A priori error bounds for general two-point boundary-value problems with arbitrarily many solutions

Nonlinew Andysir, Theory. Methods & Applicamm, Prmted in Great Bntain.

Vol. 9, No. 10. pp. 1081-1093. 1985. 0

0362-546X/85 13.00 + .oO 1985 Per&mm Press Ltd.

A PRIORI ERROR BOUNDS FOR GENERAL TWO-POINT BOUNDARY-VALUE PROBLEMS WITH ARBITRARILY MANY SOLUTIONS

Department

of Mathematics,

PETER WILDENAUER University of Kassel, Heinrich-Plett-StraRe

40, D-3500 Kassel, West Germany

(Received 14 August 1983; received for publicarion 25 February 1985) Key words and phrases:

a priori bounds, two-point

boundary-value

problems with arbitrarily many

solutions. 1. INTRODUCTION LET m

BE

a natural number. Let a, b be two real numbers with a < 6. For j = l(l)m.

assume

f, E C([a, b] x Rrn). We define the vector function f := (f,, . . . ,fm)T Let B1, B2 E R”“” and d E R”. With u=(Ur,...,U,)r a vector of m functions Uj E C’[a, b] is denoted. The component-by-component derivative of u is denoted u’. We then consider the general two-point boundary-value problem (TPBVP) u’(x) = f(x, u(x)) in (a, b) Blu(u) + B2u(b) = d

(1.1) I

of order m. We are looking for all vector functions u which solve the TPBVP (1.1) for given m, a, b, f, BI, Bz and d. Unfortunately the solutions u are known only for very few special TPBVPs (1.1). This applies to the linear initial-value problem, namely (1.1) with A E R” xm, 4j E C[U, b]

for

f(x,u):=Au+(ql BI := diag(1,.

j = l(l)m, ,...,

q,,JT,

. . , 1) and

B2:=0.

In this case the solution is unique. In general, however, the solutions u and even the number of solutions are unknown for given m, u, 6, f, B,, B2 and d. Aim of the paper

For given m, a, 6, f, BI, Bz and d we are looking for an upper and lower bound for special 1081

1082

P. WILDENAUER

solutions, classes of solutions and if possible for the set of all solutions of the TPBVP (1.1). A class of solutions could be defined with a number E E [a, b] and an interval vector [q] by the assumption

A note on the method The interval [a, b] is divided. For each subinterval [x,-r, x,] of [a, b] we interpret the differential equation system u’ = f(x, u) as a fixed-point problem. For this purpose we choose A E R”“” and map each vector function p:=(&,.

. . ,P,)7withpjEC1[X,-1,~,]

forj =l(l)m

onto the vector function r(x) with -Ap(x)in[xl-l,xll

r’(x)=Ar(x)+f(x,p(x))

G-1)

1 .

(1.2)

J

= P(x,-I).

Let u be an arbitrary solution of the TPBVP (1.1) and q(x) := f(x, u(x))

-Au(x)

for

x E [n, 61.

With the help of this interpretation we can map bounds for q in [x,_ i, xi], bounds which still have unknown variables, onto new bounds for q (cf. theorem 1). This map is based on a growth condition on f. The map is even contractive if xi - x,_ 1 is chosen sufficiently small. In this way the number of unknown variables can be reduced in the case of bounds for u on [XI-r,Xi]. Thus for any solution u of (l.l), we obtain an upper and lower bound in [xi- 1, x;]. These two bounds are now depent only on the unknown vector u(xi- 1) (theorem 2). For estimating u at the grid points we use the two-point boundary-condition Biu(a) + B*u(b) = d. and the bounds already obtained. In this way we obtain an inclusion system for u at the grid points (theorem 3). This system is then more thoroughly investigated in theorem 4. For calculating bounds for one or more solutions of (1.1). we first compute bounds at the grid points of a subdivision of [a, b] (see theorems 3 and 4) and, following this, bounds in the whole interval [a, b] (see theorem 2). The whole paper is based on a growth condition on f. We now formulate this, using the well-known interval arithmetic [I-A]. Assumption A. Let [A] = ([Ujk]) b e an m x m matrix. whose components [u,k] are real compact intervals. For j = l(l)m let y, E C[u, b] with x(x) 2 0 for every x E [a, b]. We define the vector function

For every (x, cy) E [a, b] X R” let

f(x, 4 E [Ala + [-l,l]y(x)

apply.

(1.3)

General two-point boundary-value problems

With (Y= (ai,. PIa

1083

. . , (u,) T E R” we have, as is well known

+ [--1.11+)

= ($i

.

[U,k]Uk + [-Ye,

yi(x)l>.

2. DEFINITIONS

AND NOTATION

As usual, let N be the set of natural numbers, 52 the set of real numbers and II(R) the set of real compact intervals. Let 2’ be the set R or II(R). For every natural number m, we then denote the set of m vectors and the set of m X m matrices as Z” and Zmxm respectively. The transposing of vectors and matrices is indicated by the letter T. We further define R’= := {(al,. . . , a,) T E R” ii = l( l)m

1 Uj 3

0).

Numbers from R are denoted by small letters. For vectors from R”, small bold-face letters are used. For instance, let a =

(Uj)

=

(al,.

. . ,

U,,)rfrom

R”.

Matrices from 172”“” are denoted by capital letters. For example, let 011

. . . ahI

A = (u,~) =

ER

i

( arn1.

. . amm

mxm

.

1

Numbers from II(R). vectors from II(R)m and matrices from II(R)mxm are in addition indicated by square brackets. For example. let [U] = [g, ~71= [a, b] from II(R), [a] =

([Uj])

=

([a~], . . . , [a,])‘from

and

[AI = ([ajkI> =

[a111

.*.

[al*]

[U,ll . . . In the sets introduced definitions

II(R)m

from II(R)mxm. bmml

thus far. we use the normal connections ([l-5]). [l] := (-l,l], 1 := (1,. . . ) 1>r, [I] := [l]l, I:=

, . . , l),

diag(1,

[I] : = [ 111. With the well-known Kronecker

symbol bj, :=

1 for

j=k

0

j#k

for

In addition we use the

P. WILDENAUER

1084

we define for j = l(l)m

the vectors lj I= (blj, . . . , 6,) T

and [lb:=

The modulus of an interval k, 61 HI(R)

[l]lj.

is defined by

I[a,fi]l := max()al, Ial). For every interval matrix [A] =([ Q, djk]) we define the real matrices

lPl1 := (Ikzju%cll) and R([Al) := I]AI -((ajk+

djk)/2) I

3. PREPARATIONS

We require a linear initial value problem (IVP). which we now define. Let A E R”““. For j = l(l)m let q, E C[a, b].We define 4:=(41,..

. &Lx.

In addition let two arbitrary points from [a, b] be given. We label them xl_ 1 and x,, since in all cases they will be neighboring points of a division of [a, b].Let h := Xi - x,_~> 0. Let Ui- i:=(Uri-r,.

. .U,i-l)rERm.

We define the linear IVP u’(x) = Au(x) + q(x) in (xl_,. x,) (3.1)

U(X;-1) = tl-1, For given m, xi-l, xI, A, q and Ui-i the IVP(3.1) has the unique solution U(X) = U(X)Ui_r +

’ Lr(x)U-‘(t)q(t) Ix,-1

dt,

where the matrix U(x) has the property U’(x) = A U(x). U(x,- 1) = I.

The solution u(x) is composed of V(X)

:=

:= U(X)Ui-i

V(X,X,-l,ll-1)

and w(x) : = w(x, x,-

1,

q) : =

I’

xi-1

V(x)U-*(t)q(t)

dt.

General two-point boundary-value

problems

1085

The vector function v solves (3.1) for q(x) E 0. The vector function w solves (3.1) for u;-] ..= 0. If we use the usual definition [5] of eM for matrices M, we obtain = eA(x-x,-~)> W) U(x)U-‘(f)

= eACx-‘).

The calculation of U(X) can be easily done if the Jordan normal form of A is known ([5], pages 123 and 124). Definition. For j = l(l)m

we define the vector function sj by

In

I0’

’ eA(y-nh[lb(dE

:=h

sj(_Y)

fory E [0, 11.

(3.2)

addition we define for y E [0, 11the matrix

S(Y) := MY>>* . . 3%(Y))

(3.3)

PI(Y) : = PIstY).

(3.4)

and the interval matrix

LEMMA.

(1) For j = l(l)m

and any x E[Xj-i,Xi] we have

’ eA’X-‘)[l]jdt= IXI-1

[l]sj(y)

withy := (X -Xi-,)/h.

(3.5)

(2) If for all x E [xi- i, Xi] the inclusion q(x) E[l]j is satisfied, then it follows that for every x E [Xi_i, Xi] there w(x,xi_i, (3) For j = l(l)m

q) E [l]~~y)

withy :=(x

the inclusion

-Xi-$/h.

we obtain lim max Sj(y) = 0. h-+OyE[O.1]

Proof

is

(3.6)

For 1.

Substitution: xi-1

I

0

t

X I

E

:

Xi

i

1

g=(f-~,_~)/h,t=x,_~+~h,

f = X e t$= x-t=x-X,-l

(X - Xi_ 1)/h = y, - Eh = (y - g)h, dt = h d&

eA(*-‘)[l]j d t = h

YeA(y-Q’[l]jd~=[l]Sj(y).

I0

1086

P. WILDENAUER

For 2. We get x eAcX-‘)q(t)dt E ’ eACX-‘)[lJjdt, Ixc-1 Ixi-,

w(x, Xi-l, 9) =

because of the inclusion monotony of interval arithmetic. Thus the assertion follows with (1). For 3. The interval vector [l]+j)

= I’ eA@-‘)[lJjdt xi- I

is the optimal interval inclusion of all solutions of (3.1) with ui_r := 0 and q(x) E [l]j for x E [xi-t, xi]. Thus, following [5], page 107, we obtain [l]s,{y)

EL(eqyh

-

l)[l]with

11 The assertion follows from this.

n:= max/ajkj.

n

4. THEOREMS

AND

PROOFS

Preparations

We divide the interval [a, b]. To do so, we choose a natural number n as the number of internal grid points. We define the step size h .-=b-a n+l and the equidistant

(4.1)

grid points xi:=a+ih

fori=O(l)n+l.

(4.2)

For given numbers i E (1, . . . , n + 1) and x E [x,_ 1, xi] we define for use in what follows y:=y(i,x):=(x-x,-l

(4.3)

)/A.

Let u be an arbitrary solution of the TPBVP (1.1). At the grid points xi we denote the solution values of u with IQ := u(xi). Let A E W”““. Then we define a continuous vector function q by q(x) : = f(x, u(r)) For i = l(l)n

forx E [a, b].

- Au(x)

+ 1, let cuidenote a vector from 0%“.After this preparations

(4.4)

we formulate

1. Let assumption (A) be satisfied, so that [A] and y(x) are given. We choose a matrix A with A E [A]. Then we have the following for i = l(l)n + 1:

THEOREM

VX E [Xi-

1)

Xi]:

qCx)E L1lai

vx E

[Xi_1,x;]:

q(x) E ([A] -A)[S](y)ai U(X) E eA(x-x~-l)ui-l

+

+

([A] -A)(eA(X-X1-l)ui-$+ [~]Y(x)

[S](y)cUi.

General

two-point

boundary-value

1087

problems

PKX$. For a; = (&Ii, . . . , LQ)’ E R” we get

,i

~ij[llSj(Y)

=

~[Wl(Y), . . . >Sm(Y))bi= [4(Y)

= (G(Y),

* . . > sm(Y))([ll

@>

ai

=s(Y)([ll 4.

On the basis of assumption (A) and of properties of interval arithmetic [l], we conclude the following for i = l(l)n + 1:

vx E [xl-l,x;]: VX E [Xi-l,Xi]: * + [S](y)CXi q(x) E [lILyi U(X)E eA(x-x’-‘)Ui-l + VXE [Xi-l,Xi]:

I I

q(x) : = f(x, u(x))

-Au(x)

[Alu(x) + [llY(x) -Au@)

;

Assumption (A) =

c Note on (4.5) The connection

iP1 - A)+)

([A] - A)[S](y)t~i

+ is commutative

+ ([A] -A)

+ PI Y(x) eA(x-x,-l)Ui_l) + [1]

(4.5) $x).

(4.6)

and associative. The distributivity

([Al + [Bl)u = Wlu + [Blu holds. Note on (4.6) We employ inclusion monotony and subdistributivity. and associative. The equation _

L4wl~)[Cl)

The connection

+ is commutative

= GwlmCl

holds, on the basis of [-l,lI[e

PI = [-1,1l[-max(l4, IPI),max(l4, ml.

1

Interpretation of theorem 1 and motivation of the next theorem Vectors h, . . . , u,, a,, . . . , crn+l of theorem 1 are still unknown. Thus at this point we still cannot decide whether theorem 1 is helpful for the calculation of bounds which are valid for all or special solutions of the TPBVP (1.1). In this connection the following question is important: Is it possible to give a vector Cuifor every i = l(l)n + 1 so that for all x E [x,-i, x,]

1088

P. WILDENAIJER

the bound q(x) E [ l](Yiholds, and Cuiis dependent only on the unknown vector ui_ 1? The next theorem answers this question with a “yes”. No additional assumption is necessary. However we must divide the interval [a, b] sufficiently finely, which can be done without difficulty. This situation provides subsequent justification for the division of the interval [a, b]. Conuenfions. Let [A] = ([_u,k , ~$1) and y be given by assumption (A). In the case of theorem 1 we choose a matrix A with the property A E [A]. For purposes of simplification, in what follows we always choose A := ((%k + djk)/2).

In addition, for i = l(l)n

(4.7)

+ 1 we use the definition := XE~yXl (R([A])leA’“-X’-“ui-~(

Si(ui-i)

3I

+ Y(X)).

(4.8)

2. Let assumption (A) be satisfied, and [A] and y thus given. Let interval [a. b] be so finely divided that we have the inequality

THEOREM

v:=

IIWU) y~,ylWI/

< 1

(4.9)

for a matrix norm )/aI/. Let u be an arbitrary solution of the TPBVP (1.1). Then for every i = l(l)n + 1 and x E [xi-i, xi] we have the bounds

~(x)E[~I((Z-~(IAI)~~~~,~(Y))~l~i(~~-~)~~ u(x) E e A(X-xS-l)ui-i + [S](y) [b -R([A]) Proof. For every i = l(l)n

~~~:!S(y))-‘6,(ui-*)).

(4.10) (4.11)

+ 1 we define a map g, : R~+[w1= by max S(y) (Y+ rE[O.l]

gi(a> := N[Al)

Si(Ui-1)

for (YE RF,

(4.12)

which depends on the generally unknown vector ui_ 1. On the basis of theorem 1 and rules of interval arithmetic we get the following conclusion for i = l(l)n + 1 and ai E DC!: vx E [x,_r.x;]:

vx E [Xi_1,x;]: q(x) E The maps g, are contractive unique fixed point

(

I-

R([AI),p;;lS(~I

exists for i = l(l)n

J

(4.13)

-'W-d

+ 1 an (YiE [w? v;ith q(x) E

[l]tkj

for every X E x,] we have the

Q(x) E and the rest of the assertion.

i (l(X)E [llgi(ai).

because of v < 1. Because of the Neumann lemma, they have a

Q? = In addition there [x,-i, Xi], since inclusion

[llaiI

*

n

fllai

General two-point boundary-value

1089

problems

We define the matrix Z:= Moreover

for i = l(l)n

(4.14)

R([A])

I-

+ 1 and X E [Xi-i, xi] we define [a]i(X) := e A(x-x’-l)Ui-i + [S](y)(ZSi(Ui-1)) .

(4.15)

Thus, by theorem 2, the inclusions u(X) E

(4.16)

[ali

hold. The vectors uo, . . . , un are still unknown. Thus theorem 2 provides an upper and lower bound for any solution u of the TPBVP (1. l), dependent on the unknown vectors uo = U(XO),. . . , u, = u(x,). It is thus clear that we should look for properties of the unknown vectors uo, . . . , u,. For i = l(l)n we have (4.17)

lli E [a]i(Xj) = eAhUi-i + [S](l)(Z&(Ui-1)). Furthermore

we have d E BIUO + &[a],+l(b)

= B1u0 + &(eAhun

+ [Wl)(Z4+1(u,)))

= Blue + B2eAhu, + &[S](l)(Z&+,(u,)).

Hence Biuo + &eAhu,

E &[S](l)(Z&+i(u.>)

(4.18)

+ d.

In matrix notation we have an

/

\

lsm%lt e

-I

“0

0

can

-I

.

.

.

.

csl(l)(zb,(u,))

csI(I)(z8,(u,))

“I

-

-

0

can

‘%

(4.19)

-2

B2eA”

CSl( I )(Z8,(u._,))

“.-I

“”

\

B,[Sl(I)(Z8,,,(u.))

\

+

d

/

We know that for every matrix B E R”““, matrix eB is regular, and (eB)-’ = e-’ [5, pp. 128 and 1291 holds. By multiplying the first n lines by a factor which amounts to -Ble-Ajh for the jth line, and by subsequently’adding up all n + 1 lines, we obtain the inclusion J&u. E - ,ir Ble-Ajh[S](l)(zs(uj-l))

+ &[Sl(1)(Z~,+r(u,))

+ d,

(4.20)

where B3

:=

(B1

+

B2eA(b-“))e-A(b-o)eAh

= Ble-Anh + &eAh.

(4.21)

1090

Define

the matrices

an

e

-I .

MI’

.

.

.. 0 an

e

-I

0

(4.22)

4

and

0 IV:=

(4.23) I -6,

Thus we can formulate

the following

theorem

e-J”n

as the result

of our considerations.

3. Let assumption (A) be satisfied. so that [A] and y are given. As in theorem 2, let interval [a. b] be so finely divided that u -C 1. Let u be any solution of the TPBVP (1.1). Then the vectors u. = U(Q), . . . , II, = u(x,) satisfy the inclusion system

THEOREM

Theorems 2 and 3 serve to calculate bounds which are valid for all solutions of the TPBVP (1.1) (called problem 1 in what follows). They also serve to calculate bounds which apply to special solutions (called problem 2 in what follows). In this paper we shall give further brief consideration to the first problem. Its solution requires additional assumptions, since linear eigenvalue problems are not yet excluded. and in their case there do not always exist bounds which include all solutions. On the other hand there are many TPBVPs (1 .l) with even infinitely many solutions, and in many cases problem 1 has been solved for these TPBVPs ([6-91). Problem 1 is solved if, for the vectors u 0 = U(XO), . . u, = u(x,,), one finds bounds which are valid for all solutions u of the TPBVP (1.1). The following theorem provides information on this. THEOREM

I. Matrix

4. (1) The following

B;’ exists.

two statements

are equivalent.

General

two-point

boundary-value

1091

problems

II. The problem U’(X) = AU(x)

in (a, b),

Bi U(a) + BZU(b) = I

(4.25) 1

has a regular solution U(X). (2) If B?,l exists, then the inverse of A4 exists, and we have

/

e

-an

... .-. e

-A2?l

...

e

eeeh

e

If By* exists we also have, following the assumptions

-an*

e-“‘” 8;’

-azn

-421)

e

4

-I

of theorem 3, the inclusion

Proof. For 1. The inverse of B3 exists iff B1 + BZeAcb-‘) is invertible. “I + II”. There exist linearly independent vectors vl, . . , v, E R” with (B1 + BreA(b-“))(vl,

. . > v,) = I.

The regular matrix function U(X) : = eAcx-‘)( v1 . . . . solves (4.25). “II + I”. Let U(X) be a regular solution of (4.25)

)

VA

Hence

U(x) = eA(X-o)U(a)

and I = B, U(a) + BzU(b) Thus U(a) is the inverse of B1 + B2eAcbwo).

1.

= (B, + B2eAcbma))U(a).

(4.26)

1092

P. WILDENAUER

For 2. The proof of the formula of M-’ can be easily shown by complete induction. The remainder of the assertion follows from the rules of interval arithmetic. in particular inclusion monotony and

where D, E and F are real matrices.

H

5. NUMERICAL

EXAMPLES

We now calculate bounds for all solutions of the special TPBVP -Ufl(X) = g(x, U(X))

in(O,n),u(O)

=u(n)

=0

(5.1)

with g E C([O, n] X R). On the basis of the method shown, we have developed an algorithm which calculates a bound 0 E C[O, Ed]. For every solution of the TPBVP (5.1) and for every x E [0, n] we have the inequality -zi(x) S u(x) S Li(x).

Fig. 1. Bound ri for step size h := ~17’50and

g(x. a) E 2.b+

Fig. 2. Bound ri for g(x, (u) E [13.5,14.5]cr+

1,5[l] for (x, (u) E 10, =] x [w,

2[1] for (x. (Y) E (0, x] x Iw.

General two-point boundary-value

problems

1093

REFERENCES 1. ALEF’ELDG. & HERZBERGERJ., Einfiihrung in die Intervallrechnung, B.I.-Wissenschaftsverlag, Mannheim Wien Zirich (1974). 2. MOORE R. E., Interval Analysis, Prentice-Hall, Englewood Cliffs, New Jersey (1966). 3. MOORE R. E., Methods and Applications of Interval Analysis, Prentice-Hall, Englewood Cliffs, New Jersey (1979). 4. NICKEL K., Intervall-Mathematik, Vorlesungs-Skriptum, Universitlt Freiburg (1977). 5. WALTER W., Gewtihnliche Differentialgleichungen, Springer, Berlin (1976). 6. WILDENAUER P., Domains with all solutions of non-linear problems with non-inverse-isotonic operators, Znrernational Symposium on Interval Mathematics (Edited by K. NICKEL), pp. 539-546, Academic Press, New York (1980). 7. WILDENAUER P., Construction of domains with all solutions, and the existence of extreme solutions, SIAM J. Numer. Analysis 18, 801-807 (1981). 8. WILDENAUER P., Numerische Auswertung von nichtlinearen gewijhnlichen Randwertaufgaben --x” = f(t, x), R,x = pi (i = 1,2) mit Verzweigungen, Beirr. zur Numer. Math. 11, 185-198 (1983). 9. WILDENAUER P., A new method for automatical computation of error bounds for the set of all solutions of

nonlinear boundary value problems (to appear).