FRANCESCO LERDA (*)

FORMALLY LINEAR METHODS FOR TREATING NON LINEAR ORDINARY DIFFERENTIAL EQUATIONS

SOMMARIO

Nella prima parte di questo lavoro vengono stabilite le basi di una « trattazione formalmente lineare » di equazioni differenziali non lineari, trattazione suggerita dal prof. F. G. TRICOMI in occasione di un precedente lavoro.

Si affronta poi il problema di valori al contorno relativo all'equazione:

associata a questi due tipi di condizioni:

1) y(a)=A , y(b)=B

2) y(a)-cy[o)=A , f(b) + dy(b)=B .

II problema in questione, tra l'altro gia trattato da KELLER in [5] e dall'autore in [3], viene risolto in condizioni piu general! utilizzando i citati metodi di linearizzazione forrnale.

Infine viene stabilito un teorema di confronto numerico fra due equa­zioni differenziali non lineari del tipo generale:

f'-F^y.y') , 7" = F2(*,y,y')

sotto particolari condizioni per le funzioni F^ (x, y, y') ed l\ (x, y, y'), e si estende un teorema di confronto fra una equazione ed una diseguaglianza differenziali, presentato da BAILEY, SHAMPINE e WALTMAN in [4] .

(*) Istituto di Calcoli Numerici dell'Universita di Torino. This work has been done within the C.N.R. Research Groups on Numerical Analysis

and Automatic Computation.

234 -

INTRODUCTION (**)

In chapter III of F. G. TRICOMI'S book [1] important theorems on boundary value problems for linear ordinary second order dif­ferential equations are analysed. In particular comparison theo­rems and properties of the zeros of the solutions are proved by means of very simple methods.

In [2] chapter 7 P. HENRICI proves the existence and uni­queness of the solution of the first boundary value problem:

y"=f(x,y) ; y(a)=A , y{b)=B (1)

where f(x,y) satisfies special conditions. In [3] I have extended the above theorem to the case of the

complete equation:

y"=f(x,y,y') (2)

The proof was originally obtained by means of repeated appli­cations of the mean value theorem; afterwards it was transformed into a more simple and elegant form following the indication, given to me by prof. F. G. TRICOMI : treat the non-linear equations under discussion as formally linear ones, according to the theorems presented in [ 1 ] .

The method has proved to be highly efficient, and the results have suggested a more extensive application of it.

In section 1 of the present paper the foundations of the « for­mally linear treatment» of non-linear problems are set. Sections 2 and 3 present examples which emphasize the effectiveness of the method.

The existence and uniqueness theorem given in my paper [3] for the boundary value problem of equation (2) refers to both the following boundary conditions (3) and (4) :

y(a) = A , y(b)=B (3)

/ (a) — cy(a) = A (4)

/(b) +dy(b) = B

under the hypotheses: c > 0 , d > 0 , c-J-d>0.

(**) All quantities considered in this paper are real.

— 235 —

Note that conditions (4) do not include conditions (3). In both cases (3) and (4) the hypotheses on the function

f(x,y,y) have been the following:

a - f(x,y,y') is defined and continuous for a^x^b, any y and any y

J3 - in its domain of definition f(x,y,y) satisfies a uniform LIPSCHITZ condition on the arguments y,y\ this means that there exists a non negative constant L such that, for any x, any pair y, y*, any pair y', y'* we have:

\f(x,y, y')~ f(x,y*, y'*)|<I(|y— y*\ + |y'—y'*|) (5)

y • df/dy, df/dy' exist and are continuous in the domain of definition of /, where:

f + f>° (7, By dy

In section 2 of this paper condition (7) is shown to be unne­cessary.

The theorem we have just spoken about has been proved by KELLER [5] for equation (2) and both boundary conditions (3), (4) under the following hypotheses:

a - the same as a above

0 - the same as /3 above

y - df/dy, df/dy exist, continuous, in the domain of defini­tion of /, where:

dy 3L dy'

<M (8)

M being a suitable non-negative constant. Actually the second condition in (8) is included in the fact that

f(x,y9y') is lipschitzian on y. This means that, apart from the

— 236 —

LIPSCHITZ condition, no quantitative restriction is imposed on df/dy. From this point of view Keller's treatment is more general than the one given in [ 3 ] .

On the contrary, the first condition in (8) is stronger than the first one in (6); in particular it excludes the equations of type:

/ ' = / ( * . / ) (9)

The severity of this limitation is emphasized if we consider that in mechanics it will not allow treating the extremely important problems where the forces depend only on speed.

In section 4 of this paper the theorem under consideration will be proved, for equation (2) and both boundary conditions (3) and (4), under the following hypotheses:

a" - as above in a

$' - as above in ,/3

y" - df/dy, df/dy exist and are continuous in the domain of definition of /, where:

dfjdy>0 (10)

Among all the above mentioned formulations of the theorem, this turns out to be the more general one; moreover, the proofs given in section 4 further confirm the simplicity as well as the efficiency of the formal linearisation methods discussed in section 1.

While performing the above mentioned research I received the invaluable assistance of prof. F. G. TRICOMI, whose suggestions, among other things, led to further results, presented in sections 5, 6, 7; they are:

— two comparison theorems for non-linear ordinary differential equations

— an extension of the comparison theorem treated by BAILEY,

SI-IAMPINE and WALTMAN in [4] p. 73 and ff.

— some simplifications on the proof of the last comparison theorem.

On the subject of symbology, we will make use of the following notations:

— 237 —

[a, b] = the closed interval

[a, b) = the closed/open interval

(a, b] i = the open/closed interval

(a, b) •== the open interval

1. - A MORE GENERAL INTERPRETATION OF SOME THEOREMS ON

LINEAR ORDINARY DIFFERENTIAL EQUATIONS.

The first theorem in section 19 of TRICOMI'S book [1] may be restated as follows:

THEOREM 1 :

Let p(x), P(x), y(x) be continuous functions of x on the inter­val [a, b\, where it also turns out to be:

1) p (x) > 0 , having continuous first derivative

2) y (x) twice continuously differentiable function

3) ? W < 0

4) ^[p(x)f(x)]+P(x)7(x) = 0 ax

In such a situation the function

v(x)=p(x)y{x)y'(x)

is non-decreasing in the interval [a, b].

The proof given in TRICOMI [1] is based only on the fact that P(x), p(x), y(x) are some functions of x satisfying the previous con­ditions. In particular it is not necessary that P(x), p(x) are known functions of x. On the contrary they may even depend on y(x) and its derivatives of any order. The original proof holds unmodified also in this new situation.

Furtermore the following extension of THEOREM 1 is valid:

EXTENSION OF THEOREM 1:

THEOREM 1 holds as well in the case when P(x) = — oo for either x= a or x= b or in both cases.

a^: x^ b

a^x < b

a< x^b

a < x < b

(ID

(12)

— 238 —

Proof:

Let us consider the function:

v (x)=p (x)y{x)y'(x)

this function is defined on the interval [a, b], where v (x) exists and is continuous.

On the interval (a, b) it is:

v'(x)=p{x)y'2(x) - P(x)y2(x) .

Hence v (x) is non-negative on (a, b); then, as a consequence of its continuity on the closed interval [a, b], this derivative turns out to be non-negative on this closed interval, where, finally v(x) is proved to be non decreasing.

Similar considerations are valid for the NUMERICAL COMPA­

RISON THEOREM of section 20 in TRICOMI [ 1 ] , which may be res­tated as follows:

THEOREM 2 :

Let p(x), Pi(x), P2(x), yi{x), yo{%) be continuous functions on the interval [a, 6 ] , the last two not being identically null in that interval.

If yi(x), 72(^) coincide in both value and derivative in a point x0 of the interval [a, b), then the ratio:

yi (* ) /y 2 (* )

will be an increasing function of x in a right neighborhood [x0, x0 + 8] of x(h starting from the value 1 assumed when x = x0, if the following conditions are satisfied on the interval [a, b] :

1) p(x) > 0 has a continuous first derivative;

2) yi(x), y>2(x) are twice continuously differentiate functions;

3) Pi(x)<Pi(x)

4) yi(x), y2(x) are different from zero in a right neighborhood (x0,

— 239 —

x0 + 8) of x0, the two functions being possibly null when x = xQ\

A [p(x)yl/(x)] + Pi(x)yi(x)^0 (13)

ax

-£• [pWyt'(x)-\ + Pt{x)yt(x)~Q (14) ax

It may be easily verified that also in this case the original proof given in TRICOMI [1] section 20 retains its validity, if we make the following remark:

Even under the above hypotheses, if in a certain point % ive have

yi{x)'= y\(x)== 0, y'i(x) turns out to be identically null in the

interval [a, b] (and similarly for y2(x)).

This fact is an immediate consequence of the uniqueness theorem for the solution of the initial value problem:

i.[p{x)Y'{x)] + Pl{x)Y(x)-0 ax

Y(x) = 0., Y'(x) = 0

where p(x), P(x) are the same functions which appear in (13). Furthermore the following two extensions of THEOREM 2 hold:

EXTENSION 1 :

THEOREM 2 is also valid on the left o) x0, which, in this case, may coincide with b; this means that on the left of x0 the ratio yi(x) /j'i{x) is a decreasing function of x, and:

Jim = 1 . X-+XQ y»Ax)

Proof :

As a matter of fact, if we set:

x=— t

— 240 —

equations (13), (14) become:

d - - - -

d - - - -jt\.p(t)y*'{m+PtMy*M=>o

and the left neighborhood of #0 is transformed in a right neighbor­hood of t0 •=—x0; in the above equations p{t) — p[x(t)], and simi­larly for the other functions.

EXTENSION 2 :

THEOREM 2 is also valid when at XQ either one or other of the following conditions is satisfied:

Pi(%o) = -c° , P2(^o)== + 00

Proof:

Let us consider the problem on the right side of x0. If x, Xi are points of the interval [x0, x0 + 6) such that:

XQ *C3 XI <C X

we have:

[P2(x) - Pi (x)] y^x) y2{x) = — {p {x) [y2(x)y/(x) - yi (x) y%' (x)]}

hence:

f [P%(x) - P, (x)] 7i(x) y%(x) dx= (15)

'i

=p(x) [y2(x) yi,(x)—yi{x)y%

f(x)]-p(xi) [y^xjy^xj -ydx^y^x,)]

Now, when x1 tends to x0 we have:

Xi

pM [y2{Xi) y/(*i) —yd*\) y%(xM -* o

— 241 —

As a consequence the following integral exists and it turns out to be :

X

J [P2(«) - Px(x)] yi(x) y,(x) dx=

=p(*) ly%(x) ji'(x) —yi(x) y / W ] (16)

Now, with reference to the equation of system (13), (14) in which P.f(x), i = l , 2 , is bounded, a discussion similar to the one followed in proving THEOREM 2 shows that y1[x0)

,== 72(^0) and y\.(%o)'= y^ixo) may not both be zero. Thereafter we come back to the original proof given in [ 1 ] . Similarly on the left side of x0.

Let us finally remark that the NUMERICAL COMPARISON THEO­

REM, in its original form, still holds when we have, in the interval [a,b]:

P t(*) <:/>,(*)

In this case yi(x)/y2(x) turns out to be a non-decreasing function of x on the right side of x0 and a non-increasing one on the left side (see [1] section 20). In particular, if in an arbitrarily small in­terval [#0, #0 + rj] on the right of x0 the functions Pi(x), P<j{x) are not identically equal, the ratio y1(x)/y2(x) is still an increasing fun­ction of x, and similarly on the left of x0, where it will be a decreas­ing function.

2. - ON HENRICI 'S PROOF OF A SPECIAL EXISTENCE AND UNIQUE­

NESS THEOREM.

In [2 I chapter 7 HENRICI proves the existence and uniqueness of the solution of the boundary value problem:

y"=f(x,y) ; y(a) = A , y(b) = B

where f(x,y) is continuous for a^x^b and any 7, uniformely lipschitzian on y in that domain, and such that df/dy is continuous in the same domain, where furthermore is:

16

— 242 —

In [3] I have extended the theorem to the more general problem:

/'=/(*, y,/) ; y(a)=A , y(b)=B

with f{x,y, y) continuous for a ^ x b, any y, any y9 uniformely lipschitzian on y and y in that domain, where df/dy, dfjdy' are continuous and satisfy the inequalities:

df . df — X ) — > 0 Zy d/

with the restriction that the two derivatives are never simultaneously null.

Taking into account the remarks at the end of the preceding section, this last condition (df/dy, df/dy' not simultaneously null) may be removed. As a consequence the proof given in [3] is valid even for the problem considered by HENRICI.

As a matter of fact, the proof for this theorem may more simply be organized as follows:

Let y(x, a) be the unique solution of the initial value problem:

y"=f(x,y) ; y(a)=A , y'(a) = a

If we set:

then:

and THEOREM 1 shows that yi(x) is different from zero on (a, b]. Now we set further:

x — a = y2(x)

then, obviously:

yt"(x) + Pt(x)yt{x)-0

— 243 —

with P 2 ( # ) ~ 0 on the interval [a, b]. At this point THEOREM 2

gives:

yi(x) ;> x — a

for any # of (a, 6 ] , from which the theorem immediately follows (the case df/dy identically null is included).

It seems very unlikely that the above proof could be further simplified.

3. - A SIMPLE COMPARISON THEOREM FOR AN EQUATION CONTAIN­

ING A PARAMETER.

Let us consider the initial value problem:

y"=f(x,y,y',l) ; y(a) = A , y'(a) = a (17)

where X is a parameter. Under suitable conditions, if it. is possible to foresee the dependence of / on X, it may be possible to predict the dependence of the solution of system (17) on X itself.

To be precise, let the following conditions be verified:

a - f(x,y, y\ X) is defined and continuous for a^x^: b, any y, any y, A ^ X ^ A *? the last interval being possibly infinite on either one or both sides.

ft ' f(x> !•> y'» ty satisfies a uniform LIPSCI-IITZ condition on the variables y, y.

7 " y / ^ J » ^fl^y't ^fl^ exist and are continuous in the domain of definition of f, where further:

8 ^ ° ' 81^°' df ^ ° ' di > 0 (18)

the last two inequalities holding for any set of values of y, y, X.

In such a situation, if y(x, X) is the unique solution of (17), this function and is derivative with respect to x are increasing functions of X; in other words, if X2 < X2, then we have, for any x of (a, b] (see fig. 1):

y(x,Xi)<y(x, X2) , y'{x, A4) <y'{x, l%) (19)

— 244 ~

X increasing

a Fig. 1

Proof:

We omit the details about the existence and continuity of the functions and their derivatives which will be introduced; this matter may be treated as in [ 3 ] .

From (*):

y"(x,X)=f[x,y(x,X) , /(x, X), X]

we deduce the equality:

y / W ) = | y, (*, i) +|£ y/ix. i) +f (20)

which may be written as follows:

7>- dy'y" \8y 8X yj 7x (21)

(*) As usual the primes' " indicate derivatives with respect to x; on the contrary, the index X means derivation with respect to X itself.

Furthermore:

— 245 —

y , » = 0 , y / ( o ) = 0 (22)

From (18), (20), (22) we deduce:

yx"(a,k)>0

hence in a suitable right neighborhood of a it is :

y / > o , yA>o

Now, if the above inequalities do not hold in the whole interval (a, b], there will exist a first point f from a toward b where / , (|,X):=0.

But equation (20), following the procedure indicated in TRICOMI

[1] section 18, may be written in the self-adjoint form:

j-xlp(x)yk'(x)] + P(x)n(x)=0

where:

p (x) = e J & ; P(x) = - U + | - e J ** \°y M yJ

In the interval (a, f) the function yk..is positive; hence P(x) ^ 0 on that interval, and, according to the EXTENSION OF THEOREM 1, the function v(x) = p(x) yx (x) y\(x) results as non-decreasing on [a, f J. This fact is unconsistent with the existence of the point f. We conclude that the above inequalities, and similarly inequalities (19), hold in the whole interval (a, b].

4. - GENERALISATION OF AN EXISTENCE AND UNIQUENESS THEOREM

FOR BOUNDARY VALUE PROBLEMS OF NON-LINEAR DIFFEREN­

TIAL EQUATIONS.

4.1 - The case of the first boundary problem.

We will here consider the boundary value problem:

— 246 —

y"=f(x,y,y'); y(a)~A , y(b)=B (23)

under conditions a \ ft'', y" specified in the Introduction.

The treatment will follow that given in [ 3 ] , to which reference has to be made for the details.

We start by considering the initial value problem :

/'=/(*> y> / ) ; y W=A > / («)=« <24)

The functions which will be introduced in the following exist and are continuous where and as far as necessary.

Let y(x,a) be the unique solution of (24); we have;

/ ' ( a , a)=f[x9y(x, a) , f(x9 a)] (25)

from which:

y"(x, «) = J £ y. '(*. a)M ya(x, a) (26)

where again the index a means derivation with respect to a itself. Now we set:

y.(*. « ) - * ( * ) (27)

Recalling the initial conditions of system (24) we have:

yt{a)=0, y/{a) = l (28)

Furthermore equation (26) may be written as :

^c[p(x)yl'(x)]+P{x)yi(x) = 0 (29)

where:

X X

p(x)-e J <>' , P(*)—£ e { & (30)

— 247 —

This allows us to assert that:

yi {%) > 0 for a < x < b

yi(x) > 0 for a<x<b

At this point we set :

P* (x) = 0

and search for a new function y«(x) such that:

d

7,(a) = 0, y2'{a) = l (32)

If the search is successful, the NUMERICAL COMPARISON THEO­

REM (see [ 1 ] , [3]), taking into account that P(x)^P*(x), will permit us to deduce:

yi(x) 2>y%(x) for a< x < b

The function y2 (x) may be immediately obtained from (31) which gives:

d ~T~ [pix) y%'(x)] = Q that is: p (x) y%'(x) = const. (33)

We will select the integration constants in order to fit conditions (32); so we obtain:

J By' y2'(x) = ea (34)

{ J ay' y%{x)= \{ea )dT (35)

from which:

y2(a) = 0, y 2 >) = l (36)

As a consequence:

248 —

x I df dt

yi(x)^\{ea )dx (37) ./ 8S

But:

81 ir.> - L <38) hence:

and finally:

1 _ p-L (x-a)

yjx)^ —"~j > 0 (40)

In particular when x=b equation (40) states that y1(b) = =ya (b, a) ^[l — e~L{b~a) ]/L > 0, from which we deduce that y(b, a) assumes every value exactly once when a varies from— oo to ..+ oo ; and this completes the proof.

4.2 - Extension of the theorem to boundary conditions of type y'(a) — cy(a) = A, y'(b)+dy(b) = B, c > 0, d 0, c + d>0.

The treatment follows the same lines as in subsection 4.1. In this case too the functions and derivatives we shall introduce will exist and be continuous as far as necessary in the discussion. For the omitted details reference is made again to paper [ 3 ] .

As already mentioned in the Introduction, these latter boun­dary conditions do not include the former ones considered in the preceding subsection.

We will consider again a suitable initial value problem, namely:

/ '=/(*, y, y') ; y(a) = a, y\a)=A^ca (41)

The notations introduced in the following treatment have to be interpreted as in the preceding pages.

— 249 —

Let y(x,a) be the unique solution of (41); if we set:

we obtain:

y"(*) - ~7 y,'(*) ~ ff yi(*)=o (42) dy cy

with the following initial conditions:

r i ( a ) = l , yi'(a) = c (43)

Equation (42) may be transformed in the self-adjoint form:

d

fa [p(x)y[/{x)] + P(x) y,(x) = Q (44)

by setting: X X

Bf dt _ -f-^rdt

p(x) = e a , P(x)=—7jZe a (45) dy df J dr

Ty "

from which we deduce, in the usual way:

y.L (x) > 0 for a < x < b

yl'(x)>0 for a <.%'<&

(we suppose, as in [ 3 ] , c > 0; in the case c = 0, we have d > 0, and the discussion can be done similarly starting from b toward a).

Let us now search for a function y»(x) such that:

d

fa [p(x)y*'{x)]+P*(x)y*(x)=o (46)

where P2*(x) = 0. We deduce, through a suitable selection of the integration constants:

X

i£dt ? [Mdt J dy / ./ dy

y2'(x) = cea , y2(x) = l + c {ea )dx (47)

— 250 —

from which, following once again the lines of discussion of the preceding subsection, we deduce:

\ e-L(x-a) yi'(x)^ce~L(x-a) , y^J^ l - f c j (48)

Consequently, remembering that c > 0, we have:

1 _ _ e - L ( « - a )

ya'(x, a) + dya(x, a)^.c e'L (x'al + d + dc j > 0

Hence the function y(b9 a ) + dy{b, a) has a derivative on a which turns out to be uniformely larger than a positive constant.

In the above situation the function y(x,a) satisfies condition y(a)— cy(a)==A for any a, and furthermore a unique a exists for which condition y'(b)-\-dy(b) is also satisfied.

4.3 - Extension to more general boundary conditions.

Let us consider the same problem as above with the following conditions (the minus sign inserted in the first equation simplifies the following notations):

gi ty(a), - /(a)]-0 , gly(a) , y'(a) , y(b) , y'(6)]=0 (49)

where gu g2 are functions of two and four variables respectively:

gi=gl (U, V) , g2=g2 (U, V, W, Z)

defined for any values of such variables and possessing continuous partial derivatives of the first order (*).

We start by considering the initial value problem:

/ '=• / ( * . y. / ) (50) y(a) = a, y'(a) = <p(a) such that gi[y(a), — y7(a)| = 0.

(*) This is the same problem considered in KELLER [5], p. 19; in this case too our conditions are more general than the KELLER'S ones.

— 251 —

Problem (50) has a unique solution, provided that the equation:

gi(u, v) = 0

could be uniquely solved with respect to v for any given value of u. For this, the following condition is sufficient:

8g dv

- > / i > 0 (51)

Let y(x7a) be the unique solution of (50); proceding as in subsections 4 .1 , 4.2, we obtain:

yL(a) = l , 7l'(a) = J a a

In our hypotheses dw/da exists and is given by:

Jll= a / 1 o jU = a }v=-q>(,a)l \OV )v=-(p{a) da \du )v=-wla)l \VV Jv=—ip(a)

If we suppose:

> 0 fa du

we obtain:

da

Through a discussion similar to that given in subsection 4.2 we introduce the functions:

Am J s r rim / J dy'

a

252 —

which give the inequalities:

da y 1 (* )> l +

dcp l—e-TAx-a)

da L (54)

We now consider the function:

G(a)=g*fy (&><*)> / («>«)> y(b,a), y'(b,a)]

from which, in the hypothesis that all the partial derivatives of gs are ^ 0, we obtain:

dG ?g2 dg % 2 gz da 8 u

ya (a, a) + - £ ya'{a, a) + £ f ya (b, a) + ya'(b, a) ev

Vg% , tig2 dcp dgo

ou dv da oiv

8w

dcp l-e-L{b-a)

l + T~ 1 da L

Sll *?. ,-LOra) ?z da

If we complete our hypotheses with

Bg2 du

k> 0 (55)

the derivative dG/da results uniformely bounded away from zero, and the theorem follows.

Different conditions may obvionsly be substituted for (55), as shown in KELLER [ 5 ] .

5 . - TWO COMPARISON THEOREMS FOR NON-LINEAR ORDINARY

DIFFERENTIAL EQUATIONS.

In this section we will obtain a NUMERICAL COMPARISON THEO­

REM for non-linear equations similar to that given in TRICOMI [1] for the linear case.

A first step in this direction has been represented by the following COMPARISON THEOREM 1, kindly communicated to me by prof. F. G. TRICOMI, together with the fundamentals of the proof:

— 253 —

A lemma is pre-inserted, which simplifies the discussion:

LEMMA.

Let y(x) be any solution of the following special non-linear differential equation:

y" = F(x,y,y')y' , . . (56)

•where obvious continuity conditions are supposed to be verified on the domain of definition of (56). In the interval \a, b] the sign of y'(x) is preserved.

Proof:

In fact, if y'(x0) = 0 in a certain point x0 of the interval \a, b], the uniqueness theorem for initial value problems shows that:

in other words y(x) turns out to be constant on [a, b], and y'(x) is identically null.

We come now to the

COMPARISON THEOREM 1.

Let us consider the following two special non-linear differential equations:

/ ' = F 4 (*, y, y')y', y"=F,{x, y, / ) / (57)

Under the following hypotheses:

a - for both equations the inital value problems with given values of y, y in a generical point x0 of the interval [a, b] has one and only one solution;

fi • for x£[a, 6 ] , any y and any y the functions FUF2 are continuous;

y - in the preceding domain, at least one of the two functions F1} Fo is non-decreasing on the variable y for positive values of the argument y, and at least one of the two functions under discussion is non-increasing on y for negative values of of y ;

— 254 —

8 - in the preceding domain for any set of values x, y, y it results that:

Fi(x,y>y')>f%(x>y,y') when y'> 0

when y'< 0

the two solutions yi{x),y2(x) of equations (57) corresponding to the same initial conditions on x — xQ:

yi(x0)=y%(x0), y i 'K)=y 2 ' (*o)

are such that in the whole interval [a, b] it turns out to be:

Ji(x)>yz(x) ^ x^xQ

y±(x) > y2'(x) for x > xQ

yi(x) < y2'(x) for x< x0

(58)

Proof:

We refer to the case a < x 0 < 6 ; the proof is valid also for x0 '= a and x0 = b.

Let us first consider the case:

yi'(Xo)=y*{Xo)> °

It is identically on [a, b] :

dx log

y* =Fi (x, yi9yi) — F2 (*> yt, y2') (59)

The right side of (59) is positive for .x==x0; for reasons of continuity it remains positive in a suitable complete neighborhood of XQ. Hence, in that neighborhood log (y'i/y'2), which is zero for x:= XQ, turns out to be positive, what means:

yi'{x)>y2'(x)

yi'M<yz(x)

yi(x)>y%(x)

for x> Jvr

for x < xQ

for x *£ xr

(60)

— 255 —

We now show that inequalities (60) hold on the whole inter­val [a, b].

In fact, if this should not be the case, let xx be the first point of (x0, b] starting from x0 where:

7i'(xi)=y2'(xi)

(similar considerations hold if the point Xi falls on the interval [a, x0)).

By setting:

d{x)=yi(x)-~ y%(x)

we obtain:

d(x0) = 0, d'{x0) = 0 , d'ixJ^O

d (x) > 0 for x0<x<L xi

d'(x) > 0 for x0 < x < xi

The second derivative &'(xi) must then be either negative or null; but this is not possible. In fact we have:

^"(^)=ri',K)-y//K)= = Fi[xi,yi(xi),yl'(xi)] y^x,) — F2[Xi, y^x,), y/(%)] y2'(%) >

*il>i> y%(%\),y%'(xi)] jz'M — F2\-xn y2(*i)>y2'(*i)] yAxJ> °

F^x^y^x,), y / ( ^ ) ] yAxi) — *U*i> 7i(*i)> yi'(xi)] yi'(xi) > 0

We turn now to the case:

yi'(*o)=y2'(*o)<o

By setting:

— zt{x)=yi{x) > — %(x)=y%(x)

equations (57) become:

zi =ri(x, Z), zt ) zi , 2g =r2(^5 z%, z2 ) z2

and we go back to the preceding case.

— 256 —

The above results have suggested investigating the possibility of obtaining similar inequalities for more general equations. The attempts have been successful, leading to the. following:

COMPARISON THEOREM 2.

Let us consider the following two general non-linear second order differential equations:

y" = F±(x,y,y') , y" = F%(x,y,y') (6l)

Under the following hypotheses:

a - exactly as in a of COMPARISON THEOREM 1;

yS - as in ft of COMPARISON THEOREM 1 ;

y - in the preceding domain at least one of the functions Fl9 F2

is non-decreasing on the variable y;

8 - in the preceding domain for any set of values x, y, y we have\

Fi(x,y,y')>F2(x,y,y')

the two solutions yi(x)9 y<>{x) corresponding to the same initial con­ditions for x ••= x0:

yi(*o)=y2(*o) > yi'{xo)=yi(xo).

are such that in the whole interval [a, b] it is:

y \ (x) > y% M f o r x ^ *o yi'{x) > y%'(x) for x>x0 (62)

y\\x)<y%(x) f o r x<x0

Proof:

Again we refer to the case a < x0 < b, the proof being valid in general.

Considering the function:

d(x)=yi(x) — y2(x)

257

it turns out to be :

d(x0) = 0, d'(x0) = 0

^ / W = y i " K ) - y 2/ ' ( ^ o ) = ^ i [ ^ 7i(*o)> yi(x0)l -

- F2[x0, y2(*o)> y%(x0)]>o m

As a consequence in a suitable complete neighborhood of x0

we have:

yi'(^) > y%(x) for

7iW <y%'(x) for

yi (x) > y% (x) for

X > X

X < X, 0

X ^ Xr

If the above inequalities should not hold on the whole interval [a, 6 ] , a first point xx should exist, starting from x0, either on the right side or on the left one of x0, where:

We discuss the problem at the right of x0, the conclusions being valid in general.

It results:

<5(s0) = 0 , d'(x0) = 0, ($'(*,) = 0

d (x) > 0 for x0 < x < xi

d'(x) > 0 for xQ < x < xi

Even in this case the second derivative S"'(x-i) must be either negative or null. On the contrary:

d"(x1)=y1"(x1)-y<>"(x1) =

= F1[x„ y^Xj), yiix^ — F^x^y^Xj), y / ^ ) ] ^

F±[xi9 y^x,), j 2 ' ( ^ ) l - F%[xv y2(Xi), y^x,)] > 0

. F&i, ydXi), yAxJ] — Fi[xi9 y^xj, y^x,)] > 0

and this proves the theorem.

17

— 258 —

We remark here that COMPARISON THEOREM 1 is a special case of this latter theorem. We have considered it of interest to present this section according to its chronological development.

Furthermore, COMPARISON THEOREM 2 allows us to restate the theorem of section 3 on a different basis: conditions a, ft of that section are assumed to be unmodified, while condition y becomes:

y - df/ dy, df/d A exist and are continuous on the domain of definition of f, where further:

df ?j oX dy

The differences between the two statements are such to make both instances of the theorem significant.

6. - AN EXTENSION OF THE COMPARISON THEOREM TREATED BY

BAILEY, SHAMPINE AND WALTMAN.

The theorem we will analyse in this section as well as in the following one is a comparison theorem between solutions of diffe­rential equations and solutions of differential inequalities. The sta­tement of the theorem as given in [4] p. 73 is the following:

Let h(t,u,u) be a continuous function for a^t^b, any u and any u, and let the differential equation:

u" + h t, u, w'; = 0 (63)

satisfy the folloiving conditions:

1) all initial value problems with given values of u, u in a generical point of \a,b] have a unique solution which exists throughout the interval [a, b];

2) two different solutions of (63) cannot assume the same value in more than one point of [a, b].

Let v(t) be a tivice continuously differentiable function on [a, 6 ] , satisfying:

v"{t)+h[t,v{t), v'(t)]> 0 (64)

— 259 —

/ / uto(t) is the solution of (63) which coincides with v(t) in both value and derivative at some point t0 of [a, b], then for each value t of that interval, different from t0, we have:

v(t)>utJt) (65)

Furthermore, if u(t) is the solution of (63) which coincides with v(t) in value at the points a and b, then on the interval (a, b) it turns out to be:

v (fr) <u(t) (66)

The preceding theorem is numbered 5.1 in reference [ 4 ] , where it is followed by two further theorems, numbered 5.2, 5.3, referring to different boundary conditions for the solutions of equation:

u" + h(t9 u, u') = Q

In the last two theorems, the addition of the hypothesis that h(t, u, u) is an increasing function of the argument y on its domain of definition allows us to conclude that for any x in either [a, b) or (a, b\ the following inequalities hold:

v'{t) < u^t) v'(t)> u2(t)

where ux(t), u2(t) are the considered solutions of the above diffe­rential equation.

On the contrary, such an extension is not given for theorem 5.1. We will discuss it in the following pages.

/ / we add to the above hypotheses of the theorem under discus­sion the further one that h(t, u, u) is a non-increasing junction of the argument u for t G [a , 6 ] , any u and any u, then the following inequalities hold on the whole interval \ a, b] :

uto{t)<v(t) for t^tQ

ut^t)<v'{t) for t>t0 (67)

uto'(t)>v'(t) for t<t0

— 260 —

Proof:

In fact, from:

v"(t) + h[t9v(t), v'(t)]>0

we obtain :

v"(t)+h[t, v(t), v'{t)] — 8(t) = 0 (68)

where e(t) is a continuous positive function on the interval [a,b]. By setting:

h(t, v, v') — e(^) = /i1(^ v, v') (69)

we are lead to consider the two equations:

u" = -h(t, u, u') v'^—h^v, v') (70)

where:

— h (t, u, u') < — h^t, u, ur)

for any t E [a, b], any u, any u ; furthermore —h(t , u, u) is a non-decreasing function of the argument u in its domain of definition. As a consequence, the proof of the COMPARISON THEOREM 2 is com­pletely valid also in this case.

It seems interesting to make here the following remarks: First, the above simple proof not only extends the theorem of

BAILEY, SHAMPINE and WALTMAN, but, on the hypothesis that h(t,u,u) is non-increasing in u, completely replaces the original, more complicated one presented in [4 ].

In the case when we do not introduce the above mentioned monotony assumption, some simplifications of the proof given in [4] are still possible and they will be given in the following section.

Secondly, the discussions of the preceding pages suggest a deeper investigation on the relationships between theorems which compare differential equations subjected to restrictions stated in form of inequalities and theorems which compare differential equa­tions with differential inequalities.

Let us consider the two equations:

y"=Fi(x,y,y), y"=F2(x,y,y') (71)

— 261 —

under the hypothesis:

F1(x9y,y')>Fi(x9y,y) (72)

assumed to be valid on each «point» of the definition domain of of (71): a ^ x 6, any y, any y .

Condition (72) is evidently a property of the two functions Fy, F2, which must hold independently of the specific solutions yy(x)-> J2{x) of the two equations which ive may consider.

On the contrary, if we consider an equation-inequality system of the form:

y1"(x) = F1[x,y1(x), yx'(x)], y2"(x) < Fx[x, y2(x), y2'(x)] (73)

the situation is quite different. Precisely here the inequality must be verified for the specific functions we have to insert into it.

If yi(x), y>>(x) are two functions which satisfy system (73), we can write:

yr(*)=^i[*,yi(*), yi(*)]> y / ' M ^ K ^ W , y%'(x)\ -«(*) (74)

where e(x) > 0 on [a, b]. By setting:

F2(x, y, y')=F1{x, y, y') - e(x) (75)

on the domain: a ^ x b, any y, any y', system (73) is reduced to system (71), (72).

We remark, with reference to the position (75), that:

8I±JJi, dI±Jh By 3y dy' By'

hence possible monotony properties of / \ on the variables y, y are preserverd as similar properties of F2, and this fact may be useful in some cases.

On the contrary, if yi(x), jz^x) are « solutions » of system (71), (72), from:

y1"(x)^F1 [a?, yx(x), y,'(*)] , y2"(x) = F2[x, y2(x), y2'(x)\

— 262 —

it is not possible, in general, to deduce:

y%[x) <Fi [x', y%(x), y%'(x)\

since the relationships between F1[x,y1(x),y1(x)\ and F2[x,y2(x),

y'z(x)] are intrinsecally dependent upon the actual functions

•yi(x), 72(^)5 whose behaviour is, a priori, unknown.

7. - SOME SIMPLIFICATIONS IN THE PROOF OF THEOREM 5.1 OF [4].

The statement of the theorem we will discuss here is reported in section 6 above. We will first consider inequality (65).

The original proof as given in [4] is organized according with the folloving two steps (notations are slightly different from those used in [4] in order to be in agreement with those of the preceding section 6).

a - Let uT(t) be the solution of equation (63) satisfying the initial conditions:

u{r) = v{r), U'(T) = V'(Z) (76)

where r is a generic point of the interval [a, b]. Step a shows that for every point T0€z[a, b], there exists a

80 > 0 such that, whenever the points t and r belong to the interval (To — $0) To + o)» we have:

ux(t)<v(t) (t^z) (77)

/3 - Inequality (77). is extended, in step /3, to the complete in­terval [a,b] for the function u,o(t), when i ^ i 0 .

The local property proved in a is a uniform one, since both parameters V, t are assumed as variable on the interval (r0 — So, To+ 80); proving this local property turns out to be a little com-bersome; in fact Bailey and the other authors have introduced a specific lemma to this purpose.

The treatment may be simplified as indicated in the following pages. >. , ; . '

— 263 —

1. - Proof of a non-uniform local property.

We still indicate as ur(t) the solution of equation (63) which satisfies the initial conditions (76). For any given r on the in­terval [a, b] a suitable neighborhood DT exists (*) such that for t£;Dz but different from r we have:

:uT(t)<v(t) (78)

In fact, by setting:

d(t) = uT(t)-v(t)

and taking into account (68), de deduce:

<5'(T) = U / ( T ) - I / ( T ) = 0

a"(T)=ttT"(l) — V"(T)=-E{T) < 0 .

Hence the function S(t) has a local maximum at the point / = T, and, since 8(r) = 0, the function 8(t) turns out to be negative on a complete neighborhood 2)r of r, provided that i ^ r .

2. - Extension of the above property to the whole interval \a,b].

We have proved above that, in particular, the inequality:

uto(t)<v(t) (t^t0) (79)

is valid in a suitable neighborhood of t0; we will show now that it actually holds in the whole interval [a, b] (**).

In fact, if this should not be the case, some points should exist either on the interval [a, t0) or on (£0? b] where the two functions coincide. Let us suppose that this happens on the right of t0 (we may procede similarly on the left), and let be t-i the first point

(*) A complete neighborhood if a < r < b, a right one if r = a, a left one if % — b. For simplicity we refer in the text to the case a < x < b, but the proof is valid in general.

(**) Here also we suppose a < to < b, the proof being still valid in general.

264 —

from tQ toward b where:

« j 0 w = i ; ( y (80)

If t2 is the middle point of the segment [t0, t±], in a suitable complete neighborhood of U we have:

ut (t) < v (t) (t^h)

According to the condition (2) of the statement of our theorem (section 6) at least in one of the two intervals [t0,t2), (t2,ti] a point tn should exist where (see fig. 2)

If we do iterate the above process, a sequence of points h, h, h, ••• arise; it is immediate to see that the sequence admits a unique accumulation point, T, and the following conditions must be verified:

— in a sufficiently small complete neighborhood of T the function u (t) is such that:

uT(t)<v(t) (ty*T)

I

— 265 —

— in the same neighborhood two distinct points exist, T± and T2, on the left and on the right of T respectively, such that for one of the functions:

Ut9 (0 > "*, (0 > uu W > -

which we will call here u*(t), it turns out to be (*):

uHTj^v(TJ , u*(Tt)=v{Tt)

u*(t)<v(t) on the interval (T19 T2)

As a consequence of this situation, two distinct points Ox, 62

must further exist in the interval (7\ , To) such that:

uT(01) = u*(0l) , uT(0%) = u*(02)

what is unconsistent with the condition (2) of the statement of the theorem (see fig. 3).

^ "~*"^| J AJLCt)

^--^T5^i--^>fs. J ! 4\s^ f\ \ I N « *vt)

K j! i ! i j 1 i ' ' ' ' i ! • ! ' ! ' •

' i ' i i i ! i l l ! . !

rp % T a

Fig. 3

(*) Actually there exists an infinite number of such functions.

— 266 —

Hence the inequality (79) is valid in the whole interval [a, b]. Let us finally turn to the proof of inequality (66). The function

u(t) cannot have in a the same derivative as v(t), since this fact should prevent having u(b) = v(b); neither can be u(a) < v(a), since in this case u(t) and ua(t) (the solution of (63) which coin­cides with v(t) in both value and derivative for t — a) should assume the same value in a point ^ a on the interval [a, b J (see fig. 4).

I ( I l

Fig. 4

Hence we have:

u'(a) > v (a)

therefore in the immediate neighborhoods of the point a it i s :

u(t)>v(t) .

If this inequality should not hold in the whole interval (a, b), let be U the first value of t (from a towards b) where:

u(t.2) = v(t2) .

- 267 —

In U (see fig. 5) it is:

u'(t2)<v'(t2).

Fig. 5

But the existence of such a point is unconsistent either with one or the other of conditions (1), (2) of the statement of our theorem, whose proof is so completed.

ACKNOWLEDGMENTS.

The suggestions and the assistance of prof. F. G. TRICOMI are gratefully acknowledged.

— 968 -

REFERENCES

[1.1 F. G. TRICOMI, Differential Equations, Blakie & S., Glasgow, 1961 (4a ed. italiana: Equazioni differenziali, Boringhieri, Torino, 1967).

| 2 | P. HENRicr, Discrete Variable Methods in Ordinary Differential Equations, J. Wiley, New York, 1962-65.

[3] F. LERDA, An Existence and Uniqueness Theorem for the First Bounday Value Problem for a Second Order Ordinary Differential Equation, Atti deU'Accademia delle Scienze di Torino, 103, 1968-69.

[4] P. B. BAILEY, L. F. SHAMPINE, P. E. WALTMAN, Non-Linear Two Point Boundary Value Problems, Academic Press, New York, 1968.

T5] H. B. KELLER, Numerical Methods for Two Point Boundary Value Problems, Blaisdell Publ. Co., Waltham, Mass., 1968.