Differential Inequalities and Invariance

§10. The Initial Value Problem for a System of First Order 111

(b) If y is a solution of (10) in the interval J (a, /3), then z(t) := y(t + c) is a solution in the interval = (a^— c,/3 — c).

(c) Phase Space and Trajectories (Orbits). For autonomous systems, the space is called the phase space. The curve C

y(J) :=

{y(t) t J} C C in the phase space generated by a solution y on a maximal interval of existence .1 = (a,b) is called the trajectory or the orbit of y; cf. A.I for the definition of curves. If z is another solution with z(to) E C and z(to) = y(tj) for some t1 E J, then =

y(ti

^— t0 + t) is also a solution by (b), and the relation z(to) = implies z = ^Therefore, the trajectories of y and z coincide.

If the "half trajectory" y( [c, b)) with c E J is contained in a compact subset of C, then b ^{00; a}corresponding statement holds for the interval (a, c}.

(d) The Phase Portrait. Two trajectories are either disjoint or identical.

Each point of C belongs to exactly one trajectory. The collection of all trajec- tories is called the phase portrait of the differential equation; cf. 3.V.

(e) The two differential equations y' = f(y) and y' Af(y) (with A 0)

generate the same trajectories, hence the same phase portraits (with the same orientation if A > 0).

(f) Periodic Solutions. If y is a solution and y(to) =

y(ti)

for some to then y is periodic with period p = ^{to —}ti. This follows from (c), with z = y.

The maximal interval of existence is R. A nonconstant, continuous, periodic function has a smallest period T> 0, also known as the minimal period.

(g) Critical Points. A point a E C is called a critical point (also a stationary point or equilibrium point) of f if f(a) = 0. If a is a critical point of f, then y(t) a is a solution in IR. The corresponding orbit is the singleton {a}.

(h) If the solution y exists for t to and if a urn y(t) exists and belongs t-.oo

to C, then a is a critical point; i.e. f(a) = ^0.

Proof of(h). Suppose is a real-valued C'-function and Em = a 0.

t-+oo

Ifa> 0, then for large t, > a/2. It follows that lim =^oo. Similarly,

a <

⁰ implies = —00. Now the hypotheses imply that limy'(t) = f(a). The preceding argument applied to the components yk(t) of y shows that

]imy'(t)=O,i.e.,f(a)=O. I

XII. Monotonicity and Quasimonotonicity. The function f(x, z) DC

^—,

r is

said to be increasing in y if y z implies f(x,y) f(x,z), and quasimonotone increasing in y if is increasing in for i

j, more

exactly, if for i= 1, ... ^,

y z,

(x,y),(x,z) ED implies <f2(x,z).

An n x n matrix C = (Cj3) is said to be positive if 0 for all components, and essentially positive if 0 for i j. The same terminology is used for matrices C(x) = (Cj3(X)). Inconnection with matrix products, vectors y, u,...

are always assumed to be column vectors, y = (yi,.

It is easily seen that a linear function f(x, y) =C(x)y is quasimonotone if and only if C(x) is essentially positive. If D c is open and convex and f and ôf/ôy belong to 0(D), then f is quasimonotone increasing if and only if the Jacobian af/ay is essentially positive. Without convexity, this is not true in general.

If the matrix 0(x) + Al is positive for some A > 0, then C(x) is obviously essentially positive. Conversely, if 0(x) is essentially positive and if the diagonal elements of C are bounded below, then C + Al is positive for large A. There is a similar relation between monotone and quasimonotone functions. Again it is obvious that f(x, z) is quasimonotone increasing when f(x, z) + Az is increasing for some A > 0. Conversely, if f is quasimonotone increasing and satisfies a Lipschitz condition in y, then f(x, y) + Ay is increasing in y for large A. In short, a smooth function is quasimonotone increasing if it becomes monotone increasing when a large multiple of the identity is added.

The propositions that follow can be summarized in a general

Principle. The theorems in § 9 for a single equation carry over to systems if and only if the right-hand side f(x, y) is quasimonotone increasing in y.

We use the notation Pv =v'(x) ^— f(x,v) for the defect.

Comparison Theorem. Assume that f

: D is quasimonotone in- creasing and that v, w are differentiable in J = + a]. Then

(a) v(e)

<w in J.

(b) If f(x, y) satisfies a local Lipschitz condition in y, then v(e) w(e),

Pv <

Pw

in J implies v <

w in J; moreover, the index set splits into two subsets and such that for

iEa:

^v2 < in (11)

jEf3:

^v3 = ^w2 in and in (12)

where > 0.

A simple consequence is

(c) M. Hirsch's Theorem. Assume that f(y) E

C' has an essentially positive and irreducible Jacobian c9f(y)/ay. Then two solutions v, w of y' = f(y) with v(e) w(e), v(e) w(e) satisfy v(x) <w(x) for _>

§10. The Initial Value Problem for a System of First Order 113

Proof. If (a) is false, then there is a first point x0 > such that v(x) <w(x) for x xo and v2(x0) = for some index It follows easily that

(apply the reasoning of 9.1 to v2 and w2). On the other hand, Pv < Pw

and v w,

at x xo, implies by quasimonotonicity

at

x=x0,

which is a contradiction. Here and below, the argument x or x0 is suppressed.

(b) Assume that f satisfies aLipschitz condition If(x,w+z)—f(x,w)I (maximum norm) and let p = h = E(p,.. .,p), e> ^0. Then

h' =

2Lh> f(w + h) —f(w) P(w

+ h) > Pw.

Hence, by (a), v < w+ h, which gives v w, since E > 0is arbitrary.

Last part of (b): It follows from the local Lipschitz condition that for some L> 0, f(x, z)+Lz is (componentwise) increasing in z as long as v(t) ^z w(t).

From (b) we get u =w ^—v 0, and furthermore,

= u' + Lu =f(w)+ Pw — f(v)^— Pv+ L(w —v)

>

f(w)+Lw—(f(v)+Lv)0;

i.e., each component of is increasing. This proves the last part.

I

The proof of (c) is an exercise. Hints: f is quasimonotone, hence v w.

Assume is nonempty, and write v = (va, Then

<wa, va =

implies vp) v13), and equality is excluded if ya)/'9ya ⁰ is nonzero which is a contradiction.

Sub- and Supersolutions. Maximal and Minimal Solutions. On

the basis of the preceding theorem, sub- and supersolutions for the initial value problem (6) can be defined by v' <f(x, v), and w' > f(x,w), w(e) >

(or the same with in case (b)) exactly as in 9.IV.

Furthermore, in the case where f is continuous and quasimonotone increas- ing, one can construct a maximal solution and a minimal solution of (6).

For those solutions the propositions

(d) v

f(x,

in J in J

hold. The proofs of 9.VI and 9.VIII carry over.

Our program announced in the general "principle" has been carried out now, except for the "only if" part. The fact that quasimonotonicity is a necessary prerequisite for the validity of the comparison theorem is proved in Redlieffer and Walter (1975). The theory that has been developed above goes back to M. Muller (1926, 1927) and E. Kamke (1932). In the next section we prove a theorem on differential inequalities for general systems, that also is due to M. Muller (1927). It is of a different nature inasmuch as it requires an upper bound and a lower bound simultaneously.

XIII. M. Muller's Theorem for Arbitrary Systems.

Let f(x, y) D satisfy a local Lipschitz condition in y. Let y, v, w: J

=

^{e+a] —p} be differentiable, v w in J,

w(e), y' = f(x,y),

and

< for all z E such that v(x) z w(x),

z v(x)

z w(x), =

fori=1,...,n.

Then in J.

Remark. The differential inequalities, which look quite complicated at first sight, have a simple geometric meaning. We use the following notation for

intervals in RTh:

where and

The boundary of this interval consists of 2n "faces" and defined by z E I,

= or = ^b2

(i =

1,... ,n). In the two-dimensional case lisa rectangle with sides parallel to the axes, a is the left lower and b the right upper corner point, and I' are the left and right vertical sides, while 12 and j2 ^are the lower and upper horizontal sides.

Using this notation, the differential inequality for v2 can be written in the form

f1(x,z) for z E

where 1(x) = [v(x),w(x)]. One may carry the abbreviation a step further by (i) defining inequalities between a real number s and a set A CR,

forall

aEA;

and (ii) using a familiar notation f2(x,B) =

z E

B}. Then the differential inequalities in Miller's theorem XIII can be written as

and 1(x) = [v(x),w(x)].

Proof. We again use a two-step method. Assume first that v, w satisfy strict inequalities. If the conclusion is false, there is a maximal x0 > a such

that v <

y < w in [a,xo) and, e.g., = y2(xo). Hence at xO.

On the other hand, z = y(xo)

satisfies v(xo) S z w(x0),

= and hence < y(xo)) = which is a contradiction. In the second step one may assume that F satisfies (for the arguments involved) a Lipschitz condition f(x,y) —

f(x,z)I

^{—zi, where} is the maximum norm. Then the case of weak inequalities is reduced to the case treated above by considering w6 =w+ Eh, v^—&h, where h = (p, .. . ,p), p = ^U

10. The Initial Value Problem for a System of First Order 115

Invariant Intervais. Let us consider the important special case where the functions v, w are constant. Then the assumptions in Muller's theorem are

and

where I=[v,w]

(i

=

1,...,n), and the conclusion is y(x) e^{I for} x E J.

These inequalities signify that on the boundary of I the vector field I points into I. For example, in the case n = ²

we have fi 0 on the left side and

11 <0 on the right side, and similarly 12 0 on the lower side and 0 on the upper side. When this is true, then I implies y(x) E I for x > This fact is expressed by saying that I ^is an invariant interval (invariant rectangle in the case n 2). Thus a theorem on invariant intervals is a very special case of Muller's theorem.

XIV.

The Case n =

2. Consider problem (1) for y = _(y1,y2), i.e.,

= fi(x,y1,y2),= f2(x,yl,y2), = =712, (13)

and assume for simplicity that f(x, y) =

(fi,

f2) ^is locally Lipschitz continuous in y. We discuss several monotonicity assumptions with respect to the

the two preceding sections. The behavior of Ii in the variable y2 and of 12 in the variable Yi is crucial. Let y be the (unique) solution of (13).

(a) Case (I, I). If f1(t,yi,y2) ^isincreasing in Y2 andf2(t,yi,y2) is^increas- ing in then f is quasimonotone increasing. In this case,

v'1 f1(x,vi,v2), f2(x,vi,v2) and ^<711, implies v1 yi and v2 112 in J.

(b) Case (D, D). Let fi be decreasing in 112 and f2 be decreasing in Then f is not quasimonotone, but the equivalent system for = ^{—Yi and 112,}

= —fi(x,—Vi,y2), ih =f2(x,—yl,y2),

is quasimonotone. Therefore, the statement given in (a) holds for this system.

In terms of the original system it reads as follows:

v'1

fi(x,vi,v2),

f2(x,vi,v2),

v2 a case v (v1, v2) is sometimes called an upper—lower solution.

(c) Case (D, I). If fi is decreasing in 112 and 12 is increasing in 111, it is

impossibleto enforce quasimonotonicity by a simple transformation. We employ Muller's theorem XIII. The conditions for v, w are v(e) ^ç

and

fi(x,vi,w2), fi(x,wi,vz),

Y2

S2

y1

Fitzllugh—Nagumo equations

f2(x,vi,v2), f2(X,Wl,W2),

and the conclusion is v <y w.

Example. The FitzHugh—Nagumo Equations. These equations rep- resent a simple model for the mechanism underlying signal transmission in neu- rons. Usually, these equations are given in the form

u'=sv—yu, v'=f(v)—u

where o and 'y are positive constants and f(v) =—v(v—tt)(v—1) with 0 < < 1 (typically). Here u(t) represents the density of a chemical substance and v(t) an electric potential depending on time t. For further information, see Jones and Sleeman (1983). With the notation v =yi,u = Y2 we have

(d) = fl(yl,y2) =

f(yi)

^{— Y2,} =f2(Y1,Y2) = ^—

Since fi is decreasing in 112 and f2 increasing in the conditions from (c) for a subfunction v = (v1,1)2) and a superfunction w = (Wi,w2) are

v4 f(wi)—v2,

<CYV1 7V2, 5W1 —7W2.

In the figure the curves S1 = 0 and S2 : =0 are drawn, and the side of the curve where the component is positive or negative is indicated. A rectangle [A, C] (A <C) with corners A, B, C, D is invariant if v(t) A and w(t) C satisfy the preceding inequalities; i.e. if

A is below S2, C is above S2, D is below Si, B is above Si;

Si _{Y2 =1(11')} = —yi(yi^{— —1)}

S2: y2=byi,b=cr[y>0

Si

§10. The Initial Value Problem for a S1ystem of First Order 117

for example, on the side AD we must have f2 0, which is guaranteed if A is below S2.

The straight line 52 is given by y2 = ^byi, b = o/-y. If we choose A and

Con S2, i.e., A

⁼ (a,ba),

C =

^(/3,b/3)

with a <0 <f3, then B

^{= (/3,ba),}

D = (a,b/3), and the two remaining conditions regarding S1 read

f(a)b13,

If—a =$ is a large positive number, then these inequalities are satisfied:

There are arbitrarily large invariant rectangles, which implies that all solu- tions of the FitzHugh—Nagumo equations exist for all t 0.

(e) Exercise. Find sub- and superfunctions depending on t (shrinking rect- angles). Assume that A and C, that is, v(t) and w(t), are on the line

Y2 = ^where a =^b + ^{(e >} 0)

(this line is slightly steeper than S2). Use the ansatz v(t) = ^(1,a)ae_S(t_c),

w(t) =

(a < 0 <f3, 5 > 0) and give conditions on f such that v, w satisfy the inequalities in (d) for t c.

(f) Show that there exist arbitrarily large invariant rectangles if the function f in the FitzHugh—Nagumo system satisfies f(s)/s ^—oo as s ±00.

XV.

Invariant Sets in RTh. Tangent Condition. A set M C r is

said to be (positively) invariant orflow-invariant ^with respect to the system

y' =

f(x,y) if for any solution y, y(a) e lvi implies y(x) E lvi ^for

x > a (as

long as the solution exists). The basic hypothesis for invariance is a tangent condition that roughly states that at a boundary point z E the vector f(x, z) is either tangent to lvi or points into the interior of lvi. The problem is to find a formulation that applies to arbitrary sets lvi and does not use any- smoothness of the boundary of lvi. The tangent condition is given in two forms.

It is assumed that f is defined in ^{J x M, J = [a,b]:}

h.—O+h

(z+hf(x,z),M)=0

^for

z EM, xE J,

^(Td)

(n(z),f(x,z))

0

for xE J,

z E Wi/i, where n(z) is the outer normal to lvi at z.

Here, dist (z, M) ^denotes the distance from z to lvi, ^and (y, z) is the familiar inner product y1zi + The vector n(z) 0 is said to be an outer normal to lvi ^at z E if the open ball B with center at z + n(z) and radius n(z)I has no point in common with lvi. Geometrically speaking, the ball B

touches lvi at z, z E OM fl B. The index d or i in the tangent condition

indicates that the definition employs the distance or the inner product in respectively. Naturally, (Td) istrue for z E mt M. First we give without proof some relations between these two conditions.

(a) (Ta) implies (Ti).

Tangent conditions

(b) If f is continuous in J x M, then (Ti) implies (Td), and (Ta) even holds uniformly in compact subsets of J x M.

XVII. Invariance Theorem

(with Uniqueness Condition).

The closed

set M C

^is flow-invariant with respect to the differential equation y' = f(x,y) if f satvsfles the tangent condition (T2) or (Td) and a condition of Lip- schitz type

(Yl —y2,f(x,y1) —f(x,y2)) (12)

Proof. Let y be a solution of y' =

f(x,

y), let y(a) E M and p(x) =

dlist (y(x), M). Assume that y(s)

M, i.e., p(s) >

0

for some s > a, and let z E ÔM be such that p(s) =

y(s) _{— zi. The} function cr(z) = y(z) ^{— zi} satisfies p(x) ( o(x),

p(s) =

^cr(s), and hence a'(s) is a Dini derivative). Note that o• is differentiable near s and

= = (y(x) ^— z,y'(x)) = (y(x)^— z,f(z,y(x)).

Now, n(z) =y(s)—zis an outer normal to M at z, and (Ti) implies (n(z), f(s, z)) 0. Using this inequality, we obtain for t = s,

0_c.' = (n(z),f(s,y(s))

(n(z),f(s,y) —

f(s,z))

Hence Lp(s). Since s was arbitrary, we have Lp whenever p> 0. There exists an interval [b, c] such that p(b) = 0and p> 0 in (b, c]. It follows from Theorem 9.VIII that p = ⁰ in [b, c], which is a contradiction.

I

In conclusion we state without proof another

Condition (Td) Condition (T1)

§10. The Initial Value Problem for a System of First Order 119

Invariance Theorem (Existence). Let lvi c

be closed, f(x, y) : a] x M bounded and continuous, and assume that a tangent condition (T2)

or (Td) ^holds. Then, for any E M, the initial value problem (6) has a solution y ^such

thaty(x) E Mfor

In particular, if lvi is compact and if f is continuous in oo) x M, then there exists a global solution satisfying y(x) lvi for all x

Remarks. 1. Nagumo (1942) formulated condition (Td) and proved the last theorem. The invariance problem was revitalized in the late sixties. Condition (Ti) goes back to Bony (1969), who proved Theorem XVI under the assumption

that I

f(y) is locally Lipschitz continuous; Brezis (1970) showed the same under condition (Td). The propositions XV.(a) and (b) are due to Redheffer (1972) and Crandall (1972), respectively. The paper by Hartman (1972) and other papers cited here give the impression that the authors were not aware of Nagumo's theorem. Obviously, invariance follows from this theorem if f is

continuous and if the assumptions regarding I guarantee uniqueness.

2. The invariance theorem XVI and its proof carry over to differential equa- tions in a Hilbert space H (where I and the solution have values in H and the inner product in H appears in (12)).

XVII. An Example from Ecology: Competing Species. We con-

sider nonnegative solutions (u(t), v(t)) of the autonomous system

u'—_u(3—u—2v), v'—_v(4—3u—v). (13)

In the biological model, u(t) and v(t) are the numbers of individuals in two competing populations that feed on the same limited food source, and t is time.

If v is absent, u(t) is governed by the logistic equation u' = u(3^—u)with growth rate 3—u, which diminishes to 3—u —2vin the presence of v. The same applies to V.

We discuss the global behavior of solutions as t using the phase plane.

Since both u and v are nonnegative, only the first quadrant Q = [0, inthe uv-plane is considered.

Writing the system in the form (u',v') =F(u,v)

= (f

(u, v),g(u, v)), we find

that f =

⁰ on the line BD and on the v-axis, while g =0 on the line AC and the u-axis; in the figure, the sign of f or g on the two sides of the zero line is indicated. This figure shows that there are

four stationary ^points, (0,0), B =(3,0), P= (1,1), C= (0,4) and

four regions, E1 — E4, where the signs of u' and v' do not change;

the arrows show the direction of F on the boundary of these regions. First

observation: the regions E2 and E4 arepositively invariant, and sois Q.

Consider a solution (u, v) starting (say, at t = 0) in region E3. Both com- ponents u and v decrease as t increases. The solution either stays in E3 for all

Nulldlines: u'=f=O,v'=g=O

t> 0 or enters one of E2 (through BP) or P24 (through andremains there.

Similarly, a solution starting in P21 stays there or enters E2 or E4, remaining there. Because u(t) and v(t) are eventually monotone in all cases, the limit

lirn(u(t),v(t)) =

=

EQ

exists. According to XI.(g), = ^0; i.e., every solution 0 converges to B, C, or P as t ^{00. In} the first case, the v-population dies out; in the second case it is the u-population which becomes extinct. In the third case the solution converges to a state of coexistence P. In each of P21 and E3 there is a unique solution (modulo time shift, see XI.(b), (c)) converging to P. The corresponding orbits combine to form a curve from 0 through P to infinity, which divides the first quadrant in two regions. A solution starting (at any time) in the upper region converges to C, and one starting in the lower region converges to B.

Hence this curve, also called a separatrix, categorizes the asymptotic behavior of all solutions as t —, oo. A proof (not simple) is indicated in (d) below.

Exercise. (a) Show that the regions E1 and B3 and Qarenegatively invariant and that every solution starting in E1 tends to 0 as t —00.

(b) The diagonal u = v cuts the first quadrant into a lower part Qi and an upper part ^Showthat the regions (E1 U E4) fl Q,. and (E2 U E3) nQj are positively invariant and that a solution starting in one of these regions does not

converge to P as t —4 oo.

(c) Let F be the set of points between the straight lines v = ^uand v = 2u—2.

Show that for i 1 and i = 3, the sets = fl F are negatively invariant and that a solution starting in E2 \ does not converge to P.

(d) Show that there exists a unique solution (u* (t), (t)) (modulo time shift) starting in and converging to P (same for Ei).

C

V

E3

0

B

0

B

Phase Portrait

§10. The Initial Value Problem for a System of First Order 121

Hint: Make P the origin of coordinates i.e., u =1+ü, v = Then

(13) becomes = = Consider the

differential equation for the trajectories in explicit form =ii(u) (see equation (2c) in § 3), which reads

(1+ti)(3ü+li)

dfi _— —. _)

The solutions with initial value = 2/n converge monotonically to a solution that describes the trajectory (lying in of a solution (?f, v*) of (13) converging to P. There is only one such solution in since ^is decreasing in t3 near 0 (= P), which implies that the difference of two solutions is increasing as ü 0+.

Supplement II: Differential Equations in the Sense of