Local and global bivariational gradients and singular variational derivatives of functionals on Cn[a,b]

Nonlinear Analysrs, Theory, Punted in Great Britain.

Methods

& Applications,

Vol.

17, No. 9, pp. 841-862,

1991. 0

0362-546X/91 $3.00+ .OO 1991 Pergamon Press plc

LOCAL AND GLOBAL BIVARIATIONAL GRADIENTS AND SINGULAR VARIATIONAL DERIVATIVES OF FUNCTIONALS ON C”[a, b] M. Z. NASHED Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, U.S.A.

E. P. HAMILTON Department of Mathematics, Washington College, Chestertown, MD 21620, U.S.A. (Received

7 August

1990; received for publication

20 February

1991)

Volterra derivatives, bivariational gradients, singular variational derivatives, global representation of Gateaux differentials. Key words and phrases:

1. INTRODUCTION

AND PRELIMINARIES

THE CONCEPT of variational (or functional) derivative was first introduced by Volterra [29] in 1887 in the context of the Calculus of Variations (see also [30, 311). Volterra introduced the notion of “functions of lines”, and proceeded to generalize for such functionals the classical notion of partial derivative. We quote from Dieudonne [7]: “Although, from our point of view, his definitions are not very precise (Dieudonne remarks in a footnote that this can be said of practically all mathematicians before 1906), he apparently considers the set E of C’ mappings of an interval I c IR into R3 (the “lines”), and the mapping y: E + R, continuous for the topology of uniform convergence.. . . In a manner reminiscent of the Calculus of Variations, he considers a “variation” Sy = y(+ + 0) - y(d), where the increment 0 is supposed to vanish outside of an interval [a, p], and then the quotient 6y/o, where 0 = 1: (e(t)1 dt; this should tend to a limit when p - (Y and the maximum of 101 tends to 0. With our experience of 50 years of Functional Analysis, we cannot help feeling that, without even the barest notions of general topology, these ad hoc definitions were decidedly premature. Nevertheless, they caught the fancy of Hadamard, who tried to apply similar ideas to Green’s functions and encouraged his students to work in that direction.. .“. There are errors in the statements and proofs of existence of the Volterra variational derivative for the “simplest problem” in the Calculus of Variations that occur in Volterra [29-3 11, Gelfand and Fomin [ 111, as pointed out by Bliss [2], and Hamilton and Nashed [15, 161, respectively. In fact with Volterra’s original definition, the variational derivative does not exist even for the simplest problem of the Calculus of Variations. For this setting, the situation has been rectified by modifying Volterra’s definition (see, e.g. [2, 10, 141); however, none of these modifications provide a satisfactory definition for general functionals on C[a, 61 or on C”[a, b], n 2 1. For functionals on C[a, b], the authors [15] have reformulated the definition of Volterra’s variational derivative as follows.

Definition 1 .l. exists a number

Let J: C[a, b] + R. Let x0 E [a, 61 and y E C[a, b] be fixed. Suppose there L such that for every E > 0 there exist a 6 > 0 and a neighborhood U = U(E) 841

M. Z. NASHED and E. P. HAMILTON

842

of x0, with the property

that J[Y + Al - JIYI

L

< E

5 h(x) dx for all h satisfying

the following

conditions:

(i) h E C[a, b], h(x,) # 0; (ii) h does not change sign; (iii) supp h, the support of h, is contained in U; and (iv) sup Ihl < 6. Then we say that the Volterra variational derivative of J at (Y, x0) exists (is equal to L) and denote it by 6J/6Y(x,,). In the above definition and in this paper all integrals are over [a, 61, unless otherwise We use the symbol N(x,; U, 6) to denote the class of functions satisfying conditions above. It is also convenient to introduce the notation x0 -,‘izO

stated. (i)-(iv)

G(h) = L

to mean the following: Given E > 0, there exists a 6 > 0 and a neighborhood U of x0 such that 1G(h) - LI < E for all h E N(x, ; U, 6). When x0 is fixed and is clear from context, we abbreviate this to ,IiIiO G(h) = L. Thus the Volterra variational derivative of J at (y, x0) may be written as 6J = lim J[Y + hl - J[YI h*O dY(xo) S h(x) dx * Volterra,

in essence,

sought

a formula

of the form

dJ[y; h] = relating

the linear

Gateaux

variation

of a functional

6J[y; h] := lim t-r0

to the variational continuum analog

s

6J ~ h(x) dx &Y(X) J defined

(1.1) on a function

space

J[Y + thl - JIYI t

derivative 6J/6y(x). The significance of the classical formula

of such a formula

is that

it is a

(1.2) relating the differential of a function of n real variables to its partial derivatives. Consider that Volterra’s bold attempts predate the rigorous development of even the notion of a differential of a function of several variables, and more importantly also predate measure theory, Lebzsgue integration, and the barest notions of general topology, it is not surprising that such attempts would contain some errors and fall short of the resolution of the difficulties inherent in the general case. To describe some of these difficulties consider first the case of a real-valued functional J on the space of continuous functions on [a, b] with the supremum norm, and suppose J has a linear Gateaux variation 6J[y; h] that is continuous in h in the sup norm.

Gradients

Then by Riesz’s theorem

and derivatives

of functionals

&J[y; h] has an integral ~J]Y;

hl =

on C”[a, b]

843

representation

5

h dp,r,

(1.3)

where pY is a regular Bore1 measure depending on y (see, e.g. Rudin [26]). The authors [15,22] have established conditions under which ruy is absolutely continuous and its Radon-Kikodym derivative is a continuous function. Under these conditions, the (local) Volterra variational derivative (definition 1.1) is shown to exist everywhere and to be equal to the (globally defined) Radon-Nikodym derivative. Note that in this framework, the measure p,, in the representation (1.3) is not allowed to contain atoms; the assumptions imposed rule out this possibility and thus limit the generality of these results. Moreover, while definition 1.1 is natural for functionals on C[a, b], it is inappropriate for functionals on the space of all functions with continuous nth derivative, C”[a, b], n 2 1, since it is too demanding. When restricted to functionals on C’[a, 61, this definition is too restrictive even for applications to the Calculus of Variations: with this definition, functionals of the form J[y] = jiF(x, y, y’) dw fail to have a variational derivative in general, even under additional smoothness. In this paper we introduce new notions, the regular bivariational derivatives and the regular bivariational p-gradient for a functional J on Cn[a, b]. Unlike the case of the partial derivative or earlier work on functional derivatives, we have a fundamental hierarchy or a finite scale of regular bivariational derivatives of J at the same point (y, x0), which can be tailored to the degree of smoothness one wishes to utilize. These bivariational derivatives resolve, in particular, the difficulty mentioned above for “regular” problems in the Calculus of Variations. To treat what Volterra called “exceptional” points as well as problems where there is a jump in a derivative of the function, we introduce notions of singular bivariational derivatives. Such derivatives usually exist at the endpoints of the interval [a, 61, whereas the regular variational derivative does not exist there. The new notions are extensions and modifications of similar notions for functionals on C[a, 61 studied recently by the authors [22]. The functional J[y] = 4(x1, y(xl), y’(x,)), where x1 is a fixed point in the interval [a, b] is an example of a functional with an “exceptional” point at x1. The existence of a Volterra variational derivative of J for a given y E C[a, 61 and all x E (a, b) does not necessarily imply the existence of the Gateaux variation of J at y. In this paper we will assume that the Gateaux variation of the functional exists for h in a certain class. The following new type of variational derivatives, introduced in [22], takes advantage of this assumption; we call them bivariational derivatives since they involve two limiting processes.

Definition

1.2. Let J: C[a, 61 -+ R. Let x0 E [a, b] and y E C[a, b] be fixed. Suppose

that

~J[Y; hl x0 - lim /Z+O Sh(x)dx exists. Then we say that the regular bivariational

e

introduced

of J at (y, x0) exists and denote it by

SJ[y; hl lim h-0 jh(x)dx’

6J -:=x0dY(xo) (We refer to the notation

derivative

earlier.)

844

M. 2. NASHED and E. P. HAMILTON

Remark. We use the same notation, 6J/6Y(x,), for the Volterra variational derivative and the bivariational derivative since when they both exist they are equal (see [22]). The bivariational derivative is a simpler notion than the Volterra variational derivative in two respects: (i) the latter involves a (complicated) double limit, whereas the former involves a simpler iterated limit; (ii) the bivariational derivative of J at (y, x0) can be viewed as the Volterra variational derivative of A[h] := 6J[y; h] at h = 0 and x = x0, where A is defined on N(x,; U, 6). Definition 1.3. Let J: C[a, b] -+ m. Let x0 E [a, b] and y E C[a, b] be fixed. Suppose there exists a number L with the property that for every E > 0, there exists a neighborhood U = U(E) of x,, such that the Gateaux variation

pn-:_

(J[y + t/z] - J[y]]

6Jb; h] := exists for all h E C[a, 61 whose support

is in U and 6J[Yi

hl

L

<

E

hofor all h satisfying the following conditions: (i) h E C[a, b], h[x,) # 0; (ii) the maximum of lh 1 occurs at x0 ; and (iii) supp h is contained in U. Then we say that the singular bivariational derivative (is equal to L) and denote it by dSJ/Gy(x,). It is convenient derivative at (y, x0).

of J with respect to y at x0 exists to refer to this briefly as the singular

Note that 6J[y; h]/h(x,) is homogeneous of degree zero in h and hence the condition definition 1.1 would have no effect if it were included in definition 1.3. We use the symbol H(x,; U, 6) to denote the class of functions satisfying conditions in definition 1.3 and condition (iv) of definition 1.1. We also introduce the notation x0 -

(iv) of (i)-(iii)

/imO G(h) = L

to mean the following: Given E > 0, there exists a 6 > 0 and a neighborhood U of x0 such that IG(h) - LI < E for all h E H(x, ; U, S). When x0 is fixed and is clear from context, we abbreviate this to lim, +,+,, G(h) = L. Thus the singular derivative of J at (y, x,) is 6”J ---= SY(x0) We are in 1980 a derivative singular) difficulties errors in elsewhere. functionals

lim ____ ~J[Y; hl h-0 h(x,) .

now in a position to describe briefly the objectives of this paper. The authors began research program aimed at understanding and generalizing the notion of variational (in the sense of Volterra). Our approach provides a new hierarchy of (regular and which are collectively necessary for resolving inherent variational derivatives, in Volterra’s original approach. Our approach also enables one to rectify various the concepts and existence theory of variational derivatives alluded to in [15] and In this paper we bring the most important extension of this theory, namely to on the space C”[a, b] to a satisfactory level, allowing for singular measures in the

Gradients

and derivatives

of functionals

on C”[a, b]

845

appropriate generalization of the “integral representation” of 6J[y; h]. This requires the introduction of new concepts of local bivariational p-gradients and singular bivariational derivatives (Section 2). All of these notions are local. In Section 3 we introduce an alternative global approach to the (regular and singular) bivariational derivatives via an appropriate representation of the linear Gateaux variation of a functional, based on measure-theoretic considerations. The precise connection between the local and global approaches is investigated in Section 4, where we state and prove the main theorems of this paper. In Section 5, we illustrate the computation of various (regular and singular) bivariational derivatives and boundary parts of p-gradients at regular and “exceptional” points for functionals of the Calculus of Variations. Finally, in Section 6, we make concluding remarks. 2. LOCAL

BIVARIATIONAL

GRADIENTS

VARIATIONAL

Let y E C”[a, b] and let c be any point 1 Hx) = (n - I)! Let J: C”[a, b] -+ R and define

dx)

Finally,

:=

~ (n

AND

in [a, b]; then

‘(x - t)“-‘y’“‘(t) c

.

dt +

&: C[a, 61 x R” --+ R by &[v; 1 -

x I)!

c

i

SINGULAR

DERIVATIVES

(x - t)“-‘v(t)dt

+

nc

(Y,,, . . ., (~,_i] := J(z), where

1 (x

_

c)i

Tffi*

i=O

let gno := 3, [u; 0, . . . , 01. Using this notation,

we now introduce

definition

Definition 2.1. Let y E Cn[,, b] and x0 E [a, 61 be fixed. The regular bivariational J at (y, x0) with respect to the datum point c is defined by

2.1.

n-gradient

of

a %J := an . ad, MU' aao'""aQ,_l vcYko) ( > ’ where u(x) := y’“‘(x), and (Yi := y(‘)(c). In this definition, S&/&(x,) is the previously defined regular bivariational derivative (definition 1.2) of $,[ *; 01~) . . . , a,_,] as a functional on C[a, b], where it is understood that CY~,. . . . (~,_i are held fixed. The derivatives a$, /aai, i = 0, ,.., n - 1, are ordinary partial derivatives where u is held fixed. We define 6J m: and call it the regular bivariational be simply the regular bivariational Definition 2.2. The singular and is denoted by

&.L =-

Wxo)

n-derivative of J at (y, x0). For n = 0, we define VLJ/V,y derivative of definition 1.2.

bivariational

n-derivative

of J at (y, x0) is defined

to

by 6”3,0/Su(xo),

6”J @3nO 6y'"'or,):= Wxo)' where BSgnO/6u(x,) (definition 1.3).

is the singular

variational

derivative

of gno, as a functional

on C[a, b]

M. Z. NASHED and E. P. HAMILTON

846

It is natural

to seek a definition

%J vc Y(%) ’

for

6J

6”J ~ Gy’p’(x,) ’

and

6r@‘o’

p = 0,l , . . ..n

- 1.

Following our procedure, we can make the substitution u = y@‘) ; in this case we obtain functional 3, on CnVp[a, b] x Rp. Thus our task would be completed once we improvise suitable definition for 6J/6y(xo) and 6”J/6y(xo) for a functional on Ck[a, b] for k L 1.

a a

Definition 2.3. Let J: Ck[a, b] --t R, y E Ck[a, b] and x0 E (a, 6). Suppose that there exists a neighborhood U of x0 such that 6J[y; h] exists for h vanishing outside U and is a continuous functional of h in the sup norm. Let A(h) be the continuous extension of 6J[y; h] to N(x,; U) (see Section 1). Then we define the regular bivariational derivative of J at (y, x0) by 6J

Nhl

:=x0-

lim hfxo!hdm’

6Y(Xo) Definition 6”J/6y(xo) /6J[y; h] (i) h = (ii) h = (iii) h is

bivariational derivative 2.4. Let J: Ck[a, b] + R. We say that the singular exists and equals L if and only if for each E > 0 there exists a 6 > 0 such that LI < E for all h E Ck[O, l] satisfying the following conditions: 1 on (x0 - (a/2), x0 + (d/2)), 0 for jx - x0/ 2 6, and monotonic on (x0 - 6, x0 - (d/2)) and (x0 + (a/2), x0 + 6).

We are now ready to make our fundamental definition of the variational p-gradient and the (regular and singular) p-bivariational derivatives of a function J on C” [a, b], for p = 0, 1 3 ..*, n. Definition 2.5. Let J: Cn[a, b] + I?. Let p be a fixed integer, 0 I p I n. The regular tional p-gradient of J at (y, x0) relative to the datum point c is defined by

%J :=

vcY(Xo)

(

a, . aDI,

’ aao ’ -’

ati

bivaria-

ah

aap_I > ’

where v(x) : = y@‘(x), LY~:= y”‘(c), and $p]~; 01~) . . . , ol,_i] := J[y]. We define SJ/Sy@‘(x,,) must be to be 6$,/&(x,). (Note that 3, is a map from C n-p x Rp into R so that 6$,/6v understood in the sense of definition 2.3.) The singular bivariationalp-derivative of J at (y, x0) is defined by ~~“JIYI := 6’61,]n; 0, . . . . 01 Wxo)

SY ‘p’(xo) ~J/~Y’~‘(x,) [ = p-gradient, and

Sg /&(x0)] QP

will be called

the

ad, ( will be called the boundary

aa0

principal

. part

of the

regular

bivariational

part) of the regular

bivariational

ad,

’ -’

aQp_p- >

part (or the finite-dimensional

Gradients and derivatives of functionals on C”[a, b]

p-gradient.

The bivariational

p-gradient

is defined

to be the collection

a, .

6J ad, VPJ := sy’p’oc,); G* vcY(Xo)

841

.-*’

a+

1

a”J

’ sy@)(x0) >

.

Remark. The motivations for defining 8J/6y@’ for all p = 0, 1, . . . , n rather than merely 6J/6y’“) and for the analogous definitions for singular bivariational p-derivatives are markedly different. The advantage of defining the regular bivariational derivatives 6J/dy@’ for all p = 0,l 9 ..-, n is that they allow us to adapt the notion (and consequently tailor the results) of in the problem under consideration. We bivariational derivatives to the given “smoothness” have a hierarchy or a finite scale of regular bivariational derivatives (6J/dy@’ : p = 0, 1, . . . , n) which in most cases exists under successively weaker conditions. In contrast, the singular (Dirac bivariational p-derivatives for p = 0, 1, . . . , n are needed to capture all the singularities delta functions and their derivatives) that exist in the representation of dJ[y; h]. The local approach to the singular bivariational derivatives also enables us to treat various types of “exceptional” points and nonregular functionals in the sense of Volterra. The boundary parts of the regular bivariational p-gradients contains information about boundary conditions, e.g. one can easily recover from them all the natural boundary conditions in the case of functionals of the Calculus of Variations (see Section 5). 3. GLOBAL

VARIATIONAL

GRADIENTS

VARIATIONAL

AND

SINGULAR

DERIVATIVES

In Section 2 we developed a local approach to singular bivariational derivatives of a functional on C”[a, b]. We now introduce an alternative global approach to the bivariational derivatives (both regular and singular) via an appropriate representation of the Gateaux variation of a functional, based on measure-theoretic considerations. Suppose that there exists a countable set T of points in [a, b] such that the Gateaux variation 6J[y; h] is a continuous functional of h with respect to the norm

MIT:=

Ih(x)l +

ssl$b

where A(‘) denotes the ith derivative ,u, a countable set A, and numbers

sup

XET

{lh”‘(X)l : i =

of h. Then there exists a nonatomic ai such that

6J[y; h] =

h dp + C

i XEA i=O

; We now state and prove two propositions representation.

which address

(3.1)

1, . ..) n), regular

Bore1 measure

ai(x)h(‘)(X). the existence

PROPOSITION 3.1.

and uniqueness

of this

Let A: C”[a, b] -+ IRbe linear and continuous with respect to the norm (I-II T defined by (3.1) for some countable subset T of [a, b]. Then there exists a nonatomic regular Bore1 measure ,U on [a, b], a countable subset A of [a, b], and real numbers a,(x), i = 0, 1, . . . . n, XEA, such that A(h)

=

r U Moreover,

CxcA Cyzo Iai(

< a.

h dp +

C i ai(x)h”‘(X). xeA i=o

848

M. Z.

NASHED

E. P.

and

HAMILTON

Proof Let I, := (1,2, . . . . n) have the discrete topology, let the set T have the relative topology and consider the topological sum S := [a, b] U (Z, x T). Let C(S) denote the space of continuous functions on S, equipped with the sup norm. For a function h E C(S), let h denote the restriction of h to [a, b]. Let 5 := (h E C(S) : h E C[lz,

61, 22(i,x) = h”‘(X)).

Define a functional A on 5 by k(h) := A(h). Then A is a continuous linear functional on 5. By the Hahn-Banach theorem, there is a continuous extension A, of A to C(S) with the same norm as that of A. By the Riesz representation theorem, there exists a regular Bore1 measure Dsuch that A,(h) = js k dv”. Let u be the restriction of v”to the Bore1 subsets of [a, b]. Then for h E 3, A(h) =

h du + $ 1 Ia,bl

i=l

C k(i,x)fi((l(i,x))). XET

Hence, for h E C”[Q, b] we obtain

A(h) =

h du + $ i la,bl

i=l

C h’“(x)v”(((i,x))). xeT

Let T, denote the set of points in [a, 61 which have nonzero Define a measure fi on [a, 61 by /@)

:= u(E)

-

c

xeEnT,

Let A := T, U T and define

a,(x) :=

1 L

T, is a countable

set.

W]).

iEI,,,xe

( fi(](i, 41) 0

u-measure;

T

ieI,,xeA\T

0

i=O,xeA\T,

m4)

i=O,xeTO.

Then h dP + ,cO XFA ai(x)h(‘)(x).

A(h) = I’ The finiteness

of CrEO CxE-A Iai(

Proposition 3.2. Suppose

follows

A is a functional

from the finiteness on C”[,,

h dfi, +

c

of the measure

0.

n

b] of the form i

xer, i=o

a,(x)h”‘(x),

where ,~r is a nonatomic regular Bore1 measure, Tl is a countable set, and for each x E Tl , ai # 0 for at least one i; moreover C, E r, Cyco la,(x)1 < co. Then the measure or, the set Tl, and the coefficients ai are unique.

Proof. Suppose

A(h) has another

representation

A(h) =

h d,Uz + 3

C

xeTZ

of the above

form,

2 bi(X)h”‘(X), i=O

say,

Gradients

and derivatives

of functionals

on C”[a, b]

849

where the bjs satisfy the same property as the ais. Let T := Tl U T2 and define Cj(X) :=

Let ,u : =,ul

x

-hi(x)

x E T,\T,

h d,u + c .r

i

E

x E T, fl T2.

i aAx) - hi(x)

-pu,.Then

T,/T,

ai (X)

ci(x)/z”‘(x) = 0

xeTi=O

for all h E Cn[a, b]. Let x0 E T. We claim that c,(xo) = 0. Suppose not, and choose a function g E Cn[a, b] such that (i) g(“)(x,) > 0. (ii) the maximdm of g(“) occurs at x0 ; (iii) g”‘(x,) = 0 for i = 0, 1, . , . , n - 1; and (iv) the support of g is contained in I and has length less than Ic,(x,)l/3lpj([a, b]). It is easy to show that for g E Cn[a, b] satisfying (iii), lg”‘(x)l 5 Ix - ~~j~-~sup (g’“‘(x)1 for i= 19 *.*, n - 1, and hence, under conditions (i) and (ii), Ig(‘)(x)) 5 Ix - x01”-‘g’“‘(xO).We can further choose the support of g so that j&o CFZo Ici(x)g”‘(x)) < ~lc,(xo)g(“)(xo)( where Q := (T fl Z)\(x,), and so

o=

g& + C J?Ci(Xk”‘(X) > ~Icn(xo)I IP)(Xo)I, xET i=O

which is a contradiction. Thus for each x0 E T, cn(xo) = 0. The same argument shows that = 0 for xE T and i = 1, . . ..n. Thus [hdp + C XErc,(x)h(x) = 0 for all h E C”[a, b]. Define a measure v on [a, b] by v(E) := p(E) + C x EroE co(x). Then for all h E C”[a, b] (and hence for all h E C[a, b]), j h dv = 0. By the uniqueness part of the Riesz representation theorem, v = 0. This implies that p = 0 since p is nonatomic, and also co(x) = 0 for each x E T. Hence ccl = p2 and ai = hi(x) for x E Tl fl T2. If x E T,\T,, we have ai = Ci(X) = 0, contradicting the assumption that ai # 0 for at least one i. Thus T, c T, . Similarly T, c T,, andsoT, = T,. n Ci(X)

We define the global variational derivative to be the collection ]p;ai(x),i=O,l

,...,

nandxET).

Under suitable conditions, the numbers [ai : i = 0, 1, . . . , n, x E T) will be shown to be the singular (bivariational) derivatives of J at the points of T. The singular bivariational derivatives of the functional are zero off the set T. If ,u is absolutely continuous and if the (regular) bivariational derivative exists everywhere, then the Radon-Nikodym derivative of p coincides with the (regular) bivariational derivative. The precise connection between the local and global approaches is investigated in Section 4. 4. GLOBAL

REPRESENTATION REGULAR

AND

OF THE SINGULAR

GATEAUX

DIFFERENTIAL

BIVARIATIONAL

IN TERMS

OF

DERIVATIVES

For simplicity, we first consider functionals on C’[a, b]. THEOREM4.1. Let J be a functional on @[a, b]. Suppose that there exists a finite set Tl of points in [a, b] such that the linear Gateaux differential 6J[y; h] is a continuous functional

850

M. Z. NASHED

and E.

P. HAMILTON

of h with respect to the norm (4.1) Then the following statements hold. (i) 6”J/6y’(x) exists everywhere on [a, b] and 6’J/6y’(x) = 0 for x $ Tt. set T, such that (ii) 6’J/6y(x) exists everywhere on [a, b] and there exists a countable 6sJ/6y(x) = 0 off T,. (iii) There exists a unique representation

h d/c, + C a,(x)h(x) + C a,(x)h’(x), x E

x E 70

where p0 is a nonatomic

(4.2)

T,

measure,

ai

6”J = ___ 6y”‘(X)

for x E T, i = 0, 1, and

Moreover, if J has a regular bivariational derivative on the complement of a countable p0 is absolutely continuous with respect to Lebesgue measure and its Radon-Nikodym tive is equal to 6J/6y a.e. [ml.

Proof. First we note that the existence and uniqueness of the representation proposition 3.1 and 3.2, as does the assertion C, E T0la,(x)l < 00. (i) To establish the existence of cYJ/6y’(x,), we consider ~J[Y; hI --_=-

j h $0 + C, h’(x,)

h’b.,)

E

T

Ci’=,Ui(X)h”)(X) h’(xcJ

*

set, then deriva-

(4.2) follow from

(4.3)

First we show that as h ++ 0 the first term on the right-hand side of (4.3) approaches zero. Let x1 be a point where (h( has its maximum. Then 1sh dp,I 5 Ihl(x,)lpl([a, b]). Now Ih(xl)j 5 IS::,h’(t) dI[ (since h(x,) = 0). Since sup Ih’(x)l = 1, we obtain

Shdpo ~

I I

Nsu~~h')bl(b,

h’h,)

4)

b’(x,)\

’

Since h’(x,) - 1 and m(supp h’) + 0, we see that

Since cxsTo

b&d

< co,

it follows

as above that

c x E To%(x)h(x) h’(x,)

~ _c, E ~~ l%(x)l Ih(

--f o

Ih’(x,)l

as h 8, 0. Next we show that 1, E T,,xzXOa,(x)h’(x)/h’(x,) tends to zero as h tends to zero as above. Let E > 0 be given. By the finiteness of C, f T, la,(x)], we may choose a finite set A c T, such that

Gradients

and derivatives

of functionals

on C”[a, b]

851

cXET,,A lai(x)l < E. Now choose a 6 > 0 such that (x0 - 6, x,, + 6) contains no points of A\[x,,). Then for h supported inside (x0 - 6, x,, + a), we have

Hence, if x0 $ Tl this shows that cYJ/&‘(x,) lim

h-0

6Jb; hl -= h’(x,)

exists and equals zero. If x0 E Tl, we see that

a1(xoM’(xo) = a&o). h ‘(xo)

lim

ha0

(ii) To show that 6’J/6y(xo) exists for all x0 E [a, 61 and is zero for x0 $ T,, we consider WY; hl -= h&o)

S h ho

+ C, e r, aoWW

+ Cx

l

T, WW(4

h(xo)

Since Tl is a finite set, there exists a deleted neighborhood of x0 which contains no points of Tl. Hence as h 8, 0, the term C, Er, a,(x)h’(x) eventually becomes zero. Now we show that

Since p. is nonatomic, po((xo)) = 0, and by regularity, lim 6+o+ po((xo - 6,x0 + 6)) = 0. Since the maximum of h occurs at x0, and h(x,) = 1

ISh +ol

h(x,)

5 PO(~UPPh) + 0

as h 8, 0. Next we show that lim

C ~oCW(x)

h+-OX,To

X#X(J

= o

’

h&o)

Let E > 0 be given. Since C, Er, la,(x)1 < 00, we can choose a finite set A c To such that c XET,,A la,(x)l < E. Now choose 6 > 0 such that (x0 - 6, x0 + 6) ClA\(x,l = 0. Then for L supported inside (x0 - 6, x0 + 6),

Thus lim C h*‘JxEf,,

ao( h&o)

= o ’

and so if x0 $ To if x0 E To’ Thus we have shown that S”J/Sy(xo) exists on [a, b], is zero off To, and is ao(xo) for x0 E To. (iii) Now suppose that J has a regular bivariational derivative on the complement of a countable set. Let ,u(E) = P,(E) + CXEE a,(x). The measure P can be decomposed uniquely as ,u = pPP + pCs + ,uu,,, where pPP is a pure point measure, pCs is a continuous singular

M. 2. NASHED and E. P. HAMILTON

852

measure, and pat is absolutely continuous with respect to Lebesgue measure (see, e.g. Rudin [26]). First we show that peesis zero. Let P(,‘,denote the positive part of pCs. Since ,& is singular with respect to m, D,&(x) = co a.e. [,&I, where 0~ is the derivative of the measure P. We wish to show that whenever D,I&(x) = 00 and x $ Tt, &T/dy(x) does not exist. Suppose D&&(x) = cx). Then for each positive integer N, there exists r > 0 such that (x0 - r, x,, + r) f~ T, = 0 and if Z is an open interval with length less than r, and x0 E Z, then ZI,‘,(I) < 2Nm(Z). There exists an E > 0 such that if ICX\< E, then b,‘,(Z) + oc)/m(Z) > N. Let P denote the support of P,‘,, B:= PnZ, and v:=& +p”- + m + 6,. By Lusin’s theorem, we can choose a continuous function g, supported in Z, such that 0 5 g 5 1 and g = xe except on a set of v-measure less than s/2. Then sJiv:g,=Sgd~~~gd~~-jgdlr-. Using the estimate

and 1j g dp-(

< e/2, we see that

Hence &Z/6y(x,) does not exist. Since Tr is a set of ,& measure, we see that &Z/6y(x) does not exist a.e. [z_&]. But since by assumption &Z/6y(x) exists except on a countable set, z.& is concentrated on a countable set. But P,‘, has no atoms; therefore it is the zero measure. Similarly, & = 0. Thus,

=

jkdm

+

s

c ---M-9 ~5”~ x E To &Y(X)

+

6”J c -h’(x) XE T1 dY’(X)

where f = dp,,/dm. Let x0 be a point of [a, b]\T, at which D,u,,(x,,) = f(xo) and dJ/by(x,) exists. Then by theorem 3.4 of [22], there is a neighborhood of x0 for which S”J/Sy(xo) = 0. Hence,

Let E > 0 be given. Since Do,,&,) = f(x,,), there exists r > 0 such that for any open interval containing x0 with length less than r,

m&o)

&AZ) - m(Z)

<

E

3.

Z

Gradients

and derivatives

of functionals

853

on C”[a, b]

There exists a 6, 0 < 6 < r such that if h is a continuous function that does not change sign, h(x,J # 0, and supp h c (x0- 6,x0 + a),then

We can choose a continuous (x0 - 6, x0 + 6) such that

function

Sfhdm ___Shdm

Then (D,u,,(x,) - (6J/6y(xo))j a.e. [m] and so 6J[y;

0 I h(x) I 1, h(x,) # 0,

&AX0 - 69x0 + 6) m(x, - 6,x0

+ 6)

h

supported

in

< E 3.

< E. Since E is arbitrary, D,u,,(x,) = SJ/Gy(x,). Thus U/&J = f

6J -h(x)dm dy(x)

h] =

h,

+

6”J c -h(x) x E ToSy(x)

+

6”J C -h’(x). XEr, 6y’(x)

n

We now state the generalization of theorem 4.1 to functionals on C”[u, b]. The proof is similar to that of theorem 4.1 and will be omitted. THEOREM 4.2. Let J be a functional on C’[,, b]. Suppose that there exist finite sets 7;, i= 1, *-*, n, of points in [a, b] such that the linear Gateaux differential 6J[y; h] is a continuous functional of h with respect to the norm

n

Then the following statements hold. (i) 6”J/6y”‘(x) exists everywhere on [a, b] and 6”J/6y”‘(x) = 0 for x $ T, i = 1, . . . , n. (ii) 6’J/6y(x) exists everywhere on [a, b] and there exists a countable set & such that 6”J/6y(x) = 0 off To. (iii) There exists a unique representation 6J[y; h] = S h dp, + CyZo C,, 6 ai(x)h”‘(x), where pun _ is a nonatomic measure, ai

6”J

= ~ 6y”‘(X)

forxE7;,i=O,l,...,

n,

andL=T~ lao~x)l < m.Moreover,

if J has a regular bivariational derivative on the complement of a countable set, then ,u~is absolutely continuous with respect to Lebesgue measure and its Radon-Nikodym derivative is equal to SJ/I/sy a.e. [ml. Combining theorem 4.2, propositions 3.1 and 3.2, and the techniques of Section 2, we obtain the following theorem. THEOREM 4.3. Let J be a functional on Cn[,, b]. Suppose that there exists a finite set 7;, i= 1, .**> n - p of points in [a, 61 such that the linear Gateaux differential S&,[v, a; k, /3] is a continuous functional of k with respect to the norm

M. 2. NASHEDand E. P. HAMILTON

854

Then the following statements hold. (i) SS&,/6y”‘(x) exist everywhere on [a, b] for i = 1 , ..., n - p and 6”J/6y”‘(x) = 0 forx$ T. (ii) 6’J/6y(x) exists everywhere on [a, 61 and there exists a countable set & such that 6SJ/6y(x) = 0 off T,. (iii) There exists a unique representation n-p

n

S3p[u,

p-1

C ai(X)k(‘)(X) + C VjPj*

k&o + C

a; k,Pl =

i=o

where p. is a nonatomic

j=O

X-CT;

measure, n-P

6”J

@j(X)= ~ 6y"'(X)’

C i=O

C ~~57;

laiCx)l

<

w9

and ag, I?i=Gy

j=O

,...,p - 1.

Moreover, if 3, has a regular bivariational derivative on the complement of a countable set, then p, is absolutely continuous with respect to Lebesgue measure and its Radon-Nikodym derivative is equal to &Jp/6u a.e. [ml.

Proof. This theorem follows easily from theorem 4.2 making the type of substitution discussed in Section 2 and taking into account definition 2.5. We omit the minor technical modifications needed to complete the proof. n 5. BIVARIATIONAL

GRADIENTS IN THE

AND CALCULUS

SINGULAR

5.1. In this section we illustrate the computation tional derivatives and boundary parts of p-gradients the Calculus of Variations. Let

JIYI =

a

VARIATIONAL

DERIVATIVES

OF VARIATIONS

of various (regular and singular) bivariafor two types of functionals arising from

F(x, Ye4 Y’(X), . . . , Y’“‘(X)) h,

partial derivatives where y E CZnPp for a fixed 0 I p I n, and the function F has continuous M). We also consider functionals of the form uptoordern -p + 1 withrespecttoy,y’,...,y Z[Yl = 4(Y(XOl), . . .T Y(XoA *. .; Y’(XlA . . .? Y’(Xl,)

. . .; Y%,A

. . ., Y(nkkN,

which depend on a countable number of values of y and on the values of the first n derivatives of y at a finite number of points. We assume that Z[y] is Frechet differentiable. The functional Z[y] is an example of a nonregular functional in the sense of Volterra; the Volterra variational derivative of such functionals does not exist. First we consider variational derivatives of J. Let Y(x) denote the vector (Y(x), Y’(X), . . . >Y’“‘W).

Gradients

and derivatives

of functionals

855

on C”[a, b]

Let c E [a, 61, and define the following functions on [a, 61: (x

(P -

G,(x, t) =

t)P-m-’

-

for t between c and x

m - I)!

0

otherwise

for m=O,l,..., p-l. Let v=y@‘, (%, **., e+_ i). Then we can easily obtain

CY~=~(‘)(C) for

s s b

P-m

Qp-1

G,Ax, MO dt + 1

yCm’(x)=

(p -

i=l

0

i=O,l,...,

Wl

(x

p-l,

and

cy=

C)P-m-i

-

i)!

-

for m = 0, . . . , p - 1, and y@‘(x) = u(“-~) for m = p, . . . , n. Using this expression for yCm’,we obtain by a straightforward differentiation

ad=

acxj-

bj

a Jo

m

u%(x)~Y(x))

‘;,-“, Wl!

hi

j=O

9

. .

..P

-

1,

where F,,, = aF/ay (W. To compute 6J/6y@‘(x,) we must consider the limit of &&[~,~;kOl

kdx

ii

Now

ask*Oatx,. b a

G,,,(x, t)k(t) dt dx

Fj+p(X, Y(X))k”‘(X) b

p-l

=

n-P

C,Ax,

j=O

YOKZAx,

0

k(t)

dt

c II b

IS (-

+C

+

b

c m=O

.i[a

dx

~)j

s

Fj+p(x,

YO)k(x)

dx

a

n-p

j-l

j=O

I=0

c c

=

(-I)‘$&j+,cx, Y(x))k’j+‘-“(x)j;I,b &At,

Y(t))G,(t,

If xoE(a,6), then k-0 at x0, k’j-‘-“(a) and k”-‘-“(b) 0 at x0, &J,,[u, a; k, O]/j k dx converges to

xl

dt

are eventually zero. Hence as

k e

n-P

Fm(t, W))G,(t,

xo)

dt +

c WY’ $e+,(xo,

j=O

Wo)).

M. Z. NASHEDand E. P. HAMILTON

856

Thus n-p

6J Mf,

W)G,(f,

x0)

dt

+

c j=O

sy@‘o=

(-1)j$;+,(x,,

Y(x,))

for x0 E (a, 6). In general the singular derivatives of J will be nonzero at a and b and so dJ/dy@‘(x) will not exist at these points. We note that S”J/Syp(x) = 0 for x E (a, b). Next we compute d”J/dy”‘(a). This is clearly zero for i > n - p - 1; for i = 0, . . . , n - p - 1, we have 6”J

___

= a-

dy”‘(a) From the final expression

lim k(‘)8*0

S$,[Q a; k, 01 /P(a) .

for 6,$p[v, a; k, 0] above, n-P -,=F+,

Using a similar argument r = 0, 1, . . . , n - p - 1,

at

~ 6”J = 6y(‘)(a) 6”J ~ 6y(“(b)

(-l)j-1-r

b

the last limit is clearly equal to

dJY1-’ duJ-l-‘~+p(~,

and

summarizing

Y(a)). our

results,

we

see

that

for

-j_~+l(_lp~-~C~ tiJ-i_r J+p(Q* Y(4), n-P

fl-P

= j=F+,

(-1y-1-r

dJ:‘-’ dx’-‘-‘~+p(b,

Y(b)),

and 6”J ---= 6y”‘(X) Now consider

the variational

o

derivatives

for x f a, b.

of I. For simplicity,

we consider

the functional

Z[Yl = 4(Y(X,), Y’(Xz)). The treatment is similar for the more general First we let p = 0. We have

functional

dZ[Y, hl = hfw,)

given at the beginning

of this section.

+ 42ffca

where the partial derivatives $Q and C#Qare evaluated at (y(xi), y’(xz)). It is clear that 61/6y(x) does not exist for x = x1 or x, and is zero otherwise. Also, ifx=Xi,i=

1,2

otherwise. Next, we let p = 1. Then &l*]Q a; k, PI = &PO + 41

40 df + &M-Q)

and so

adI

-=

aa0

and

41

-

=

CYZ = dY’(X)

0

x #

42

x = x2

L

x $ k,xJ x E (c, Xl).

x2

.

Gradients

and derivatives

of functionals

857

on C”[a, b]

If x = c or xi, then 61/6Y’(x) does not exist unless also x = a or b; in that case 61/6y’(x) = I& again. Finally, we consider the case p = 2. Then &[v, al = 4

+

Xl

qtx -

4 +

c

S&[u, a; k, PI = &PO + (XI - CMIPI + 41

(x1 - t)u(t) dt, crl +

XI (x1 - WW dt + &PI +

x2 $2

40

dt,

c

c

and so

and 6J -

~Y”W

=

61

-

.wu[c,x,~(x)

+

~2x[c,x&)

provided that x $ (c, xi, x2]. If x E (c, xi, x2], then 61/6y”(x) does not exist unless x also happens to be an endpoint of [a, b]; in this case SUSy”(x) is still given by the same expression as above. 5.2. We now consider simple applications of variational derivatives to extremal problems in the Calculus of Variations. For illustration, we consider the case of “free” variational problems in which there are no prescribed boundary conditions or other constraints on the class of admissible functions, In this case, since dJ[y; h] must vanish for all h at an extremizing function y, we have by the uniqueness result in proposition 3.2 that the regular and singular bivariational derivatives and the boundary parts must all vanish. First we show how the natural boundary conditions arise from the singular variational derivatives at the endpoints. For the simplest problem of the Calculus of Variations, b

J[ul =

a

w9 Y(X), Y ‘(XI) d-&

our formulas give, in the case p = 0: for x E (a, b)

6”J -= MY

-ma, Y(a)9Y’(4) F,(b, Y(b)9 Y’(b)) 0

atx=a atx=b otherwise.

Equating these expressions to zero, we obtain the Euler equation and the natural boundary conditions. Now consider the functional b

K[YI = J[YI + 11~1 =

e-G Y>Y’) dx + ddY(X,), (I

Y’(X2)).

858

M. Z. NASHEDand E. P. HAMILTON

Then we obtain (again taking p = 0):

-

6K

SY(x)

$3(x,Y(X),Y'(X))

= F2(-% Y(X), Y'(X)) -

atx=a

-F2(@Y(4, Y'(4) 6°K

Mb,

= dY(X)

6°K

atx=b

Y(b), Y’(b))

ifx=x,,

91(Y(XI), r’(x2)) 0

_

42(YW,

dY’(X)

Y’(X2))

forx$(a,xI,x2,b)

otherwise. x = x2 x #X2.

I0

Setting 6K/dy(x) to 0, we find that Euler equation is satisfied on each of the intervals (a, x,), (x1, x2), and (x2, b). Since we have assumed y to be of class C2, this would imply that the Euler equation is in fact satisfied on (a, b). Setting the singular derivatives equal to zero we obtain for x = a or 6, F2(x, Y(X),Y’(X)) = 0 &(YW

Y’(X2))

= 0,

and

dz(YW

Y’(X2))

= 0.

Since the Euler equation is only second order and we have four boundary conditions, the problem in general will be overdetermined, that is, a solution will not exist. Since we first determined only that the Euler equation was satisfied on the intervals (a, x,), (x1, x2), and (x2, b), it is natural to relax our conditions of smoothness on y by requiring only that it be of class C2 on each of these intervals and that it be continuous at x1 and x2. Then we have three Euler equations (one on each interval) and so we expect to have six supplementary conditions to give a unique solution. Because y is no longer assumed to be of class C2, the variational derivatives must be recomputed. The same analysis as above gives -

6K

SY (4

= wx>

Y(X), Y’(X))

6”k = dY(X)

-

$3(xv

Y(X), Y’(X))

*mx, Y(X), Y’(X))

forxfEhx,,x,,

WI

for x = a or b

and 6°K -= au ‘62)

42(Y(Xd

Y’(X2)).

The calculation of d”k/6y(x,), however, utilized an integration by parts. By repeating the derivation of dSk/6y(xl), integrating by parts only on the intervals (a, x,), (x1, x2), and (x2, b), we obtain 6°K

-

~YW

= &(Y(xr)9 Y’(X2))

+ 4(x1

9 Y(XA

Y’W)

-

ax,

9 Y&l),

Y’Wh

Gradients

and derivatives

of functionals

859

on C”[a, b]

Thus the extremizing function should satisfy F2-iF3=0

on

4(x,

(a, x1), (x1, x2) and (x2, N,

Y(X), Y’(4)

for x = a or b.

= 0

&(Y(XA Y’(Q)) = 0 &(Y(X,), Y’(G) + &(x1 9Y(Xl), Y’ca

- 6(x,

9Y(XA Y’ca

= 0,

and should be continuous at x1 and x2. To illustrate by a simple example the need for relaxing the smoothness conditions on Y when the functional has a nonzero singular bivariational derivative at an interior point, consider the functional 1

Jfrl =

I’0

[Y’(-w~

+ YW

= OYI + mYI

on C2[0, l] with the boundary condition y(0) = 0. Then $(y(xr), y’(x2)) = y(x,) and $r = 1, so that the condition 6”K/6y(x,) = +r = 0 cannot be met for any function y. If we require y to be of class C2 only on the intervals (0, x1) and (x1, l), then the singular variational derivative of Z[x] at x1 is nonzero and we obtain the conditions y”

= 0

for x E (0, l), x# x1, y’(1) = 0,

and &(x1, Y(Xl), Y’&))

- &(x1, Y&l), Y’(Xl+))+ 1 = 2(Y’(G)

- Y’(Xl+))+ 1 = 0.

The function that satisfies these conditions is: y(x) =

1

o
-t

-z

1

x1

5x5

1.

It may be observed that the local notions of variational gradients capture the natural boundary conditions, interface conditions, and the Euler-Lagrange equations independently, and without using a derivative higher than that explicitly contained in the functional of the variational problem. Our approach also subsumes and generalizes what Lanczos [19] called “an integral approach” to the Calculus of Variations. In his review of Lanczos’ paper [Math. Reviews 26(5445), (1963)], M. R. Hestenes states that the purpose of that paper is “to show that one can give a satisfactory treatment of the first necessary condition for variational problems without reference to a derivative higher than that explicitly contained in the formula defining the variational problem. The method used is essentially an application of Weierstrass in an effective way and is closely related to the maximum principle enumerated by Pontryagin . . .“. The approach using variational gradients and variational derivatives is a more direct and general approach. 6. CONCLUDING

REMARKS

AND

GLIMPSES

OF RELATED

TOPICS

6.1. The local bivariational derivatives (either regular or singular) introduced in this paper preserve two important features in Volterra’s original scheme: (i) any bivariational derivative of a functional J on Cn[u, b] at a given y E C*[,, b] and a fixed x0 E [a, b] is a number; and (ii) the variational derivatives generalize the notion of a partial derivative of a function of

860

M. 2. NASHED and E. P. HAMILTON

several variables. This contrasts with global approaches that use integral representations of 6J[y; h] via Schwartz distributions or measure theory (see [8] and remarks and references in [15]). For example, in the representation (1.3) one can call ,uYa “variational derivative” of J, but then this “global” variational derivative is a measure rather than a function; moreover, in general, it cannot be localized at x0 E [a, 61 to obtain a number 6J/6y(x0). A similar remark applies to the approach using Schwartz distributions. In these approaches, the subtleties of the local definition of Volterra-type variational derivatives are not addressed. 6.2. The notion of variational derivative introduced by Volterra precedes all other notions of differentiability in infinite dimensional spaces, such as those introduced by Gateaux, Frechet, Hadamard, Fan, Levy and others. The essence of these approaches is to generalize the notion of the directional derivative or the total differential of a function of several variables. For historical and mathematical overviews/comparisons of the different kinds of differentials, see [l, 20, 21 and 91. Notions of variational derivatives are in many ways more subtle than those in other approaches (see [15] for a more detailed discussion). Variational derivatives can be more finely tuned to functionals on function spaces to reflect or “resonate” a local behavior that cannot be directly studied using other notions of differentials on abstract spaces. For this reason, appropriate definitions of variational derivatives have to be tailored to the specific function space under consideration, such as C”[a, 61 or Sobolev spaces. These definitions involve intricate limits that use more than just the norms on these spaces. 6.3. In this paper we provided useful sufficient conditions for the existence of the various bivariational derivative and gradients introduced. We also established a “global” representation theorem for the linear Gateaux differential of a functional J on C”[,, b] as the sum of an integral and a series whose “kernels” were identified with the various locally defined notions of (regular or singular) bivariational derivatives and gradients. This representation theorem provides, for the first time, connections between the Gateaux differential and Volterra-type variational derivatives when the latter are not necessarily continuous and “exceptional” points occur. Prior to this paper, no modification of Volterra’s local variation derivative has been proposed to treat singular and “exceptional” points for functionals on C’[,, b]. Donsker and Lions [S] have introduced a notion of Frechet-Volterra derivative for functionals on C”(T) (or C(T)), the space of infinitely differentiable (or continuous) functions on an open interval T,with compact support. These spaces are endowed with the topology of Schwartz (or with the topology of uniform convergence on every compact subset of T).Their notion allows one to consider simultaneously the ordinary and “exceptional” points of Volterra. However, in their setting the variational derivative is a distribution, and their notion is global. 6.4. Since variational derivatives hinge upon and utilize the rich structure of function spaces, they provide information which is not directly obtainable from, say, the Frechet differential, of the Euler-Lagrange equations (see, e.g. such as invariance properties [ill) and the differential-form characterization of causality in quantum field theory in terms of the variational derivative (see, e.g. [3]). Various variational derivatives also play an important role, or have useful connections with inverse problems and variational principles in function spaces (see, e.g. [l, 23, 28, 20]), in certain boundary value problems of partial differential equations (e.g. [5, 6, 8]), in quantum mechanics, relativistic quantum field theory, statistica], hydrodynamics, nonlinear elasticity, etc. (e.g. [3, 12, 17, 19,25, 27]), and in some fundamental

Gradients

and derivatives

of functionals

on C”[a, b]

861

aspects of function space integrals (Wiener measure and integral, stochastic and Markovian processes, diffusion equations, etc.; e.g. [8, 12, 13, 17, 18, 61). Finally, we point out that the von Mises calculus for statistical functionals (see [33, 341) is based on the Volterras model for variational derivatives in function spaces. An extension of the notions and results of this paper to statistical functionals resolves several problems in this area (see, e.g. [24, Section 5 entitled “Open Problems”]). These issues will be addressed in another paper. 6.5. In some of the areas mentioned in 6.4, especially in mechanics, quantum field theory, and variational methods, ad hoc integral representations involving the functional derivative are used. Also, functional derivatives are defined using the same quotient as in the definition of the Even operaGateaux variation with h(x) : = 6(x - x0), where 6 is the Dirac delta function. tionally, the approach is both imprecise and limited since nonlinear functionals of the Dirac delta functions are not defined. The setting and results of this paper and those of [22] provide a mathematical foundation and extensions of the integral representations used in the contexts mentioned above, which will be considered elsewhere. REFERENCES 1. AVENBUKH V. I. & SMOLYANOV 0. B., The theory of differentiation in linear topological spaces, Usp. Mat. Nauk 22, 201-260 (1967) (in Russian); Russian Math. Surveys 22, 201-258 (1967) (English translation). 2. BLISS G. A., A note on functions of lines, Proc. natn. Acad. Sci. U.S.A. 1, 173-177 (1915). 3. BOGLIUBOV, N. N. & SHIRKOV D. V., Introduction to the Theory of Quantized Fields. Interscience, New York (1959). 4. CAMERON R. H., The first variation of an indefinite Wiener integral, Proc. Am. math. Sot. 2, 914-924 (1951). 5. DALETSKI Yu. L., Elliptic operators with functional derivatives and diffusion equations associated with them, Dokl. Akad. Nauk SSSR 171, 21-214 (1966) (in Russian); Soviet Math. Dokl. 7, 1399-1402 (1966) (English translation). 6. DALETSKI Yu. L., Differential equations with functional derivatives and stochastic equations for generalized stochastic processes, Dokl. Akad. Nauk SSSR 166, 1035-1038 (1966) (in Russian); Soviet Math. Dokl. 7, 220-223 (1966) (English translation). 7. DIEUDONNE J., History of Functional Analysis. North-Holland, Amsterdam (1981). 8. DONSKER M. D. & LIONS J. L., Frechet-Volterra equations, boundary value problems, and function space integrals. Acta Math. 108, 148-228 (1962). 9. FLETT T. M., Differential Analysis. Cambridge University Press, London (1980). 10. FRBCHET M., Existence de la differentielle d’une integrale du calcul de variations, C. r. hebd. Seanc. Acad. Sci. Paris 240, 2036-2038 (1955). 11. GELFAND I. M. & FOMIN S. V., Calculus of Variations. Prentice-Hall, Englewood Cliffs, NJ (1963). 12. GELFAND I. M. & YAGLOMA. M., Integration in function spaces and its applications in quantum physics, Uspekhi Math. Nauk 11, 77-l 14 (1956) (in Russian); J. math. Phys. 1, 48-69 (1960) (English translation). 13. GROSS L., Integration and nonlinear transformations in Hilbert space, Trans. Am. math. Sot. 94, 404-440 (1960). 14. HAMILTON E. P., A new definition of variational derivative, BUN. Aust. math. Sot. 22, 205-210 (1980). 15. HAMILTON E. P. & NASHED M. Z., Global and local variational derivatives and integral representations of Gateaux differentials, J. funct. Analysis 49, 128-144 (1982). 16. HAMILTON E. P. & NASHED M. Z., Variational derivatives andp-gradients of functionals on spaces of continuously differentiable functions, Math. Methods appt. Sci. 5, 530-543 (1983). 17. HOPF E., Statistical hydromechanics and functional calculus, J. Rat. mech. Anal. 1, 87-123 (1952). 18. KAC M., On distributions of certain Wiener functionals, Trans. Am. math. Sot. 65, 1-13 (1949). 19. LANCZOS C., An integral approach to the calculus of variations, in Studies in Mathematical Analysis and Related Topics (Essays in honor of George Polya; edited by D. GILBARG, H. SOLOMONand others), pp. 191-198. Stanford University Press, CA (1962). 20. NASHED M. Z., Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in nonlinear functional analysis, in Nonlinear Functional Analysis and Applications (Edited by L. B. RALL), 103-o-309. Academic Press, New York (1971). 21. NASHED M. Z., Some remarks on variations and differentials, Am. Math. Monthly 73, Slaught Memorial Papers, 63-76 (1966).

M. Z. NASHED and E. P. HAMILTON

862

22. NASHED M. Z. & HAMILTON E. P., Bivariational

and singular variational

derivatives,

J.

London math. Sot. 41,

526-546. 23. POMRANING G. C., A derivation 24. 25. 26. 21. 28. 29.

of variational principles for inhomogeneous equations, Nucl. Sci. Engng 29, 220-236 (1967). REY W. .I. J., Robust Statistical Methods. Springer, New York (1978). ROMAN P., Introduction to Quantum Field Theory. Wiley, New York (1969). RUDIN W.. Real and Complete Analvsis, 2nd edition. McGraw-Hill, New York (1974). SCHWEB&S. S., Relativiskc Quantum Field Theory. Row, Peterson, Evanston (1961). TELEGA J. J., On variational formulation for nonlinear nonpotential operators, J. Inst. math. Applic. 24, 175-195 (1979). VOLTERRA V., Sopra le funzioni the dipendono da altre funzioni. Nota 1, Rend. Lincei Ser. 4 3, 97-105 (1887); Nota 2, 3, 141-146 (1887); Nota 3, 3, 151-158 (1907). See also OpereMatematische, Vol. 1, pp. 294-314. Accad.

Nazi&ale dei Lincei, Rome (1954). Paris (1913). 30. VOLTERRA V., Lecons sur les Fonctions de Lignes. Gauthier-Villars, 31. VOLTERRA V., Theory of Functionals and of Integral and Integro-Differential Equations. Blackie, London (1931). 32. VOLTERRA V. & HOSTINSKP B., Operations lnfinitesimales Lineaires, Applications aux Equations Differentielles et Fonctionnelles. Gauthier-Villars, Paris (1938). of differentiable statistical functions. Annls math. Stat. 18, 309-348 33. MISES R. VON, On the asymptotic distributions

(1947). 34. MISES R. VON, Mathematical Theory of Probability and Statistics (Edited published

posthumously).

Academic

Press, New York

(1964).

and complemented

by H. GEIRINGER;