Abstract
In a recent work (2016), the first author proved the fuzzy sum rule for the V-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 spaces as well as the sequence spaces , the Sobolev spaces , and the Schatten trace ideals .
1. Preliminaries
Let be a Banach space with dual a function, . Throughout the paper, denotes the closed unit ball in and is the dual pairing between and its dual . We define (see [1, 2]) the analytic (resp., the geometric) -proximal subdifferential of at as follows: (resp. , where is the -proximal normal cone associated with at . Here, is the normalized duality mapping and is a functional defined by
We recall, respectively, the well-known concepts of proximal subdifferential and Fréchet subdifferential (see, for instance, [3]): (i) if and only if there exist such that (ii) if and only if for any , there exists such that
The Fréchet and proximal normal cones are defined as and . Notice that (see [3–5]) Fréchet and proximal subdifferential can be defined geometrically by the formulas
Using the same terminology used in Ioffe [6], we will say that is a -proximal trustworthy space provided that for any , any two functions and any such that is lower semicontinuous and is Lipschitz around , the following fuzzy sum rule holds:
Here, and denotes the closed unit ball in . It has been proved in Theorem 2.3 in [1] that if X is a Banach space with an equivalent norm such that (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 normsuch that(for some) is-differentiable on, and letbe the functional associated to that norm. For any, any two functionsand anysuch thatis lower semicontinuous andis Lipschitz around, the following fuzzy sum rule holds:
For a positive measure space (, , ), we denote by , the Banach space with its canonical norm .
We recall the following result from Theorem 1.1 in Section 5.1 in [7] (see also [8]).
Theorem 2. (i)Ifis an even integer, thenis-differentiable on(ii)Ifis an odd integer, thenis-smooth on(iii)Ifis not integer, thenis-smooth on, whereis the integer part of
The following corollary follows directly from Theorem 2.
Corollary 3. For the canonical norm of, we havewhich is-differentiable on, for any.
Unfortunately, for the case of , the function is not -differentiable on 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 is -proximal trustworthy, for any . In the sequel of this section, we assume that with . It is well-known that is 2-uniformly convex (see, for instance, [7]), that is, there is a constant such that
The following lemma is taken from [9].
Lemma 4. Ifis a uniformly convex Banach space, then the inequalityholds for allandin, where.
In page 51 in [9], the following parallelogram inequality is presented:
It follows that and so the previous lemma with the inequality (8) yields the following important result.
Proposition 5. For some positive constant, we have for any x and y in(),
Using this proposition, we get obviously, in our setting (), the inclusions
However, the inclusions have been proved, in [1], for Banach spaces with the assumption: For some and ,
This assumption is valid whenever the space is -uniformly convex Banach spaces. For reason of completeness, we present their proofs.
Proposition 6. Assume that there exist constantsandsuch that 1.8 holds. Then,
Proof. We prove only the first relation; the second one can be conducted similarly. Let . Then, by Proposition 3.4 in [2], there exists such that
Now, let be given and take . Then, by (15),
for , one has
Hence, for any >0, there exists such that , for any , that is and so .
Using Remark 7.7 in [9] and the fact that is -uniformly smooth, we obtain and hence, for any and any , we can write where depends on , and . Consequently, as a direct corollary of Proposition 6, we have, in our setting of spaces (), both inclusions that hold for any lower semicontinuous function and any nonempty closed set .
Now, we recall from Ioffe [10] and Fabian [11] the following two important results.
Proposition 7. Suppose thatis uniformly convex and thatis a closed subset of. Letand. Then, for any, there existandsuch thatand.
Theorem 8. Letbe an Asplund space. For any, any two functionsand anysuch thatis lower semicontinuous andis Lipschitz around, the following fuzzy sum rule holds:
Now we are ready to prove the main result of this section.
Theorem 9. Let. For any, any two functionsand anysuch thatis lower semicontinuous andis Lipschitz around, the following fuzzy sum rule holds:
Proof. Fix any . Let . Then, by (21) we have . Then, by Theorem 8, there exist and , such that . Thus, , . Let . We use now Proposition 7 to get , and such that Clearly, and so , for . Thus, by the inclusions (5) and (13), we obtain Set and . Then, we have Also, we have Thus, Therefore, for any , there exist , , , and such that This completes the proof.
Remark 10. (i)An inspection of the previous proof shows that Theorem 9holds for any 2-uniformly convex and-uniformly smooth. Indeed, all the assumptions of Propositions6–7and Theorem 8are satisfied whenever the space is 2-uniformly convex and-uniformly smooth. Consequently, all the spacesforare-proximal trustworthy. For the proof of the uniform convexity and uniform smoothness of the spaces, we refer, for instance, to Remark 1.6.9 in [9] and for the Schatten trace ideals, we refer to [12](ii)For the case, we have the following:(i)All Sobolev spacesandare-proximal trustworthy for anyby combining Theorem 2 and Theorem1.(ii)The spacesforare-proximal trustworthy by combining Theorem1.and the following result proved in Theorem 2 in [13].
Theorem 11. (i)Ifis an even integer, then the norm inis-differentiable away from zero(ii)Ifis an odd integer, then the norm inis-differentiable away from zero and is not-differentiable(iii)Ifis not integer, then the norm inis-differentiable away from zero and is not-differentiable
2. Approximate Mean Value Theorem
This section is concerned with the approximate mean value theorem for both analytic and geometric -proximal subdifferentials in -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 -proximal subdifferential is the smallest one in some important cases (for instance, with ), 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 -proximal subdifferential ∂ in -proximal trustworthy spaces. For the geometric -proximal subdifferential , the AMVT follows directly from the inclusion .
Theorem 12 (Approximate mean value theorem). Letbe a smooth Banach space which is-proximal trustworthy and let, be l.s.c. function finite at two distinct points. Letwith. Then, there existsand a sequencewithf(xn) ⟶ f(c) andsuch that
Proof. Since is smooth, we can take so that . Set . Clearly, and so attains its minimum at some point and so . Using the fuzzy sum rule for in Theorem 9, we get a sequence with , , and such that . Then, for sufficiently large and hence,
Hence, (by (32)).
Taking the liminf on both sides of the previous inequality yields
and hence, (i) is proved. Let us show (ii). Since , we can write for a sequence converging to some satisfying . Hence, and so . Hence,
To prove (iii), we use the fact that is the minimum of on and hence, ; that is, .
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). Letbe a smooth Banach space which is-proximal trustworthy and letbe an open convex subset of. Let f: be a l.s.c. function with. The following two assertions are equivalent:(1)For some positive constant, we have ≤ L, for all(2)is Lipschitz onwith rank
Proof. The implication follows directly from the fact that is always included in the Clarke subdifferential. So, we have to prove the implication . Let and let . Applying Theorem 12 to get a point and a sequence with and such that Let . By the definition of the liminf, there exist and such that By taking , we obtain and by exchanging the roles of x and y we arrive to This completes the proof.
Data Availability
Not applicable.
Conflicts of Interest
The authors declare that they have no competing interests.
Authors’ Contributions
All authors contributed equally to this work.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding the work through the research group Project No. RGP-024.