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Abstract and Applied Analysis A version of Zhong's coercivity result for a general class of nonsmooth functionals
A version of Zhong's coercivity result for a general class of nonsmooth functionals
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巻:
7
年:
2002
言語:
english
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Abstract and Applied Analysis
DOI:
10.1155/s1085337502207058
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A VERSION OF ZHONG’S COERCIVITY RESULT FOR A GENERAL CLASS OF NONSMOOTH FUNCTIONALS D. MOTREANU, V. V. MOTREANU, AND D. PAŞCA Received 27 October 2001 A version of Zhong’s coercivity result (1997) is established for nonsmooth functionals expressed as a sum Φ + Ψ, where Φ is locally Lipschitz and Ψ is convex, lower semicontinuous, and proper. This is obtained as a consequence of a general result describing the asymptotic behavior of the functions verifying the above structure hypothesis. Our approach relies on a version of Ekeland’s variational principle. In proving our coercivity result we make use of a new general PalaisSmale condition. The relationship with other results is discussed. 1. Introduction In this paper, we deal with the class of nonsmooth functionals I : X → R ∪ {+∞} on a Banach space X of the form I = Φ + Ψ, (1.1) with Φ : X → R locally Lipschitz and Ψ : X → R ∪ {+∞} convex, lower semicontinuous (l.s.c.), and proper (i.e., ≡ +∞). For the functional I as in (1.1), it was given in [9] the following definition of PalaisSmale (PS) condition. Definition 1.1. The functional I : X → R ∪ {+∞} in (1.1) satisfies the PS condition if every sequence (un ) ⊂ X with I(un ) bounded and for which there exists a sequence (n ) ⊂ R+ , n → 0+ , such that Φ0 un ;v − un + Ψ(v) − Ψ un ≥ −n v − un , contains a (strongly) convergent subsequence in X. Copyright © 2002 Hindawi Publishing Corporation Abstract and Applied Analysis 7:11 (2002) 601–612 2000 Mathematics Subject Classification: 58E30, 58E05, 49K27 URL: http://dx.doi.org/10.1155/S1085337502207058 ∀v ∈ X, ∀n, (1.2) 602 A version of Zhong’s coercivity result The notation Φ0 in (1.2) stands for the generalized directional derivative of the locally Lipschitz functional Φ : X → R introduced by Clarke [5] as follows: Φ0 (u;v) = limsup w→u t →0+ 1 Φ(w + tv) − Φ(w) , t ∀u,v ∈ X. (1.3) In the case where Φ ∈ C 1 (X; R) and Ψ = 0, Definition 1.1 reduces to the usual PS condition. If Φ is locally Lipschitz and Ψ = 0, Definition ; 1.1 expresses the PS condition in the sense of Chang [4]. If Φ ∈ C 1 (X; R) and Ψ is convex, l.s.c., and proper, Definition 1.1 represents the concept of PS condition introduced by Szulkin [10]. A diﬀerent extension of the usual PS condition is the following one. Definition 1.2 (Zhong [11]). Let h : [0,+∞) → [0,+∞) be a continuous nonde+∞ creasing function such that 0 (1/(1 + h(r)))dr = +∞. An l.s.c. proper function Φ : X → R ∪ {+∞} which is Gâteaux diﬀerentiable on its eﬀective domain satisfies the PS condition if every sequence (un ) ⊂ X with Φ(un ) bounded and Φ un 1 + h un −→ 0 as n −→ +∞ (1.4) has a (strongly) convergent subsequence in X. If h = 0, Definition 1.2 reduces to the classical PS condition. In the case where h(t) = t, for all t ≥ 0, Definition 1.2 coincides with the PS condition in the sense of Cerami [3]. It is natural to look for a concept of PS condition for functionals of type (1.1) incorporating simultaneously the two definitions above. Definition 1.3. The functional I : X → R ∪ {+∞} in (1.1) satisfies the PS condition if every sequence (un ) ⊂ X with I(un ) bounded and for which there exists a sequence (n ) ⊂ R+ , n → 0+ , such that Φ0 un ;v − un + Ψ(v) − Ψ un ≥ − n v − un , 1 + h un ∀v ∈ X, ∀n, (1.5) contains a (strongly) convergent subsequence in X. If h = 0, Definition 1.3 reduces to Definition 1.1. If Φ ∈ C 1 (X; R) and Ψ = 0, Definition 1.3 coincides with Definition 1.2 since relation (1.5) becomes (1.4) in this case. We point out that for h(t) = t, for all t ≥ 0, Definition 1.3 expresses the extension of the PS condition in the sense of Cerami [3] to the class of nonsmooth functionals in (1.1). A problem that has been extensively studied was the relationship between the PS condition and coercivity. We recall that a functional I : X → R ∪ {+∞} is said to be coercive if the following property holds: I(u) −→ +∞ as u −→ +∞. (1.6) D. Motreanu et al. 603 The basic assertion in this direction is that, generally, the PS condition implies the coercivity. The first such result is the one of Čaklović et al. [2] who established this property for a functional I : X → R which is l.s.c., Gâteaux diﬀerentiable, and satisfying the classical PS condition (see also Brézis and Nirenberg [1] for continuously diﬀerentiable functionals). The first result of this type, for nondiﬀerentiable functionals, is due to Goeleven [7] who has shown the coercivity property in the case where the functional I : X → R ∪ {+∞} has the structure (1.1) with Φ l.s.c. and Gâteaux diﬀerentiable, and Ψ convex, l.s.c., and proper such that the PS condition in the sense of Szulkin [10] is satisfied. For nonsmooth functionals of the general form (1.1), an analogous result has been obtained in [8] making use of the PS condition stated in Definition 1.1. The corresponding property for nonsmooth functionals, satisfying the PS condition formulated in Definition 1.2, has been given by Zhong [11]. The aim of this paper is to prove the coercivity for the nonsmooth functionals verifying (1.1) together with the PS condition given in Definition 1.3. In this paper, the coercivity assertion is obtained as a consequence of Theorem 2.3 below expressing the asymptotic behavior of a nonsmooth functional of type (1.1). Specifically, the coercivity property is derived from Theorem 2.3 by assuming the PS condition as formulated in Definition 1.3. Theorem 2.3 cannot be deduced from Zhong’s corresponding result [11, Theorem 3.7] because, generally, the functionals of type (1.1), which we consider, are not Gâteaux diﬀerentiable. However, Theorem 2.3 is not an extension of Theorem 3.7 in Zhong [11] because a Gâteaux diﬀerentiable, l.s.c., and proper functional is not necessarily of form (1.1). Our Theorem 2.3 represents the version of Zhong’s corresponding result for a nonsmooth functional fulfilling the structure assumption (1.1). The method of proof for Theorem 2.3 relies, as in the case of [11, Theorem 3.7], on Zhong’s variational principle [11, Theorem 2.1] which is an extension of Ekeland’s variational principle [6]. The proof of Theorem 2.3 takes into account, essentially, the structure of functionals in (1.1). To the end of rigourously proving Theorem 2.3, we slightly extend Zhong’s variational principle [11, Theorem 2.1] in Theorem 2.1 below. The main idea is to allow the reference point x0 to be in a larger space. Precisely, this extension is necessary because in Theorem 2.3 we encounter the situation where x0 = 0 does not belong to the space M0 , on which the variational principle must be applied. Our argument corrects a small gap in the proof of [11, Theorem 3.7] concerning the mentioned diﬃculty. Furthermore, in comparison with Zhong’s paper [11], our approach in Theorem 2.3 makes other improvements, among them, the accurate treatment of the passage from the (N − 1)th to the Nth step. Moreover, our hypotheses in Theorem 2.3 and Corollary 2.4 either are slightly weaker (see (2.8)) or give the correct requirement for making the proof (see (2.5)). The rest of the paper is organized as follows. Section 2 is devoted to the statements of the results. Section 3 contains the proof of the main result. 604 A version of Zhong’s coercivity result 2. Main results We start with a slight extension of Zhong’s variational principle in [11]. Theorem 2.1. Let h : [0,+∞) → [0,+∞) be a continuous nondecreasing function such that +∞ 0 1 dr = +∞. 1 + h(r) (2.1) endowed with the metric d, Let M be a closed subset of a complete metric space M let a point x0 ∈ M, and let f : M → R ∪ {+∞} be an l.s.c., proper function which is bounded from below. Then, for all > 0, v ∈ M with f (v) < inf f + , (2.2) M and λ > 0, there exists a point w,λ ∈ M such that f w,λ ≤ f (v), d w,λ ,x0 ≤ r + r0 , d u,w,λ , f (u) ≥ f w,λ − λ 1 + h d x0 ,w,λ ∀u ∈ M, (2.3) where r0 = d(x0 ,v) and r verifies r0 +r r0 1 dr ≥ λ. 1 + h(r) (2.4) so M becomes Proof. We endow M with the metric induced by the one on M, a complete metric space. A careful examination of the proof of [11, Theorem 2.1] shows that the argument therein can be carried out with any point x0 in M. Following the same lines as in the proof of [11, Theorem 2.1] (which goes back to Ekeland [6]), we achieve the stated conclusion. Remark 2.2. The classical Ekeland’s variational principle [6] is obtained from = M, h = 0, x0 = v, and r = λ. Theorem 2.1 taking M Our main result is the following theorem. Theorem 2.3. Let X be a Banach space and let a functional I : X → R ∪ {+∞} be of type (1.1), that is, I = Φ + Ψ with Φ : X → R locally Lipschitz and Ψ : X → R ∪ {+∞} convex and l.s.c. If α := liminf I(v) ∈ R, v →+∞ (2.5) then for every sequence (n ) ⊂ R+ with n → 0+ , there exists a sequence (un ) ⊂ X satisfying (1.5) (with a function h : [0,+∞) → [0,+∞) nondecreasing, continuous, D. Motreanu et al. 605 and verifying (2.1)), un −→ +∞ as n −→ +∞, (2.6) I(un ) −→ α as n −→ +∞. (2.7) The proof will be done in Section 3. Now we apply Theorem 2.3 for studying the coercivity of nonsmooth functionals in (1.1). Corollary 2.4. Assume that the functional I : X → R ∪ {+∞} satisfies the structure hypothesis (1.1) with Φ : X → R locally Lipschitz and Ψ : X → R ∪ {+∞} convex, l.s.c., and proper. If I verifies the PS condition in Definition 1.3 and liminf I(v) > −∞, v →+∞ (2.8) then I is coercive, that is property (1.6) holds. Proof. Suppose, by contradiction, that I is not coercive. In view of (2.8), this is equivalent to (2.5) which enables us to apply Theorem 2.3. Corresponding to a sequence n → 0+ , we find a sequence (un ) ⊂ X fulfilling (1.5), (2.6), and (2.7). On the basis of (1.5), (2.7), and PS condition in the sense of Definition 1.3, it follows that there exists a subsequence of (un ) which is strongly convergent in X. Thus, we arrived at a contradiction with (2.6). The proof is complete. 3. Proof of Theorem 2.3 Note that the condition α ∈ R, imposed in (2.5), implies that the functional I is proper outside every ball in X. We fix a positive number ≤ 1/3. Define the function m(r) = inf I(u), r > 0. u ≥r (3.1) The function m in (3.1) is nondecreasing and it satisfies lim m(r) = α. r →+∞ (3.2) From (3.2), there exists a number r = r() ≥ 1/ such that α − 2 ≤ m(r), ∀r ≥ r. (3.3) Using property (2.1) of the function h, it follows that +∞ r 1 dr = +∞. 1 + h(r) (3.4) Therefore, we can choose r ∗ > r such that r∗ r 1 dr ≥ 1. 1 + h(r) (3.5) 606 A version of Zhong’s coercivity result Corresponding to > 0, the definition of the integral yields a partition r = rN < rN −1 < · · · < r1 < r0 = r ∗ for which one has ∗ N −1 r 1 1 dr − rk − rk+1 < . r 1 + h(r) 1 + h r k k =0 (3.6) We consider the following sets: Mk = u ∈ X : u ≥ rk , 0 ≤ k ≤ N. (3.7) = X, f = I, and The requirements in Theorem 2.1 are fulfilled with M = M0 , M x0 = 0. To justify this, we notice that M0 is a closed subset of the Banach space X. The functional I : X → R ∪ {+∞}, expressed in (1.1), is l.s.c. and proper on M0 . Finally, by (3.1) and (3.3) we derive that I(u) ≥ m u ≥ α − 2 , ∀u ∈ X, u ≥ r, (3.8) which ensures that the functional I is bounded from below on M0 since r < r0 . Then, Theorem 2.1 (with replaced by 2 and λ = ) provides a point w0 = w0 () ∈ M0 such that I w0 < m r0 + 2 ≤ α + 2 (3.9) (see also (3.2)) and I(u) ≥ I w0 − u − w0 , 1 + h w0 ∀ u ∈ M0 . (3.10) Assume that w0 > r0 (3.10) holds for every u ∈ M1 . or (3.11) For an arbitrary w ∈ X, it is permitted to set in (3.10) u = w0 + t(w − w0 ) with t > 0 suﬃciently small. Then, we can write Φ w0 + t w − w0 + Ψ w0 + t w − w0 ≥ Φ w0 + Ψ w0 − t w − w0 . 1 + h w0 (3.12) On the basis of convexity of Ψ : X → R ∪ {+∞}, the inequality above yields − Φ w0 + t Ψ(w) − Ψ w0 t w − w0 . ≥− 1 + h w0 Φ w0 + t w − w0 (3.13) D. Motreanu et al. 607 Dividing by t > 0 and then passing to the upper limit as t → 0+ , we obtain that limsup t →0+ 1 Φ w0 + t w − w0 − Φ w0 + Ψ(w) − Ψ w0 t ≥− (3.14) w − w0 . 1 + h w0 We deduce that Φ0 w0 ;w − w0 + Ψ(w) − Ψ w0 ≥ − w − w0 , 1 + h w0 ∀w ∈ X. (3.15) Since w0 ∈ M0 , it is known that w0 ≥ r0 = r ≥ 1/ . From (3.8) and (3.9) we infer that α − 2 ≤ I(w0 ) < α + 2 . Taking into account that the assertions above are valid for an arbitrary ∈ (0,1/3), we see that properties (1.5), (2.6), and (2.7) are proved under the additional assumption (3.11). We may, thus, suppose that ∗ w0 = r0 (3.16) and there exists some u1 ∈ M1 \ M0 such that I u1 < I w0 − u1 − w0 . 1 + h w0 (3.17) Using the construction leading to [11, relation (2.7)] with M1 , 2 and in place of M, and λ, respectively, and choosing u11 = w0 , there exists a sequence (u1n ) ⊂ M1 such that I u1n+1 ≤ I u1n − u1n − u1n+1 , 1 1+h u ∀n ≥ 1. (3.18) n Arguing as in the proof of [11, Theorem 2.1], on the basis of relation (3.18), we deduce the existence of a point w1 ∈ M1 with the properties lim u1 n→+∞ n = w1 , (3.19) u − w1 , I(u) ≥ I w1 − 1 + h w1 ∀ u ∈ M1 . (3.20) In addition, by (3.19) and the lower semicontinuity of I in conjunction with (3.18) and (3.9), we obtain that I w1 ≤ liminf I u1n = lim I u1n ≤ I u11 < α + 2 . n→+∞ n→+∞ (3.21) 608 A version of Zhong’s coercivity result Relations (3.21) and (3.20) show that we arrived at a situation which is similar to the one described in the previous step, that is, (3.9) and (3.10). In this respect, if w1 > r1 (3.20) holds for every u ∈ M2 or (3.22) (an assertion analogous to (3.11)), we complete the proof as above. It remains to consider the case where (3.22) does not hold, that is, w1 = r1 (3.23) and there exists some u2 ∈ M2 \ M1 such that I u2 < I w1 − u2 − w1 . 1 + h w1 (3.24) Thanks to relations (3.23) and (3.24), we are in the same situation as (3.16) and (3.17), hence, we may pass to the next step. Continuing the process, by the construction around [11, relation (2.7)], we find a convergent sequence (ukn ) ⊂ Mk , 1 ≤ k ≤ N, with uk1 = wk−1 , satisfying I ukn+1 ≤ I ukn − ukn − ukn+1 , k 1+h u ∀n ≥ 1. (3.25) n Moreover, there exists lim uk n→+∞ n = wk (3.26) with the property I(u) ≥ I wk − u − wk , 1 + h wk ∀ u ∈ Mk . (3.27) Now, two situations could arise: either wk > rk or (3.27) holds for every u ∈ Mk+1 (3.28) wk = rk (3.29) (where we set MN+1 = X) or and there is uk+1 ∈ Mk+1 \ Mk with I uk+1 < I wk − uk+1 − wk . 1 + h wk (3.30) Now we prove that for at least one 0 ≤ k ≤ N we are in the situation described in (3.28). Clearly, this will accomplish the proof by means of a reasoning similar to the one below relation (3.11). D. Motreanu et al. 609 We argue by contradiction. Suppose that wk satisfies (3.27), (3.29), and (3.30), for every 0 ≤ k ≤ N. (3.31) The contradiction will be achieved through the inequality wk − wk+1 ≤ I wk − I wk+1 , 1 + h wk ∀k, 0 ≤ k ≤ N − 1. (3.32) Assume for a moment that (3.32) is valid. According to (3.32), (3.9), wN ∈ MN , and (3.3), we derive that N −1 k =0 wk − wk+1 ≤ I w0 − I wN < α + 2 − m rN ≤ 22 . (3.33) 1 + h wk Using (3.5), (3.6), (3.29), (3.31), and (3.33), it turns out that 1− ≤ = r∗ r N −1 k =0 ≤ N −1 k =0 N −1 1 1 rk − rk+1 dr − < 1 + h(r) 1 + h rk k =0 1 wk − wk+1 1 + h wk (3.34) 1 wk − wk+1 ≤ 2. 1 + h wk Thus, we get that > 1/3. This contradicts the choice ≤ 1/3 and completes the proof provided that (3.32) is true. In order to check (3.32), we fix k with 0 ≤ k ≤ N − 1. Note that if k+1 u ≤ wk = rk , j ∀ j ≥ 1, (3.35) the claim in (3.32) is proved. Indeed, by (3.26), the lower semicontinuity of I and (3.25), we may write I wk+1 ≤ liminf I uk+1 ≤ I uk+1 m n m→+∞ −1 n ≤ I uk+1 − 1 j =1 k+1 uk+1 j − u j+1 , 1 + h uk+1 j ∀n ≥ 2. (3.36) 610 A version of Zhong’s coercivity result In addition, by means of equality uk+1 1 = wk , (3.26), and the triangle inequality, we have wk − wk+1 = lim uk+1 − uk+1 1 n n→+∞ ≤ lim n→+∞ = n −1 j =1 k+1 u − uk+1 j j+1 (3.37) +∞ k+1 u − uk+1 . j =1 j j+1 Making use of monotonicity of h in conjunction with (3.35) and (3.36), the inequality above leads to +∞ k+1 wk − wk+1 ≤ uk+1 j − u j+1 1 + h wk 1 + h wk j =1 +∞ k+1 k+1 uk+1 j − u j+1 j =1 1 + h u j ≤ I wk − I wk+1 . ≤ (3.38) This means that if (3.35) holds, inequality (3.32) is checked. Property (3.35) is true for j = 1 being verified with equality. Thus, for completing the proof it is suﬃcient to verify k+1 u < wk = rk , j ∀ j ≥ 2. (3.39) The proof of (3.39) is done by recurrence. Considering first the case j = 2, we point out that the set u ∈ Mk+1 : I(u) < I wk − u − wk 1 + h wk (3.40) is nonempty because it contains at least uk+1 (see (3.30) and (3.31)). Taking into k+1 account the equality wk = uk+1 1 , by the construction of the sequence (un ), it is k+1 known that u2 belongs to the set described by (3.40). Comparing (3.40) and k+1 < rk . Assertion (3.39) is, (3.27), it follows that uk+1 2 ∈ Mk which reads as u2 thus, checked for j = 2. <rk with 2 ≤ n ≤ j − 1. Assume inductively that for some j >2, we have uk+1 n We must verify that uk+1 < r . Arguing by contradiction, suppose uk+1 ≥ rk . k j j k+1 k+1 This ensures that u j = u j −1 , so by (3.25) with n = j − 1 and (3.31), we infer D. Motreanu et al. 611 that < I uk+1 I uk+1 j j −1 − k+1 k+1 uk+1 j −1 − u j 1+h u j −1 j −1 ≤ I uk+1 − 1 n =1 k+1 uk+1 n − un+1 . k+1 1+h u (3.41) n ≥ rk and using (3.27), we may write Since we admitted uk+1 j I uk+1 ≥ I wk − j uk+1 j − wk . 1 + h wk (3.42) = wk , (3.42), the triangle inequality, uk+1 < rk = wk Relations (3.41), uk+1 1 n for 2 ≤ n ≤ j − 1, and the increasing monotonicity of h yield j −1 n =1 k+1 k+1 uk+1 − I uk+1 j n − un+1 < I u1 k+1 1+h u n ≤ k+1 k+1 uk+1 j − u1 1+h u 1 ≤ j −1 k+1 u − uk+1 k+1 n n+1 1+h u n =1 1 j −1 ≤ n =1 (3.43) k+1 uk+1 n − un+1 . k+1 1+h u n < rk . The inductive process is The achieved contradiction implies that uk+1 j accomplished, thus (3.39) holds true. The proof is complete. References [1] [2] [3] [4] [5] [6] [7] [8] H. Brézis and L. Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 89, 939–963. L. Čaklović, S. J. Li, and M. Willem, A note on PalaisSmale condition and coercivity, Diﬀerential Integral Equations 3 (1990), no. 4, 799–800. G. Cerami, An existence criterion for the critical points on unbounded manifolds, Istit. Lombardo Accad. Sci. Lett. Rend. A 112 (1978), no. 2, 332–336. K.C. Chang, Variational methods for nondiﬀerentiable functionals and their applications to partial diﬀerential equations, J. Math. Anal. Appl. 80 (1981), no. 1, 102– 129. F. H. Clarke, Optimization and Nonsmooth Analysis, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, New York, 1983. I. Ekeland, On the variational principle, J. Math. Anal. Appl. 47 (1974), 324–353. D. Goeleven, A note on PalaisSmale condition in the sense of Szulkin, Diﬀerential Integral Equations 6 (1993), no. 5, 1041–1043. D. Motreanu and V. V. Motreanu, Coerciveness property for a class of nonsmooth functionals, Z. Anal. Anwendungen 19 (2000), no. 4, 1087–1093. 612 [9] [10] [11] A version of Zhong’s coercivity result D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optimization and Its Applications, vol. 29, Kluwer Academic Publishers, Dordrecht, 1999. A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), no. 2, 77–109. C.K. Zhong, A generalization of Ekeland’s variational principle and application to the study of the relation between the weak P.S. condition and coercivity, Nonlinear Anal. 29 (1997), no. 12, 1421–1431. D. Motreanu: Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France Email address: motreanu@univperp.fr V. V. Motreanu: Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France Email address: viorica@univperp.fr D. 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