# V-Proximal Trustworthy Banach Spaces

Authors: Messaoud Bounkhel; Mostafa Bachar
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In a recent work (2016), the first author proved the fuzzy sum rule for theV-proximal subdifferential under some natural assumptions on an equivalent norm of the Banach spaces. In the present paper, we are going to prove that the class of Banach spaces satisfying the fuzzy sum rule is very large and contains all Lp spaces 1<p<∞ as well as the sequence spaces lp1<p<∞, the Sobolev spaces Wp,n1<p<∞, and the Schatten trace ideals Cp1<p<∞.


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## Body

## 1. Preliminaries

LetX be a Banach space with dual X∗,f:X⟶R∪∞ a function, x¯∈domf:x∈X:fx<∞. Throughout the paper, B denotes the closed unit ball in X and ·,· is the dual pairing between X and its dual X∗. We define (see [1, 2]) the analytic (resp., the geometric) V-proximal subdifferential of f at x¯ as follows:
(1)∂Aπfx¯≔x∗∈X∗:∃δ,σ>0,x∗,x−x¯≤fx−fx¯+σVJx¯,x,∀x∈x¯+δB(resp. ∂Gπfx¯≔x∗∈X∗:x∗,−1∈Nπepif;x¯,fx¯, where NπS;u≔∂AπψSu is the V-proximal normal cone associated with S at u∈S. Here, J:X⟶X∗ is the normalized duality mapping and V:X∗×X⟶R is a functional defined by
(2)Vx∗,x=x∗2−2x∗,x+x2,foranyx∗∈X∗,x∈X.We recall, respectively, the well-known concepts of proximal subdifferential and Fréchet subdifferential (see, for instance, [3]):
(i)
x∗∈∂Pfx if and only if there exist δ,σ>0 such that x∗,x′−x≤fx0−fx+σx′−x2,foranyx0∈x+δB(ii)
x∗∈∂Ffx if and only if for any ε>0, there exists δ>0 such that(3)x∗,x′−x≤fx′−fx+εx′−x,foranyx′∈x+δBThe Fréchet and proximal normal cones are defined asNFS;x¯=∂FψSx¯ and NPS;x¯=∂PψSx¯. Notice that (see [3–5]) Fréchet and proximal subdifferential can be defined geometrically by the formulas
(4)∂Ffx¯=x∗∈X∗:x∗,−1∈NFS;x¯,(5)∂Pfx¯=x∗∈X∗:x∗,−1∈NPS;x¯.Using the same terminology used in Ioffe [6], we will say that X is a V-proximal trustworthy space provided that for any ε>0, any two functions f1,f2:X⟶R∪∞ and any u∈X such that f1 is lower semicontinuous and f2 is Lipschitz around u, the following fuzzy sum rule holds:
(6)∂Aπf1+f2u⊂∪∂Aπf1u1+∂Aπf2u2:ui∈Ufiu,ε,i=1,2+εB∗.Here,Ufiu,ε≔x∈u+εBsuchthatfix−fiu<εand B∗ denotes the closed unit ball in X∗. It has been proved in Theorem 2.3 in [1] that if X is a Banach space with an equivalent norm ⋅ such that ·s (for some s ≥2) is C2-differentiable on X \{0} and let V be the functional associated to that norm ⋅, then X is a V -proximal trustworthy space. Indeed, we have proved the following result.Theorem 1.
Let X be a Banach space with an equivalent norm⋅such that·s(for somes≥2) isC2-differentiable onX\0, and letVbe the functional associated to that norm⋅. For anyε>0, any two functionsf1,f2:X⟶R∪∞and anyu∈Xsuch thatf1is lower semicontinuous andf2is Lipschitz aroundu, the following fuzzy sum rule holds:(7)∂Aπf1+f2u⊂∪∂Aπf1u1+∂Aπf2u2:ui∈Ufiu,ε,i=1,2+εB∗.For a positive measure space (Ω, Σ, μ), we denote by Lp,p∈1,∞, the Banach space LpΩ,Σ,μ with its canonical norm x=∫Ωxwpdμw1/p.We recall the following result from Theorem 1.1 in Section 5.1 in [7] (see also [8]).Theorem 2.
(i)
Ifpis an even integer, then·pisC∞-differentiable onLp0(ii)
Ifpis an odd integer, then·pisCp−1-smooth onLp\0(iii)
Ifpis not integer, then·pisCp-smooth onLp\0, wherepis the integer part ofpThe following corollary follows directly from Theorem 2.Corollary 3.
For the canonical norm ofLp, we have·pwhich isC2-differentiable onLp\0, for anyp≥2.Unfortunately, for the case ofp∈1,2, the function ·p is not C2-differentiable on Lp\0 and so the fuzzy sum rule cannot be covered by Theorem 1. This our objective in the next few lines, and so we obtain that Lp is V-proximal trustworthy, for any p∈1,∞. In the sequel of this section, we assume that X≔Lp with p∈1,2. It is well-known that X is 2-uniformly convex (see, for instance, [7]), that is, there is a constant c>0 such that
(8)δXε≥cε2,forall0<ε≤2.The following lemma is taken from [9].Lemma 4.
IfEis a uniformly convex Banach space, then the inequality(9)VJx,y≥8C2δEx−y4Cholds for allxandyinE, whereC=x2+y2/2.In page 51 in [9], the following parallelogram inequality is presented:
(10)2x2+2y2−x−y2≥p−1x+y21<p≤2foranyx,y∈X.It follows that(11)x−y4C≤12,foranyx,y∈X,and so the previous lemma with the inequality (8) yields the following important result.Proposition 5.
For some positive constantc>0, we have for any x and y inX(X=Lp1<p≤2),(12)VJx,y≥c2x−y2.Using this proposition, we get obviously, in our setting (X=Lp1<p≤2), the inclusions
(13)∂Pfx¯⊂∂Aπfx¯,(14)NPS;x¯⊂NπS;x¯.However, the inclusions∂Aπfx¯⊂∂Gπfx¯⊂∂Ffx¯andNπS;x¯⊂NFS;x¯ have been proved, in [1], for Banach spaces with the assumption: For some σ¯,δ¯>0 and q∈1,2,
(15)VJx¯,x≤σ¯x−x¯q,forallx∈x¯+δ¯B.This assumption is valid whenever the space isq-uniformly convex Banach spaces. For reason of completeness, we present their proofs.Proposition 6.
Assume that there exist constantsσ¯,δ¯>0andq∈1,2such that 1.8 holds. Then,(16)NπS;x¯⊂NFS;x¯,∂Aπfx¯⊂∂Gπfx¯⊂∂Ffx¯.Proof.
We prove only the first relation; the second one can be conducted similarly. Letx∗∈NπS;¯x. Then, by Proposition 3.4 in [2], there exists σ>0 such that
(17)x∗,x−x¯≤σVJ¯x,x,forallx∈S.Now, letε>0 be given and take μ=1/2minδ¯,ε/σσ¯a/q−1. Then, by (15),forx∈S∩x¯+µB⊂x¯+δ¯B, one has
(18)x∗,x−x¯≤σVJx¯,x≤σσ¯x−x¯q=σσ¯x−x¯q−1x−x¯<εx−x¯.Hence, for any>0, there exists μ>0 such that x∗,x−x¯≤εx−x¯, for any x∈S∩x¯+µB, that is x∗∈NFS;x¯ and so NπS;x¯⊂NFS;x¯.Using Remark 7.7 in [9] and the fact that X=Lpp∈1,2 is p-uniformly smooth, we obtain
(19)VJx,y≤βρX4x−yR≤β4px−ypRp,∀x,y∈RBand hence, for any x¯∈X and any δ>0, we can write
(20)VJx¯,x≤σ¯x−x¯p,∀x∈x¯+δB,where σ¯ depends on x¯,δ, and β. Consequently, as a direct corollary of Proposition 6, we have, in our setting of Lp spaces (p∈1,2), both inclusions
(21)NπS;x¯⊂NFS;x¯,(22)∂Aπfx¯⊂∂Gπfx¯⊂∂Ffx¯that hold for any lower semicontinuous function f and any nonempty closed set S.Now, we recall from Ioffe [10] and Fabian [11] the following two important results.Proposition 7.
Suppose thatXis uniformly convex and thatSis a closed subset ofX. Letx¯∈Sandx∗∈NFS;¯x. Then, for anyε>0, there existxε∈Sandxε∗∈NPS;xεsuch thatxε−x¯<εandxε∗−x∗<ε.Theorem 8.
LetXbe an Asplund space. For anyε>0, any two functionsf1,f2:X⟶R∪∞and anyu∈Xsuch thatf1is lower semicontinuous andf2is Lipschitz aroundu, the following fuzzy sum rule holds:(23)∂Ff1+f2u⊂∪∂Ff2u2:ui∈Ufiu,ε,i=1,2+εB∗.Now we are ready to prove the main result of this section.Theorem 9.
LetX=Lpp∈1,2. For anyε>0, any two functionsf1,f2:X⟶R∪∞and anyu∈Xsuch thatf1is lower semicontinuous andf2is Lipschitz aroundu, the following fuzzy sum rule holds:(24)∂Aπf1+f2u⊂∪∂Aπf1u1+∂Aπf2u2:ui∈Ufiu,ε,i=1,2+εB∗.Proof.
Fix anyε>0. Let x∗∈∂Aπf1+f2u. Then, by (21) we have x∗∈∂Ff1+f2u. Then, by Theorem 8, there exist vi∈Ufiu,ε/2 and xi∗∈∂Ffivi, i=1,2 such that x∗−x1∗+x2∗<ε/2. Thus, xi∗,−1∈NFepifi;vi,fivi, i=1,2. Let δ∈0,minε/2,ε/41+maxx1,x2. We use now Proposition 7 to get i=1,2, xi,δ,ri,δ∈epifiand x∗i,δ,−αi,δ∈NPepifi;xi,δ,ri,δ such that
(25)xi,δ,ri,δ−vi,fivi<δ2,xi,δ∗,−αi,δ−xi∗,−1<δ2.
Clearly,αi,δ>0 and so ri,δ=fixi,δ, for i=1,2. Thus, by the inclusions (5) and (13), we obtain
(26)xi,δαi,δ∈∂Pfixi,δ⊂∂Aπfixi,δxi,δ,fixi,δ−vi,fivi<δ2,xi,δ∗,−αi,δ−xi∗,−1<δ2.
Setyi∗≔xi,δ∗/αi,δ and ui≔xi,δi=1,2. Then, we have
(27)ui−u≤ui−vi+vi−u≤δ2+ε2<ε,fiui−fiu≤fiui−fivi+fivi−fiu<δ2+ε2<ε.
Also, we have(28)yi∗−xi∗≤yi∗−αi,δ−1xi∗+αi,δ−1xi∗−xi∗≤αi,δ−1xi,δ∗−xi∗+αi,δ−1−1xi∗≤δ2αi,δ+1−αi,δαi,δxi∗≤δ2αi,δ1+xi∗<δ1+xi∗≤ε4.
Thus,(29)x∗−y1∗+y2∗≤x∗−x1∗+x2∗+∑i=12yi∗−xi∗≤ε2+ε2=ε.
Therefore, for anyε>0, there exist u1∈Uf1u,ε, u2∈Uf2u,ε, y1∗∈∂Aπf1u1, and y2∗∈∂Aπf2u2 such that
(30)x∗∈y1∗+y2∗+εB∗.
This completes the proof.Remark 10.
(i)
An inspection of the previous proof shows that Theorem 9holds for any 2-uniformly convex andp-uniformly smooth. Indeed, all the assumptions of Propositions6–7and Theorem 8are satisfied whenever the space is 2-uniformly convex andp-uniformly smooth. Consequently, all the spaceslp,Cp,Wp,m,Lpforp∈1,2areV-proximal trustworthy. For the proof of the uniform convexity and uniform smoothness of the spaceslp,Lp,Wp,m, we refer, for instance, to Remark 1.6.9 in [9] and for the Schatten trace idealsCp, we refer to [12](ii)
For the casep≥2, we have the following:(i)
All Sobolev spacesWp,mandLpareV-proximal trustworthy for anyp≥2by combining Theorem 2 and Theorem1.(ii)
The spacesCpforp≥2areV-proximal trustworthy by combining Theorem1.and the following result proved in Theorem 2 in [13].Theorem 11.
(i)
Ifpis an even integer, then the norm inCpisC∞-differentiable away from zero(ii)
Ifpis an odd integer, then the norm inCpisCp−1-differentiable away from zero and is notCp−1-differentiable(iii)
Ifpis not integer, then the norm inCpisCp-differentiable away from zero and is notCp+1-differentiable


## 2. Approximate Mean Value Theorem

This section is concerned with the approximate mean value theorem for both analytic and geometricV-proximal subdifferentials in V-proximal trustworthy spaces. It is well-known that approximate mean value theorem (AMVT) and its variants are very important in nonsmooth analysis and optimization. It has been proved for all the existing subdifferentials (see, for instance, [14]). Since the analytic V-proximal subdifferential ∂Aπ is the smallest one in some important cases (for instance, Lp,lp,Cp,Wp,m with p≥2), we cannot deduce that the AMVT form the ones proved for the other existing subdifferentials ([14]). For this reason, it is needed to prove the AMVT for the analytic V-proximal subdifferential ∂Aπ in V-proximal trustworthy spaces. For the geometric V-proximal subdifferential ∂Gπ, the AMVT follows directly from the inclusion ∂Aπf⊂∂Gπf.Theorem 12 (Approximate mean value theorem).
LetXbe a smooth Banach space which isV-proximal trustworthy and letf:X⟶R∪∞, be l.s.c. function finite at two distinct pointsa≠b. Letr∈Rwithr≤fb−fa. Then, there existsc∈a,band a sequencexn⟶cwithf(xn) ⟶ f(c) andxn∗∈∂Aπfxnsuch that(31)liminfnxn∗,c−xn≥0,liminfnxn∗,b−α≥rfc−fa≤r.Proof.
SinceX is smooth, we can take x∗∈X∗ so that r≔x∗,a−b. Set gx≔fx+x∗,x+ψa,bx. Clearly, gb≥ga and so g attains its minimum at some point c∈a,b and so 0∈∂Aπf⋅+x∗,⋅+ψa,b⋅c. Using the fuzzy sum rule for ∂Aπ in Theorem 9, we get a sequence xn⟶c with fxn⟶fc, yn⟶c, and xn∗∈∂Aπfxn,yn∗∈Nπa,b;yn such that xn∗+yn∗+x∗<1/n. Then, ynn⊂a,b for n sufficiently large and hence,
(32)yn∗,z−yn≤0,∀z∈a,b,∀n.
Hence,xn∗,c−xn=xn∗+x∗+yn∗,c−xn+−yn∗,c−yn+−yn∗,yn−xn+−x∗,c−xn≥xn∗+x∗+yn∗,c−xn+−yn∗,yn−xn+−x∗,c−xn(by (32)).
Taking the liminf on both sides of the previous inequality yields(33)liminfnxn∗,c−xn≥liminfnxn∗+x∗+yn∗,c−xn+−yn∗,yn−xn+−x∗,c−xn≥0,and hence, (i) is proved. Let us show (ii). Since yn∈a,b, we can write yn≔tnb+1−tna for a sequence tnn⊂0,1 converging to some t0∈0,1 satisfying c=t0b+1−t0a. Hence, b−yn=1−tnb−a and so b−a=1/1−tnb−yn. Hence,
(34)liminfnxn∗,b−a=x∗,a−b+liminfnxn∗+x∗,b−a=r+liminfn−yn∗,b−a=r+11−t0liminfnyn∗,yn−b≥by29.To prove (iii), we use the fact thatc is the minimum of g on a,b and hence, gc≤ga; that is, fc−fa≤x∗,a−c≤1/1−t0x∗,a−b=1/1−t0r≤r.We close this section with the following result which is an application of the AMVT. Many other applications of the previous version of AMVT can be proved.Theorem 13 (Lipschitz behavior).
LetXbe a smooth Banach space which isV-proximal trustworthy and letUbe an open convex subset ofX. Let f: X⟶R∪+∞be a l.s.c. function withU∩domf6=∅. The following two assertions are equivalent:(1)
For some positive constantL>0, we havesupx∗:x∗∈∂Aπfx ≤ L, for allx∈U(2)
fis Lipschitz onUwith rankLProof.
The implication2⇒1 follows directly from the fact that ∂Aπfx is always included in the Clarke subdifferential. So, we have to prove the implication 1⇒2. Let x,y∈U and let r≔fy−fx. Applying Theorem 12 to get a point c∈x,y⊂U and a sequence xn⟶c with fxn⟶fc and xn∗∈∂Aπfxn such that
(35)fy−fx≤liminfnxn∗,y−x.
Letε>0. By the definition of the liminf, there exist z∈U and z∗∈∂Aπfz such that
(36)fy−fx≤z∗,y−x+ε≤z∗y−x+ε≤Ly−x+ε.
By takingε⟶0, we obtain
(37)fy−fx≤Ly−x,and by exchanging the roles of x and y we arrive to
(38)fy−fx≤Ly−x,forallx,y∈U.
This completes the proof.



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*Source: 4274160-2020-06-05.xml*