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George J Minty

Minty

Minty, George J

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 11, 434-439 (1965) 
A Theorem on Maximal Monotonic Sets in Hilbert Space* 
GEORGE J. MINTY 
University of Michigan, Ann Arbor, Michigan 
Submitted by C. L. Do&h 
1. INTRODUCTION 
Let H be a Hilbert space, with real or complex scalars. For X, y E H, we 
denote by (x, y) the real part of the inner product. A set E C (H x H) is 
called monotonic (see [I, 21) provided that for all (x1 , yr), (xs , ya) E E, we 
have (x1 - xs , yr - ys) 3 0, and is called maximal if it is not properly 
contained in another monotonic set. 
The zero vector of H will be denoted by 0. The set S C H will be said to 
surround f3 provided every ray R, = {tx : t > 0} (for x # 0) contains a point 
of S. The closed convex hull of S is denoted by K(S). (By a well known theo- 
rem, K(S) is also weakly closed.) 
The main theorem of this paper is: 
THEOREM 1. Let E be a maximal monotonic set in H x H satisfying 
(i) There exists a bounded set C C H surrounding 0 such that, for any 
x E C, there exists y  E H with (x, y) E E and (x, y) > 0. 
(ii) There exists a bounded set D C H surrounding 0 such that, for any 
y  E D, there exists x C H with (x, y) E E and (x, y) > 0. 
Then for any closed linear subspace X, with orthogonal complement Y, the 
set E n (X x Y) is nonempty. Moreover, these conditions are su$%ient for the 
set E n (X x Y) n [K(C) x K(D)] to be nonempty. 
This theorem seems to be applicable to “orthogonal projection” methods 
for the solution of differential equations; see the papers of F. E. Browder’[3-71 
for indications of these directions. 
We shall call a function f  with domain in H and range in H monotonic 
if the graph off is a monotonic set, and strongly monotonic if there exists 
c > 0 such that for any x1 , x2 in the domain off, 
(Xl - x2 ,f(Xl) -f(xz)> 2 c II Xl - x2 l12. 
* This research was supported by the Institute of Science and Technology of the 
University of Michigan, N. S. F. Grant G-23830, and U. S. Air Force Contract 
AF-33(637)-11501. 
434 
MAXIMAL MONOTONIC SETS IN HILBERT SPACE 435 
Also, f will be called hemicontinuous provided its restriction to line segments in 
H is continuous with respect to the weak topology in the range. A special case 
of Theorem 1 in which X = H, Y = {e} is proved in an earlier paper of the 
writer [2] in a Banach-space context. (The writer has not yet been successful 
in transferring the present theorem to such a context.) In the main theorem 
of [2], there is a continuity-assumption which is used, however, only to 
guarantee the maximality of the set E. 
2. TOOLS 
LEMMA 1 ([I], Theorem 3). Let H be a HiZbert space, and E C H x H a 
maximal monotonic set. Then the map (x, y) -+ x + y  is a homeomorphism of E 
onto H. 
LEMMA 2 ([2], Lemma 1). Let f  : H + H be a hemicontinuous function, 
and x0 , y,, E H. Then su.cient for y,, = f(xJ is: fm any x, 
(x0 - x, yo -f(x)> 3 0. 
LEMMA 3 ([5]). Letf: H + H be a continuous, strongly monotonic function. 
Then f  has a continuous inverse dejined on all of H. 
(Lemma 3 can be easily proved from Lemmas 1 and 2 by consideration 
of the continuous monotonic function [c-‘f(x) - xl.) 
LEMMA 4. Let f  : H -+ H be a continuous, strongly monotonic function. Let 
X and Y be orthogonal complements in H. Then there exists a unique x E X 
with f(x) E Y. 
PROOF. Let P be the projection map of H onto X. Then g = Pf is a 
continuous strongly monotonic function carrying X into X. By Lemma 3, 
g(x) = 6’ has a unique solution. Since Pf(x) = 8, we have f(x) E Y. 
REMARKS. Lemma 4 is a kind of primitive form of Theorem 1, as can be 
seen by thinking of the set E of Theorem 1 as the graph off. Notice that the 
proof of the conclusion on g depends very strongly on the assumption of 
continuity (or hemicontinuity) off; it is, in fact, the main object of this 
paper to show that such an assumption is not really necessary. (One may refer 
to a later section of this paper for the motivations.) 
3. PROOF OF MAIN THEOREM 
Let H be a Hilbert space. For S C H, we denote by N,(S) the closed 
c-neighborhood of S, i.e. the set (x + y : x E CZS, 1) y 11 < ~1. 
436 MINTY 
LEMMA 5. Let EC H x H be a maximal monotonic set satisfying the 
hypotheses (i), (ii) of Theorem 1. Let X and Y be orthogonal complements in H. 
Then there exists a sequence (p, , qn) E X x Y such that, for any (x,, , yO) E E 
and any E > 0, we have for sufficiently large n: 
(Pn 3 qn) E Qo 3 Yo > 4 = {(x1 Y) :XEX,yE Y,(x,-x,y,-yy) 3 -6, 
x E N,(K(C)), Y E ~GwNl. 
PROOF. For each fixed n > 0: let E, = {(x, y/n) : (x, y) E E). It is trivial 
to show that E, is a maximal monotonic set. Thus by Lemma 1 applied to E, , 
for each u e H there exists a unique point (x, y) of E with u = x + y/n and 
moreover, x and y/n depend continuously on U. Hence y + x/n depends 
continuously on U. It follows that the set 
J% = {(x + r/n, Y + x/n) : (x, Y> E E} 
is the graph of a continuous function F defined on all of H. We have, for 
(x1 , yl) and (x2 , y.J any two elements of E, 
(Xl + xi4 - (x2 + Y2/4, (Yl + 44 - (Y2 + x2/4 
= II Xl -- x2 II2 + II Yl - Y2 !I2 
n + (1 + l/n21 (Xl - X2,% - Y2) 
> II Xl - *2 ll”h 
and hence F is a strongly monotonic function. 
By Lemma 4, there is a point of X which is mapped into Y by F, and hence 
there exists (x,, , y,J E E with p, = x, + y&z E X, and qn = yn + x,/n E Y. 
Form the real part of the inner product: 
(p q ) = 11 xn I2 + HY?l I2 + (1 + Iin”) cx n, n n 
y > = () 
VI* n (3.1) 
We shall show (by an argument used in [2]) that x, E K(C). For if not, then 
since C surrounds 0, x, # 0, and there is a point (z,, , w,) E E with z,, E C, 
<-% 9 w,J > 0, x, = h,z, , A, > 1. Thus by monotonicity of E, 
Substituting X,z, for x, and cancelling (A, - l), we obtain 
MAXIMAL MONOTONIC SETS IN HILBERT SPACE 437 
Multiplying by X, , we see finally (x~ , yn) 3 0, in contradiction with 
Eq. (3.1). Now, by the analogous argument, yn E K(D). Hence, 
as n+ co. 
II xn/n II-+0 and IIY& II-0 (3.2) 
It follows from relations (3.2) that p,, = x, + y&r and qn = ym + x;/n 
lie in N,(K(C)) and N,(K(D)) respectively for sufficiently large 11. 
It remains to be shown that for sufficiently large n, 
<x0-Ppn,Yo-Qn) 2 --* 
We have 
(x0 - 6% + m/n), yo - (Yn + %a> 
= (x0 - 2, , y. - m> + <Y&, y. -Y& + <x0 - x, , x,/n> + <m/n, x&> 
> - II Y& II llyo -Ye II - II x0 - *, II ll4n II - Ilmln II II x,/n II 
and the conclusion follows from relations (3.2) and the fact that x, E K(C), 
3/n E K(D), with C and D bounded, so that x, and y,, remain bounded. 
PROOF OF THEOREM 1. Let E stand for a real number between zero and 1 
( i.e., 0 < E < 1). Consider the two-parameter family of sets in H x H, 
parametrized by (x0 , yo) E E and E: 
wxo 7 YOL 4 = I(% Y) :xEX,yEY,(x,-x,y,-y)>-•, 
x E NM% Y E N,VW))l. 
By Lemma 5, the intersection of any finite subcollection of these sets is non- 
empty. Now, these sets are all subsets of T = N,(K(C)) x N,(K(D)), which 
is a closed convex bounded set in H x Hand hence is weakly compact. Also, 
S((xo , yo), c) can be written as 
and is thus clearly a weakly closed subset of T. It follows from the finite 
intersection property of compact sets that the intersection of the entire 
family of sets is nonempty. Letting (x, y) be in this intersection, we have 
x E X, y E Y, x E K(C), y E K(D), and: for all (x0, yo) E E, 
(x0-Xx,Yo-Y) 20. 
The conclusion follows from the hypothesized maximality of E. 
438 MINTY 
4. AN APPLICATION TO NONLINEAR ELECTRICAL NETWORKS 
Let R be the real numbers. 
Following [8], we call a maximal monotonic set in R x R a resistor. Let 
X and Y be a pair of orthogonal complementary subsets of R”, and let I$ 
(for i = 1, ..., n) be resistors. 
THEOREM 2. Suppose it is possible to find 
(XIO, ..‘Y x,“) E x and (ylo, . . ..yno) E Y 
such that the resistors Eio = {(x, y) : (x - xio, y  - yio) E Ei} each contain a 
point of the open first quadrant, and a point of the open third quadrant, in R2. 
Then there exist (x1 , ..., x,) E X and (yl , ‘.., yJ E Y such that for each i, 
(xi , ri) E 4 . 
REMARKS. I f  X and Y are the current-space and voltage-drop-space of 
an electrical network, with “branches” numbered 1, “‘, n, a very slightly 
stronger theorem (Theorem 8.1 of [S]) holds-namely, the same theorem with 
the weaker hypothesis that each Eio contains a point of the horizontal axis 
and a point of the vertical axis. Another minor variant is shown in [9] by 
means of the Kuhn-Tucker optimality conditions of nonlinear (convex) 
programming. The relationship between the present “monotonicity” method 
and the variational method (first used by D&in [IO] for the present problem) 
is clear on examination of [l l] and [9]. Considering that the problem has 
been so thoroughly studied, it would seem unnecessary to give yet another 
proof. Our excuse is that the present treatment seems to provide the synthesis 
of nonlinear electric network theory and nonlinear boundary-value problems 
called for by Birkhoff and Diaz in [13]. 
PROOF OF THEOREM 2. Let H be Rn. Let EC H x H be the set 
{((Xl 7 . ..Y 4 (Y1 , ..., y,J) : for each i, (xi , yi) E E,O). 
It is easily verified that the sets Eio are resistors, and that the set E is a mono- 
tonic set. Applying Lemma 1 to each set ES0 individually, we see that the 
map (x, y) -+x + y  carries E onto H, and then applying the lemma to E, 
we see that E is a maximal monotonic set. From the hypothesis that each 
EiO contains points of the open first and third quadrants, the existence of 
the sets C and D necessary for application of Theorem 1 is easily established. 
Hence there exist (a, , ..., Q E X and (Ti, . . ..jQ E Y such that for each i, 
(n, , yi) E Eio. Thus 
(Xl I . . . . x,)=(x,O+t,;..,x,O+~~)EX 
MAXIMAL MONOTONIC SETS IN HILBERT SPACE 439 
and 
(YI , *.*, Ynn) = (rl” +?I 2 *-,yn” +sJ E y  
yield the conclusion of the theorem. 
A similar theorem can be stated for complex scalars, using H = C”, 
for “resistors” in C x C of the type 
Ei = {(x, y) : x = ziy + ei) 
where zi , ei are complex constants and Re zi > 0. The constant zi is of 
course “impedance”; Lemma 4 (or well-known results on linear dissipative 
operators) should be used for the existence-proof instead of Theorem 1. 
V. ADDENDUM 
Since this paper was written, F. E. Browder has transmitted to the writer 
a proof of a very closely related theorem [7] which offers the possibility of 
working with a reflexive Banach space rather than a Hilbert space, under 
slightly different hypotheses. The writer is also indebted to Browder for 
pointing out a redundant hypothesis is an earlier version of Theorem 1. 
REFERENCES 
1. G. J. MINTY. Monotone (non-linear) operators in Hilbert space. Duke Math. J. 
29 (1962), 341-346. 
2. G. J. MINTY. On a “monotonicity” method for the solution of nonlinear equations 
in Banach spaces. Proc. Nutl. Acad. Sci. U.S. 50 (1963), 1038-1041. 
3. F. E. BROWDER. Nonlinear elliptic boundary value problems. Bull. Am. Math. 
Sot. 69 (1963), 862-874. 
4. F. E. BROWDER. Variational boundary value problems for quasi linear elliptic 
equations, III. Proc. Natl. Acad. Sci. U. S. 50 (1963), 794-798. 
5. F. E. BROWDER. The solvability of nonlinear functional equations. Duke Math. J. 
30 (1963), 557-566. 
6. F. E. BROWDER. Strongly non-linear parabolic boundary value problems. Am. 
J’. Muth. 86 (1964), 339-357. 
7. F. E. BROWDER. Multi-valued monotone nonlinear mappings and duality mappings 
in Banach spaces. Unpublished manuscript. 
8. G. J. MINTY. Monotone networks. Proc. Roy. Sot. (London), A257 (1960), 194- 
212. 
9. C. BERGE, AND A. GHOUILA-HOURI. “Programmes, Jeux, et Reseaux de Trans- 
port.” Dunod, Paris, 1962. 
10. R. J. DUFFIN. Nonlinear networks II. Bull. Am. Math. Sot. 53 (1947), 963-971. 
11. G. J. MINTY. On the monotonicity of the gradient of a convex functional. Pacific 
J. Math., in press. 
12. G. BIFWHOFF, AND J. B. DIAZ. Nonlinear network problems. Quart. Appl. Math. 
13 (1956), 431-443. 


A theorem on maximal monotonic sets in Hilbert space 

Let H be a Hilbert space, with real or complex scalars. For X, y E H, we denote by (x, y) the real part of the inner product. A set E C (H x H) is called monotonic (see [I, 21) provided that for all (x1, yr), (xs, ya) E E, we have (x1- xs, yr- ys) 3 0, and is called maximal if it is not properl

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