), pp. 580-594]. For nonlinear boundary conditions, the results appear to be new. A geometric explanation of the above result of Opial is given.

Introduction. Solutions of some boundary value problems for nonlinear ordinary differential equations can be shown geometrically to exist by analysis of the funnel of solutions. This method of proof of existence of solutions is more elementary and geometrically more intuitive than use of the fixed-point theorems of Banach or Schauder. Even if existence of solutions must be proved rigorously in another way, insight into a boundary value problem can often be obtained by examining geometrically the flow of solutions from one boundary condition to the next boundary condition.

In ? 1 existence of solutions for nonlinear second order ordinary differential equations with linear boundary conditions at two points is proved by showing that the funnel of solutions starting at one boundary condition is intersected by the other boundary condition. In ? 2 similar conclusions which can be proved by the same method are presented for higher order differential equations, for systems of differential equations, and for nonlinear boundary conditions.

The results in ?? 1 and 2 are compared with theorems proved by other methods in ? 3. This paper is partly expository since the conclusions obtained here for problems with linear boundary conditions are contained in a very general result of Opial [6], which is proved by making a careful choice of topologies and then using the Schauder fixed-point theorem. In ? 4 geometrical interpretations of this important result of Opial are given.

1. Existence theorems for second order equations with linear boundary conditions. The principal tool used in our proof is the lemma below which is a corollary to the Kneser-Hukuhara funnel theorem [3, p. 22]. This funnel theorem has been used in the proofs of other results for boundary value problems (e.g., [2], [9]).

LEMMA. Let f(t, x) be bounded and continuous in S: [a, b] x Rn where n is the dimension of the vector x, and let X1 be the set of values at t = t1 of all solutions of x'(t) = f(t, x) which have initial values in the set XO at time t = to. If XO is compact and connected, then X1 is compact and connected.

* Received by the editors April 26, 1971. t Department of Mathematics, University of Iowa, Iowa City, Iowa 52240.

121



122 HERBERT W. HETHCOTE

We first consider the two-point boundary value problem for nonlinear second order equations:

(1) x"(t) - f(t, x, x') = 0,

(2) alx(a) + a2x'(a) = X, la1l + 1a21 =# 0,

(3) blx(b) + b2x'(b) = A1, lbll + Ib21 =A 0-

THEOREM 1. If f(t, x, y) is bounded and continuous in the region S: [a, b] x R2 and

(4) A = a1b2- a2b1 + a1bl(b - a) =# O,

then there exists at least one solution of (1), (2), (3). Moreover, these solutions satisfy

Ix(t)I ? D1 + D2(b - a)M,

Ix'(t)l < D3 + D4(b - a)M,

where M is a bound on If I in S and the Di are constants which involve a, b, a,, a2,

b1, b2, oc, ,B and A. Proof The system of two one-dimensional equations corresponding to (1)

with y = x' is

(6) v'(t) = ( ) = (ft( )) = F(t, v).

Consider the vector initial condition

(7) v(a) = c = . C2

Since F(t, v) is continuous, at least one local solution of(6), (7) exists. These solutions satisfy

t

y(t) = C2 + ,ff(s, x(s), y(s)) ds, (8)

x(t) = C1 + C2(t- a) + ff(s, x(s), y(s)) ds dr.

If v(t) were not continuable to b, then v(t) would be unbounded [3, p. 16], but this contradicts (8) since f is bounded. We cannot apply the lemma yet since the first component of F(t, v) is unbounded in S.

Let P(n) be the square region IXI _ n, IyI _ n, and let M be a bound on If I in S. From (8) we see that c E P(n) implies

(t,xIY)eQn = [a,b] x P(n + (b - a)(M + n) + (b - a)2M).

Let Fn(t, v) = F(t, v) for (t, v) E Qn, and let Fn(t, v) be a bounded continuous extension ofF in S - Qn. This could be done by defining the value of Fn at a point (t1, v1) in S - Qn as the value of F at that point on aQn where a line through (t1, 0) and (t1, v1) intersects aQn.



GEOMETRIC EXISTENCE PROOFS 123

Let Xa(n) be the segment of the line (2) in P(n). Let Xb(n) be the set of all solutions evaluated at t = b of (6) with F replaced by Fn and (7) with c E Xa(n). Since Fn is bounded and continuous in S and Xa(n) is compact and connected, the image Xb(n) is also compact and connected by the lemma. Since c E Xa(n), then (t, v) E Qn and Fn = F. Consequently, the set Xb(n) is also the image of Xa(n) under the differential equation (6). Clearly Xb(n) c Xb(n + 1) and Xb = Un= 1 Xb(n) is connected.

We find by using (8), (2), and I fi ? M that

(9) lalx(b) + [a2 - al(b - a)]y(b)-oel <-

where

(10) = [lall(b - a) + 1a2l](b - a)M.

Thus Xb, which is the image of the line (2) under the differential equation (1), is contained in the region (9) which is bounded by two parallel lines. Moreover, from (8) we see that Xb extends to infinity in both directions in the parallel slit region (9) as we go to infinity in both directions on the line (2). Because of the determinant condition (4), the line (3) will intersect the parallel lines bounding the region (9) and, consequently, will intersect Xb. Hence solutions of (1), (2), (3) exist.

These solutions at t = b lie on the segment of the line (3) in the parallel slit region (9). Points on this line segment have coordinates

x(b) = (a2f/ - alf(b - a) - b2yc - b2y)/A,

y(b) = x'(b) = (b1 oc - al + b17)/A.

From (1) we find that ob ob

x(t) = x(b) + x'(b)(t - b) + J J'f(s, x(s), x'(s)) ds dr,

(12) b

y(t) = x'(t) = x'(b) - f (s, x(s), x'(s)) ds.

Using If I ' M, (11) and (12), we see that

x(t) _ a2f3 - alf(b - a) - b2xC + (t - b)(bloc - a,l) A

(13) < [Ib2I + lbll(b - a)]y + (b -a)2M - \ \ A

x blot- a,1f< Ib1Iy b-aM l()- A A ?baM

The proof is completed by noting that (5) follows from (13). We remark that if line (3) is contained in the region (9), it could intersect Xb

even if condition (4) is not satisfied. Thus condition (4) is sufficient but not necessary. The same determinant condition (4) would be obtained if we proved the theorem by going from b to a instead of from a to b. Our proof is related geometrically to the numerical method of shooting to find an approximate solution of a boundary




value problem [5]. Indeed, points on the segment of the line (3) in the parallel slit region (9) are appropriate starting points for shooting from t = b to t = a. Clearly, appropriate starting points at t = a would lie on the segment of the line (2) in the parallel slit region determined by transporting the line (3) back to t = a by means of (12).

The f(t, x, x') term can be considered as a perturbation term in the differential equation (1) since the image Xb of (2) under (1) is only a perturbation of the image of (2) under x" = 0. In the next theorem it is shown that f is still a perturbation term under a weaker condition. The proof uses the technique of assuming the solution is in a certain region and then showing that it stays in that region if that region is sufficiently large.

THEOREM 2. Let f(t, x, y) be continuous and satisfy

m (14) If(t,x,y)I _ K1 + K21XI' + K3IyI + E Lijxjyj

in S: [a, b] x R2, where , i, I, qi, Ki, Li and m are positive constants. Let 0 = max {I, 1, Ci + ,ji}. If (4) is satisfied and either 0 < 1 or b - a is sufficiently small when 0 = 1, then there exists at least one solution of (1), (2), (3).

Proof Consider the region Sn = [a, b] x [-n, n] x [-n, n]. Let fn = f in Sn and let fn be the bounded continuous extension of f in S - Sn constructed as in the proof of Theorem 1. If K = K2 + K3 + MU1 Li, then I fn(t, x, y)l < K1 + Kno in S. By Theorem 1, there exists at least one solution of (2), (3) and (1) with f replaced by fn. Moreover, these solutions satisfy

Ix(t)l < D1 + D2(b - a)(K1 + Kn0),

Ix'(t)l ? D3 + D4(b - a)(K1 + Kn0).

If 0 < 1, then the right-hand sides of (15) are less than n for n sufficiently large. If 0 = 1, the right-hand sides of (15) are less than n for n sufficiently large if b - a is sufficiently small; that is,

(16) Dj(b - a)K < 1

for j = 2 and 4. Thus foX n sufficiently large, x(t) and x'(t) are in Sn so that fn = f and x(t) is a solution of (1), (2), (3).

To determine in a particular problem with 0 = 1 how small b - a must be to obtain existence, the conditions (16) above could be improved by using (13) instead of (5) and by letting K be the sum of those coefficients in (14) corresponding to first powers. For certain problems better a priori estimates than (13) can be obtained by considering the differential equations x" + M = 0 with the boundary conditions (2), (3).

2. Existence theorems for other equations and boundary conditions. The methods of proof of Theorems 1 and 2 can be used to prove similar theorems for the differential equation X(n)(t) = f(t, x, x', * , I X(n- 1)) of order n greater than two when n - 1 linear boundary conditions Ax(a) = o are given at t = a and one linear boundary condition Bx(b) = ,B is given at t = b. Geometrically, n - 1 linear boundary conditions determine a straight line in Euclidean n space which is




transported by the differential equation to a connected set bounded by n - 1 sets of parallel hyperplanes at t = b if f is only a perturbation term. If the determinant condition det [A*, B] =# 0 is satisfied, where

* -( 1)(b - a)' j= k k

k = O

then the hyperplane corresponding to the one boundary condition at t = b intersects the image of the boundary condition at t = a; and, consequently, at least one solution exists. Existence of solutions can also be shown for problems with one boundary condition at t = a and n - 1 boundary conditions at t = b by merely reversing the positive time direction.

Under the hypotheses of Hukuhara's theorem, the cross sections of solution funnels need not be simply connected [7]. Indeed, the question of which conditions on the differential equation imply that the cross sections of solution funnels are simply connected seems to be open. Consequently, the method of proof of Theorem 1 does not work if one of the boundary conditions is not a line since the image of other surfaces could be connected and still have a hole at the origin through which the surface defined by the other condition could pass.

The methods of proof of Theorems 1 and 2 can be used to prove similar theorems for the nth order system x' = f(t, x). If the boundary conditions are Ax(a) = o and Bx(b) = ,B, where A is an (n - 1) x n matrix of rank n - 1 and B is a row vector, then the determinant condition is det [A, B] =A 0. The methods could also be used on a system with a linear part and a nonlinear perturbation part as discussed in ? 4.

If (3) is replaced by the nonlinear boundary condition x'(b) = b1 sin (x(b) + c) and condition (4) is replaced by a, =A 0, then Theorems 1 and 2 can be proved by the same methods since the image of (2) will intersect the sine curve unless the image is parallel to the x-axis. If (2) is replaced by x'(a) = x3(a) and condition (4) is removed, then Theorems 1 and 2 can be proved as before since the image at t = a of the line (3) will intersect x'(a) = x3(a). For given separated nonlinear boundary conditions one can determine if the method of proof will work by graphing boundary conditions and images of boundary conditions.

3. Comparisons with other results. If f(t, x, y) is Lipschitz continuous in x and y, then it satisfies the growth condition (14) with 0 = 1 and Theorem 2 implies existence if the interval b - a is sufficiently small. This corollary of Theorem 2 is related to the well-known existence and uniqueness theorems which assume f is Lipschitz continuous and b - a is sufficiently small (see [1, p. 34]). The proofs of these theorems use Picard's method of iteration which is equivalent to using the Banach contraction mapping theorem. The proof of Theorem 2 indicates why the restriction that the interval be sufficiently small is needed for linear growth (0 = 1) off and is not needed for sublinear growth (0 < 1).

The results obtained here contain some results which are proved by use of the Schauder fixed-point theorem. For example, Theorem 1 contains a theorem due to Scorza-Dragoni [8] which deals with (1), (2),(3) if a = a2 2 = =o = ,B = 0. Theorem 2 contains a theorem due to Jackson [4, p. 310] which considers (1), (2), (3) ifa2 = b2 = 0 and If(t,x,y)I < K1 + K2IxI"2.




If f(t, x, y) satisfies the growth condition (14) in Theorem 2 with 0 < 1, then it also satisfies

I b (17) lim inf - fJc(t) dt = 0,

C-o 00 C

where

ftc(t) = sup {lf(t, x, y)l: IXI + IYI < c}.

Consequently, the result of Theorem 2 with 0 < 1 follows from a result of Opial [6, p. 588] whose proof uses the Schauder fixed-point theorem. By using

rb rb JIlf (s, x(s), x'(s))j ds < J Bc(s) ds

for lxI + IYI ? c throughout our proofs instead of

rb

I f(s, x(s), x'(s))l ds < (b - a)M,

we could replace condition (14) in Theorem 2 by condition (17). The result of Theorem 2 with 0 = 1 follows from a remark in the same paper of Opial [6, p. 592]. Our conclusions on problems with nonlinear boundary conditions are not covered in Opial's paper [6] since he considers only linear problems. For existence of solutions in particular problems with 0 = 1, the allowable length b - a computed using the remark following Theorem 2 will be larger than the length b - a allowed using Opial's restriction [6, p. 592].

4. Geometric interpretations. In this section we first present some develop- ments from Opial's paper [6] and then give a geometric explanation of them. Consider

(18) x'(t) = A(t)x + f(t, x).

A linear problem (or generalized boundary value problem) consists of (18) and

(19) Tx(t) =r,

where T is a linear continuous map from C'(A) to R' and t E A c R. If

(20) x'(t) = A(t)x,

which is the linear part of (18), has a fundamental solution V(t) such that V(0) = I, then

rt (21) x(t) = V(t)c + V(t) V- '(s)f(s, x(s)) ds

is a solution of (18). IffA I A(t)l dt < oo and f(t, x) satisfies condition (17), then Opial [6, p. 588] shows that solutions of (18), (19) exist if the problem (20) and

(22) Tx = 0

have only the trivial solution. This latter condition on (20), (22) is shown to be equivalent to det T'V(t) :A 0, where T'V(t) is the result of T operating on each




column of V(t). We remark that the methods of proof of Theorems 1 and 2 could be used on (18) if A(t) is continuous. Opial actually lets A in (18) be a function of both t and x. The principal resulting change is that the theorem hypotheses must hold for a set of A's instead of for a single A.

Now we give a geometrical explanation using the boundary value problem (1), (2), (3) as a first example. Condition (17) implies thatf is a perturbation term in (18) so that solutions of (18) behave essentially like solutions of (20) and the linear problem (19), (20) can be considered instead of the problem (18), (19). The fundamental matrix V(t) indicates the rotating action of (20) as a function of time. For example, the hyperplane Bx(to) = oc, where B is a row vector, is transported to the hyperplane

BV(to)V'(t )x(t1) =Lx

attimet = t1. If (1) is converted into the vector form (18), then the fundamental solution of

(20) is

-1 t- V(t)=[ j,

and (21) corresponds to (8). The boundary condition (19) corresponding to (2), (3) is T(x, y) = (o, f,), where T:(x, y) -* (a,x(a) + a2y(a), b1x(b) + b2y(b)). If no solutioni of (19), (20) corresponding to (1), (2), (3) exists, then the image of line (2) under (20) at t = b is a line parallel to the line (3). If these lines are parallel, then the lines at t = b corresponding to (22) coincide and the homogeneous problem (20), (22) has many nontrivial solutions. Thus the problem (20), (22) has no nontrivial solutions implies that the lines at t = b are not parallel, and hence (19), (20) must have a solution corresponding to the intersection point of the lines. Opial's condition det T'V 0 0 states that the combination of the boundary condition and the rotating action of (20) must yield a consistent set of equations so that the problem (20), (22) has only the trivial solution. In the problem (1), (2), (3), the condition

det TnV = al ala + a2 Ib, blb + b2 =1

is equivalent to the condition (4) in Theorem 1. As a second example, we consider a three-point boundary value problem:

(23) x"' -f(t,x,x',x") = 0,

(24) a1x(a) + a2x'(a) + a3x"(a) = ,

(25) b1x(b) + b2x'(b) + b3x"(b) =,B,

(26) C1X(C) + C2X'(C) + C3X"(C) = y.

The plane (24) at t a is transported by x"' = 0 to a plane at t = c given by

F27) alx() + +W +1(c - alx"(c)= )2] (27) alx(c) + [a2 al(c - a)]x'(c) + [a3 a a2c - a) + 2l( j a x1(c) = cc.




Similarly the plane (25) at t = b is transported by x"' = 0 to a plane at t = c given by

(28) b1x(c) + [b2- b1(c - b)]x'(c) + b3 -b2(c- b) + b 2 (c = bt.

The condition that the planes (26), (27), (28) have a unique intersection point is

(c - a)2-

a, a,-(c-a)al a3-a2(c-a) + a 2

(29) det b b2 -(c- b)b b3 - b2(c- b) +

b1, )2

)

C1 C2 C3

As before, if f is only a perturbation term in (23), then (29) implies that there exists at least one solution of (23)-(26). Using the linear map T and fundamental solution V corresponding to (23)-(26), we find that Opial's condition for existence is

a, ala + a2 al2 + a2a + a3

12

(30) det T'V= det b, blb + b2 b ? 2 + b2b + b3 0 , 2

c1 ClC + C2 Cl2 + C2C + C3

which can be verified to be equivalent to (29).

REFERENCES

[1] P. B. BAILEY, L. F. SHAMPINE AND P. E. WALTMAN, Nonlinear Two Point Boundary Value Problems, Academic Press, New York, 1968.

[2] J. W. BEBERNES AND R. WILHELMSEN, A technique for solving two-dimensional boundary value problems, SIAM J. Appl. Math., 17 (1969), pp. 1060-1064.

[3] W. A. COPPEL, Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965. [4] L. K. JACKSON, Subfunctions and second-order ordinary differential inequalities, Advances in Math.,

2 (1968), pp. 307-363. [5] H. B. KELLER, Numerical Methods for Two-Point Boundary Value Problems, Blaisdell, Waltham,

Mass., 1968. [6] Z. OPIAL, Linear problems for systems of nonlinear differential equations, J. Differential Equations,

3 (1967), pp. 580-594. [7] C. C. PUGH, Cross-sections of solution funnels, Bull. Amer. Math. Soc., 70 (1964), pp. 580-583. [8] G. SCORZA-DRAGONI, Sul problema dei valori ai limiti per i systemi di equazioni differenziali del

secondo ordine, Boll. Un. Mat. Ital., 14 (1935), pp. 225-230. [9] P. WALTMAN, Existence and uniqueness of solutions to a nonlinear boundary value problem, J. Math.

Mech., 18 (1968), pp. 585-586.



geometric existence proofs for nonlinear boundary value problems

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