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Solution of a non-linear discrete boundary value problem by a continuous analog of Newton's method
SOLUTION OF A NON-LINEAR DISCRETE BOUNDARY VALUE PROBLEM BY A CONTINUOUS ANALOG OF NEWTON'S METHOD* E. P. ZHIDKOV
and I. V. PUZYNIN Dubna 16 July 1968)
(Received
IN this paper we consider the non-linear discrete boundary value problem obtained from a finite difference approximation of the boundary value problem for the ordinary differential equation cp(yj = v”+f(x, Y(O) = y(l)
Y) = 0, = 0.
(0.1) (0.2)
In [ll a continuous analog of Newton’s method was used to find an approximate solution of problem (0.1)~(0.2). The existence of these solutions is assumed. The proposed computational scheme is a realization of Euler’s method of solving a Cauchy problem for a certain differential equation in B-space. It reduces to the solution for a fixed value of the parameter t of a boundary value problem for a linear ordinary differential equation, and to a subsequent advancement along the parameter by a simple difference relation. The solution of the boundary value problem for fixed t is assumed to be found numerically, for example, by the use of difference approximations. It is easy to see that this computational scheme is a numerical realization of the continuous analog of Newton’s method of solving the difference operator equation approximating to the original boundary value problem (O.lk(O.2). In the present paper a justification of this method is given: a proof is given of the existence of a solution of the discrete boundary value problem which is the difference analog of problem (0.1)~(0.2), and the convergence of the approximate solution to the solution of the original problem is proved. The result obtained in [2l is used in proving the existence of the solution of the discrete boundary value problem. * Zh. vychisl.
Mat. mat. Fiz.
9, 2, 442-447,
254
1969.
Solution of a non-linear discrete
boundary value problem
255
Similar problems were considered, for example, in [3l, pp. 347-388, [41, pp. 59-67. Thus, in [4l a proof is given of the existence and method of finding an approximate solution of problem (O.l)-(0.2) by Newton’s method. However, by connecting the questions of the existence and uniqueness of the solution of the original problem with the method of the approximate solution, the author obtains extremely restricted conditions for the applicability of the method. 1
Approximations
and fundamental
assumptions
Let N be a natural number and h = N-‘. We form a uniform subdivision of the real axis segment CO,11 by means of the finite subset @t} c IO, 11, containing the points zi = ih, i = 1, . . . , N - 1. Let C’2’[0, 11 be a set of real functions satisfying the condition (0.2) which are twice differentiable in the interval 10, 11. We define the net image yh of the function Ytx) G C@)[O, 11 as the (N + 1)-dimensional vector
Yh =
(0, yi, .-.,
YN_l,
O),
with the components yk = ytx,). We denote by O1 the set of all the net images of the space C’[O, 13. It is easy to show that (I)‘1is a B-space, if the norm of yk is defined by
We also consider the set of real functions continuous in (0, 11, and denote it by C [O, 11. In the sm way we construct the set of net images of the elements of C[O, 11, and denote it by 0,. Introducing the norm of the element as yk E % llyhll~ = maxlyi(, it can be shown that (I)2will be a B-space. we consider the linear operators &y(s)
=
h-1 [Y (3: + IL) .-- Y(2)1, A,,? =
We notice that if y(r) E COG’,
For the functions
y(z) E @I, 5 E {ziJ
V,,Y(~1 = h-’ IY (4 - Y (5 - h)l,
h-‘(A,, - \:h).
E. P. Zhidkov
Ah%(t)
=
and 1. V. Puzynin
f’(c)+
-
h2 12
&J”‘(E).
(1.2)
Bearing in mind the last relation, we approximate the boundary value problem (0.1)~(0.2) using the non-linear difference problem %(Y) = A~‘Y@)+ f(x, Y(X)) = 0,
5 E
{Xf).
Y=
a,
(1.3)
which represents a non-linear system of N - 1 algebraic equations for the N - 1 unknowns y@i), . . . , y(rnr_i). We assume that the assumptions of Theorem 2 of [ll are satisfied. 1. Let the solution y* of the boundary value problem (O.l)-(0.2) exist and in the case of non-uniqueness can be localized, that is, it is possible to construct functions z(n) and 2 (x) twice continuously differentiable in [O, 11, which satisfy the boundary condition (0.2) but do not have tangents in common with y* (2~)at the points x = 0 and x = 1, z(x) < Z(X) for 0 < x < 1, so that within the domain G
there is only one solution y*, the inequalities
Z(X) < y’(z) <
Z(s)
being satisfied
for any X, 0 < x < 1. We suppose that f(x, y) has in G continuous derivatives to the second order inclusive.
up
2. The boundary value problem 0” -l- fu'(x,
v(0) = v(1) = 0
Y)" = 0,
has only a trivial solution for any continuously
differentiable
(1.4)
function y (5) E G.
If condition 1 of the localization of the solution is satisfied, it is easy to see that a K > 0 can be found such that in the sphere D c C(z) [O, 11 Ilv -
~*llc(J
f
K
there is only one solution y* of problem (0.1)~(0.2). We suppose that the domain of definition of the operator cph is the &pace
Solution
of a non-linear
discrete
boundary
value
257
problem
a,, and that its domain of values in the B-space OZ. For sufficiently small h it is possible tp construct a sphere
Di c
cPi
Ki > 0,
(1.5)
such that it contains only one net image of the solution of problem (O.lMO.2) and all the yh E D, are net images of y(s) ED. In D, the operator 4,(y) has a Frkhet derivative 4;(y) and a linear Gateaux derivative c$‘i(yl bounded in the neighbourhoodof every point of the sphere D,. The Frkchet derivative of the operator ~J~(Y)is a linear operator, and vh’(yj
=
{Ah2
+
fv’12? ytz))h
z
E
{Z(}.
We first prove some auxiliary propositions. 2. Lemmas Lemma 1 If condition (l-2) of section 1 are satisfied, the operator 4;(y) has an inverse operator $h(y)-’ in D, for sufficiently small h. Proof.To show that the operator +h(y) has an inverse operator 4;;(y)-’ in D,,it is sufficient to prove that the discrete boundary value problem Ah% t- fv’(q
Y
(~1) CJ =
0,
has only a trivial solution for all sufficiently small h. We assume the opposite: no matterhow small h,, it is possible to find an h < h, and a y E D such that the system of linear algebraic equations (2.1) has a norrtrivial solution. We consider the natural series i, 2.
. . . , m, . . . .
(2.21
We establish a correspondence between each element m of (2.21 and the pair (h m, ym) for which (2.1) has a non-trivial solution. The sequence (hm 1converges to zero. By ArzelA’s theorem it is possible to choose from the sequence of functions {y,,,}, I/,,, (5) E D, a convergent subsequence, the limit function y’(x)
E. P. Zhidkov
258
and I. V. Puzynin
being continuously differentiable in [O, 11 and belonging to the domain G. We will consider that for every pair (hm, Y,), m = 1, 2, . . . , where hm and y, are elements of the corresponding convergent sequences, the system of linear algebraic equations (2.1) has a linearly independent system of non-trivial solutions. From each such system we select any non-trivial solution v,,,(z), Ii urn 112= 1. From the z E {~i}m, fill d multiply it by a constant such that sequence of points 3tm E [O, 11 at which the equation 1v, (xm) 1 = I, is satisfied, it is possible to choose a convergent subsequence x,,, -* f E 10,1], m + 00. By equation (2.1) the sequence of functions v, has uniformly bounded second It is easy to show that the first divided differences divided differences A,%,. Ahv, are also uniformly bounded. For each m we continue the functions v, and Ahvm = w, onto the whole segment IO, 11 by linear interpolation at the adjacent nodal points. The continuous functions V,(X) and W,(x) are uniformly bounded and equicontinuous. By Arzelh’s theorem it is possible to select from them uniformly convergent subsequences, the limits of which will be functions u (xl and w (xl continuous in CO,11. By using the method described, for example, in IX, pp. 39-43, it can be shown that du/dx= w and u (xl satisfies the boundary value problem (1.41. From this we obtain a contradiction with condition 2 of section 1: the solution of problem (1.4) for y’ E G at a point x’ E [O,11 equals 1 in modulus. The contradiction
proves the lemma.
Lemma2 Let {g(x)1 and Ir(xl1 be families of uniformly bounded functions and {v(x)}, x E {xi], a set of solutions of the difference equation Ah+(x)
=
g(x)v(x)
+ r(x)
(2.3)
with the fixed initial condition
u(0) -
a,
v(k)
= bh+a.
(2.41
Then the set of functions u (xl, uniformly bounded in the segment [O, 11 in the sense of the norm, is defined by equation (1.1) (the estimate is independent of h). The lemma is almost obvious: to prove it, it is sufficient to construct the difference analog of Picard’s method for calculating the solutions of problem (2.3M2.4) by successive approximations. It can be shown that the successive
Solution
of a non-linear
discrete
259
boundary value problem
approximations are uniformly bounded and converge to the solution of the original problem. This implies the statement of the lemma. Lemma 3 The operator +‘,(y)-’ is bounded in norm in the sphere D, defined by the inequality (1.5). Proof.
To prove the inequality Ii (Fh'(Yj-' 11Q B,
B > 0,
in D,,it is sufficient to show that if u W is a solution of the discrete boundary value problem
then IIc(z) II, < B.
(2.6)
Letu,(x) and u,kx) be linearly independent solutions of the homogeneous equation AA~u(x) +fa’(x,
Y(x))u(x)
= 0,
X E
{Xi),
y(xj ED,
satisfying the initial conditions a,(O)
=Y
1,
h(h)
= 1;
G(O) = 0,
u,(h)
= h.
Also, let u’(x) be a solution of the inhomogeneous difference At,*v(x) + Iv’(x, u(x))ti(x)
= r(x),
5 E
(~1,
y(r) ED,
equation
II r(z)
112<
with the initial conditions 5(O) = a,
U(h) =
bh+u.
Then the solution of problem (2.5) can be represented
in the form
1,
260
E. P. Zhidkov
u(x)=
-
-aui(z)+
and 1. V. Puzynin
c(l)+
au1
-(I)
(1) ‘h(t)+
S(S).
(2.71
By Lemma2, the functions C(x), U,(X)and u,(x) are uniformlybounded with respect to the normdefined by equation (1.1). We show that
inf 1u2(1) I=
A > 0.
(2.fv
YED
We assume the opposite: for all sufficiently small h inf 1ua(l)j=O. llED
A sequence u,,(l) can be chosen for which
lim (rz2k (,I) 1 = 0. A sequence
Yk(x) ED corresponds to it. By Arzela’s theore::
is possible to select from
it a convergent subsequence which in [O, 11 converges uniformlyto a continuously differentiable function y(s) E G. The solution of the limiting difference equation
Ah21z(s)-i-fu’(x,
jj(x))u(x)
= 0,
x E cd
with the initial conditions u(0) = 0,
u(h) = h,
not being identically zero, will vanish at x = 1. Passing to the limit as h -, 0, it is easy to obtain a contradiction with condition 2 of section 1. Therefore, the relation (2.8) is proved. Taking account of the latter, the required inequality (2.6) is easily obtained from expression (2.7). The lemmas is proved. 3. Fundamental results As was shown in Lemma3, section 2, the operator (ph’(yh)-i in normin the sphere D, (see (1.5) ), that is, the inequality 11(ph’(Yh)
-’
11 <
B,
is satisfied. We consider the sphere
Yh
E
Di,
B > 0.
is bounded
Solution
of a non-linear
discrete
boundary
11 YA - YA*It < B II ~~(~fi*~
value problem
II.
261
(3.1)
Noticing that the solution y* of problem (O.lk(O.2) is four times continuously differentiable in the segment [O, 11, and also the relations (1.2) and (1.31, it is easy to see that for sufficiently small h the sphere (3.1) will be situated within the sphere I),. Therefore, for sufficiently small h the conditions of Theorem 1 of [21 are satisfied for the operator $h(yh). This implies that the differential equation
has a solution y,(t) for values of t in the interval 0 < t < M, its values lying in exists and is the root of equation the sphere (3.11. The limit limyh(t)= yr,, t-+m (1.3). Since yh is situa~ in the sphere (3.1) and the relation 11 qh(yh*) /i -+ 0, is satisfied as h -+0, this implies that lim IIYh - Yh*ll= 0. h-0
The last relation implies the convergence of the solution of the difference problem (1.3) to the solution of the original problem (O.lL(O.2). Thus we have proved the following theorem.
If the conditions l-2 of section 1 are satisfied, the solution of the discrete value problem (1.3) exists for sufficiently small h, and converges to the solution of problem (O.lMO.2) as h + 0. Having proved the existence theoremof the solution of the non-linear difference problem (1.3), we find ourselves with conditions for which Theorem 1 of 111is applicable. Therefore, to find the solution of equation (1.3) we apply the continuous analog of Newton’s method with an appropriatechoice of the initial ~pro~mation in the sphere D,. In its n~eric~ realization presented in (11, the convergence of the method is ensured by the steps in the variable x and time t tending to zero independently. Note. In 141,which is a generalization of [31, the existence of the solution
262
E. P. Zhidkov
and I. V. Puzynin
of a similar difference problem and its convergence to the exact solution, are obtained on the assumption that sup fv’ = Tj < J-s,
O
--m
The latter condition ensures the existence and uniqueness of the solution of the original boundary value problem (O.lMO.2). The assumption of the existence the constraints on the quantity f’,.
of solutions of problem (O.lMO.2) reduces
Translated
by J. Berry
REFERENCES 1.
ZHIDKOV, E. P. and PUZYNIN, I. V. A method of introducing a parameter in the solution of boundary value problems for non-linear second-order differential equations. Zh. vj;chisl. Mat. mat. Fiz. 7, 5. 1086-1095, 1967.
2.
GAWRIN, M. K. Non-linear functional equations and continuous methods. Izv. Vuzov. Matematika. 5, 6, 18-31, 1956.
analogs
3.
HENRICI,
Equations.
P. Discrete
New York,
Variable Methods
in Ordinary Differential
of iterative
Wiley,
1962.
4.
LEES, M. Discrete methods for non-linear twepoint boundary value problems. solution of partial differential equations. Proc. Symp. held Univ. Maryland, Park, Maryland, May 3-6, 1965. Academic Press, New York, 1966.
Numerical College
5.
PETROVSKII, I. G. Lectures on the theory of ordinary differential equations (Lektsii po teorii obyknovennykh differentsial’nykh uravneniil. pp. 39-43, Nauka, Moscow, 1964.
and I. V. PUZYNIN Dubna 16 July 1968)
(Received
IN this paper we consider the non-linear discrete boundary value problem obtained from a finite difference approximation of the boundary value problem for the ordinary differential equation cp(yj = v”+f(x, Y(O) = y(l)
Y) = 0, = 0.
(0.1) (0.2)
In [ll a continuous analog of Newton’s method was used to find an approximate solution of problem (0.1)~(0.2). The existence of these solutions is assumed. The proposed computational scheme is a realization of Euler’s method of solving a Cauchy problem for a certain differential equation in B-space. It reduces to the solution for a fixed value of the parameter t of a boundary value problem for a linear ordinary differential equation, and to a subsequent advancement along the parameter by a simple difference relation. The solution of the boundary value problem for fixed t is assumed to be found numerically, for example, by the use of difference approximations. It is easy to see that this computational scheme is a numerical realization of the continuous analog of Newton’s method of solving the difference operator equation approximating to the original boundary value problem (O.lk(O.2). In the present paper a justification of this method is given: a proof is given of the existence of a solution of the discrete boundary value problem which is the difference analog of problem (0.1)~(0.2), and the convergence of the approximate solution to the solution of the original problem is proved. The result obtained in [2l is used in proving the existence of the solution of the discrete boundary value problem. * Zh. vychisl.
Mat. mat. Fiz.
9, 2, 442-447,
254
1969.
Solution of a non-linear discrete
boundary value problem
255
Similar problems were considered, for example, in [3l, pp. 347-388, [41, pp. 59-67. Thus, in [4l a proof is given of the existence and method of finding an approximate solution of problem (O.l)-(0.2) by Newton’s method. However, by connecting the questions of the existence and uniqueness of the solution of the original problem with the method of the approximate solution, the author obtains extremely restricted conditions for the applicability of the method. 1
Approximations
and fundamental
assumptions
Let N be a natural number and h = N-‘. We form a uniform subdivision of the real axis segment CO,11 by means of the finite subset @t} c IO, 11, containing the points zi = ih, i = 1, . . . , N - 1. Let C’2’[0, 11 be a set of real functions satisfying the condition (0.2) which are twice differentiable in the interval 10, 11. We define the net image yh of the function Ytx) G C@)[O, 11 as the (N + 1)-dimensional vector
Yh =
(0, yi, .-.,
YN_l,
O),
with the components yk = ytx,). We denote by O1 the set of all the net images of the space C’[O, 13. It is easy to show that (I)‘1is a B-space, if the norm of yk is defined by
We also consider the set of real functions continuous in (0, 11, and denote it by C [O, 11. In the sm way we construct the set of net images of the elements of C[O, 11, and denote it by 0,. Introducing the norm of the element as yk E % llyhll~ = maxlyi(, it can be shown that (I)2will be a B-space. we consider the linear operators &y(s)
=
h-1 [Y (3: + IL) .-- Y(2)1, A,,? =
We notice that if y(r) E COG’,
For the functions
y(z) E @I, 5 E {ziJ
V,,Y(~1 = h-’ IY (4 - Y (5 - h)l,
h-‘(A,, - \:h).
E. P. Zhidkov
Ah%(t)
=
and 1. V. Puzynin
f’(c)+
-
h2 12
&J”‘(E).
(1.2)
Bearing in mind the last relation, we approximate the boundary value problem (0.1)~(0.2) using the non-linear difference problem %(Y) = A~‘Y@)+ f(x, Y(X)) = 0,
5 E
{Xf).
Y=
a,
(1.3)
which represents a non-linear system of N - 1 algebraic equations for the N - 1 unknowns y@i), . . . , y(rnr_i). We assume that the assumptions of Theorem 2 of [ll are satisfied. 1. Let the solution y* of the boundary value problem (O.l)-(0.2) exist and in the case of non-uniqueness can be localized, that is, it is possible to construct functions z(n) and 2 (x) twice continuously differentiable in [O, 11, which satisfy the boundary condition (0.2) but do not have tangents in common with y* (2~)at the points x = 0 and x = 1, z(x) < Z(X) for 0 < x < 1, so that within the domain G
there is only one solution y*, the inequalities
Z(X) < y’(z) <
Z(s)
being satisfied
for any X, 0 < x < 1. We suppose that f(x, y) has in G continuous derivatives to the second order inclusive.
up
2. The boundary value problem 0” -l- fu'(x,
v(0) = v(1) = 0
Y)" = 0,
has only a trivial solution for any continuously
differentiable
(1.4)
function y (5) E G.
If condition 1 of the localization of the solution is satisfied, it is easy to see that a K > 0 can be found such that in the sphere D c C(z) [O, 11 Ilv -
~*llc(J
f
K
there is only one solution y* of problem (0.1)~(0.2). We suppose that the domain of definition of the operator cph is the &pace
Solution
of a non-linear
discrete
boundary
value
257
problem
a,, and that its domain of values in the B-space OZ. For sufficiently small h it is possible tp construct a sphere
Di c
cPi
Ki > 0,
(1.5)
such that it contains only one net image of the solution of problem (O.lMO.2) and all the yh E D, are net images of y(s) ED. In D, the operator 4,(y) has a Frkhet derivative 4;(y) and a linear Gateaux derivative c$‘i(yl bounded in the neighbourhoodof every point of the sphere D,. The Frkchet derivative of the operator ~J~(Y)is a linear operator, and vh’(yj
=
{Ah2
+
fv’12? ytz))h
z
E
{Z(}.
We first prove some auxiliary propositions. 2. Lemmas Lemma 1 If condition (l-2) of section 1 are satisfied, the operator 4;(y) has an inverse operator $h(y)-’ in D, for sufficiently small h. Proof.To show that the operator +h(y) has an inverse operator 4;;(y)-’ in D,,it is sufficient to prove that the discrete boundary value problem Ah% t- fv’(q
Y
(~1) CJ =
0,
has only a trivial solution for all sufficiently small h. We assume the opposite: no matterhow small h,, it is possible to find an h < h, and a y E D such that the system of linear algebraic equations (2.1) has a norrtrivial solution. We consider the natural series i, 2.
. . . , m, . . . .
(2.21
We establish a correspondence between each element m of (2.21 and the pair (h m, ym) for which (2.1) has a non-trivial solution. The sequence (hm 1converges to zero. By ArzelA’s theorem it is possible to choose from the sequence of functions {y,,,}, I/,,, (5) E D, a convergent subsequence, the limit function y’(x)
E. P. Zhidkov
258
and I. V. Puzynin
being continuously differentiable in [O, 11 and belonging to the domain G. We will consider that for every pair (hm, Y,), m = 1, 2, . . . , where hm and y, are elements of the corresponding convergent sequences, the system of linear algebraic equations (2.1) has a linearly independent system of non-trivial solutions. From each such system we select any non-trivial solution v,,,(z), Ii urn 112= 1. From the z E {~i}m, fill d multiply it by a constant such that sequence of points 3tm E [O, 11 at which the equation 1v, (xm) 1 = I, is satisfied, it is possible to choose a convergent subsequence x,,, -* f E 10,1], m + 00. By equation (2.1) the sequence of functions v, has uniformly bounded second It is easy to show that the first divided differences divided differences A,%,. Ahv, are also uniformly bounded. For each m we continue the functions v, and Ahvm = w, onto the whole segment IO, 11 by linear interpolation at the adjacent nodal points. The continuous functions V,(X) and W,(x) are uniformly bounded and equicontinuous. By Arzelh’s theorem it is possible to select from them uniformly convergent subsequences, the limits of which will be functions u (xl and w (xl continuous in CO,11. By using the method described, for example, in IX, pp. 39-43, it can be shown that du/dx= w and u (xl satisfies the boundary value problem (1.41. From this we obtain a contradiction with condition 2 of section 1: the solution of problem (1.4) for y’ E G at a point x’ E [O,11 equals 1 in modulus. The contradiction
proves the lemma.
Lemma2 Let {g(x)1 and Ir(xl1 be families of uniformly bounded functions and {v(x)}, x E {xi], a set of solutions of the difference equation Ah+(x)
=
g(x)v(x)
+ r(x)
(2.3)
with the fixed initial condition
u(0) -
a,
v(k)
= bh+a.
(2.41
Then the set of functions u (xl, uniformly bounded in the segment [O, 11 in the sense of the norm, is defined by equation (1.1) (the estimate is independent of h). The lemma is almost obvious: to prove it, it is sufficient to construct the difference analog of Picard’s method for calculating the solutions of problem (2.3M2.4) by successive approximations. It can be shown that the successive
Solution
of a non-linear
discrete
259
boundary value problem
approximations are uniformly bounded and converge to the solution of the original problem. This implies the statement of the lemma. Lemma 3 The operator +‘,(y)-’ is bounded in norm in the sphere D, defined by the inequality (1.5). Proof.
To prove the inequality Ii (Fh'(Yj-' 11Q B,
B > 0,
in D,,it is sufficient to show that if u W is a solution of the discrete boundary value problem
then IIc(z) II, < B.
(2.6)
Letu,(x) and u,kx) be linearly independent solutions of the homogeneous equation AA~u(x) +fa’(x,
Y(x))u(x)
= 0,
X E
{Xi),
y(xj ED,
satisfying the initial conditions a,(O)
=Y
1,
h(h)
= 1;
G(O) = 0,
u,(h)
= h.
Also, let u’(x) be a solution of the inhomogeneous difference At,*v(x) + Iv’(x, u(x))ti(x)
= r(x),
5 E
(~1,
y(r) ED,
equation
II r(z)
112<
with the initial conditions 5(O) = a,
U(h) =
bh+u.
Then the solution of problem (2.5) can be represented
in the form
1,
260
E. P. Zhidkov
u(x)=
-
-aui(z)+
and 1. V. Puzynin
c(l)+
au1
-(I)
(1) ‘h(t)+
S(S).
(2.71
By Lemma2, the functions C(x), U,(X)and u,(x) are uniformlybounded with respect to the normdefined by equation (1.1). We show that
inf 1u2(1) I=
A > 0.
(2.fv
YED
We assume the opposite: for all sufficiently small h inf 1ua(l)j=O. llED
A sequence u,,(l) can be chosen for which
lim (rz2k (,I) 1 = 0. A sequence
Yk(x) ED corresponds to it. By Arzela’s theore::
is possible to select from
it a convergent subsequence which in [O, 11 converges uniformlyto a continuously differentiable function y(s) E G. The solution of the limiting difference equation
Ah21z(s)-i-fu’(x,
jj(x))u(x)
= 0,
x E cd
with the initial conditions u(0) = 0,
u(h) = h,
not being identically zero, will vanish at x = 1. Passing to the limit as h -, 0, it is easy to obtain a contradiction with condition 2 of section 1. Therefore, the relation (2.8) is proved. Taking account of the latter, the required inequality (2.6) is easily obtained from expression (2.7). The lemmas is proved. 3. Fundamental results As was shown in Lemma3, section 2, the operator (ph’(yh)-i in normin the sphere D, (see (1.5) ), that is, the inequality 11(ph’(Yh)
-’
11 <
B,
is satisfied. We consider the sphere
Yh
E
Di,
B > 0.
is bounded
Solution
of a non-linear
discrete
boundary
11 YA - YA*It < B II ~~(~fi*~
value problem
II.
261
(3.1)
Noticing that the solution y* of problem (O.lk(O.2) is four times continuously differentiable in the segment [O, 11, and also the relations (1.2) and (1.31, it is easy to see that for sufficiently small h the sphere (3.1) will be situated within the sphere I),. Therefore, for sufficiently small h the conditions of Theorem 1 of [21 are satisfied for the operator $h(yh). This implies that the differential equation
has a solution y,(t) for values of t in the interval 0 < t < M, its values lying in exists and is the root of equation the sphere (3.11. The limit limyh(t)= yr,, t-+m (1.3). Since yh is situa~ in the sphere (3.1) and the relation 11 qh(yh*) /i -+ 0, is satisfied as h -+0, this implies that lim IIYh - Yh*ll= 0. h-0
The last relation implies the convergence of the solution of the difference problem (1.3) to the solution of the original problem (O.lL(O.2). Thus we have proved the following theorem.
If the conditions l-2 of section 1 are satisfied, the solution of the discrete value problem (1.3) exists for sufficiently small h, and converges to the solution of problem (O.lMO.2) as h + 0. Having proved the existence theoremof the solution of the non-linear difference problem (1.3), we find ourselves with conditions for which Theorem 1 of 111is applicable. Therefore, to find the solution of equation (1.3) we apply the continuous analog of Newton’s method with an appropriatechoice of the initial ~pro~mation in the sphere D,. In its n~eric~ realization presented in (11, the convergence of the method is ensured by the steps in the variable x and time t tending to zero independently. Note. In 141,which is a generalization of [31, the existence of the solution
262
E. P. Zhidkov
and I. V. Puzynin
of a similar difference problem and its convergence to the exact solution, are obtained on the assumption that sup fv’ = Tj < J-s,
O
--m
The latter condition ensures the existence and uniqueness of the solution of the original boundary value problem (O.lMO.2). The assumption of the existence the constraints on the quantity f’,.
of solutions of problem (O.lMO.2) reduces
Translated
by J. Berry
REFERENCES 1.
ZHIDKOV, E. P. and PUZYNIN, I. V. A method of introducing a parameter in the solution of boundary value problems for non-linear second-order differential equations. Zh. vj;chisl. Mat. mat. Fiz. 7, 5. 1086-1095, 1967.
2.
GAWRIN, M. K. Non-linear functional equations and continuous methods. Izv. Vuzov. Matematika. 5, 6, 18-31, 1956.
analogs
3.
HENRICI,
Equations.
P. Discrete
New York,
Variable Methods
in Ordinary Differential
of iterative
Wiley,
1962.
4.
LEES, M. Discrete methods for non-linear twepoint boundary value problems. solution of partial differential equations. Proc. Symp. held Univ. Maryland, Park, Maryland, May 3-6, 1965. Academic Press, New York, 1966.
Numerical College
5.
PETROVSKII, I. G. Lectures on the theory of ordinary differential equations (Lektsii po teorii obyknovennykh differentsial’nykh uravneniil. pp. 39-43, Nauka, Moscow, 1964.