The Equivalence of Several Basic Theorems for Subdi ... - CiteSeerX

The Equivalence of Several Basic Theorems for Subdierentials Qiji J. Zhu Department of Mathematics and Statistics Western Michigan University Kalamazoo, MI 49008 e-mail:[email protected]

Abstract. Several dierent basic properties are used for developing a system of calculus for

subdierentials. They are a nonlocal fuzzy sum rule in [5, 25], a multidirectional mean value theorem in [7, 8], local fuzzy sum rules in [14, 15] and an extremal principle in [19, 21]. We show that all these basic results are equivalent and discuss some interesting consequences of this equivalence.

Keywords: Subdierentials, mean value inequalities, local fuzzy sum rules, nonlocal fuzzy sum rules, extremal principles and Asplund spaces.

AMS (1991) subject classi cation: Primary 26B05.

1

1 Introduction Smooth subdierentials play important roles in nonsmooth analysis for two reasons. They characterize many important generalized dierentials and results in terms of smooth subdifferentials often require very little technical assumptions. Currently there are several ways of developing a set of basic theorems for subdierentials so that they can be conveniently applied to a wide range of problems. The dierence lies on the starting point. Borwein, Treiman and Zhu [5] use the nonlocal fuzzy sum rule [25]; Clarke, Ledyeav, Stern and Wolenski [8] starts with the Clarke-Ledyeav multidirectional inequality [7]; Ioe [15] begins with the local fuzzy sum rule and Mordukhovich and Shao [21] choose the Kruger-Mordukhovich extremal principle [17]. The main purpose of this note is to observe that all the aforementioned basic results are equivalent. Combining this equivalent result and Fabian's results on the equivalence of the fuzzy sum rule of Frechet subdierential for Lipschitz functions and the Asplund property [1] of a Banach space yields that the nonlocal fuzzy sum rule [25] and the Clarke-Ledyeav mean value inequality [7] for the Frechet subdierential also equivalent to the Asplund property of a Banach space which expands the list of equivalent characterizations of Asplund spaces given by Mordukhovich and Shao in [20]. Also, this equivalent result shows that several versions of the local fuzzy sum rules with dierent assumptions on the summand functions (see, e.g. [2, 6, 10, 14, 15, 16]) are actually equivalent. We state our main results and comment on some of their applications in the next section. The proofs are contained in Section 3.

2 The Main Result and some of its Consequences 2.1 The main result

It turns out that the relations among the basic results discussed in the introduction become transparent when they are applied in product spaces. For any positive integer N and Banach space X we will use X N to denote the Euclidean product space of N copies of X . Our main result is

Theorem 2.1 Let X be a Banach space. Then the following are equivalent: a. for any positive integer N , X N has a -local Lipschitz fuzzy sum rule; b. for any positive integer N , X N has a -nonlocal fuzzy sum rule; c. for any positive integer N , X N has a -multidirectional mean value inequality; d. for any positive integer N , X N has a -local fuzzy sum rule; e. for any positive integer N , X N has a -extremal principle.

2

Since [a.]-[e.] in Theorem 2.1 are equivalent we will say that a Banach space X is subdiferential compatible if any of them holds for X . Now let us make precise the concepts used in Theorem 2.1. Let X be a Banach space with closed unit ball BX and with continuous real dual X  . For a point x 2 X and a set Y  X , we denote d(x; Y ) := inf fkx ? yk : y 2 Y g and [x; Y ] := fx + t(y ? x) : y 2 Y; t 2 [0; 1]g. The diameter of points xn; n = 1; :::; N in X is de ned by diam(x1 ; :::; xN ) := maxfkxn ? xmk : n; m = 1; :::; N g. We use R to denote the extended real line R [ f+1g. Let us recall the de nition of smooth subdierentials and normal cones. De nition 2.2 Let f : X ! R be a lower semicontinuous function and S a closed subset of X . We say f is -subdierentiable and x is a -subderivative of f at x if there exists a locally Lipschitz -smooth function g such that rg(x) = x and f ? g attains a local minimum at x. We denote the set of all -subderivatives of f at x by D f (x). We de ne the -normal cone of S at x to be N (x; S ) := D S (x) where S is the indicator function of S de ned by S (x) = 0 for x 2 S and +1 otherwise. In De nition 2.2 \ -smooth" represents various smoothness concepts of dierent strengths. It can be any of the bornological smoothness or s-Holder smoothness concepts. A bornology of a Banach space X is a family of closed bounded and centrally symmetric subsets of X whose union is X , which is closed under multiplication by scalars and is directed upwards (that is, the union of any two members of is contained in some member of ). We will denote by X  the dual space of X endowed with the topology of uniform convergence on -sets. Given a function g on X , we say that g is -dierentiable at x and has a -derivative rg(x) if g(x) is nite and

t?1 (g(x + tu) ? g(x) ? thrg(x); ui) ! 0 as t ! 0 uniformly in u 2 V for every V 2 . We say that a function g is -smooth at x if rg : X ! X  is continuous in a neighborhood of x. Some well known examples are: Gateaux smooth ( = G): the function g is Gateaux dierentiable and rg is weak-star continuous in a neighborhood of x; Frechet smooth ( = F ): the function g is Frechet dierentiable and rg is norm continuous in a neighborhood of x. Similarly, for s 2 (0; 1], we say g is s-Holder smooth ( = H (s)) if the function g is s-Holder dierentiable and rg is norm continuous in a neighborhood of x. In a Hilbert space, DH (1) is the proximal subderivative [22]. We refer to [4, 9, 23] for details. By shifting a constant we may always assume that the -smooth function g in De nition 2.2 satis es g(x) = f (x). De nitions below are motivated by [5, 25] (the nonlocal sum rule), [7, 25] (the multidirectional mean value inequality), [2, 6, 13, 14, 15, 16] (the local fuzzy sum rule and the local Lipschitz fuzzy rule) and [17, 18, 20] (the extremal principle).

3

De nition 2.3 We say that X has a -nonlocal fuzzy sum rule if for lower semicontinuous functions f1 ; :::; fN : X ! R bounded below and any " > 0, there exist xn; n = 1; :::; N and xn 2 D fn(xn ) satisfying diam(x1 ; :::; xN ) < ", (1) diam(x1 ; :::; xN )
max(kx1 k; :::; kxN k) < "; and

N N X X fn(xn ) < lim inf f fn (xn) : diam(x1 ; :::; xN )  g + " !0 n=1

n=1

such that

02

(2)

N X xn + "BX  :

n=1

De nition 2.4 We say that X has a multidirectional mean value theorem if for any nonempty, closed and convex subset Y of X , any element x 2 X and any lower semicontinuous function f : X ! R bounded below on a neighborhood of [x; Y ] and r < lim inf f (y) ? f (x); !0 y2Y +BX given " > 0, there exist z 2 [x; Y ] + "B and z  2 D f (z ) such that

r < hz ; y ? xi + "ky ? xk for all y 2 Y: Further, we can choose z to satisfy

f (z ) < lim inf f + jrj + ": !0 [x;Y ]+B X

De nition 2.5 We say that X has aPlocal fuzzy sum rule if, for lower semicontinuous functions f1; :::; fN : X ! R such that Nn=1 fn attains a local minimum at x and satis es, for some  > 0

N N X X inf fn (y)  "lim inf f fn(xn ) : kxn ? xmk  "; xn ; xm 2 x + BX ; n; m = 1; :::; N g; (3) !0 y2x+BX n=1

n=1

and any " > 0, there exist xn; n = 1; :::; N satisfying (xn; fn (xn)) 2 (x; fn(x)) + "BX R and xn 2 D fn(xn ) such that N X 0 2 xn + "BX  : n=1

De nition 2.6 We say that X has a Lipschitz local fuzzy sum rule if condition (3) in De nition 2.5 is replaced by all but one of the functions are locally Lipschitz at x. 4

Remark 2.7 Obviously condition (3) is satis ed if all but one of the functions fn are uni-

formly continuous in a neighborhood of x. In particular, a local fuzzy sum rule implies a local Lipschitz fuzzy sum rule. Let S1 and S2 be closed sets in a Banach space X and let x 2 S1 \ S2 . Then x is called a locally extremal point of the set system fS1; S2 g if there are a neighborhood U of x and a sequence fak g  X , such that ak ! 0 and (S1 ? ak ) \ S2 \ U = ; 8k = 1; 2; : : : (4) De nition 2.8 . We say that X has a -extremal principle if, for every local extremal point x of closed sets S1 and S2 in X and any " > 0 there exist xi 2 Si \ (x + "BX ) and i 2 N (xi ; Si ) + "BX  ; i = 1; 2, such that k1k = k2k = 1 and 1 + 2 = 0: (5)

2.2 A sucient condition

Note that if X has a -smooth Lipschitz bump function then so does X N for each positive integer. It is known that for a Banach space with a -smooth Lipschitz bump function a -local fuzzy sum rule holds (see Borwein and Ioe [2] and the re nement in Borwein and Zhu [6]), a -nonlocal fuzzy sum rule and a -multidirectional mean value inequality holds (see [25]). Thus, we have Theorem 2.9 Let X be a Banach space with a -smooth bump function. Then X is subdiferential compatible.

2.3 Characterizations for Asplund spaces

In this subsection when refer to a Frechet-subdierential we use the following limiting definition. Let f : X ! R . The limiting Frechet-subdierential f at x 2 dom(f ) is the set f (x + h) ? f (x) ? h; hi  0g: f 2 X  : lim inf khk!0 khk When X has a Frechet-smooth bump function this limiting de nition and the viscosity Frechet-subdierential de nition we used so far are the same [9] but they may dier in general. We note that althrough our equivalent result is proved for viscosity subdierentials the proof is valid for the limiting Frechet-subdierential with obvious modi cations. In [13] Fabian proved that a Banach space is Asplund if and only if X has a Frechet Lipschitz fuzzy sum rule. In [20] Mordukhovich and Shao showed that an extremal principle in terms of the Frechet normal cone characterizes Asplund spaces. Observe if X is Asplund then so is X N for each positive integer N . Combining Fabian's and Mordukhovich and Shao's results and Theorem 2.1 we obtain X is Asplund if and only if X is Frechet subdierential compatible. That is: 5

Theorem 2.10 Let X be a Banach space. Then the following are equivalent: a. X is an Asplund space; b. for any positive integer N , X N is a Asplund space; c. for any positive integer N , X N has a Frechet-local Lipschitz fuzzy sum rule; d. for any positive integer N , X N has a Frechet-nonlocal fuzzy sum rule; e. for any positive integer N , X N has a Frechet-multidirectional mean value inequality; f. for any positive integer N , X N has a Frechet-local fuzzy sum rule; g. for any positive integer N , X N has a Frechet-extremal principle;

Remark 2.11 Theorem 2.10 give three new characterizations d., e. and f. of Asplund

spaces. It shows that for the purpose of discussing the Frechet-subdierential with a variational analysis approach the usual assumption that the underlying Banach space admit an equivalent Frechet-smooth norm or a Frechet-smooth bump function (associated with the use of smooth variational principles, see [4, 9]) can be weakened to Asplund property (every continuous convex function is densely Frechet dierentiable). We should note that the smooth variational principles still play a crucial role here because the genes of the proofs of these equivalent relations is the smooth variational principle combined with a separable reduction method in Fabian [13]. In [21] Mordukhovich and Shao discussed in detail calculus rules for a coderivative in Asplund spaces under certain additional constraint quali cation conditions required by the limiting process in de ning the coderivative. The equivalence theorem in this section shows that parallel calculus rules for Frechet-subdierentials and Frechet-coderivatives can be extablished in Asplund spaces without quali cation conditions.

2.4 Local fuzzy sum rules

Local fuzzy sum rules are very useful in discussing optimization problems and as a basic result for deriving other calculus rules for subdierentials. They have been the focus for many researches. Ioe rst proved a local fuzzy sum rule in nite dimensional spaces in [14]. In nite dimensional generalizations are discussed in [15] under the assumption that all but one summand functions are locally Lipschitz. The local Lipschitz assumption is weakened to uniform continuity by Deville and Haddad in [10]. Borwein and Ioe [2] and Ioe and Rockafellar [16] weakened the requirement further to a sequential form of the uniform lower semicontinuity condition. Then the slightly weaker topological form of the uniform lower semicontinuity (3) was used in Borwein and Zhu [6] where the fuzzy sum rule was also re ned to contain an estimate of the sizes of the subderivatives involved. It is somewhat surprising that according to the equivalence theorem in Section 2 all these dierent versions of the fuzzy sum rules are equivalent. On the other hand examples in Deville and Ivanov [11] and 6

Vanderwer and Zhu [24] show that local sum rule does not hold for lower semicontinuous functions without additional assumption while an example in [26] shows that the sucient conditions mentioned above are not necessary.

3 Proof of Theorem 2.1

We will show (a) ) (b) ) (c) ) (d) ) (e) ) (a). The implication (b) ) (c) has been established in [25]; for proximal subderivatives the implication (c) ) (d) has been established in [8] and for Frechet subdierentials implications (d) ) (e) ) (a) has been established in [20] in Asplund spaces. The methods for proving those special cases apply to the general case with minor modi cations. The missing link is (a) ) (b) which we establish below. For completeness we also include the proofs for the general case of the other implications.

3.1 (a) ) (b)

Consider extended-valued function f and Lipschitz function ' on X N de ned by

f (y) = f (y1 ; :::; yN ) := and

'(y) = '(y1; :::; yN ) :=

De ne, for any real number i > 0,

N X f (y )

n=1

n n

N X kyn ? ymk:

n;m=1

wi (y) := f (y) + i'(y) and Mi := inf wi . Then Mi is an increasing sequence. Note that , for each i,

Mi  inf ff (y) + i'(y) : diam(y1; ::; yN )  g  infff (y) + iN 2 : diam(y1; ::; yN )  g  infff (y) : diam(y1; ::; yN )  g + iN 2

(6)

Taking limits when  ! 0 we obtain inf ff (y) : diam(y1; ::; yN )  g: Mi  lim !0

(7)

Thus, Mi converges. Let M = limi!1 Mi . For each i, applying the Ekeland variational principle [12] to function wi , we obtain that there exist ui = (ui1; :::; uiN ) such that

wi (y) + "ky ? ui kX N = f (y) + i'(y) + "ky ? ui kX N 7

attains a minimum at y = ui and

wi (ui ) < inf wi + 1=i < M + 1=i: (8) Note that both i' and "k
k are Lipschitz. By the Lipschitz fuzzy sum rule (a) there exist xi ; z i 2 ui + ("=i2 )BX N with f (xi )  f (ui ) + "=i2 such that 0 2 D f (xi ) + D (i')(z i ) + ("=N )BX N  : (9) Let  i = (1i ; :::; Ni ) 2 D f (xi ) and  i = (1i ; :::; Ni ) 2 D (i')(z i ) satisfy 0 2  i +  i + ("=N )BX N  :

(10)

Then ni 2 D fn(xin ). Note that, for any h 2 X and t > 0 we have '(z1i + th; :::; zNi + th) = '(z1i ; :::; zNi ). Therefore,  i = (1i ; :::; Ni ) 2 D (i')(z i ) implies that, for any h 2 X ,

i'(z1 + th; :::; zN + th) ? i'(z1 ; :::; zN ) = 0; h1i + ::: + Ni ; hi  tlim + !0 t i

that is,

i

i

i

1i + ::: + Ni = 0:

Thus,

1i + ::: + Ni 2 "BX  :

By the de nition of Mi we have

Mi=2  wi=2 (ui ) = wi (ui ) ? 2i '(ui )  Mi + 1i ? 2i '(ui ):

(11)

Rewriting (11) as yields

i'(ui )  2(Mi ? M1=2 + 1i ) lim i i!1

Therefore,

(12)

N X kxin ? ximk = 0:

(13)

n;m=1

Since kxi ? ui k  "=i2 we also have lim i i!1

N X kuin ? uimk = 0:

n;m=1

lim diam(xi1 ; :::; xiN ) = 0:

i!1

8

Since ' is a Lipschitz function with a Lipschitz constant N 2 , k i k  iN 2. It follows from (13) that lim diam(xi1; :::; xiN )
max(k1i k; :::; kNi k) = 0: i!1 Thus, N X fn (xn) : diam(x1 ; :::; xN )  g n=1 N N X i )  lim inf X f (ui )  lim inf f ( x n n n n i!1 i!1

M  lim inf f !0 =

that is It remains to take

n=1 lim inf w (ui )  M; i!1 i

M = lim inf f !0 xn = xin

3.2 (b) ) (c)

n=1

N X fn(xn ) : diam(x1; :::; xN )  g:

n=1 and n = ni ,

n = 1; :::; N for a suciently large i.

1. A special case. We begin by considering the special case when

lim inf f (y) > f (x) and r = 0:

!0 y2Y +BX

Let f = f + [x;Y ]+hBX . Then f is bounded below on X . Fix a h 2 (0; h=2) such that inf y2Y +2hBX f (y) > f (x). Without loss of generality we may assume that

" < minfy2Y inf f (y) ? f (x); h g: +2hB X

Applying (b) to f1 = f and f2 = [x;Y ] we obtain that there exist z; u with kz ? uk < ",

z  2 D f(z ) = D f (z ) and u 2 N (u; [x; Y ]) satisfying

max(kz  k; ku k)
kz ? uk < "

(14)

f (z ) < lim inf f + "  f (x) + " !0 [x;Y ]+B

(15)

kz + uk < ":

(16)

and X

such that

9

Since [x; Y ] is convex N (u; [x; Y ]) coincides with the normal cone of [x; Y ] at u in the sense of convex analysis. Thus, u 2 N (u; [x; Y ]) implies that hu; w ? ui  0; 8w 2 [x; Y ]: (17) Combining (16) and (17) yields 0 < hz ; w ? ui + "kw ? uk; 8w 2 [x; Y ]nfug: (18) Moreover, we must have d(u; Y )  h for otherwise we would have d(z; Y )  2h and f (z )  inf y2Y +2hBX f (y) > f (x)+ " which contradicts (15). Let u = x + t(y ? x). Then h  ku ? yk = (1 ? t)kx ? yk implies 1 ? t > 0. Clearly x 62 Y . For any y 2 Y set w = y + t(y ? y) 6= u in (18) yields 0 < hz ; y ? ui + "ky ? uk; 8y 2 Y: (19) 2. The general case. We now turn to the general case. Consider X  R with the euclidean product norm. Take an "0 2 (0; "=2) small enough so that

lim inf f (y) ? f (x) > r + "0 and de ne F (z; t) := f (z ) ? (r + "0 )t. Obviously F is lower semicontinuous on X  R and is bounded below on [(x; 0); Y  f1g] + hBX R . Moreover, lim inf F = lim inf f ? (r + "0) > f (x) = F (x; 0): !0 Y f1g+B !0 Y +B !0 y2Y +BX

X

XR

Since X R is a subspace of X X we can apply the special case proved above with f , x and Y replaced by F , (x; 0) and Y f1g to conclude that there exist (z; s) 2 [(x; 0); Y f1g]+"BX R , z  2 D f (z ) satisfying f (z ) ? (r + "0 )s < lim inf (f (w) ? (r + "0 )t) + "0 !0 (w;t)2[(x;0);Y f1g]+B i.e.,

XR

f (z ) < lim inf (f (w) ? (r + "0 )(t ? s)) + "0 !0 (w;t)2[(x;0);Y f1g]+BXR  lim inf f + jrj + " !0 [x;Y ]+B X

such that, for all y 2 Y , q 0 < hz  ; y ? xi ? (r + "0 ) + "0 ky ? xk2 + 1  hz; y ? xi ? r + "0ky ? xk  hz; y ? xi ? r + "ky ? xk: This completes the proof. 10

3.3 (c) ) (d)

Let y = (y1; :::; yN ) 2 X N , f (y) := f1 (y1) + f2(y2) + ::: + fN (yN ) : X N ! R . Let " be an arbitrary positive number. Since fn are lower semicontinuous there exists a positive number 0 < " such that u 2 x + 0 BX implies that f (u) > f (x) ? "=2N . Let  < 0 =4N be as in the uniform lower semicontinuity condition (3) and Y := f(y1 ; y2; :::; yN ) : yn 2 x + BX ; n = 1; 2; :::; N g. Then lim inf f (u) ? f (x)  0 h!0 u2Y +hB XN

where x = (x; x; :::; x). Note that [x; Y ]  Y . Applying the multidirectional mean value theorem (c) with "0 = " min(1=4N; 0=2) and r = ?2 yields that there exist z = (x1; x2 ; :::; xN ) 2 Y + "0 BX N and z  = (x1 ; :::; xN ) 2 D f (z ), i.e., xn 2 D fn (xn) such that

?2 ? "0ky ? xkXN < hz; y ? xi; 8y 2 Y

and

(20)

f (z ) < hlim inf f (u) + "0 : !0 u2Y +hB N X

Observing that limh!0 inf u2Y +hBX N and fn(xn) > fn (x) ? "=4N we arrive at (xn; fn (xn)) 2 (x; fn (x)) + "BX R . For any w 2 BX , let y = (x + w; :::; x + w) in (20) we obtain N p X 2 0 ? ? " N kwk < h xn; wi:

f (u) + "0 < f (x) + "=4N

It follows that

n=1

N p X k x k < "0 N +   ": n=1

n

3.4 (d) ) (e)

Let x 2 S1 \ S2 be an extremal point of (S1; S2 ). Then there exists a neighborhood U of x such that, for any " > 0, there exists an a 2 X with kak < "2=4 and (S1 + a) \ S2 \ U = ;. We may choose a smaller " if necessary to ensure that x + 2" BX  U . Consider function f : X 2 ! R de ned by

f (x1 ; x2) := kx1 ? x2 + ak + (S1 \U )(S2 \U ) (x1; x2 ): Then

kak = f (x; x) < "4  inf f + "4 : 2

2

By the Ekeland variational principle [12] there exists a point (y1; y2) 2 (x; x) + 2" B such that (x1; x2 ) ! f (x1 ; x2) + " k(x1; x2 ) ? (y1 ; y2)k 2 11

attains a minimum at (x1; x2 ) = (y1; y2 ). Note that we must have y1 2 S1 and y2 2 S2. Therefore, ky1 ? y2 + ak > 0. Since both kx1 ? x2 + ak and 2" k(x1 ; x2) ? (y1; y2)k are Lipschitz these two function together with (S1 \U )(S2 \U ) (x1; x2 ) satisfy the condition of the local fuzzy sum rule (see Remark 2.7). Applying the local fuzzy sum rule (d) to the above functions, there exist (xi1 ; xi2) 2 (y1 ; y2) + 2" B  (x; x) + "B such that 0 2 D (kx11 ? x12 + ak) + N (x21; S1 )  N (x22; S2 ) + "B(X 2 ) : We may choose (x11; x12 ) closer to (y1; y2) if necessary to make kx11 ? x12 + ak > 0. Then it is easy to deduce that all the elements of D (kx11 ? x12 + ak) is of the form (; ? ) with k k = 1. Thus, there exists  in the unit ball of X  with (; ? ) 2 N (x21; S1 )  N (x22; S2 ) + "B(X 2 ) : It remains to take 1 =  and 2 = ? .

3.5 (e) ) (a)

Note that we need only to prove the Lipschitz local fuzzy sum rule for two functions because the general result can be deduced then by induction. Let f1 be a Lipschitz function with rank L and f2 be a lower semicontinuous function. Assume, without loss of generality that f1 + f2 attains a minimum at 0 and f1(0) = f2(0) = 0. De ne S1 := f(x; ) 2 X  R : f1 (x)  g and S2 := f(x; ) 2 X  R : f2 (x)  ?g: Then it is easy to check that (0; 0) is an extremal point for (S1; S2 ). For any " > 0, let "0 = minf"=4(1+ L)2; 1=4(1+ L)g. By the extremal principle (e), there exist (xi ; i ) 2 Si \ "BX R , and (i ; ?i) 2 X   R, i = 1; 2 such that (i ; ?i) 2 N ((xi ; i ); Si );

k(i ; ?i)k  1 ? "0  1=2; i = 1; 2 and

(21)

j1 + 2j < "0 k1 + 2k < "0: Let g be a -smooth function on X  R such that S1 ? g attains a minimum 0 at (x1; 1 ) and rg(x1 ; 1 ) = (1 ; ?1). Note that we must have S1 (x1 ; 1) = 0 so that g(x1 ; 1 ) = 0. Since f1 is Lipschitz with rank L, for any h 2 X with khk = 1 we have S1 (x1 +th; 1 +tL) = 0

for t > 0 suciently small. Therefore, g(x1 + th; 1 + tL) ? g(x1 ; 1)  0: t 12

Taking limits as t ! 0 we arrive at

h; hi ? L  0; 8khk = 1:

(22)

Combining (21) and (22) we get 1 > 2(1+1 L) > 0 and 2 < "0 ? 1 < ? 4(1+1 L) < 0 which forces 1 = f1(x1 ) and 2 = f2 (x2). By Proposition 2.3 in [3] we have x1 := 1 =1 2 D f1 (x1) and x2 := ?2=2 2 D f2(x2). Then, observing that kx1k  L we obtain kx1 + x2k = k 1 ? 2 k = k 1(1 + 2) ? 1 + 2 k 1 2 1 2 2  +  k  +  k  kx1k
j 1  2 j + 1j j 2  4L(1 + L)"0 + 4(1 + L)"0 = 4(1 + L)2"0 < ": 2 2

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