Some results on integration of subdifferentials

Nonlinear Analysis 39 (2000) 955 – 976 www.elsevier.nl/locate/na Some results on integration of subdi erentials Zili Wu, Jane J. Ye ∗ Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada V8W 3P4 Received 13 January 1998; accepted 28 June 1998 Keywords: Nonsmooth analysis; Integration of subdi erentials; Clarke subdi erential; Michel– Penot subdi erential; Lower Dini subdi erential; Frechet subdi erential; m-subdi erential and @∗ -subdi erential 1. Introduction Integration of subdi erentials is a fundamental problem in nonsmooth analysis. The problem is whether or not the condition that the subdi erential of f contains the subdi erential of g implies that f and g di er only by a constant. For this problem, perhaps the earliest result is Theorem 24.9 in Rockafellar [15], which asserts that if f and g are two closed proper convex functions such that @c g(x) ⊆ @c f(x) ∀x ∈ R n ; where @c is the subdi erential in the sense of convex analysis, then f and g di er by a constant. Some progress on the study of integration of subdi erentials for nonconvex functions has been made over the last decade or so. Rockafellar [16] proved that if f; g : R n → R are locally Lipschitz and f is Clarke regular, and if the Clarke subdi erential @g(x) ⊆ @f(x) ∀x ∈ R n , then g(x) = f(x) + C for some constant C. This result was generalized to upper–upper and upper–lower regular functions in any Banach space by Correa and Thibault [10]. Some classes of nonconvex functions which can be determined by their Clarke subdi erentials (i.e., any two functions from the same class di er only by a constant if their Clarke subdi erentials coincide) have also been found. Qi [14] showed that a primal function whose domain is connected is determined by its Clarke subdi erential up to an additive constant. Poliquin [13] found a class of functions called primal lower nice functions which can be determined by their Clarke subdi erentials (hence their proximal subdi erentials since they coincide for this class Corresponding author. Tel.: +1-250 -721-7437; fax: +1-250 -721-8962. E-mail address: janeye@uvic.ca (J.J. Ye) ∗ 0362-546X/00/$ - see front matter ? 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 2 6 3 - 6 956 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 of functions) up to a constant. Other results on integration of subdi erentials include those of Borwein [2], Correa and Jofre [9], Thibault and Zagrodny [17]. Although most of these integration results concern the Clarke subdi erential, integrating subdi erentials other than that of Clarke is also important. Clarke and Redhe er [6] proved that if g : R n → (−∞; ∞] is a lower semicontinuous function, f is a continuously twice di erentiable function on R n and @ g(x) ⊆ @ f(x), where @ denotes the proximal subdi erential, then f(x) = g(x)+C for some constant C. They also indicated that if f is di erentiable and @F g(x) ⊆ @F f(x), where @F is the Frechet subdi erential, then f and g di er only by a constant. In this paper, we study primarily the integration of @∗ -subdi erentials, a class of subdi erentials which includes the lower Dini subdi erential, the Frechet subdi erential and the m-subdi erential. We prove that if lower semicontinuous functions f : X → R, g : X → (−∞; ∞] are such that g − f is lower semicontinuous on a nonempty open convex subset U of X , and for each x in U , @∗ f(x) ∪ @∗ (−f)(x) is nonempty and bounded, then g di ers from f by a constant on U if and only if for some subdi erential @0 and any x in U; @∗ g(x) ⊆ @∗ f(x) for x with @∗ f(x) 6= ∅; @∗ (−f)(x) ⊆ @∗ (−g)(x) for x with @∗ f(x) = ∅ and @0 (g − f)(x) ⊆ @∗ (g − f)(x), where @0 h is a subdi erential having the property that @0 h(x) ⊆ {0} on U implies that h is constant on U . This result serves to unify and extend several results in the literature. In particular, a consequence of the above result is a generalization of Rockafellar’s result on integrability of regular functions, i.e., if f is a locally Lipschitz function and g is an extended-valued lower semicontinuous function on an Asplund space or a separable Banach space and if f is Michel–Penot regular (weaker than Clarke regular) and @− g(x) ⊆ @− f(x), where @− denotes the lower Dini subdi erential (weaker than @g(x) ⊆ @f(x)) for all x, then f and g di er by a constant. We also obtain similar results for a separately regular function, i.e., a bivariate function on a product of two Asplund spaces (or two separable Banach spaces) X × Y which is Clarke regular as a function of x and y separately. Our main result also extends Clarke and Redhe er’s result by allowing f to be any local Lipschitz function whose proximal subdi erential or Frechet subdi erential is nonempty everywhere. We organize the paper as follows. In Section 2, we give conditions under which some @∗ -subdi erentials are nonempty and some results on the calculus of these subdi erentials. In Section 3, we discuss our integration results in detail. Throughout this paper, X is a real Banach space whose open unit ball and dual space are denoted by B and X ∗ , respectively. H denotes a real Hilbert space. 2. Calculus of subdi erentials We brie y review some well-known notions of subdi erentials. Let f : X → R be Lipschitz of rank L near x ∈ X . • The Clarke derivative of f at x in the direction v is de ned by f◦ (x; v) := lim sup y→x t→0+ f(y + tv) − f(y) : t Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 957 • The Michel–Penot derivative of f at x in the direction v is de ned by f⋄ (x; v) := sup lim sup y t→0+ f(x + ty + tv) − f(x + ty) : t • The Clarke subdi erential of f at x is the set @f(x) := { ∈ X ∗ : h; vi ≤ f◦ (x; v) ∀v ∈ X }: • The Michel–Penot subdi erential of f at x is the set @⋄ f(x): = { ∈ X ∗ : h; vi ≤ f⋄ (x; v) ∀v ∈ X }: Since the Clarke and Michel–Penot derivatives are both sublinear functions of v and their values f◦ (x; v) and f⋄ (x; v) are bounded by Lkvk, their corresponding subdi erentials are nonempty. Let f : X → R ∪ {∞} be lower semicontinuous (l.s.c.) at x ∈ dom f := {x ∈ X : f(x)¡ ∞} and v be in X . • The lower Dini derivative of f at x in the direction v is de ned by f− (x; v) := lim inf + t→0 f(x + tv) − f(x) : t • The lower Dini subdi erential of f at x is the set @− f(x) := { ∈ X ∗ : h; vi ≤ f− (x; v) ∀v ∈ X }: • The Frechet subdi erential of f at x is the set ( @F f(x) :=  ∈ X ∗ : f(y) − f(x) + (ky − xk) ≥ h; y − xi for some (t) with lim+ t→0  (t) = 0; ¿0 and any y in x + B : t To unify certain notions of subdi erentials, we de ne the m-subdi erential as in Clarke et al. [7]. De nition 2.1. Let f : X → R ∪ {∞} be l.s.c. at x ∈ dom f. Let m : [0; ∞) → [0; ∞) be a modulus function (m-function), that is, m is nondecreasing and limt→0+ m(t) = m(0) = 0. The m-subdi erential of f at x is the set @m f(x) := { ∈ X ∗ : f(y) − f(x) + Mm(ky − xk)ky − xk ≥ h; y − xi for some M ¿0; ¿0 and any y in x + B}: Common m-subdi erentials include the Fenchel subdi erential of f at x: @c f(x) := { ∈ X ∗ : f(y) − f(x) ≥ h; y − xi ∀y ∈ x + B}; 958 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 the proximal subgradient of f at x: @ f(x) := { ∈ X ∗ : f(y) − f(x) + M ky − xk 2 ≥ h; y − xi for some M ¿0; ¿0 and any y in x + B} and the s-Holder subdi erential (s¿0) of f at x: @HS f(x) := { ∈ X ∗ : f(y) − f(x) + M ky − xk1+s ≥ h; y − xi for some M ¿0; ¿0 and any y in x + B} whose special case (where s = 1) is Lipschitz smooth @LS f(x) which in Hilbert space coincides with Rockafellar’s proximal subgradient. Remark 2.2. From the de nition, it is easy to see that the following relations hold: (i) Let f : X → R ∪ {∞} be l.s.c. at x ∈ dom f. Then for any m-function m(t), @c f(x) ⊆ @m f(x) ⊆ @F f(x) ⊆ @− f(x): Moreover, if f : X → R ∪ {∞} is convex, then @c f(x) = @m f(x) = @F f(x) = @− f(x): (ii) Let U be a nonempty open convex subset of X and f : U → R bounded above on a neighborhood of some point of U . Then f is convex if and only if for each x ∈ U , f is Lipschitz near x and for any m-function m(t), @c f(x) = @m f(x) = @F f(x) = @− f(x) = @⋄ f(x) = @f(x): Generally, if f is Lipschitz near x, then @c f(x) ⊆ @m f(x) ⊆ @F f(x) ⊆ @− f(x) ⊆ @⋄ f(x) ⊆ @f(x): (iii) A function f : X → R is Frechet di erentiable at x ∈ X if and only if for some m-function m(t), both @m f(x) and @m (−f)(x) are nonempty, if and only if both @F f(x) and @F (−f)(x) are nonempty. In this case, @m f(x) = @F f(x) = {Df(x)}; where Df(x) is the Frechet derivative of f at x. By the de nition of subdi erentials, it is easy to prove the following scalar multiplication rule and sum rule. Proposition 2.3. Let f; g : X → R be l.s.c. at x ∈ dom f ∩ dom g: Then (i) for any ¿0, @∗ (f)(x) = @∗ f(x): (ii) @∗ f(x) + @∗ g(x) ⊆ @∗ (f + g)(x); where @∗ denotes @− , @m or @F and by convention A + B = ∅ if either A or B is empty. Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 959 Analogous to Gâteaux and Frechet di erentiability, we de ne m-di erentiability as follows: De nition 2.4. For some m-function m(t), a function f : X → R is m-di erentiable at x ∈ X if there exist  in X ∗ , M ¿0 and ¿0 such that |f(y) − f(x) − h; y − xi | ≤ Mm(ky − xk)ky − xk ∀y ∈ x + B: Clearly, f is m-di erentiable at x ∈ X if and only if both @m f(x) and @m (−f)(x) are nonempty. It turns out that when one of two functions in the sum rule is “di erentiable”, the inclusion in the sum rule becomes equality. We omit the proof which is similar to that of [7, Lemma 2.2]. Proposition 2.5. Let f; g : X → R ∪ {∞} be l:s:c: at x ∈ dom f ∩ dom g: (i) If g is m-di erentiable at x, then @m (f ± g)(x) = @m f(x) ± @m g(x): (ii) If g is Gâteaux di erentiable at x, then @− (f ± g)(x) = @− f(x) ± @− g(x): (iii) If g is Frechet di erentiable at x, then @F (f ± g)(x) = @F f(x) ± @F g(x): Remark 2.6. From Proposition 2:5, we conclude that adding a “di erentiable” function does not change nonemptiness of the subdi erentials, that is, if g : X → R ∪ {∞} is m-di erentiable (Gateaux di erentiable or Frechet di erentiable) at x, then @m f(x) ˆ  − m (@ f(x) or @F f(x)) is nonempty if and only if @ (f + g)(x) (@− (f + g)(x) or @F (f + g)(x)) is nonempty. Unlike the Clarke subdi erential of a function which is nonempty when the function is locally Lipschitz, the subdi erentials considered above may be empty. We now investigate some sucient conditions for nonemptiness of subdi erentials. The following is one for the nonemptiness of lower Dini subdi erential. Proposition 2.7. Let f : X → R be Lipschitz near x. Suppose that there exists  in X ∗ satisfying lim inf {h − ; vi:  ∈ @f(y) and y →v x} ≥ 0 ∀v ∈ X; where y →v x denotes that y goes to x in the direction v. Then  must be in @− f(x): Proof. Let ¿0 be small enough such that f is locally Lipschitz on x + B: For any nonzero vector v, let 0¡t¡=kvk: Then y := x + tv is in x + B: By Lebourg 960 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 Mean-Value Theorem [5, Theorem 2.3.7], there exist  in (0; 1) and  in @f(x + tv) such that f(y) − f(x) = h; y − xi: Thus f(x + tv) − f(x) − h; vi = h − ; vi: t Denote z := x + tv. Then  is in @f(z) and by assumption, lim inf {h − ; vi:  ∈ @f(z) and z →v x} ≥ 0 ∀v ∈ X: Hence, lim inf + t→0 f(x + tv) − f(x) − h; vi ≥ 0 t ∀v ∈ X; which implies that  is in @− f(x): Recall that f : X → R is said to be Clarke regular at x provided that for all v ∈ X , the usual one-sided derivative f′ (x; v) exists and is equal to the Clarke derivative f◦ (x; v) (see e.g. [5]). Clearly, if f is Lipschitz near x, then f is Clarke regular at x if and only if @− f(x) = @f(x): In this case, since the Clarke subdi erential is nonempty, so is the lower Dini subdi erential. Similarly, we de ne Michel–Penot (m-C or m-MP) regular function as below. De nition 2.8. Let f : X → R be Lipschitz near x. • f is said to be Michel–Penot regular at x if @− f(x) = @⋄ f(x). • f is said to be m-C regular at x if @m f(x) = @f(x) for some m-function m(t). • f is said to be m-MP regular at x if @m f(x) = @⋄ f(x) for some m-function m(t). Obviously, if a function is Clarke regular, then it is Michel–Penot regular. But the reverse is not true. For example, the function f is de ned by ( 2 x · sin 1x if x 6= 0; f(x) = 0 if x = 0 is Michel–Penot regular at 0 but not Clarke regular at this point since @− f(0) = @⋄ f(0) = {0} = 6 [−1; 1] = @f(0): The following proposition gives a sucient condition for m-subdi erential to be nonempty. Proposition 2.9. Let f : X → R be Lipschitz near x. If for some m-function m(t),  in X ∗ satis es lim inf y→x ∈@⋄ f(y) h − ; y − xi ¿−∞; m(ky − xk)ky − xk (1) Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 961 then  is in @m f(x): And hence if f is m-MP hypomonotone at x, i.e., inequality (1) holds for all  in @⋄ f(x), then f is m-MP regular at x. Proof. Let ¿0 be small enough such that f is locally Lipschitz on x + B: Then for any y in x + B; by the Mean-Value Theorem [3, Theorem 19], there exist  in (0; 1) and  in @⋄ f(x + (y − x)) such that f(y) − f(x) = h; y − xi: Denote z = x + (y − x). Then h − ; y − xi f(y) − f(x) − h; y − xi = m(ky − xk)ky − xk m(ky − xk)ky − xk = h − ; (z − x)=i m(ky − xk)(kz − xk=) = m(kz − xk) h − ; z − xi · : m(ky − xk) m(kz − xk)kz − xk By assumption, lim inf z→x ∈@⋄ f(z) h − ; z − xi ¿−∞: m(kz − xk)kz − xk Thus lim inf y→x f(y) − f(x) − h; y − xi ¿−∞; m(ky − xk)ky − xk which implies that  is in @m f(x): Furthermore, if inequality (1) holds for each  in @⋄ f(x), then by Remark 2.2(ii), we have @m f(x) = @F f(x) = @⋄ f(x): Therefore, f is m-MP regular at x. By Proposition 2.9 and the fact that @⋄ f(x) ⊆ @f(x) (see Remark 2.2(ii)), we can easily obtain the following result which implies that for an m-C hypomonotone function, which is introduced as a generalization of hypomonotone function de ned in Rockafellar [16], its m-subdi erential is always nonempty. Corollary 2.10. Let f : X → R be Lipschitz near x. If for some m-function m(t),  in X ∗ satis es lim inf y→;x ∈@f(y) h − ; y − xi ¿ − ∞; m(ky − xk)ky − xk (2) 962 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 then  is in @m f(x): And hence if f is m-C hypomonotone at x, i.e., inequality (2) holds for all  in @f(x), then f is m-C regular at x. Note that a hypomonotone function is an m-C hypomonotone function when m(t) = t: Based on Corollary 2.10, for hypomonotone function f, we have @ f(x) = @f(x), which was proved by Poliquin [13] when X = R n . In nonsmooth analysis, we often deal with the pointwise maximum function of several functions in the form of f(x) = max{f1 (x); : : : ; fl (x)}. For such a function, we present two cases in which the subdi erential of f can be expressed by those of fi ’s and hence its subdi erential is nonempty. Proposition 2.11. Let X be a real re exive Banach space. Suppose that fi : X → R is continuous for i = 1; : : : ; l and f(x) := max{f1 (x); : : : ; fl (x)}. Denote I (x) := {1 ≤ i ≤ l: fi (x) = f(x)} ∀x ∈ X: If for each i in I (x), fi is Frechet di erentiable at x ∈ X , then for some m-function m(t), @m f(x) = @F f(x) = co{Dfi (x): i ∈ I (x)}; where co denotes the convex hull, Dfi (x) is the Frechet derivative of fi at x for each i in I (x): Proof. According to Remark 2.2(i), @m f(x) ⊆ @F f(x). Hence it suces to prove that for some m-function m(t), @F f(x) ⊆ co{Dfi (x): i ∈ I (x)} ⊆ @m f(x): By Remark 2.2(iii), it is straightforward to show that for some m-function m(t), co{Dfi (x): i ∈ I (x)} ⊆ @m f(x): Hence we only need to prove @F f(x) ⊆ co{Dfi (x): i ∈ I (x)}. By contradiction suppose that  is in @F f(x) but not in co{Dfi (x): i ∈ I (x)}: Then there exist M ¿0, 1 ¿0 and (t) with limt→0+ (t)=t = 0 such that f(y) − f(x) + (ky − xk) ≥ h; y − xi ∀y ∈ x + 1 B: Denote S1 = {} and   X  X i Dfi (x): i ≥ 0; i = 1 : S2 =   i∈I (x) (3) i∈I (x) Obviously, S1 and S2 are two disjoint closed convex subsets of X ∗ . Also S1 is compact. By the separation theorem [8, Theorem 3.9, Ch. IV], S1 and S2 can be strictly separated, that is, there exist a nonzero vector p∗∗ in X ∗∗ and ¿0 such that kp∗∗ k = 1 and h p∗∗ ; i ≥  + sup{h p∗∗ ; i:  ∈ S2 }: 963 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 Since X is re exive, there is p in X satisfying kpk = kp∗∗ k = 1 and h p∗∗ ; x∗ i = hx∗ ; pi ∀x∗ ∈ X ∗ : Thus, h; pi ≥  + sup{h; pi:  ∈ S2 }: (4) On the other hand, by the continuity of fi for i = 1; : : : ; l, there exists 2 ¿0 such that f(y) = max{fi (y): i ∈ I (x)} ∀y ∈ x + 2 B: For each i in I (x), the Frechet di erentiability of fi at x implies fi (y) = fi (x) + hDfi (x); y − xi + ky − xk i (y − x) ∀y ∈ X; where limy→x i (y − x) = 0 for each i in I (x). Therefore for any y in x + 2 B, f(y) = max{fi (y): i ∈ I (x)} = max{fi (x) + hDfi (x); y − xi + ky − xk i (y − x): i ∈ I (x)} = max{f(x) + hDfi (x); y − xi + ky − xk i (y − x): i ∈ I (x)} = f(x) + max{hDfi (x); y − xi + ky − xk i (y − x): i ∈ I (x)}: Taking  = min{1 ; 2 } and y = x + (1=n)p, then by inequality (3), we have     1 1 1 1 + h; pi for n¿ : f x + p ≥ f(x) − n n n  And, hence,  max hDfi (x); pi + i      1 1 p : i ∈ I (x) ≥ h; pi − n n n 1 for n¿ :  Letting n → ∞ yields max{hDfi (x); pi: i ∈ I (x)} ≥ h; pi: Thus, sup {h; pi:  ∈ S2 } = sup  X  i hDfi (x); pi: i ≥ 0; i∈I (x) X i∈I (x) ≥ max{hDfi (x); pi: i ∈ I (x)} ≥ h; pi; which contradicts inequality (4). Therefore, @F f(x) ⊆ co{Dfi (x): i ∈ I (x)} and the proof of the proposition is complete.   i = 1  964 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 The following proposition shows that the class of m-C regular functions is closed under ( nite) pointwise maximization. Proposition 2.12. Let fi : X → R be Lipschitz near x for i = 1; : : : ; l and f(x) = max{f1 (x); : : : ; fl (x)}: Denote I (x) := {1 ≤ i ≤ l: fi (x) = f(x)} ∀x ∈ X: If for some m-function mi , fi is mi -C regular for each i in I (x); then f is m-C regular and @m f(x) = co{@m fi (x): i ∈ I (x)}; where m(t) = max{mi (t): i ∈ I (x)}: Proof. Since @mi fi (x) ⊆ @m fi (x) ⊆ @fi (x) and @mi fi (x) = @fi (x) for each i in I (x), @m fi (x) = @fi (x): By Remark 2.2(ii) and [5, Proposition 2.3.12], we have @m f(x) ⊆ @f(x) ⊆ co{@fi (x): i ∈ I (x)} = co{@m fi (x): i ∈ I (x)}: So it suces to show that co{@m fi (x): i ∈ I (x)} ⊆ @m f(x): We suppose that i is in @m fi (x) for i in I (x): Then there exist M ¿0 and ¿0 such that fi (y) − fi (x) + Mm(ky − xk)ky − xk ≥ hi ; y − xi ∀y ∈ x + B: Thus, f(y) − f(x) + Mm(ky − xk)ky − xk ≥ hi ; y − xi ∀y ∈ x + B; m m which implies P that i is in @ f(x) for any i in I (x): Since @ f(x) is convex, for any i ≥ 0 with i∈I (x) i = 1; X i i ∈ @m f(x): i∈I (x) This is what we need to prove. The next proposition indicates that the sum of two m-C regular functions is still an m-C regular function and the m-subdi erential of sum function is equal to the sum of m-subdi erentials of these two functions. Proposition 2.13. If f; g : X → R are m-C regular at x, then f +g is also m-C regular at x and @m (f + g)(x) = @m f(x) + @m g(x): Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 965 Proof. Since @m f(x) = @f(x) 6= ∅; @m g(x) = @g(x) 6= ∅; by Proposition 2.3, Remark 2.2(ii) and Proposition 2.3.3 in [5], we have @m f(x) + @m g(x) ⊆ @m (f + g)(x) ⊆ @(f + g)(x) ⊆ @f(x) + @g(x) = @m f(x) + @m g(x): It follows that f + g is also m-C regular and the equality stated holds. The following is a chain rule for m-subdi erentials. Proposition 2.14. Let gi : X → R be Lipschitz of rank L near x0 in X for i = 1; : : : ; l and f : Rl → R ∪ {∞} be l.s.c. Suppose that g(x0 ) := (g1 (x0 ); : : : ; gl (x0 ))t is in dom f and i ≥ 0 (i = 1; : : : ; l) whenever  := (1 ; : : : ; l )t is in @m f(g(x0 )) for some m-function m(t). Then l X i @m gi (x0 ) ⊆ @m1 (f ◦ g)(x0 ); i=1 where m1 (t) = max{m(t); m(lLt)}: Proof. Let  be in @m f(g(x0 )). Then for some m-function m(t), there exist M1 ¿0 and 1 ¿0 such that f(y) − f(g(x0 )) + M1 m(ky − g(x0 )k)ky − g(x0 )k ≥ l X i (yi − gi (x0 )) ∀y ∈ g(x0 ) + 1 Bl ; i=1 where y = (y1 ; : : : ; yl )t and Bl is the open unit ball in Rl . Let g be Lipschitz of rank L near x0 . Then for 1 , there is 2 ¿0 such that |gi (x) − gi (x0 )| ≤ Lkx − x0 k¡1 =l ∀x ∈ x0 + 2 B: If i is in @m gi (x0 ) for i = 1; : : : ; l; then there exist M2 ¿0 and (2 ¿)¿0 such that gi (x) − gi (x0 ) ≥ hi ; x − x0 i −M2 m(kx − x0 k)kx − x0 k ∀x ∈ x0 + B: Since for any x in x0 + B; lLkx − x0 k ≥ l X |gi (x) − gi (x0 )| ≥ kg(x) − g(x0 )k; i=1 we have f(g(x)) − f(g(x0 )) + M1 lLm(lLkx − x0 k)kx − x0 k ≥ f(g(x)) − f(g(x0 )) + M1 m(kg(x) − g(x0 )k)kg(x) − g(x0 )k 966 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 ≥ l X i (gi (x) − gi (x0 )) i=1 ≥ * l X i i ; x − x0 i=1 + − l X i M2 m(kx − x0 k)kx − x0 k; i=1 which implies that l X i i ∈ @m1 (f ◦ g)(x0 ): i=1 Hence, l X i @m gi (x0 ) ⊆ @m1 (f ◦ g)(x0 ): i=1 3. Integration of subdi erentials In this section, we prove our main results on integration of subdi erentials and show that our results unify and extend many results in the literature. First consider the following example from Poliquin [13]. The lower semicontinuous functions de ned by f(x) = ( 0 if x ≤ 0; 1 if x¿0 and g(x) = ( 0 if x ≤ 0; 2 if x¿0 do not di er by the same constant on the intervals (−∞; 0] and (0; ∞) even though they have the same proximal subgradient everywhere. Observe that g − f is l.s.c. and @ f(x) is nonempty for all x but @ f(0) = [0; ∞) is not bounded. However, if g − f is l.s.c., @ f(x) is nonempty and bounded, and @ g(x) ⊆ @ f(x), then f and g must di er by only a constant and the result is actually true for the class of @∗ -subdi erentials de ned as follows. De nition 3.1. Let f; g : X → R ∪ {∞} be lower semicontinuous at x ∈ dom f. (1) The @∗ -subdi erential of f at x, denoted by @∗ f(x), is a subset of X ∗ satisfying the following properties: (p1 ) @∗ f(x) = @c f(x) whenever f is convex. (p2 ) @∗ f(x) ⊆ @f(x) when f is Lipschitz near x. (p3 ) 0 ∈ @∗ f(x) when x is a local minimum of f. (p4 ) @∗ f(x) + @∗ g(x) ⊆ @∗ (f + g)(x). ˜ ˜ (2) The @-subdi erential of f at x, denoted by @f(x), is a subset of X ∗ satisfying the following properties [1]: Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 967 ˜ @f(x) = @c f(x) whenever f is convex. ˜ 0 ∈ @f(x) when x is a local minimum of f. ˜ + g)(x) ⊆ @f(x) ˜ ˜ @(f + @g(x) whenever g is a real-value convex continuous ˜ ˜ ˜ function which is @-di erentiable at x, i.e., both @g(x) and @(−g)(x) are nonempty. (3) The @∗ -subdi erential of f at x, denoted by @∗ f(x), is a subset of X ∗ satisfying the following properties [17]: (P1 ) @∗ f(x) = @c f(x) whenever f is convex. (P2 ) @∗ f(x) = @∗ g(x) whenever f = g on a neighbourhood of x. (P3 ) 0 ∈ @∗ f(x) when x is a local minimum of f. (P4 ) @∗ (f + g)(x) ⊆ @∗ f(x) + @∗ g(x) for any continuous convex function g. (i) (ii) (iii) By Remark 2.2, Propositions 2.3 and 2.5, the lower Dini subdi erential, the Frechet ˜ subdi erential and the m-subdi erential are both @∗ -subdi erentials and @-subdi erentials while the Clarke subdi erential and the Michel–Penot subdi erential are ˜ both @-subdi erentials and @∗ -subdi erentials. ˜ De nition 3.2 (Aussel et al. [1]). A norm k · k on X is said to be @-smooth if the ˜ following functions are @-di erentiable: 2 min{kx − ck 2 : c = a + (1 − )b;  ∈ [0; 1]} where a; b ∈ X . (i) d[a; b] (x) := P P (ii) △2 (x) := n n kx − n k 2 , where n n = 1; n ≥ 0, and (n ) converges in X . ˜ ˜ A Banach space X admits a @-smooth renorm if it has an equivalent @-smooth norm. − It was pointed out in [1] that a norm is @F -smooth (resp. @ -smooth, @LS -smooth) if and only if it is Frechet di erentiable (resp. Gâteaux di erentiable, Lipschitz smooth) o the origin and that re exive Banach spaces (resp. separable Banach spaces, Hilbert spaces and Lp (2 ≤p¡∞) spaces) admit a @F -smooth renorm (resp. @− -smooth renorm, @LS -smooth renorm). We use @0 h to denote any subdi erential of h for which the inclusion @0 h(x) ⊆ {0} on a nonempty open convex subset U of X implies that h is a constant on U . For ˜ example, @∗ -subdi erentials in a Banach space, @-subdi erentials in a Banach space ˜ with a @-smooth renorm and the Frechet subdi erential in an Asplund space (that is, the Banach space on which every continuous convex function is generically Frechet di erentiable) are all @0 -subdi erentials according to the following lemma which is very useful to our result. Lemma 3.3. Let U be a nonempty open convex subset of X and h : U → (−∞; ∞] be l:s:c: Then h is constant if and only if (i) @∗ h(x) ⊆ {0} ∀x ∈ U when X is a Banach space [17, Theorem 1.2]. ˜ ˜ (ii) @h(x) ⊆ {0} ∀x ∈ U when X is a Banach space with a @-smooth renorm [1, Theo− rem 5.2]. (In particular, h is constant if and only if @ h(x) ⊆ {0} ∀x ∈ U when X is a separable Banach space and h is constant if and only if @ h(x) ⊆ {0} ∀x ∈ U when X is a Hilbert space [7, Corollary 3.9] or a Lp (2 ≤ p¡∞) space.) (iii) @F h(x) ⊆ {0} ∀x ∈ U when X is an Asplund space [12, Corollary 8.9]. To prove Theorem 3.5 we need the following cancellation rule. 968 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 Lemma 3.4. Let A and B be nonempty subsets of X ∗ , and let B be bounded. Suppose A + B ⊆ B. Then A = {0}: Proof. Suppose that there exists a nonzero element a in A. Since a + b is in B for any b in B, na + b is in B for any n in N (natural number set) and any b in B: By the boundness of B, there is a positive M such that kbk∗ ≤ M for all b in B: In particular, kna + bk∗ ≤ M ∀n ∈ N and b ∈ B: On the other hand, since kak∗ 6= 0, for any n in N with n¿2M=kak∗ , we have kna + bk∗ ≥ nkak∗ − kbk∗ ≥ nkak∗ − M ¿M; which contradicts the boundness of B: Now, we state and prove our main results in this section as below. In the following context we assume that (−f) is l.s.c. at x whenever @∗ (−f)(x) is used. Theorem 3.5. Let U be a nonempty open convex subset of X; f : U → R and g : U → (−∞; ∞] be such that g − f is lower semicontinuous. Suppose that for each x in U , @∗ f(x) ∪ @∗ (−f)(x) is nonempty and bounded. Then for some constant C and any x in U , f(x) = g(x) + C if and only if for some @0 -subdi erential, @∗ g(x) ⊆ @∗ f(x) ∀x ∈ U s:t: @∗ f(x) 6= ∅; (5) @∗ (−f)(x) ⊆ @∗ (−g)(x) (6) ∀x ∈ U s:t: @∗ f(x) = ∅ and @0 (g − f)(x) ⊆ @∗ (g − f)(x) ∀x ∈ U: (7) Proof. The “only if ” part is straight from the de nition of @∗ -subdi erential and Lemma 3.3. We prove the “if ” part of this result as follows. Suppose that Eqs. (5) –(7) hold. Consider the function h : U → (−∞; ∞] de ned by h(x) = g(x) − f(x) ∀x ∈ U: For any x in U with @∗ f(x) 6= ∅, @0 h(x) + @∗ f(x) ⊆ @∗ h(x) + @∗ f(x) (by Eq: (7)) ⊆ @∗ g(x) (by (p4 ) of De nition 3:1) ⊆ @∗ f(x) (by Eq: (5)) Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 969 and for any x in U with @∗ f(x) = ∅, since @∗ (−f)(x) 6= ∅, @0 h(x) + @∗ (−g)(x) ⊆ @∗ h(x) + @∗ (−g)(x) (by Eq: (7)) ⊆ @∗ (−f)(x) (by (p4 ) of De nition 3:1) ⊆ @∗ (−g)(x) (by Eq: (6)); which implies by Lemma 3.4 that @0 h(x) ⊆ {0} ∀x ∈ U . Thus by the de nition of @0 -subdi erential, for some constant C and any x in U , h(x) = C, that is, f(x) = g(x) + C: Remark 3.6. It is natural to ask whether the condition that @∗ f(x) ∪ @∗ (−f)(x) is nonempty and bounded everywhere in Theorem 3.5 can be weaken to the condition that it is satis ed on a set that is big in the sense of either measure or category. This question is answered partially in the case @∗ = @HS by Borwein and Wang [4], who gave an example where f is Lipschitz continuous on R and @HS f(x) ∪ @HS (−f)(x) is densely nonempty but there exist in nitely many Lipschitz functions g on R di ering by more than a constant, such that @HS g(x) ⊆ @HS f(x) for all x ∈ R and @HS (−g)(x) ⊆ @HS (−f)(x) for all x ∈ R. Using the continuity of functions, we can extend Theorem 3.5 to a connected subset of a Banach space. Corollary 3.7. Let Ui be a nonempty open convex subset of X for i = 1; : : : ; l and Sl V := i=1 U i be connected, where U i is the closure of Ui . Suppose that f; g : V → R are continuous and that for any x in Ui (i = 1; : : : ; l); @∗ f(x) ∪ @∗ (−f)(x) is nonempty and bounded. Then for some constant C and any x in V , f(x) = g(x) + C if and only if for some @0 -subdi erential and any x ∈ Ui (i = 1; : : : ; l), @∗ g(x) ⊆ @∗ f(x) for x with @∗ f(x) 6= ∅; @∗ (−f)(x) ⊆ @∗ (−g)(x) for x with @∗ f(x) = ∅ and @0 (g − f)(x) ⊆ @∗ (g − f)(x): Example 3.8. Let U1 = {(x; y): x¿0; y¿0}, U2 = {(x; y): x¡0; y¿0}, U3 = {(x; y): x¡0; y¡0} and U4 = {(x; y): x¿0; y¡0}. By applying Corollary 3.7 to Ui (i = 1; 2; 3; 4) and @0 = @∗ = @ , we see that the following locally Lipschitz function f : R 2 → R de ned by f(x; y) = |x| − |y| 970 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 is the uniquely locally Lipschitz  {(1; −1)}       {(−1; −1)} @ f(x; y) =  {(−1; 1)}      {(1; 1)} function satisfying f(0; 0) = 0 and if (x; y) ∈ U1 ; if (x; y) ∈ U2 ; if (x; y) ∈ U3 ; if (x; y) ∈ U4 : Note that since @ f(0; 0) = ∅ and @ (−f)(0; 0) = ∅, we can not directly apply Theorem 3.5 to U = R 2 to get the above conclusion. From the proof of Theorem 3.5, if @∗ f(x) is nonempty and bounded on U , then we will not need the information of @∗ (−f). Theorem 3.9. Let U be a nonempty open convex subset of X , f : U → R and g : U → (−∞; ∞] be such that g−f is lower semicontinuous. Suppose that @∗ f(x) is nonempty and bounded for each x in U . Then for some constant C and any x in U , f(x) = g(x)+ C if and only if for some @0 -subdi erential and any x in U , @∗ g(x) ⊆ @∗ f(x) and @0 (g − f)(x) ⊆ @∗ (g − f)(x): Remark 3.10. (i) For a locally Lipschitz function f, due to the fact that @f(x) is nonempty and bounded and contains @∗ f(x), @∗ f(x) is also bounded. So the condition that @∗ f(x) is bounded in Theorem 3.9 can be omitted if f is locally Lipschitz. (ii) According to Lemma 3.3, we can take @ (resp. @− , @F ) as @0 in Theorems 3.5 and 3.9 when X is a Hilbert space (resp. a separable space, an Asplund space). In Hilbert space, the inclusion @ (g − f)(x) ⊆ @∗ (g − f)(x) is satis ed automatically by many @∗ -subdi erentials including Frechet subdi erentials, lower Dini subdi erentials and m-subdi erentials for all modulus functions m which satisfy lim inf + t→0 m(t) ¿0: t Next, we apply Theorems 3.5 and 3.9 to obtain some results with particular @∗ subdi erentials and particular spaces and explore the relationship between our results and the existing results. The rst consequence of Theorem 3.9 is an integration result on pointwise maximum function of nite number of functions. Corollary 3.11. Let U be a nonempty open convex subset of a real re exive Banach space X . Let g : U → (−∞; ∞] be lower semicontinuous, fi : U → R be continuous for i = 1; : : : ; l and f(x) := max{f1 (x); : : : ; fl (x)}. If for each i in I (x), fi is Frechet di erentiable at x ∈ U . Then for some constant C and any x in U , f(x) = g(x) + C if and only if for any x in U , @F g(x) ⊆ @F f(x): Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 971 Proof. By Proposition 2.11, for all x ∈ U , @F f(x) = co{Dfi (x): i ∈ I (x)}, which means that @F f(x) is nonempty and bounded at each x ∈ U . Recall that a re exive Banach space is an Asplund space. By Lemma 3.3(iii), one can take @0 = @F . So taking @∗ = @0 = @F in Theorem 3.9 completes the proof. The following question was asked on [11, p. 104]: Let f; g : R n → R be locally Lipschitz with f(0) = 0 = g(0) and @ g(x) ⊆ @ f(x) for all x. Does it follow that f and g are identical? The negative answer was given by Borwein and Wang [4] who found in nitely many Lipschitz functions g and a Lipschitz function f satisfying @ g(x) ⊆ @ f(x) for all x ∈ R but f and g di er by more than a constant. In their example, @ f(x) is not always nonempty. However, if @ f(x) is nonempty for every x, then we can apply Theorem 3.9 to the proximal subgradient and obtain the following corollary which not only gives a positive answer under the condition that @ f(x) is nonempty but also extends a result of Clarke and Redhe er [6] in which H = R n and f ∈ C 2 (U ). Corollary 3.12. Let U be a nonempty open convex subset of a Hilbert space H or a Lp (2 ≤ p¡∞) space, f : U → R locally Lipschitz and g : U → (−∞; ∞] lower semicontinuous. Suppose that for any x in U , @ f(x) is nonempty. Then for some constant C and any x in U , f(x) = g(x) + C if and only if @ g(x) ⊆ @ f(x). Proof. By Lemma 3.3(ii), @ can be taken as @0 . Hence taking @∗ = @0 = @ in Theorem 3.9 completes the proof. Similarly, by Lemma 3.3(iii) and taking @∗ = @0 = @F in Theorem 3.9, we can obtain the following corollary with a weaker condition than that of Clarke and Redhe er [6] in which X = R n and f is di erentiable. Corollary 3.13. Let U be a nonempty open convex subset of an Asplund space X , f : U → R locally Lipschitz and g : U → (−∞; ∞] lower semicontinuous. Suppose that for any x in U , @F f(x) is nonempty. Then for some constant C and any x in U , f(x) = g(x) + C if and only if @F g(x) ⊆ @F f(x). Recall that @F f(x) ⊆ @− f(x) and that f is Michel–Penot regular at x means that @ f(x) is nonempty and bounded. By Lemma 3.3(ii), one can take @0 = @F when the space is an Aspund space and @0 = @− when the space is a separable Banach space. Taking @∗ = @− in Theorem 3.9 yields the following result. − Corollary 3.14. Let U be a nonempty open convex subset of an Asplund space (or a separable Banach space), f : U → R locally Lipschitz and g : U → (−∞; ∞] lower semicontinuous. Suppose that f is Michel–Penot regular at any x in U . Then for some constant C and any x in U , f(x) = g(x) + C if and only if one of the following inclusions holds: (i) @− g(x) ⊆ @− f(x) for all x in U . (ii) g is locally Lipschitz on U and @⋄ g(x) ⊆ @⋄ f(x) for all x in U . 972 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 Proof. (i) It is a special case of Theorem 3.9 where @0 = @F and @∗ = @− . (ii) The necessity follows directly from the de nition of Michel–Penot subdi erential. Suciency is from (i) and the following inclusions @− g(x) ⊆ @⋄ g(x) ⊆ @⋄ f(x) = @− f(x) ∀x ∈ U: Note that the class of m − MP regular functions and m − C regular functions are Michel–Penot regular. According to Proposition 2.9 and Corollary 3.14, the m − MP hypomonotone functions can be determined by their @− -subdi erentials up to a constant. By Lemma 3.3(ii), one can take @0 = @ when the space is a Hilbert space. Taking @∗ = @ in Theorem 3.9 yields the followng result which slightly improves Poliquin’s result on integrability of hypomonotone functions on R n [13] by not requiring the function g to be a hypomonotone function and with weaker assumption @ g(x) ⊆ @ f(x). Corollary 3.15. Let U be a nonempty open convex subset of a Hilbert space, f : U → R locally Lipschitz and g : U → (−∞; ∞] lower semicontinuous. Suppose that f is hypomonotone at any x in U . Then for some constant C and any x in U , f(x) = g(x) + C if and only if one of the following inclusions holds: (i) @ g(x) ⊆ @ f(x) for all x in U . (ii) g is locally Lipschitz on U and @g(x) ⊆ @f(x) for all x in U . In the following result, we will apply Corollary 3.14 to the function f which is de ned on a nonempty open convex set of X × Y and is regular separately in x and y or f is regular in x and −f is regular in y. These functions generalize the convex– convex functions and the convex–concave functions. Denote f1⋄ ((x; y); u) := sup lim sup x′ t→0+ f(x + tx′ + tu; y) − f(x + tx′ ; y) ; t @⋄1 f(x; y) := { ∈ X ∗ : h; ui ≤ f1⋄ ((x; y); u) ∀u ∈ X } the partial Michel–Penot derivative of f with respective to x at (x; y) in the direction u and the partial Michel–Penot subdi erential of f with respective to x at (x; y). Similarly we use f2⋄ ((x; y); v) (f1− ((x; y); u) and f2− ((x; y); v)) and @⋄2 f(x; y) (@− 1 f(x; y) and f(x; y)) to denote the corresponding limit and subdi erential. @− 2 Corollary 3.16. Let X and Y be Asplund spaces (or separable Banach spaces) and U be a nonempty open convex subset of X × Y . Suppose that f : U → R is a locally Lipschitz function and g : U → (−∞; ∞] is a lower semicontinuous function such that − − ⋄ ⋄ ⋄ f satis es @− 1 f(x; y) = @1 f(x; y) and @2 f(x; y) = @2 f(x; y) (@2 (−f)(x; y) = @2 (−f) (x; y)) at any (x; y) in U . Then for some constant C and any (x; y) in U , f(x; y) = g(x; y) + C if and only if for any (x; y) in U , one of the following conditions holds: − − − − (i) @− 1 g(x; y) ⊆ @1 f(x; y) and @2 g(x; y) ⊆ @2 f(x; y) (g is continuous and @2 (−g) − (x; y)⊆ @2 (−f)(x; y)). (ii) @⋄1 g(x; y) ⊆ @⋄1 f(x; y) and @⋄2 g(x; y) ⊆ @⋄2 f(x; y) (g is locally Lipschitz and @⋄2 (−g) (x; y) ⊆ @⋄2 (−f)(x; y)). Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 973 ⋄ Proof. “only if ” part is directly from the de nition of @− i and @i (i = 1; 2). We only prove “if ” part. Since − ⋄ ⋄ @− 1 g(x; y) ⊆ @1 g(x; y) ⊆ @1 f(x; y) = @1 f(x; y) and − ⋄ ⋄ @− 2 g(x; y) ⊆ @2 g(x; y) ⊆ @2 f(x; y) = @2 f(x; y) − ⋄ ⋄ (@− 2 (−g)(x; y) ⊆ @2 (−g)(x; y) ⊆ @2 (−f)(x; y) = @2 (−f)(x; y)); condition (ii) implies condition (i). Therefore, we only need to show that condition (i) is sucient. − − − Let for any (x; y) in U , @− 1 g(x; y) ⊆ @1 f(x; y) and @2 g(x; y) ⊆ @2 f(x; y) (resp. − − @2 (−g)(x; y) ⊆ @2 (−f)(x; y)) and denote for any x ∈ X and y ∈ Y , U (x) := {y ∈ Y : (x; y) ∈ U } and U (y) := {x ∈ X : (x; y) ∈ U }: Then U (x) and U (y) are open convex subsets in X and Y , respectively, and (x; y) ∈ U if and only if both U (x) and U (y) are nonempty. By Corollary 3.14, f(x; y) = g(x; y) + c(y) ∀x ∈ U (y): Note that c(y) is continuous on U (x) for any x ∈ X such that U (x) is nonempty and that − − − @− 2 f(x; y) + @ (−c)(y) ⊆ @2 g(x; y) ⊆ @2 f(x; y) − − − (@− 2 (−f)(x; y) + @ c(y) ⊆ @2 (−g)(x; y) ⊆ @2 (−f)(x; y)): By Lemma 3.4, @− (−c)(y) ⊆ {0} (@− c(y) ⊆ {0}). Thus c(y) = C for some constant C. Therefore, f(x; y) = g(x; y) + C ∀(x; y) ∈ U: Remark 3.17. (i) Note that if f is Clarke regular and @g(x) ⊆@f(x) at any x ∈ U ⊆ X , then f is Michel–Penot regular and @− g(x) ⊆ @− f(x). By Corollary 3.16, f = g + C on U for some constant C. Therefore, Corollary 3.16 extends Rockafellar’s result on integrability of regular functions from R n to any Asplund space with weaker condition @− g(x) ⊆ @− f(x). (ii) The result of Corollary 3.16 is di erent from that of [10, Proposition 3.7] which states that if f is upper–upper (resp. upper–lower) regular at any (x; y) ∈ U , then @g(x; y) ⊆ @f(x; y) ∀x ∈ U if and only if f = g + C on U . By [10, Proposition 1.4], f is upper–upper (resp. upper–lower) regular at (x; y) if and only if f◦ ((x; y); (u; 0)) = f− ((x; y); (u; 0)) and f◦ ((x; y); (0; v)) = f− ((x; y); (0; v)); (−f)◦ ((x; y); (0; v)) = (−f)− ((x; y); (0; v)); 974 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 for any u ∈ X and v ∈ Y , from which we have f1◦ ((x; y); u) = f1− ((x; y); u) and f2◦ ((x; y); v) = f2− ((x; y); v) ((−f)◦2 ((x; y); v) = (−f)− 2 ((x; y); v)): Thus for any (x; y) ∈ U; − − ⋄ ⋄ ⋄ @− 1 f(x; y)= @1 f(x; y) and @2 f(x; y)= @2 f(x; y)(@2 (−f)(x; y)= @2 (−f)(x; y)): Hence that f is upper–upper regular at (x; y) implies that f is Clarke regular in x and y, respectively, at this point. So [10, Proposition 3.7] has stronger assumptions on the function f. However, the conclusions of [10, Proposition 3.7] and Corollary 3.16 are di erent. Since there is no exact relationship between the partial subdi erential and the Clarke subdi erential, Corollary 3.16 cannot be compared with [10, Proposition 3.7]. Since the composite function of a strictly di erentiable map and a Clarke regular map is Clarke regular, Corollary 3.16 has the following consequence about integrability of composite functions. Corollary 3.18. Let U be a nonempty open convex subset of an Asplund space (or a separable Banach space) X . Suppose that F is a strictly di erentiable map from U to Banach space Y and g is a real-valued function de ned on Y and Clarke regular at F(x) for each x ∈ U . Then for any l.s.c. function f : U → (−∞; ∞], f(x) = g ◦ F(x) + C ∀x ∈ U if and only if @− f(x) ⊆ @g− (F(x)) ◦ Ds F(x) ∀x ∈ U; where Ds F(x) is the strict derivative of F at x. Proof. By [5, Theorem 2.3.10], f1 := g ◦ F is Clarke regular and @− f1 (x) = @f1 (x) = @g(F(x)) ◦ Ds F(x) = @− g(F(x)) ◦ Ds F(x) at each x in U . The result follows from Corollary 3.16. Using Corollary 3.16, we can recover the result of Rockafellar which says that a closed proper convex function is integrable in the case where int(dom f) is nonempty. Corollary 3.19. Let f; g : R n → R ∪ {+∞} be l.s.c. convex functions such that int(dom f)(the interior of dom f) is nonempty. Suppose that @c g(x) ⊆ @c f(x) for each x ∈ R n . Then f = g + C on R n for some constant C. Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 975 Proof. Denote U := int(dom f). Then U is nonempty, open and convex and f is convex and locally Lipschitz on U . And hence f is Clarke regular on U: By Corollary 3.16, f = g + C on U for some constant C which also means that int(dom f) ⊆ int(dom g): Now for any y ∈ cl(dom f); let x ∈ int(dom f): Then (1 − )x + y ∈ int(dom f). By [15, Corollary 7.5.1], f(y) = lim f((1 − )x + y) →1− = lim g((1 − )x + y) + C − →1 = g(y) + C: Since g is l.s.c. convex and @c g(x) ⊆ @c f(x) ∀x ∈ R n , and for any x ∈ int(dom g), @ g(x) is nonempty, c int(dom g) ⊆ int(dom @c g) ⊆ int(dom @c f) ⊆ int(dom f): Thus, int(dom f) = int(dom g): And, hence, cl(dom f) = cl(dom g): Therefore for any y not in cl(dom f), f(y) = ∞ = g(y): The proof is completed. References [1] D. Aussel, J. Corvellec, M. Lassonde, Mean value property and subdi erential criteria for lower semicontinuous functions, Trans. Amer. Math. Soc. 347 (1995) 4147– 4161. [2] J. Borwein, Minimal cuscos and subgradients of Lipschtiz functions. [3] J.M. Borwein, S.P. Fitzpatrick, J.R. Giles, The di erentiability of real functions on normed linear space using generalized subgradients, J. Math. Anal. 128 (1987) 512–534. [4] J.M. Borwein, X. Wang, Lipschitz functions on the line with prescribed Holder subdi erentials, preprint. [5] F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, 1983; reprint, Classics in Applied Mathematics, vol. 5, SIAM, Philadelphia, PA, 1990. [6] F.H. Clarke, R.M. Redhe er, The proximal subgradient and constancy, Canad. Math. Bull. 36 (1993) 30–32. [7] F.H. Clarke, R.J. Stern, P.R. Wolenski, Subgradient criteria for monotonicity, the Lipschitz condition, and convexity, Canad. J. Math. 45 (1993) 1167–1183. [8] J.B. Conway, A Course in Functional Analysis, Springer, New York, 1990. [9] R. Correa, A. Jofre, Tangentially continuous directional derivatives in nonsmooth analysis, J. Optim. Theory Appl. 61 (1989) 1–20. 976 Z. Wu, J.J. Ye / Nonlinear Analysis 39 (2000) 955 – 976 [10] R. Correa, L. Thibault, Subdi erential analysis of bivariate separately regular functions, J. Math. Anal. Appl. 148 (1990) 157–174. [11] P.D. Loewen, Optimal Control Via Nonsmooth Analysis, CRM Proc. & Lecture Notes, American Mathematical Society, Providence, RI, 1993. [12] B.S. Mordukhovich, Y. Shao, Nonsmooth sequential analysis in Asplund spaces, J. Math. Anal. Appl. 148 (1990) 157–174. [13] R.A. Poliquin, Integration of subdi erentials of nonconvex functions, Nonlinear Anal. Theory, Methods Appl. 17 (1991) 385–398. [14] Liqun Qi, The maximal normal operator space and integration of subdi erentials of nonconvex functions, Nonlinear Anal. Theory, Methods Appl. 13 (1989) 1003–1011. [15] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970. [16] R.T. Rockafellar, Favorable classes of Lipschitz-continuous functions in subgradient optimization, in: Progress in Nondi erentiable Optimization, E. Nurminski (Ed.), IIASA Collaborative Proceedings Series, International Institute of Applied Systems Analysis, Laxenburg, Austria, 1982, pp. 125–144. [17] L. Thibault, D. Zagrodny, Integration of subdi erentials of lower semicontinuous functions on Banach spaces, J. Math. Anal. Appl. 189 (1995) 33–58.