A two-point boundary value problem for systems of ordinary non-linear differential equations and the lemma of Visik

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXI (1979)

A. La da (Bydgoszcz)

A two-point boundary value problem

for systems of ordinary non-linear differential equations and the lemma of Visik

1. Introduction. The lemma of Visik [4], [5] has been used for proving the existence of solutions of quasi-linear partial differential equations boundary value problems (see [1], [4], [5]). The aim of the paper is to apply the lemma for solving the two-point boundary value problem. A sufficient condition for existence of solution is given. The condition has an integral inequality form, which is convenient for verifying in a special case.

2. Preliminaries. Denote by C1 [a, b] the set of continuously differenti­

able functions on the interval [a ,b ]. We have

b

(1.1) max (|w(t)|: t e [ a ,b ] } ^ y jb — a ( { \u' (s)|2 ds) 1 / 2 + \u (a)|,

a

where u e С1 [а, b].

Let H l [a, b] denote the completion of the linear space of the functions C1 [a, b] = { u e C1 [a, b~\: и (a) = u'(a) = u{b) = u'(b) = 0} in the norm

ь b

\\u\\ = ( j u2(s)ds+ j \u'(s)\2ds)112.

a a

It is known [6] that H1 [a, b] is a separable Hilbert space and the norm || • ||

is equivalent to

IMIi = ( j \u'{s)\2ds)112.

a

There exists a countable, linearly dense in H1 [a, b] sequence {щ}

a C1 [a, b] orthonormal under the scalar product

b

(u, rh = j u'(s)u'(s)ds,

where the derivative is understood in the generalized sense. We denote

n

by Hi [a, b] the Cartesian product Y[ [<T h] with the norm

i = 1

П

INL 2 =

INI?’

= (мь •••’

i = 1

n

So, there exists a sequence {w*} <= C1 [a, b] orthonormal under the

i = 1

scalar product

П (u, v)„ = Y N vi)l

i — 1

and linearly dense in Hl[a, b~\.

Now we recall some results of [6] and [3].

Pr o p o s i t i o n 2.1. Let и: [a, b] -> R be continuous. Suppose that и has the generalized derivative u' e L2 [a, b] and is equal to zero at a and b. Then

u eH1 [a, b].

Th e o r e m 2.1. A closed sphere in a Hilbert space is weakly sequentially compact.

The following is due to Visik :

Le m m a 2.1. Let A: Rp -> Rp be a non-linear continuous mapping such that <Л(х), x ) ^ M \x\1+d — К for all x that |x| ^ L, where M > О, К ^ 0, d > 0, L ^ 0 are constants. Then there exists ùt least one solution of the equation A(x) = h, where h e R p is a constant vector.

By <•,■>, M we denote, respectively, the scalar product and the Eucli­

dean norm in the space Rp. A generalization of Lemma 2.1 is given in [1].

3. A two-point boundary value problem for systems of non-linear ordinary differential equations. Let f : Rn+1^ R n be a continuous mapping and P, Q e R n. We will consider a problem of existence of the mapping x: [0, 7 ] -> Rn and a constant vector ceR" such that

(3.1) x(t) = f ( t , x ( t ) ) + c, x(0 ) = P, x{T) = Q.

Th e o r e m 3.1. Let the mapping f satisfy the condition

(V) I ( f ( t , u ( t ) + T - ‘f + t 9 Ù( t ) \ dt «: K JkIH + Kj,

where u e H l [ 0 , T] and Xj > 0, K2 > 0, 0 ^ r < 2 are constants. Then there П

exists at least one solution (x0, c0)e Y[ C1 [0> x X" satisfing (3.1). Moreover,

i = 1

{ T - t ) P + tQ

--- г --- ’ ^°e

1 i= 1

* o(0 = Уо(0 + - У о е П ^ Ч о . г ]

Two-point boundary value problem 159

and

1 T

c0 = — - - J / ( s , x0(s))ds.

* о P roof. Note that

, ч , ( T - t ) P + t Q \ P - Q F(t, y{t)) = . / ( t , y { t ) + --- --- l+-

T T

satisfies condition (V) with P = Q = 0. Thus problem (3.1) is equivalent to (3.2) y ( t ) = F(t,.v(r)) + c, y(0) = y ( T) = 0.

If (y0, c0) is a solution of (3.2), then

( T - t ) P + tQ ко (0-1---J ---» c0

is the solution of (3.1). Take the set {wfc} described in Section 2. We put

y j(0 = Z akWk(t)

k= 1

and consider the system

(3.3) J ( ÿ j ( t ) ~ F {t, yj{t)), wk(t)} dt = 0, 1 ^ к ^ j , о

of non-linear equations in relation to aj = (al 5..., aj ) e Rj. By orthonormality of {wk|, this system has the form

(3.4) Tak- $ ( F ( t , y j ( t ) ) , wk( 0 ) dt = 0, 1 ^ к ^ j.

о

We apply the lemma of ViSik for proving the existence of zero of the mapping A: RJ- + Rj, where

M a j) = Tak— J < F( t , y j ( t )), wk( t ) )dt , A = {Au ..., Aj).

о

From condition (V) we conclude that

< З Й У > = T\a*\2 — j ( F ( t , yj(t)), ÿj(t)y dt > Т \а > \* -К Л у,\\'п -И2

0

/ IS rf rl2 \

= г й 2- к, г '2И ' -к2 = И г ( г — j ^ p - ) - к 2.

Thus, according to the inequality 2 — r > 0, condition (2.1) is satisfied, from what it follows that for each j ^ 1 there exists at least one solution aj0

of equation (3.3). Therefore for

j y°j = Z ak w

fc = l к

we have by (3.3)

(3.5) \\y]\\2n - (t, yj{t)), ÿj(t )} dt = 0, 1 < oo.

о

Condition (V) implies \\у*}\\% ^{^} К г \\y<j\\rn + K2 and therefore from 2 — r > 0 we conclude that the sequence {у®} is bounded:

(3.6) WIL < K 3, 1 ^ 7 < 00.

Thus from inequalities (1.1) and (3.6) it follows that {у®} is jointly bounded and identically continuous on the interval [О, Т]. By the Ascoli-Arzeli’s theorem exists a subsequence (y® J uniformly convergent to a continuous y0.

From Theorem 2.1 and inequality (3.6) it follows that exists a subsequence n

{ÿyks} c= (y?) weakly convergent in the space Y[ L2[0, Г ]. Thus the weak i= 1

limit is equal to y0 in the generalized sense. It is obvious that To(0)

= y0(T) = 0. Therefore, according to Proposition 2.1, y0eHj, [О, Т ].

Putting 7 = j ks in (3.3) and letting j -> oo, we obtain

T

(Уо, wk)n- J ( F ( t , y0(t)), wk{ t ) }dt = 0, 1 ^ к < go, о

and consequently

(3.7) $ ( F ( t , y0( t ) ) - ÿ0( t ) , v{ t ) } dt = 0 о

for all v e H„ [0, Г ]. We recall that

T

(3.8) j ( c, v(t)) dt = 0

о

for a v e H ^ l 0, Г] and a constant vector c e R n. Notice that y0 ( t ) - F ( t , y0(tj)

d dt

t

yo(t ) ~ J F ( s , y0(s))ds + о

t

~T

T

f F ( s , y0(s))ds

0

- zr$- F( s, y0{s))ds

1 0

and put it into (3.7). Making use of (3.8) we obtain

<M

0 ^ 0

, v (t)\ dt = 0

Two-point boundary value problem 161

for all v e H l [ 0 , Т ]. Observe that

y o( t ) +- ^r $ F (s, y0 (s)) d s - $ F (s, у о (s)) ds e Я* [0, Г]

' 0 0

what in view of (3.9) gives

1 r

ÿo(t) = F (t, v0(0) — — $ F ( s , y0{s))ds, te\_0, Т ].

1 0

n

Finally, y 0e П С1 [0, Г] by continuity of F and y 0. The proof is complete.

i= 1

One ought to find such T, P, Q that inequality (V) remains true in a special problem.

Co r o l l a r y 3.1. Assume that f is periodic in the variable t with the period T and satisfies conditions of Theorem 3.1 with P = Q. Then there exists at least one solution (x0, c0) o f problem (3.1) and x0 is periodic with the period T.

References

[1] Yu. A. D u b in s k ii, Quasi-linear elliptic and parabolic equations of arbitrary degree, Usp.

Mat. Nauk, t. 23, 1 (1968), p. 45-90 (Russian).

[2] R. F a u re , Sur l'application d'un théorème de point fixé a l'existence de solutions pério­

diques, C. R. Acad. Sci. Paris, t. 283, Sér. A (1976), p. 1295-1298.

[3] K. M a u rin , Methods of Hilbert spaces, Warszawa 1967.

[4] M. I. V isik, Solution of a quasi-linear system of the divergent form equations, under periodic boundary conditions, Dokl. Acad. Nauk SSSR, t. 137, 3 (1961), p. 502-505 (Russian).

[5] — Quasi-linear strong elliptic systems o f divergent form differential equations, Trudy Mosc.

Mat. Ob-wa, 12 (1963), p. 125-184 (Russian).

[6] L. P. V ole vie, В. P. N a n e y a k h , Some generalized function spaces and the embeddings theorems, Usp. Mat. Nauk, t. 20, 1 (1965) p. 3-74 (Russian).

INSTITUTE OF MATHEMATICS AND PHYSICS

TECHNICAL AND AGRICULTURAL ACADEMY. BYDGOSZCZ

II — Prace Matematyczne 21.1