Generalized differential equations for maps of Banach spaces

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXX (1990)

Ta d e u s z Po r e d a ( t ô d z )

Generalized differential equations for maps of Banach spaces

Abstract. We consider a natural generalization of differential equations (which have the properties of ordinary differential equations) for maps of one Banach space into another.

1. Introduction. Let X , Y be real Banach spaces and let L(X, Y) be the space of continuous linear maps from X into Y.

De f in it io n

1. Let U, V be open subsets of X and Y, respectively, and let h be a map from U into X, and H a map from U x V into Y. A differential equation of the form

(1) Df(x)(h(x)) = H(x, f(x)) for x e U will be the subject of our considerations.

This equation will be called a generalized differential equation of the first order, and a function / (differentiable at each point x of U in the direction of the vector h(x)) defined on U with values in V, satisfying equation (1), will be called its solution.

R em ark 1. Equations of this kind can be found among others in [3], [4], [7], [8]. They appear there as consequences of some geometrical properties of holomorphic maps. However, in these papers no analysis of such differential equations is made.

De f in it io n

2. Let U be an open set in X, а Ф 0 a fixed point of X, g a map from U into Y, A a map from U into L(X, Y). A generalized differential equation of the form

(2) Df(x){a) + A(x)(/(x)) = g(x) for

is

called a generalized linear differential equations of the first order.

R em ark 2. For X = Y = R, (2) is an ordinary linear differential equation.

When X = R, Y = Rn, solving (2) amounts to solving a system of linear

differential equations.

When X = Rm, Y = Rn, (2) is just a linear system of partial differential equations of the first order (see e.g. [6]).

In the first part of this paper we shall give an initial condition such that with suitable assumptions concerning the maps h and H, there exists exactly one solution of equation (1) satisfying this initial condition. In the second part we shall deal with the form and properties of the solution of the generalized linear differential equations.

While discussing equations (1) and (2) we shall need the following notations.

Let a be a fixed point of X and La = [xa: xeR ). By X a we shall denote any complementary subspace to La (for the definition see [1], p. 61). Further, let Xa° = {x + x0; x e X a} where x0 is a fixed point of X. By B(x0, r) we shall denote the ball with radius r and centre x0, and Хд°(г) = I J ° n B ( i 0, r).

De f in it io n

3. The maps p : X - > X ai t : X ^ R such that

(3) x = p(x) + t(x)a for x e X

will be called projection operators. (These maps are obviously linear opera­

tions). The correctness of this definition follows from Th. 4.6 from [1], p. 60.

2. Cauchy problem for the generalized differential equation of the first order. Let U, V be open subsets of X and Y, respectively, h a map from U into X and H a map from U x V into Y. Let x0 be any point of U, X h{Xo) a certain (fixed in further considerations) complementary subspace to Lh(xo) and let f 0 be a function from X h{xo)n U into V.

With the above notations we can formulate the following

Th e o r e m

1. I f the maps h, H and f 0 are continuously differentiable and the projection operators are continuous then for any x 0eU such that h(x0) ф 0 there exists a neighbourhood U0 of x0 such that the equation

(4) Df{x)(h(x)) = H(x, f{x)) for x e U 0 has exactly one solution satisfying the condition

(4') / (*) = /о (*) for x

X%X0) n U 0.

P ro o f. First, consider the differential equation (5) dv

with the initial condition y(0, ÿ) = ÿ where ÿ e X xg{xo) n U. By Theorem 10.8.1

from [2] there exist e > 0, r > 0 and a function v = v{t, y), where (Г, ÿ)e( — s, e)

х Х ь (°Хо)(г), which is a solution of equation (5), continuously differentiable on

( —e, e) x Xfc(°Xo)(r). Let us introduce the auxiliary function v = v(t, x0 + j;) for

(Г, y)e( —e,

)

1 { ,

)(

). Notice that (dv/dy)(0, 0) = /, where I is the identity

operator on X h(Xo), and (dv/dt){0, 0) = h(x0). Since h(x0) Ф 0 therefore DiT(0, 0) is a linear homeomorphism from R x X h(xo) onto X. In virtue of Theorem 10.2.5 from [2] there exist e0 > 0, r0 > 0 and a neighbourhood Ü0 of x0 (Ü0 a U) such that v is a diffeomorphism of class C1 from ( — e0,

x X°(xo)(r0) onto Ü0. From the above follows the existence of an inverse map to v such that i>_1(x) = (T(x), P(x)), for x e Ü 0, is continuously differentiable -on Ü0 and maps Ü0 onto ( - £ 0, e0) x ^ (°Xo)(r0).

Next, consider the differential equation

(6) — (s, x) = H(v{s, x), w(s, x)+ f0(xj). dw

with the initial condition w(0, x) = 0 for x e I ^ o)(r0). By Theorems 10.8.1 and 10.8.2 from [2] there exist ^ > 0 (e1 < e0), r1 > 0 (rl < r0) such that (6) has exactly one solution w = w(s, t) defined and continuously differentiable on

and such that w(0, x) — 0.

Now, define

(7) w(s, x) = w(s, x )+ /0(x) for (s, x ) e ( - 8 u e j x This function is differentiable and satisfies the equation (8) dw — (s, x) = H(v{s, x), w(s, x)) for (s, x ) e ( - e 1, x

with the initial condition w(0, jc) = f 0(x).

Now, let U0 be the image of ( — by

We define a function / by

(9) f(x ) = w(T(x), P(x)) for x e U 0, where w is defined by (7).

We now prove that / satisfies equation (4). By definition of / and (8) we obtain

л

(10) Df(x) = H(v(T(x), P(x)),f(x))DT(x) + -£(T{x), P(x))DP{x) for x e U0.

Let x be a fixed point of U0. In virtue of the definition of the maps T, P we see that for t e T ( U 0)

(11) T{v(t,P(x))) = t, P{v{t,P(x))) = P(x).

Differentiating (11) with respect to t we obtain for t = T(x) (12) DT{x)(h(x)) = 1, DP(x)(h(x)) = 0.

Thus equation (10) has the form

(13) Df(x){h(x)) = H{v(T(x), P(x)), f(x)) for x e U 0.

Since v(T(x), P(x)) = .x for

therefore

(14) Df (x)(h(x)) = H ( x , f (x)j for x e U 0.

Hence / satisfies equation (4) and it is obvious that / fulfils the initial condition.

It remains to show uniqueness. Suppose that two different maps f x, f 2 satisfy equation (4) on U0 and condition (4'). Hence there exist b e U 0 such that M b) Ф f 2(b). Let b = P(b), w1(s,b) = Ми(Б,Ь)) and w2(s, b) = f 2(v(s, b)) when N < £i • Then the functions wx and w2 are also different and satisfy (8),

which contradicts the uniqueness of the solution of (8). This ends the proof.

3. Generalized linear differential equation of the first order. Let U be an open set in X, a a fixed point of X different from 0, g a map from U into У, and A a continuous map from U into L(X, Y).

Let x be a fixed point of X , X a a certain complementary space to La and let p, t be the projection operators (see Def. 3).

Assume, additionally, that U is convex in the direction of La (i.e. for every

x e U

the segment [x, p(x)] is contained in U). \ ^

Observe that the set J p{x) = {

; p(x) + zcieU} is an interval in R and if x e U then t(x )e J p(x).

Let K(-, f(x), p(x)): J pix)-* L (X , Y) be the solution of the equation (15) — ( dR

, t(x), p(x)) = А(р(х) + ат)(Я{т, f(x), p{x)))

satisfying the initial condition R(t(x), r(x), p(x)) = I Y, where I Y is the identity operator on Y (By Theorem IX. 10 from [6], such a function exists and is uniquely determined).

With the above notations we can formulate the following.

Th e o r e m 2.

Let a be a nonzero point of X, U an open subset of X including 0 and convex in the direction of La, and f 0 a map from U n X a into Y. I f the maps A, g are continuous then there exists exactly one solution of the equation (16) Df(x) (a) = A (x) (f(x)) + g (x) for

U

satysfying the condition

( 16') / (x) = /о (x) for x e f n X a, and it is of the form

(17) /(x ) = R(t(x), 0, p(x))(f0(p(x)))

t(x)

+ R(t(x), 0, p{x))( J' R(0, s, p(x))g(p(x) + as)ds) о

for

U.

P roof. We first prove that if / is a solution of (16), (16') then it has the form (17). Let x be a fixed point of U. From (16) it follows that

(18) Df(p (x) + ax) (a) = A(p (x) + ax) ( f (p (x) + ax)) + g(p(x) + ax) for

Set

= f(p(x) + ax) for x

Jp(x).

Then (18) can be written in the form

(19) Æ*>(T) = A(p(x) + ax) (fp(x)( x)) + g(p(x) + ax) for x e J p(x).

From (16') it follows that f p(x)(0) = f 0(p(x)). Hence, by the general form of the solution of a linear differential equations (see e.g. [6], p. 242) the function f p(x) can be represented in the form

(20) f p{x)(x) = R(x, 0, p(x))(/0(p(x)))

T

+ R(x, 0, p(x))(j R(0, s, p(xj)g(p(x) + as)ds) о

for X E Jp(x) •

Since t(x)eJp(x) for x e U , therefore f(p(x) + at(x)) = f P(X)(t(x)) for xeU . (20) and the fact that x = p(x) + at(x) for

U now imply (17) for

U.

Now, we prove that the function defined by (17) satisfies equation (16).

Let x be a fixed point of U. Set h(x) = f (x + ax) for |

| < £, where e > 0 is so small that the definition makes sense. Observe that

h(x) = R(t{x) + x, 0, p(x))(/0(p(x)))

t ( x ) + T

+ R(t(x) + x, 0, p(x))( j R(0, s, p{x))g(p{x) + as)ds) о

for

< 6. It is not difficult to show that this function is differentiable and (21) h'{0) = 4(p(x) + at(x))(K(t(x), 0, p(x))(/0(p(x))))

t(x)

+ A(p(x) + at(xj)(R(t(x), 0, p(x))( f R(0, s, p(xj)g(p{x) + as)ds)) о

+ R(t(x), 0, p(x))(K(0, t(x), p(x))(g(p(x) + at{x)))y

Hence it follows immediately that / has a derivative at x in the direction of a and Df(x){a) = h'(0). Since R(t(x), 0, p(x))oR(0, t(x), p(x)) = / (see Th. IX.

11 from [6]) and x = p(x) + at(x) for x e U , (21) takes the form Df{x)(a) = А(

)(Д(£(

), 0, p(x))(/o(p(x))))

t(x) + T

+ A{x)(R(t(x),0, p{x))( j R(0, s, p(x))g(p{x) + as)ds)y

Ю — Commentationes Math. 30.1

This implies directly that / fulfils (16). The fulfilment of (16') is obvious.

R em ark 3. If 0 does not belong to U, then Theorem 2 can still be used after a suitable translation.

With the assumptions and notations of Theorem 2 the following corol­

laries are true.

Co r o l l a r y 1.

I f there exists a space X a complementary to La such that two solutions of equation (16) agree on U n X a then these solutions are identical on U.

Co r o l l a r y 2.

I f we assume in addition that the projection operators are continuous and that the maps A , f 0, g are differentiable then the solution of

equation (16) defined by equality (17) is differentiable on U.

References

[1] A. A le x ie w ic z , Functional Analysis, PWN, Warsaw 1969 (in Polish). \ ^

[2] J. D ie u d o n n é , Foundations of Modern Analysis, Academic Press, New York and London 1960, Russian transi. Moscow 1964.

[3] K. J. G u r g a n u s , ф-like holomorphic functions in Cn and Banach spaces, Trans. Amer. Math.

Soc. 205 (1975), 389 -406.

[4] E. K u b ic k a and T. P o r e d a , On parametric representation of starlike maps of the unit ball in C" into C", Demonstratio Math. 21(2) (1988), 345-355.

[5] H. M a r c in k o w s k a , Introduction to the Theory of Partial Differential Equations, PWN, Warsaw 1972 (in Polish).

[6] K. M a u r in , Analysis, Part I, PWN, Warsaw 1971 (in Polish).

[7] J. A. P f a lt z g r a f f and T. J. S u ffr id g e , Close-to-starlike holomorphic functions o f several variables, Pacific J. Math. 57 (1975), 271-279.

[8] T. J. S u ffr id g e , Starlike and convex maps in Banach spaces, ibid. 46 (1973), 575-589.

INSTYTUT MATEMATYKI, POLITECHNIKA LÔDZKA

INSTITUTE OF MATHEMATICS, TECHNICAL UNIVERSITY OF LÔDZ AL. POLITECHNIKI 11, 93-590 LODZ, POLAND