Quasi-Differential Calculus

Chapter 4

QUASI-DIFFERENTIAL CALCULUS We introduce in this book, starting from this chapter, a new concept for differentiability of mappings between locally bounded F-spaces which we have called"Quasi-differential, or pq-dij~ferential ''. We shall use it in all this chapter and in other chapters. I t is d i s c o v e r e d in 1 9 9 5 while we were working to have a new form of L a g r a n g e M e a n - Value t h e o r e m in some non locally convex spaces and for more f o u n d a t i o n s o f the t h e o r y o f H o l o m o r p h y w i t h o u t c o n v e x i t y c o n d i t i o n ; it is also called Bayoumi differential by some Mathematicians, see [20], [21], [26] ~r So all the results dealing with this quasi-differentiability are due to the author, see bibliography.

4.1

QUASI-DIFFERENTIABLE MAPS

One of the fundamental concepts of the theory of functions between normed spaces is differentiability where the differentials of Fr~chet and G~teaux appear to be the most appropriate concepts. In this part we study a new and different concept of differentiability. This is what we have called, Q u a s i - D i f f e r e n t i a b i l i t y or p q - D i f f e r e n t i a b i l i t y , for maps f : E --, F between locally bounded F-spaces E, F. In fact, our Quasi differentiability is totally different from the Fr~chet one although it has the same advantage of the Fr~chet one. T h a t is, it is linear and continuous mapping. This new concept of differentiability was discovered by the author in 77

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78

4 QUASI-DIFFERENTIAL

CALCULUS

1995 while he was working on finding theorems for the mean-value in real and complex locally bounded F-spaces, see the author [20], [21] and [35]. So all the results of this chapter are due to the author. 4.1.1

Q u a s i - D i f f e r e n t i a b l e M a p s in F - S p a c e s

Definition respectively f , g : U --. pq-tangent

27 (0 F, to

( 1 9 9 5 ) Let < p,q <_ 1), and a E U, each other at lim

IIf(x)

x---4a

E and F be p-normed and q-normed spaces and U a nonempty open subset of E. For we say that f and g are q u a s i t a n g e n t or a if -

g(x)lll/q/llx

-

all 1/p

-

0

.

That is, for any e > 0 there exists 6 > 0 such that

0 < Ilx- all < 6 =>

Ill(x)- g(x)ll 1/q ~ EIIx- all

9

(4.2)

This is equivalent to 0 < I I x - all < 5 =~ I [ f ( x ) - g(x)ll p <_ el l l x - all q,

s

-

-

(4.3)

s

from which we choose the word pq-tangency in our definition.

Notice t h a t if f, g are quasi-tangent to each other at a then ( f - g) is continuous at a and f ( a ) - g(a); thus if one of these functions is continuous at a the other must be continuous at a. Moreover f, g are quasi-tangent at a if and only if and only if ( f - g) is quasi-tangent to 0 at a. Now it is easy to verify t h a t quasi-tangency at a is an e q u i v a l e n c e r e l a t i o n on the vector space of all mappings from U to F which are continuous at a. In fact, if f and g are pq-tangent at a, and g and h are pq-tangent at a, then f and h are pq-tangent at a. Notice t h a t

ilf(x) _ h(x)lll/q <_ ~(llf(x) - g(x)[] 1/q + ilg(x) - h(x)lll/q),

~ >__1 (4.4)

as a consequence of which II" IIlzq is a quasi-norm. The notation of pq-tangency depends only on the topologies of E not on the p-norm and, q-norm used to define the topologies.

and

4.1 QUASI-DIFFERENTIABLE

MAPS

79

28 [20] ( 1 9 9 5 ) Let E and F be p-normed and q- normed spaces, respectively (0 < p, q < 1), and U a n o n e m p t y open set in E. A mapping f : U ~ F is said to be q u a s i - d i f f e r e n t i a b l e (or p q - d i f f e r e n t i a b l e ) at a C U, if there exists a linear map Ta E L ( E , F ) , such that f and the continuous affine linear map x E E ~ f (a) + Ta(x - a) are pq-tangent at a, that is, Definition

lira ( l l f ( x ) x----+a

f(a)

-

T~(x

-

a)llP/llx

Hence,

for every ~ > O, there exists 5 > O"

II/(x)

f(a)

-

-

Ta(x - a)ll <_ ellx

-

-

(4.5)

all q) - o.

0 < IIx- all < 5 =:>

all q/p

29 I f f " U --, F is pq-differentiable at each point of U, is said to be p q - d i f f e r e n t i a b l e on U.

Definition

f

then

R e m a r k 11 It is to be noted that if E and F are both p-normed spaces or quasi- normed spaces with the same quasi-norm constants then the condition (~.5) turns out to be as the following classical one: lim ( l l f ( z ) -

x---~a

4.1.2

Properties

f(a)

-

T~(x

-

a)ll/llz

- all) - 0.

of Quasi-Differentials

We show in what follows the linearity of the quasi-differentiable mappings. Theorem

39

[21]

The set of the quasi-differentiable mappings at a C U represents a vector space. P r o o f . In fact, if f and g are pq-differentiable at a C U, then there exist linear m a p p i n g s T~, Tg E L ( E , F) such t h a t [If(x) - f ( a ) - T~(x - a)[[ 1/q ~ ~[[X -- all 1/p Iig(x)

-- g(a)

--

T * ( x - a)ll 1/q ~ ~ I I x - a]] 1/p

for a given c > O. Since [[. I]1/q is a quasi norm, we have

II(f + g ) ( x ) - ( f + g ) ( a ) - (T~ + T ~ ) ( x - a)ll ~/q ~_ a ( l l f ( x )

-

f(a)

- T~(x

- a)ll ~/q + ]lg(x) - g ( a ) - T 2 ( x

- a)lll/q).

CHAPTER

80

4 QUASI-DIFFERENTIAL

CALCULUS

So by dividing b o t h sides by I I x - a[[ lip and taking the limit as x ---, a we obtain

D ( f + g)(a) - D r ( a ) + Dg(a)

(4.6)

D()~ f)(a) - AD f (a).

(4.7)

Also Notice t h a t IIA(f(x) - f ( a ) - T ~ ( x - a))ll x/q - I I ( A f ) ( x ) - ( ) ~ f ) ( a ) - A T ~ ( x - a)ll ~/q

So if we divide b o t h sides by I I x - a[[ 1/p and take the limit as x --~ a we o b t a i n the required equality, i T h e following t h e o r e m extends the classical Lipschitzian property to locally b o u n d e d spaces, see [21]

Theorem

40 ( L i p s c h i t z i a n property)J21]

Let E

and F be p-normed and q-normed spaces, respectively, (0 < and U a non-empty open subset of E. If f : U --~ F is pqdifferentiable at a E U, then there exist c > 0 and 5 > 0 such that

p , q <__ 1)

IIf(x) - f ( a ) l l <_ clJx - all q/p fo~ x c u,

I I x - all < ~.

Let T IIf(z) - f ( a ) ] ] -

Proof.

I n particular, it f o l l o w s that f

(4.8) is c o n t i n u o u s at a.

D f ( a ) and A(x) -- f ( z ) - f ( a ) - T ( x - a ) for x E U. T h e n l i T ( x - a ) + A(x)II <_ IlTllqllz-a[Iq/P + IIA(x)[i because IlT(x - a)l [ _< Ilrl[qllx -- al[q/p,

see (1.3). Since l i m z ~ a IlA(x)iiP/]]x- allq -- O, and A ( a ) -- 0, given e -- 1, there exists S > 0 such t h a t if IIx - all _< 5, x E U, t h e n

ilZX(x)ll _< IIx- all ~/~. Hence

I I f ( x ) - f(a)] I <_ (IITIIq +

1)llx-ailq/p

= ~ l l x - all ~/~ where c-

IITll q + 1. i

T h e following t h e o r e m has useful applications.

4.1 Q U A S I - D I F F E R E N T I A B L E

MAPS

81

41 [21] Let f : U --~ F be pq- differentiable at a E U and let T E L ( F , G) where E, F and U are as in the preceding theorem and G is a q - n o r m e d space. Then ( T o f ) is pq- differentiable at a and Theorem

D ( T o f ) (a) Proof.

=

T o D f (a).

(4.9)

Since

[IT[f(x)- f(a)-Df(a)(x-a)]II

1/q <_ I l T i l l / q l l f ( x ) - f ( a ) - D f ( a ) ( x - a ) l l

1/q,

dividing b o t h sides by IIx - a l l ~/p a n d t a k i n g t h e limit as x --+ a we o b t a i n II(T o f ) ( x ) - ( T o f ) ( a ) - (T o D f ) ( a ) ( x - a)lll/q/ll x - all ~/p <_

IITlll/qllf(x) - f ( a ) - D f ( a ) ( x - a)ill/q/llx - all 1/p -+ 0 i.e. D ( T o f ) ( a ) = ( T o D f ) ( a ) . I 30 A f u n c t i o n f : E --~ F is said to be l o c a l l y c o n s t a n t i f f is a constant f u n c t i o n on every neighborhood of each point of E . B y a constant map o n E we mean a map x E E --~ b E F, for a fixed b in F. Definition

Theorem

Let E p,q<_l).

4 2 [21]

and F

be p - n o r m e d and q- normed spaces respectively, (0 <

(a) If f : E ~ F

is a constant map, then D f = 0; i.e.D f (x) = 0 Vx E E.

(b) I f A E L ( E , F ) ,

then D A DA(x)

Proof. we have

is the constant map satisfying :

A,

(a) Let f be c o n s t a n t on E . IIf(a)

-

- Df(x)(a -

Vx E

(4.10)

E.

H e n c e for its q u a s i - d i f f e r e n t i a b i l i t y

x)lll/q/llx-

-+ 0

i.e. I I D f ( x ) ( a - x)lll/q/llx - all 1/p < IlDf(x)llll x - aJlllP/llx - all lIp --+ 0 to have IIDf(x)ll -+ o from which D f ( x ) = 0 for all x E E .

82

CHAPTER

4 QUASI-DIFFERENTIAL

CALCULUS

(b) Let A E L ( E , F ) . Then ]]A(x) - A ( a ) - D A ( a ) ( x - a ) i i l / q / I I

= ]IA(x - a) - D A ( a ) ( x - a ) i i l / q / I I = IIA - DA(a)l[ii(x - a ) i i l / p / i i x

x x -

ai] I / p aiI 1 / p

- aiI l i p --+ o

i.e. D A ( a ) - A for a l l a C E . I We shall write for simplicity the constant map D A " x C E --, A E L ( E , F ) by D A A. However the rule D A A should not be confused with the formula d(**) = e ~ in calculus. In fact the rule D A - A should be compared dx with d (cx) -- c by noticing that K ' - K .

T h e o r e m 43 ( M a p p i n g i n t o a p r o d u c t s p a c e ) [ 2 1 ] ( 1 9 9 6 ) Let U C E be open in a p - n o r m e d space E and Fn be p n - n o r m e d spaces (n = 1 , . . . , m ) . A mapping f : U --, F1 • • Fn is quasi-differentiable at a if and only if each coordinate map fn = 7rn o f is quasi-differentiable at a. In this case D f ( a ) = ( D f l ( a ) , . . . , D fro(a)) where 7rn is the projection f r o m F1 x . . . x Fm

(4.11)

onto F~.

Proof. Let Fn be pn-normed spaces for n - i , . . . , m , and let F = F1 x ... x Fro. Then F is an F-slJace and the topology induced by an F - n o r m on F is the product topology. Let 7rn " F ~ Fn

be the projection onto the nth factor F~ and let u~ " F ~ - - , F

be the natural embedding map defined by Un(Xn)

--

( O , O . . . , O, X n ,

O, O, . . . , O )

Then both 7rn and un are continuous linear maps and 7~n O U n

-~Un O Trn

-

-

-

-

1F~ 1F

(the identity map on Fn) (the identity map on F).

4.1 Q U A S I - D I F F E R E N T I A B L E MAPS

83

Let U be open subset of E and let f 9U --~ F; and let f n be the nth coordinate map. Then m

7r,~o f 9 U - ~ Fn

m

s-

o s-

o

n=l

(Sl,...,sm).

n--1

So if f is quasi-differentiable at a, then (u~ o 7r~) is quasi-differentiable at a by Theorem 41 and we obtain D f ( a ) -- ~

Un o

D f n ( a ) -- ( D f l ( a ) , . . . ,

Df,~(a)).

Conversely, if each f , is differentiable at a, then f is clearly differentiable at a. This completes the proof, m 4.1.3

Quasi-Differentials of Multilinear M a p s

We note that the product rule (fg)' - f' g + fg'

can be derived from the following theorem. Observe that the map x e U ~ f(x)g(x) e K

is a composition of the following two maps: x e U ~ ( f ( x ) , g ( x ) ) e K 2, (Yl, y2) C K 2 ~ ylY2 E K , where the first map taking values in the product space K 2 and the second being a continuous bilinear map(see subsection 2.1.3, p. 3a).

Theorem 44

[29] (2000)

Let E 1 , E 2 , . . . , E r a be p~-normed spaces, (i = 1 , . . . , m ) and F be a qnormed space. I f A C L ( E 1 , . . . , Era; F) is a continuous m-linear mapping, then A is quasi-differentiable at ( a l , . . . , a m ) . Furthermore

m

DA(al,

...,am)(Xl, ...,Xm)

-

A ( a l , ...ai-1, x~, a~+l,..., am),

-

1

i.e.

D A ( a ) " E1 x ... x Em ~ L ( E 1 , . . . E m , F)

(4.12)

CHAPTER 4 QUASI-DIFFERENTIAL CALCULUS

84

P r o o f . For m = 1, the result is given by T h e o r e m 42(b). If m = 2, we want to claim

D A ( a l , a2)(Xl, x2) = A ( x l , a2) + A ( a l , x2) Since

A ( x l , x 2 ) = A(al, a2) + A ( h l , a2) + A(al, h2) + A ( h l , h2) where h i = Xl - a l ,

h2 = x2 - a 2 ,

IlA(xl,x2)

we get

- A ( a l , a 2 ) - A ( x l - al,a2) - A ( a l , x 2 - a2)[I 1/q

< IlAll[[Xl - allll/m[lx2 - a2111/p2 < where we used the F - n o r m El x E 2 . If we write

Ilxll

IIAllllx-

- sup{llXlll lips,

all 2

IIx2ll 1/p~ }

for x - ( x l , x 2 ) e

Aa(Xl,X2) = A ( x l , a 2 ) + A ( a l , x 2 ) . t h e n Aa C

L(E1

IIA(x)

-

x E2; F) and

A(a) - Aa(x - a)ll 1/q <

IlAIIIIx

-

all~/P~llx2

-

a2111/p2

<_ IIAllllx-all 2. Hence

D A ( a ) = Aa. T h e p r o o f for the general case can be c o m p l e t e d by induction. II

4.1.4

Quasi-Differentials of P o l y n o m i a l s

T h e following t h e o r e m shows t h a t the quasi-differential of a p o l y n o m i a l is a linear m a p .

Theorem 45 [29] (2000,) If P c p ( m E , F)

and P -

A,

then

D P ( x ) - m A x "~-1

i . e . , D P " iF_,---, L(E; F) - P ( 1 E " F).

(4.13)

4.1 Q U A S I - D I F F E R E N T I A B L E

MAPS

85

P r o o f . We use the multinomial formula (Lemma 1) which in particular has the binomial formula (2.4)

A ( x + y)'~ = A ( x + y , . . . , x

Axm-iy i

+ y) = i=0

where x, y E E. Now

lira [[A(x +

y ) ' ~ - A x "~

- mAx'~-lylll/q/lly[ll/P - 0

y--~0

and if we compare it with lira IIP(x + y ) - P ( x ) - D P ( x ) ( y ) l l / l l y l l

y---~O

1/p - 0

we get D P ( x ) - m A x m-1. m

4.1.5

Inverse Mapping Theorem

Let U, V be open subsets of E and F p-normed and q-normed spaces respectively. If the mapping f : U ~ V is a homeomorphisms and is quasidifferentiable on U, it does not necessarily follow that f is a diffeomorphism, i.e. the inverse f - 1 of f is quasi-differentiable on V. Consider for example, the function f : ~ --~ ~ defined by f(x)

-

x 3.

Then f is a homeomorphism of class C 1 but the inverse g = f - 1 quasi-differentiable at the origin.

is not

The following theorem gives a sufficient condition to have inverse quasidifferentiable maps between locally bounded F-spaces.

T h e o r e m 46

( I n v e r s e m a p p i n g theorem)[29](2000) Let U and V be open subsets of E and F, p-normed and q-normed spaces respectively, and let f : U ~ V be a homeomorphism. A s s u m e that f is pq-differentiable at a point a C U. Then the inverse g - f - 1 is qp-differentiable at the point b - f ( a ) if and only if D r ( a ) C L ( E , F ) is a homeomorphism of E onto F; i.e. D f ( a ) is a topological isomorphism. In that case,

86

CHAPTER

4 QUASI-DIFFERENTIAL

Dg(b) - ( D f ( a ) ) -1.

CALCULUS

(4.14)

P r o o f . If g is qp-differentiable at b = f ( a ) then by the chain rule and T h e o r e m 42, we have g o f = 1v, f o g = 1v and

Dg(b) o D f (a) = 1E,

D f(a) o Dg(b) = l r

which proves t h a t D r ( a ) is a topological isomorphism of E onto F . Conversely, assume t h a t D f ( a ) is topological isomorphism from E into F . We want to show t h a t 9 is qp-differentiable at b = f(a). We will first prove t h a t g has the Lipschitzian property at b. For simplicity, let A = D r ( a ) and A ( x ) = f (x) - f (a) - A ( x - a) for x E U. we obtain

Since A -1 E L ( F ; E ) ,

x-

a-

by applying A -1

A-l(f(x)

(4.15) to b o t h sides of (4.15)

- f(a) - A ( x ) )

I I x - all ~/p ~< ~c(llf(x) - f(a)ll 1/q + II/X(x)ll l/q)

(4.16) (4.17)

with c - I I A - 1 1 1 . Since f is pq-differentiable at a, for e - 1 / 2 a , there is by definition, r > 0 such t h a t B(a, r) C U and IIAxll 1/q <_ I l x - alll/p/2ccr for x E B ( a , r ) . Hence from (4.17) we get

IIx - all 1/p <_ 2~cllf(x)

- f ( a ) l l 1/q

(4.18)

Since g is also continuous, for some s > 0, g(B(b, s)) C B(a, r). If we set in (4.18), we have

x = g(y)

IIg(Y) - g(b)ll lip <- 2crclly - bll 1/q.

(4.19)

for y C B(b, s), which shows t h a t g is Lipschitzian at b. Now we show t h a t g is qp-differentiable. From the relation (4.16), we have

IIx-

a-

A-l(f(x)

- f(a))ll 1/p < cllAxl] 1/q.

(4.20)

4.1 Q U A S I - D I F F E R E N T I A B L E M A P S

87

Since f is pq-differentiable at a, for any e > 0 there exists r > 0 such t h a t B(a,r) C U, and if x e B(a,r), IIAxlI1/q< e i l x - all 1/p. Hence (4.20) becomes ]Ix - a - A - l ( f ( x ) - f(a))]] lip < c llx - all lip for x E B(a,r). Now choose a > 0 such t h a t B ( b , a ) C U and g(B(b, 5 ) ) C T h e n if y E B(b, 5) and x = g(y), using (4.19) we obtain

IIg(Y) - g(b) - A - l ( y - b)ll 1/p <_ coe]lg(y ) - g(b)]l 1/q ~

2o-c2~lly

-

B(a,r).

b]l 1/q.

This completes the proof. I

4.1.6

R e a l and C o m p l e x Cases

If E and F are two complex F - n o r m e d spaces, they can be considered as real F - n o r m e d spaces since tg is a subfield of (F. If we denote these real F - n o r m e d spaces by E~ and F~, then L(E; F) becomes a subspace of L(E~; F~) since a complex linear m a p is a f o r t i o r i a real linear map. Let U be an open subset of E, f : U --~ F and let a C U. We say t h a t f is complex (respectively real) quasi-differentiable at a if f is quasi-differentiable at a for the vector space s t r u c t u r e over ~T (respectively /~). Since L ( E ; F ) C L(E~;E~), it is clear t h a t if f is complex quasidifferentiable at a then it is real quasi-differentiable at a and its quasidifferential in the real sense is equal to its quasi-differentiable D f ( a ) in the complex sense. However, the converse is not true as shown by the following example. If we let E - F - C , then E~ - F~ - t g 2, and the function f " z C ~ --+ 2 C (/' is real quasi-differentiable, but not complex quasi-differentiable. If f is real quasi-differentiable at a and D f ( a ) C L(E~; E~) then for f to be complex quasi-differentiable at a it is necessary and sufficient t h a t Dr(a) should belong to the subspace L(E, F) of L(E~, F~). A function which is complex quasi-differentiable on an open set is said to be q u a s i - h o l o m o r p h i c . We will s t u d y such functions in Ch.5.