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Evolution hemivariational inequalities with hysteresis
Nonlinear Analysis 57 (2004) 323 – 340
www.elsevier.com/locate/na
Evolution hemivariational inequalities with hysteresis Leszek Gasi$nski Institute of Computer Science, Jagiellonian University, ul. Nawojki 11, Cracow 30072, Poland Accepted 6 February 2004
Abstract In this paper we study evolution hemivariational inequalities containing a hysteresis operator. For such problems we establish an existence result by reducing the order of the equation and then by the use of the time-discretization procedure. ? 2004 Elsevier Ltd. All rights reserved. MSC: 47J20; 49J40 Keywords: Hemivariational inequality; Clarke subdi4erential; Hysteresis operator; Time-discretization method; Gronwall inequality
1. Introduction Theory of variational inequalities provides us with an appropriate mathematical model to describe many physical problems. This theory was started in 1960s by Baiocchi, H. Brezis, G. Duvaut, G. Ficher, D. Kinderlehrer, J.L. Lions, G. Stampacchia and many others. In 1980s, Panagiotopoulos introduced the so-called hemivariational inequalities (see [13,14]), using the notion of Clarke’s subdi4erential (see [3]), which is deAned for locally Lipschitz functions. Hemivariational inequalities are an e4ective tool to treat problems with nonmonotonicity and multivaluedness. For a great amount of examples of applications we refer to Naniewicz and Panagiotopoulos [10] and Panagiotopoulos [12–14]. Parallel to the theory of hemivariational inequalities, the theory of hysteresis has been developed. The latter is based on the notion of the hysteresis operator, introduced
This paper was partially supported by the KBN Grants no 2 P03A 003 25 and no 7 T07A 047 18. E-mail address: [email protected] (L. Gasi$nski).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.02.016
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by Krasnoselskii and Pokrovskii [8] and developed e.g. by Brokate and Sprekels [2], Visintin [15] and others. In this paper we study evolution hemivariational inequality with a memory hysteresis operator which occurs in the source term. The unknown function u may represent the displacement and w describe a force term characterized by continuous hysteresis cycles. The existence and uniqueness results for the corresponding semilinear parabolic and hyperbolic equations with hysteresis in the lower term can be found in Visintin [15, Chapter X.1]. Similar approach to the one exploited in this paper can be done when the hysteresis operator depends on the time derivative of the unknown function. We mention that in other problems with hysteresis arising in continuous mechanics, hysteresis operators may also appear in constitutive laws. For such situations we refer to Brokate and Sprekels [2] and Visintin [15]. For the evolution hemivariational inequalities similar to the one presented in this paper (but not containing hysteresis operator), we refer to the papers of Gasi$nski [4,5] and Gasi$nski and Sma lka [6,7]. In all these papers the proofs of the existence theorems are based on the surjectivity result for coercive pseudomonotone operators. In our case, because of the presence of the hysteresis operator, this method does not work any more. The method of the proof in the present paper is based on the “reduction of the order” of the inequality and then on the time discretization method and the surjectivity result for a stationary problem (cf. [2, Chapter 3.3]; [15,9]).
2. Preliminaries For the convenience of the reader we recall some basic deAnitions and notions from the theory of pseudomonotone operators, the theory of Clarke’s subdi4erential and the theory of hysteresis operators. Let X be a Banach space and X its topological dual. By · X we will denote a norm in X and by ·; ·X the duality brackets for the pair (X; X ). Let X and Y be two Banach spaces. If the embedding X ⊆ Y is continuous, then by cYX we denote the continuity constant, i.e. such a constant that for all x ∈ X , we have that xY 6 cYX xX . A single valued operator A : X → X is said to be pseudomonotone, if (a) A is bounded; (b) if x n → x weakly in X and lim supAx n ; x n − xX 6 0, then for all y ∈ X , we have that Ax; x − yX 6 lim inf Ax n ; x n − yX .
A multivalued operator A : X → 2X is said to be pseudomonotone, if (a) for all x ∈ X , set Ax is nonempty, convex and weakly compact in X ; (b) for every Y , Anite-dimensional subspace of X , operator A|Y is upper semicontinuous into X furnished with the weak topology (i.e. if U ⊆ X is weakly open, then the set {x ∈ Y : Ax ⊆ U } is open in Y );
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(c) if x n → x weakly in X , x∗n ∈ Ax n for n ¿ 1 and lim supx∗n ; x n − xX 6 0, then for every y ∈ X , there exists x∗ (y) ∈ Ax such that x∗ (y); x − yX 6 lim inf x∗n ; x n − yX . If A is bounded (i.e. maps bounded sets into bounded sets) and satisAes condition (c), then it satisAes condition (b) too. An operator A : X → 2X is said to be generalized pseudomonotone, if x n → x weakly in X, x∗n → x∗ weakly in X , x∗n ∈ Ax n for n ¿ 1 and lim supx∗n ; x n − xX 6 0, imply that x∗ ∈ Ax and x∗n ; x n X → x∗ ; xX . We recall some basic facts concerning pseudomonotone operators (for the proofs see [10, Theorems 2.2–2.4, 2.6]).
Theorem 2.1. If X is a re4exive Banach space, A : X → 2X is a bounded generalized pseudomonotone operator and for each x ∈ X , Ax is a nonempty, closed and convex subset of X , then A is pseudomonotone.
Theorem 2.2. If X is a re4exive Banach space and A : X → 2X is a maximal monotone operator, which is everywhere de9ned, then A is pseudomonotone.
Theorem 2.3. If X is a re4exive Banach space and A; B : X → 2X are two pseudomonotone operators, then A + B is pseudomonotone.
Theorem 2.4. If X is a re4exive Banach space, A : X → 2X is a pseudomonotone operator, which is coercive in the following sense: there exists a function C : R+ → R with limr→+∞ mC(r) = +∞, such that for all x ∈ X and x∗ ∈ X , we have x∗ ; xX ¿ C(xX )xX , then A is surjective. In analogy with the directional derivative of a convex function, we deAne the generalized directional derivative of a locally Lipschitz function ’ : X → R at x ∈ X in the direction h ∈ X , by ’(x + th) − ’(x ) df ’0 (x; h) = lim sup : t x →x t0
The function X h → ’0 (x; h) ∈ R is sublinear, continuous and by the Hahn–Banach theorem it is the support function of a nonempty, convex and w∗ -compact subset of X , deAned by df
@’(x) = {x∗ ∈ X : x∗ ; hX 6 ’0 (x; h) for all h ∈ X }:
The multifunction X x → @’(x) ∈ 2X is called generalized or Clarke’s subdi:erential of ’ at x. Now we recall the basic facts and notions from the theory of hysteresis operators. Let T ¿ 0. A function v : [0; T ] → R is called piecewise monotone if there exists a partition 0 6 t0 ¡ t1 ¡ · · · ¡ tn = T of interval [0; T ] (called monotonicity partition), such that v|[ti−1 ;ti ] is monotone, for all i = 1; : : : ; n. Such a partition is called standard monotonicity partition if the number of subintervals n is minimal. By Cpm ([0; T ]), we denote the space of all continuous and piecewise monotone functions on [0; T ].
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Functional H : Cpm ([0; T ]) → R is called rate independent if and only if H[v ◦ ’] = H[v];
(2.1)
for all v ∈ Cpm ([0; T ]) and all continuous increasing functions ’ : [0; T ] → R satisfying ’(0) = 0 and ’(T ) = T . It can be easily seen that only the local extremal values of v have an inNuence on H[v]. By S we denote the set of all Anite strings of real numbers and by SA , the set of all alternating strings, i.e. df
SA = {(s0 ; : : : ; sn ) ∈ S: (si+1 − si )(si − si−1 ) ¡ 0; 1 6 i 6 n − 1}
(2.2)
We deAne the restriction operator A : Cpm ([0; T ]) → SA , as follows df
A (v) = (v(t0 ); v(t1 ); : : : ; v(tn ));
(2.3)
for all v ∈ Cpm ([0; T ]), where 0 = t0 ¡ t1 ¡ · · · ¡ tn = T is the standard monotonicity partition of v. By A : SA → Cpm ([0; T ]), we denote the so-called prolongation operator, which maps any string s = (s0 ; s1 ; : : : ; sn ) into the linear interpolate function v : [0; T ] → R of the points (iT=n; si ) for i = 0; 1 : : : ; n. Using these operators, we obtain : SA → R and rate independent the bijective correspondence between functionals H functionals H : Cpm ([0; T ]) → R, as follows ◦ A ; H=H
= H ◦ A : H
(2.4)
An operator W : Cpm ([0; T ]) → C([0; T ]) is said to be a hysteresis operator on Cpm ([0; T ]) if there exists a rate independent functional H called a generating functional of W, such that W[v](t) = H[vt ] where
df
vt ( ) =
∀t ∈ [0; T ]; ∀v ∈ Cpm ([0; T ]);
v( )
for 0 6 6 t;
v(t)
for t ¡ 6 T:
(2.5)
(2.6)
: SA → S is called a hysteresis operator on SA , if An operator W 0 ); H(s 0 ; s1 ); : : : ; H(s)) = (H(s W(s)
∀s = (s0 ; s1 ; : : : ; sn ) ∈ SA ;
(2.7)
where H=H◦ A is called a generating functional of W and H is a rate independent functional on Cpm ([0; T ]). are called 9nal value mappings and The unique generating functionals H and H f , respectively. Due to (2.4), we also have a bijective are denoted by Wf and W deAned correspondence between hysteresis operators W deAned on Cpm ([0; T ]) and W and also Wf for both on SA . We will use the same notation W for both W and W f . Finally, we can extend the hysteresis operators deAned on Cpm ([0; T ]) Wf and W to the set of all continuous functions C([0; T ]), by the use of the density of the embedding Cpm ([0; T ]) ⊆ C([0; T ]). For more details on hysteresis operators we refer to Brokate and Sprekels [2, Chapter 2], Krasnoselskii and Pokrovskii [8] and Visintin [15, Chapter I].
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3. Formulation of the problem Let ∈ RN be an open, bounded set with Lipschitz boundary @ and T ¿ 0. Let H = L2 ( ). By V we denote a Hilbert space such that the embedding V ⊆ H 1 ( ) is dense and continuous. In our evolution case we will need the following spaces: H = L2 (0; T ; H ) = L2 ((0; T ) × ); V = L2 (0; T ; V ); Y = H 1 (0; T ; H ) ∩ L∞ (0; T ; V ): We consider the following evolution hemivariational inequality: Find u ∈ C([0; T ]; V ) with u ∈ Y and % ∈ H; such that u (t) + w(t) + A(u (t)) + B(u(t)) + %(t) = f(t) in V u(0) = 0 ; u (0) = 1 in (HVI) w(t; x) = W[u(·; x); x](t) for a:a: (t; x) ∈ (0; T ) × %(t; x) ∈ @j(x; g(u(t; x); u (t; x))) for a:a: (t; x) ∈ (0; T ) × : Our hypotheses on the right-hand side f, initial conditions 0 , 1 , the hysteresis operator W, the operators A and B, the nonsmooth potential function j and the function g are the following: H (f; ) f ∈ H, 0 ; 1 ∈ V . H (W) W[·; x] is a family of hysteresis operators (indexed by x ∈ ), such that (i) for all x ∈ , operator W[·; x] : C([0; T ]) → C([0; T ]) is continuous; (ii) the parameterized Anal value mapping x → Wf (s; x) ∈ R is measurable for all n ∈ N and all s = (s0 ; s1 ; : : : ; sn ) ∈ S; we also have |Wf (s; x)| 6 aW (x) + cW max |sk |; 06k6n
∀x ∈ ; s ∈ S; n ∈ N;
where aW ∈ H and cW ¿ 0; (iii) Wf ( 1 (·); ·) ∈ H . H (g) g : R × R → R is a function, such that (i) g is continuous; (ii) for all ; , ∈ R, we have |g( ; ,)| 6 -g | | + .g |,|, where -g ; .g ¿ 0. H (j) j :
× R → R is a function, such that
(i) for all ∈ R, the function x → j(x; ) ∈ R is measurable; (ii) for almost all x ∈ , the function R → j(x; ) ∈ R is locally Lipschitz;
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(iii) for almost all x ∈ , all ∈ R and all / ∈ @j(x; ), we have that |/| 6 cj (1 + | |), where cj ¿ 0. H (A) A : V → V is a linear operator, such that (i) for all v; w ∈ V , we have that Av; wV 6 -A vV wV , with some -A ¿ 0. (ii) for all v; w ∈ V , we have Av; wV = Aw; vV ; (iii) for all v ∈ V , we have Av; vV ¿ .A v2V − 0A v2H , with some .A ¿ 2(cHV )2 cj .g and 0A ¿ 0. H (B) B : V → V is a linear operator, such that (i) for all v; w ∈ V , we have Bv; wV 6 -B vV wV , with some -B ¿ 0; (ii) for all v ∈ V , we have Bv; vV ¿ 0. Now we can formulate our main result. Theorem 3.1. If hypotheses H (W), H (A), H (B), H (j), H (g), H (f; ) hold, then the problem (HVI) admits a solution. 4. Auxiliary results Now we will give some existence result for an elliptic hemivariational inequality, which will be useful in the sequel. Theorem 4.1. Let AQ : V → V be a pseudomonotone operator, which is weakly coercive in the following sense: there exist two constants .A ¿ 0 and 0A ¿ 0 such that Q vV ¿ .A v2V − 0A vV . Let fQ ∈ V and z ∈ V . Finally, let for all v ∈ V , we have Av; g and j satisfy hypotheses H (g) and H (j), respectively. Then for all h ¿ 0 such that h(h-g + .g ) ¡ .A =2(cHV )2 cj the following problem Find u ∈ V and %Q ∈ H; such that Q + h%Q = fQ in V (EHVI) Au %(x) Q ∈ @j(x; g((z + hu)(x); u(x))) for a:a: x ∈ admits a solution. Remark 4.2. Compare the above theorem with the result of Naniewicz and Panagiotopoulos [10, Theorem 4.25, p. 120]. In our formulation we use a stronger coercivity condition but we do not assume any sign condition (cf. [10, (4.3.27), p. 120]). Proof of Theorem 4.1. First let us introduce a multivalued operator G : V → 2H , deAned for all v ∈ V , by df
Gv = {/ ∈ H : /(x) ∈ @j(x; g((z + hv)(x); v(x))) for a:a: x ∈ }:
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Using hypotheses H (j)(iii) and H (g)(ii) for any v ∈ V and / ∈ Gv, we have 2 2 2cj2 (1 + |g((z + hv)(x); v(x))|2 ) d x /H = |/(x)| d x 6 6 2cj2
(1 + 2-g2 |z(x)|2 + 2(-g2 h2 + .g2 )|v(x)|2 ) d x = c1 + c2 v2H ;
df
(4.1)
df
where c1 = 2cj2 (| | + 2-g2 z2H ) and c2 = 4cj2 (-g2 h2 + .g2 ). Using operator G, we can rewrite the problem (EHVI) as follows: Find u ∈ V; such that Q Q + hGu f: Au
(4.2)
In order to obtain the existence of a solution for (4.2), we examine the range of the multivalued mapping AQ + hG. First, let us note that G has nonempty, weakly compact and convex values. Next we will check that operator G is generalized pseudomonotone. To this purpose, let us choose two sequences {vn }n¿1 ⊆ V and {/n }n¿1 ⊆ H , such that w−V
w−V
vn → v ∈ V , /n → / ∈ V , /n ∈ Gvn for n ¿ 1 and lim sup/n ; vn − vV 6 0. From the H compactness of the embedding V ⊆ H , we have that vn →v. So, from estimate (4.1), w−H Q Using the convergence passing to a subsequence if necessary, we get that /n → /. theorem of Aubin–Cellina (cf. e.g. [1, Theorem 7.2.2, p. 273]), we can easily show that /Q ∈ Gu. From the uniqueness of the weak limit in V , we know that / = /Q and so also that / ∈ Gv. Finally, we also have /n ; vn V = (/n ; vn )H → (/; v)H = /; vV : Thus G is generalized pseudomonotone. From Theorem 2.1, we get that G is pseudomonotone and from Theorem 2.3, we have that AQ + hG is also pseudomonotone. To verify the coercivity of AQ + hG, let us choose any v ∈ V and / ∈ Gv. Using the coercivity of A and (4.1), we have Q + h/; vV ¿ .A v2V − 0A vV + h(/; v)H Av √ √ ¿ .A v2V − 0A vV − h( c1 + c2 vH )vH √ ¿ (.A − 2hcj (cHV )2 (h-g + .g ))v2V − (0A + h c1 cHV )vV : Q From the choice of h ¿ 0, we get that operator A+hG is coercive. Now using Theorem 2.4, we also see that operator AQ + hG is surjective and so there exists u ∈ V , such that Q Q + hGu f. Au 5. The proof of the main theorem Proof of Theorem 3.1. First let us introduce the following operator: t df (Ku)(t) = u(s) ds + 0 : 0
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Using the operator K, we can rewrite the problem (HVI) as the following problem (P): Find u ∈ Y; and % ∈ H such that u (t) + w(t) + A(u(t)) + B(z(t)) + %(t) = f(t) u(0) =
1
in
in V ;
(5.1)
;
w(t; x) = W[z(·; x); x](t)
(5.2) for a:a: (t; x) ∈ (0; T ) × ;
%(t; x) ∈ @j(x; g(z(t; x); u(t; x)))
(5.3)
∀t ∈ [0; T ]; for a:a: x ∈ ;
(5.4)
z = Ku:
(5.5)
Now, u is a solution of (P) if and only if Ku is a solution of (HVI). For problem (P), we will use time-discretization method (cf. [2, Chapter 3.3; 15,9]). For any m ∈ N, let h = T=m. We deAne 1 ih fhi (x) = f(t; x) dt ∀i = 1; : : : ; m; (5.6) h (i−1)h uh0 (x) =
1 (x);
(5.7)
zh0 (x) =
0 (x):
(5.8)
We formulate the following semilinear problem (Ph ): Find uhi ∈ V and whi ; %hi ∈ H (for i = 1; : : : ; m); such that uhi − uhi−1 + whi + Auhi + Bzhi + %hi = fhi h whi (x) = Wf (zh0 (x); zh1 (x); : : : ; zhi (x); x) %hi (x) ∈ @j(x; g(zhi (x); uhi (x)))
in V ;
(5.9)
for a:a: x ∈ ;
(5.10)
for a:a: x ∈ ;
(5.11)
zhi = zhi−1 + huhi :
(5.12)
In order to solve problem (Ph ), let us assume that we have already solve it for k = 1; : : : ; i − 1, so we already know functions uh0 ; uh1 ; : : : ; uhi−1 ∈ V (and thus also zh0 ; zh1 ; : : : ; zhi−1 ∈ V ) and let us solve it for k = i. We rewrite problem (Ph ) as follows: Find uhi ∈ V and whi ; %hi ∈ H; such that hAuhi + h2 Buhi + uhi + hwhi + h%hi = hfhi + uhi−1 − hBzhi−1
in V ;
(5.13)
whi (x) = Wf (zh0 (x); zh1 (x); : : : ; zhi−1 (x); (zhi−1 + huhi )(x); x) for a:a: x ∈ ; %hi (x) ∈ @j(x; g((zhi−1 + huhi )(x); uhi (x)))
(5.14) for a:a: x ∈ :
(5.15)
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331
Let us introduce the following operators and functions: df AQ = A1 + A2 ; df
A1 v = hAv + h2 Bv + v; df
(A2 v)(x) ≡ b(x; v(x)) = hWf (zh0 (x); zh1 (x); : : : ; zhi−1 (x); (zhi−1 + hv)(x); x); df fQ = hfhi + uhi−1 − hBzhi−1 ∈ V :
Now we can rewrite problem (5.13)–(5.15) as follows: Find uhi ∈ V and %hi ∈ H; such that Q hi + h%hi = fQ Au
in V ;
(5.16)
%hi (x) ∈ @j(x; g((zhi−1 + huhi )(x); uhi (x)))
for a:a: x ∈ :
(5.17)
To solve problem (5.16)–(5.17), we will apply Theorem 4.1. First, we assume that h 6 1=0A (or h is arbitrary if 0A = 0), so that A1 is linear and from hypotheses H (A)(iii) and H (B)(ii), it is coercive in the following sense: A1 v; vV ¿ h.A v2V
∀v ∈ V
(5.18)
and thus A1 is maximal monotone. As A1 is everywhere deAned, using Theorem 2.2, we get that A1 is pseudomonotone. From hypotheses H (W), we see that b : × R → R is a Caratheodory function satisfying the following growth condition: |b(x; )| 6 haW (x) + h2 cW | |
∀(x; ) ∈
× R;
(5.19)
df
with aW ∈ H , namely aW (x) = |aW (x)| + cW max06k6i−1 |zhk (x)|. Thus A2 is a continuous and bounded Nemytski operator from H to H (see e.g. [16, Proposition 26.7, p. 563]) and so also a pseudomonotone operator from V to V . From Theorem 2.3, we get that AQ is pseudomonotone. From (5.18) and (5.19), for any v ∈ V , we get Q vV ¿ h.A v2V − A2 vV vV Av; ¿ h.A v2V − (haW V + h2 cW vV )vV ¿ h(.A − h(cHV )2 cW )v2V − hcHV aW H vV ; so if h ¡ .A =((cHV )2 cW ), then we have that AQ is coercive in the sense of Theorem 4.1, with .A = h(.A − h(cHV )2 cW ). Because of hypothesis H (A)(iii), we can choose h ¿ 0 so small, that .A =2(cHV )2 cj = h(.A − h(cHV )2 cW )=2(cHV )2 cj ¿ h.g . Thus, decreasing again h ¿ 0, if necessary, we may assume that h(h-g + .g ) ¡ .A =(2(cHV )2 cj ). Applying Theorem 4.1, we obtain uhi ∈ V , a solution of (5.16)–(5.17).
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Next we obtain some a priori estimate for the solutions of (5.9)–(5.12). We multiply (5.9) by uhi − uhi−1 and sum it from i = 1 to l (with 1 6 l 6 m). Then we estimate separately each term. First, we have
l l uhi − uhi−1 i 1 i i−1 = uh − uhi−1 2H ; uh − uh h h i=1
i=1
H
1 ¿ T ¿
l
2 (uhi H
−
i=1
uhi−1 H )
1 l 2 1 uh H − 1 2H : 2T T
(5.20)
From hypothesis H (W)(ii) and (5.10), we get |whi (x)| 6 a (x) + cW h
i
|uhk (x)|
for a:a: x ∈ ;
(5.21)
k=1 df
with a ∈ H , namely a (x) = |aW (x)| + cW | 0 (x)|. Using the Cauchy–Schwarz inequality, the Young inequality and estimate (5.21), we have
1=2 l
1=2 l l 1 i i−1 i−1 2 i i i 2 (wh ; uh − uh )H ¿ − h wh H uh − uh H h i=1
i=1
¿−
¿−
i=1
l
l
i=1
i=1
h i 2 8 i wh H − uh − uhi−1 2H 48 h l i 2 Th2 cW T a 2H − uhk 2H 28 28 i=1 k=1
l
−
8 i uh − uhi−1 2H : h i=1
Because of l
1 l 2 1 1 i uh − uhi−1 2H uh H − 1 2H + 2 2 2 i=1
l
=
1 (uhi 2H − uhi−1 2H + uhi − uhi−1 2H ) 2 i=1
l (uhi − uhi−1 ; uhi )H = i=1
(5.22)
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333
and the Young inequality, we get l
1 i 1 l 2 uh − uhi−1 2H uh H + 2 2 i=1
6
l
l
i=1
i=1
8 i h i 2 1 1 2H + uh H + uh − uhi−1 2H : 48 h 2
Using hypotheses H (A) and the last inequality, we have l i=1
Auhi ; uhi − uhi−1 V l
=
1 (Auhi ; uhi V − Auhi−1 ; uhi−1 V + A(uhi − uhi−1 ); uhi − uhi−1 V ) 2 i=1
l
1 1 1 A(uhi − uhi−1 ); uhi − uhi−1 V = Auhl ; uhl V − Auh0 ; uh0 V + 2 2 2 i=1
¿
l 1 .A i .A l 2 uh V + uh − uhi−1 2V − A 1 ; 2 2 2
1 V
i=1
l 0A l 2 0A i − uh H − uh − uhi−1 2H 2 2 i=1
l .A l 2 .A i -A ¿ uh V + uh − uhi−1 2V − 1 2V 2 2 2 i=1
−
l
l
i=1
i=1
0A 0A h i 2 0A 8 i uh H − uh − uhi−1 2H : 1 2H − 2 48 h
(5.23)
Next, from (5.12), hypotheses H (B) and the Young inequality, we have l i=1
Bzhi ; uhi − uhi−1 V
=Bzh1 ; uh1 −
1 V
+
l i=2
(Bzhi ; uhi V − Bzhi ; uhi−1 V )
l l (Bzhi ; uhi V − Bzhi−1 ; uhi−1 V ) − h Buhi ; uhi−1 V + Bzh1 ; uh1 − = i=2
i=2
1 V
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=Bzhl ; uhl V − Bzh1 ;
1 V
−h
l i=2
=B 0 ; uhl −
1 V
+h
l i=1
Buhi ; uhi−1 V
Buhi ; uhl − uhi−1 V
¿− -B 0 V (uhl V + 1 V ) − h-B
l i=1
uhi V (uhl V + uhi−1 V )
-B -B 0 2V − -B 8uhl 2V − ( 0 2V + 1 2V ) 2 48 l l h-B 1 i 2 l 2 uh V + 8uh V − (uhi 2V + uhi−1 2V ) −h-B 48 2
¿−
i=1
i=1
l−1
h h h-B (1 + 48) i 2 + ¿− uh V − -B 8(1 + T ) + 48 2 48 −-B
1 1 + 48 2
uhl 2V
i=1
0 2V −
-B (1 + h) 1 2V : 2
(5.24)
As zhi 2H 6 2 0 2H + 2Th
i
uhk 2H
∀i = 1; : : : ; m;
k=1
so using the Cauchy–Schwartz inequality, the Young inequality, (5.11), hypotheses H (j)(iii) and H (g)(ii), we have l i=1
(%hi ; uhi − uhi−1 )H ¿ −
¿−
l
l
i=1
i=1
h i 2 8 i %h H − uh − uhi−1 2H 48 h l cj2 h (1 + 2-g zhi 2H + 2.g uhi 2H ) 28 i=1
l
−
8 i uh − uhi−1 2H h i=1
¿−
l i 2cj2 -g Th2 cj2 T uhk 2H (1 + 4-g 0 2H ) − 28 8 i=1 k=1
−
l l cj2 .g h 8 i uhi 2H − uh − uhi−1 2H : 8 h i=1
i=1
(5.25)
L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
335
Using the Cauchy–Schwartz inequality, the Young inequality and (5.6), we obtain l
(fhi ; uhi
−
i=1
l
8 i 1 uh − uhi−1 2H : 6 f2H + 48 h
uhi−1 )H
(5.26)
i=1
Putting together all the above estimates (5.20), (5.22)–(5.26), we get .A hc4 h 2 c6 − 8c3 − − hc5 − uhl 2V 2 8 8 +
l l uhi − uhi−1 2H .A i uh − uhi−1 2V + (1 − 38 − 0A 8) 2 h i=1
i=1
hc6 c10 c8 + hc9 + h -B + + 6 c7 + 8 8 8
l−1 i l−1 h 2 c6 k 2 × uh V uhi 2V + 8 i=1
(5.27)
i=1 k=1
df -B df 1 V 2 2 2 , c6 = 2 T (cH ) (cW + df df c7 = 12 (-A + -B ) 1 2V + 02A 1 2H + -2B 0 2V , c8 = 14 (2T a 2H + -B 0 2V + df df B + 4-g 0 2H ) + f2H ), c9 = 1 2V , c10 = 14 (0A (cHV )2 + -B ) + cj2 .g (cHV )2 . df
df
with c3 = -B (1 + T ), c4 = 14 (-B + (cHV )2 (0A + 4cj2 .g )), c5 = 4cj2 -g ), 2cj2 T (1
2 First, let us choose 0 ¡ 8 ¡ min{1=(3+0A ); .A =(2c3 )}. Then, we see that there exists h0 ¿ 0 small enough, such that for all 0 ¡ h ¡ h0 , the constant in front of uhl 2V is positive. Applying Theorem 6.1, we obtain max uhi 2V 6 const
16i6m
for 0 ¡ h 6 h0 :
(5.28)
From (5.27), we have m i=1
uhi − uhi−1 2V 6 const
m uhi − uhi−1 2H 6 const h
for 0 ¡ h 6 h0 ;
for 0 ¡ h 6 h0 :
(5.29)
(5.30)
i=1
As zhi V 6 0 V + h
i
k=1
max zhi V 6 const
16i6m
uhk V , so from (5.28), we have for 0 ¡ h 6 h0 :
(5.31)
From (5.12) and (5.28), we see that zhi − zhi−1 V 6 const 16i6m h max
for 0 ¡ h 6 h0
(5.32)
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L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
and m i=1
zhi − zhi−1 V 6 const
for 0 ¡ h 6 h0 :
(5.33)
From (5.11), hypotheses H (j)(iii), H (g)(ii), (5.28) and (5.31), we have max %hi H 6 const
16i6m
for 0 ¡ h 6 h0
(5.34)
and from (5.21) and (5.28), we obtain max whi H 6 const
16i6m
for 0 ¡ h 6 h0 :
(5.35)
For s ∈ (0; 1] and i = 1; : : : ; m, let us deAne the following interpolates: df
uh ((i − 1 + s)h; x) = (1 − s)uhi−1 (x) + suhi (x); df
uh ((i − 1 + s)h; x) = uhi (x); df
zh ((i − 1 + s)h; x) = (1 − s)zhi−1 (x) + szhi (x); df
zh ((i − 1 + s)h; x) = zhi (x); df
wh ((i − 1 + s)h; x) = whi (x); df
%h ((i − 1 + s)h; x) = %hi (x); df
fh ((i − 1 + s)h; x) = fhi (x): Using these deAnitions, we can rewrite the problem (Ph ), as follows: uh (t) + wh (t) + Auh (t) + Bzh (t) + %h (t) = fh (t) wh (t; x) = W[zh (·; x); x](t)
in V ∀t ∈ (0; T );
for a:a: (t; x) ∈ (0; T ) × ;
%h (t; x) ∈ @j(x; g(zh (t; x); uh (t; x)))
for a:a: (t; x) ∈ (0; T ) × :
(5.36) (5.37) (5.38)
From (5.28)–(5.34), we know that all uh L2 (0; T ;H ) ;
uh L∞ (0; T ;V ) ;
zh L∞ (0; T ;V ) ;
wh L∞ (0; T ;H ) ;
uh L∞ (0; T ;V ) ;
zh W 1; ∞ (0; T ;V ) ;
%h L∞ (0; T ;H )
are bounded, for 0 ¡ h ¡ h0 . Thus, we can choose subsequences and limit functions, such that uh → u
weakly in H 1 (0; T ; H ) and weakly∗ in L∞ (0; T ; V );
(5.39)
uh → uQ
weakly∗ in L∞ (0; T ; V );
(5.40)
zh → z
weakly in H 1 (0; T ; V ) and weakly∗ in W 1; ∞ (0; T ; V );
(5.41)
L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
zh → zQ
337
weakly∗ in L∞ (0; T ; V );
(5.42)
weakly∗ in L∞ (0; T ; H );
wh → wQ
(5.43)
weakly∗ in L∞ (0; T ; H ):
%h → %Q
(5.44)
From (5.29), we deduce that uh − uh 2L2 (0; T ;H ) =
m i=1
ih
(i−1)h
i−
t 2 i uh − uhi−1 2H dt h
m h i = u − uhi−1 2H 6 hconst 3 h i=1
and so from (5.39) and (5.40), we see that u = u. Q In the same manner, we obtain that z = z. Q As zh = uh , so also Ku = z and thus (5.5) holds. Passing to the limit in (5.36) we obtain (5.1). From (5.39) and the fact that uh (0) = 1 , we conclude that (5.2) holds. From the compactness of the embedding Y ⊆ L2 ( ; C([0; T ])) (see e.g. [2, p. 129; 15, p. 265]] and from (5.41), we get zh → z
in L2 ( ; C([0; T ]))
(5.45)
and consequently zh (·; x) → z(·; x)
uniformly in [0; T ] for a:a x ∈ :
(5.46)
Let us deAne df
wh∗ (t; x) = W[zh (·; x); x](t); df
w∗ (t; x) = W[z(·; x); x](t); df
:h (x) =
2 sup |wh∗ (t; x) − w∗ (t; x)| ;
06t6T
2 2 df ;h (x) = 2 aW (x) + cW sup |zh (t; x)| + 2 aW (x) + cW sup |z(t; x)| ; df
;(x) = 4 aW (x) + cW
06t6T
06t6T
2 sup |z(t; x)| :
06t6T
From H (W), we obtain sup |wh∗ (t; x)| 6 aW (x) + cW sup |zh (t; x)|
06t6T
06t6T
sup |w∗ (t; x)| 6 aW (x) + cW sup |z(t; x)|
06t6T
06t6T
for a:a: x ∈ ; for a:a: x ∈
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L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
and thus |:h (x)| 6 |;h (x)|
for a:a: x ∈
∀0 ¡ h 6 h0 :
(5.47)
From (5.46), we see that ;h (x) → ;(x)
for a:a: x ∈
and from (5.45), we also have ;h (x) d x → ;(x) d x
(5.48) as h → 0+ :
(5.49)
From hypothesis H (W)(i) and (5.46), we have wh∗ (·; x) → w∗ (·; x);
uniformly in [0; T ] for a:a: x ∈
and thus :h (x) → 0
for a:a: x ∈ :
(5.50)
Properties (5.47)–(5.50) allow us to use the generalized majorized convergence theorem (see [16, Appendix, Theorem 19a, p. 1015]) and obtain that lim+ :h (x) d x = lim+ :h (x) d x; h→0
h→0
what in the connection with (5.50), gives that :h (x) d x → 0; thus wh∗ → w∗
in L2 ( ; C([0; T ])):
As, for t = ih, with i = 1; : : : ; m, we have wh (t; x) = wh∗ (t; x) = W[uh (·; x); x](t); so wh is the piecewise constant interpolate of wh∗ and sup |wh (t; x)| 6 aW (x) + cW sup |zh (t; x)|; 06t6T
06t6T
wh → w ∗ ;
for a:a: x ∈
uniformly in [0; T ] for a:a: x ∈ :
Arguing in the same way as before, we obtain that wh → w ∗
in L2 ( ; C([0; T ]))
and from (5.43) we have that wQ = w∗ , so (5.3) holds. Finally from (5.40), (5.42), (5.44) and hypothesis H (g)(i), we have that g(zh (t; x); uh (t; x)) → g(z(t; Q x); u(t; Q x)) %h → %Q
for a:a: (t; x) ∈ (0; T ) ×
weakly in L1 ((0; T ) × );
so from (5.38), using the so-called convergence theorem of Aubin and Cellina (cf. e.g. [1, Theorem 7.2.2, p. 273]), we get (5.4).
L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
339
Remark 5.1. Note that if we assume the dependence of W on the time derivative u of the unknown function u (and not on the function u itself) then the argumentation will be similar (in place of hypothesis H (W)(iii), we will need that Wf ( 0 (·); ·) ∈ H ). Remark 5.2. Note that if the function g depends only on the Arst variable (i.e. there is no dependence of @j on the derivative u ), then the additional assumption on the coercivity constant .A in hypothesis H (A)(iii) is no more needed (i.e. it is enough to assume that BW ¿ 0). Remark 5.3. It seems to be interesting to have a similar result to this of Theorem 3.1, but with a noncoercive operator A (see [7] for analogous hemivariational inequality) and in particular for wave-type hemivariational inequality with hysteresis (i.e. with A = 0; see [6]). Remark 5.4. Another interesting generalization of problem (HVI) would be one with a nonlinear operator A. Appendix We formulate and proof a generalized discrete version of the Gronwall inequality (see e.g. [11] for a continuous version). Theorem 6.1. Let {ak }k¿1 ⊆ R be a sequence such that ak ¿ 0 for all k ¿ 1 and there exist numbers A; B; C; h ¿ 0, such that ak 6 A + hB
k−1
ai + h 2 C
i=1
k−1 i
aj
∀k ¿ 1:
(A.1)
i=1 j=1
Then ak 6 Aeh(k−1)B+h
2
(k−1)2 C
∀k ¿ 1:
(A.2)
Proof. Case 1: A = 1. Let us denote the right-hand side of (A.1) by Rk . For any k ¿ 2, we have ak 6 Rk = 1 + hB
k−1
ai + h 2 C
k−1 i
i=1
= Rk−1 + hBak−1 + h2 C
aj
i=1 j=1 k−1
aj
j=1
6 Rk−1 + hBRk−1 + h2 (k − 1)C max aj j=1;:::; k−1
6 Rk−1 + hBRk−1 + h2 (k − 1)CRk−1 = (1 + hB + h2 (k − 1)C)Rk−1 :
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L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
As R1 = 1, we obtain ak 6 (1 + hB + h2 (k − 1)C)k−1 =[(1 + hB + h2 (k − 1)C)1=(hB+h
2
(k−1)C) (k−1)(hB+h2 (k−1)C)
]
Exploiting the estimate (1 + x)1=x 6 e for all x ¿ 0, we obtain ak 6 eh(k−1)B+h
2
(k−1)2 C
∀k ¿ 1
and this Anishes the proof in Case 1. Case 2: A ¿ 0. We divide (A.1) by A and obtain that k−1
k−1
i=1
i=1 j=1
i
ai aj ak + h2 C 6 1 + hB A A A
∀k ¿ 1:
Let us deAne bk =ak =A for k ¿ 1. Applying Case 1 to the sequence {bk }k¿1 , we obtain that bk 6 eh(k−1)B+h
2
(k−1)2 C
∀k ¿ 1;
so (A.2) holds. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
J.P. Aubin, H. Frankowska, Set-Valued Analysis, BirkhSauser, Boston, 1990. M. Brokate, J. Sprekels, Hysteresis and Phase Transitions, Springer, Berlin/New York, 1996. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983. L. Gasi$nski, Existence result for hyperbolic hemivariational inequalities, Nonlinear Anal. 47 (2001) 681–686. L. Gasi$nski, Existence of solutions for hyperbolic hemivariational inequalities, J. Math. Anal. Appl. 276 (2002) 723–746. L. Gasi$nski, M. Smolka, An existence theorem for wave-type hemivariational inequalities, Math. Nachr. 242 (2002) 79–90. L. Gasi$nski, M. Smolka, Existence of solutions for wave-type hemivariational inequalities with noncoercive viscosity damping, J. Math. Anal. Appl. 270 (2002) 150–164. M.A. Krasnoselskii, A.V. Pokrovskii, Systems with Hysteresis, Springer, Heidelberg, 1989. M. Miettinen, P.D. Panagiotopoulos, Hysteresis and hemivariational inequalities: semilinear case, J. Glob. Optim. 13 (1998) 269–298. Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel-Dekker, New York, 1995. B.G. Pachpatte, A note on Gronwall–Bellman inequality, J. Math. Anal. Appl. 44 (1973) 758–762. P.D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, BirkhSauser, Basel, 1985. P.D. Panagiotopoulos, Coercive and semicoercive hemivariational inequalities, Nonlinear Anal. 16 (1991) 209–231. P.D. Panagiotopoulos, Hemivariational Inequalities. Applications to Mechanics and Engineering, Springer, New York, 1993. A. Visintin, Di4erential Models of Hysteresis, Springer, Berlin, New York, 1994. E. Zeidler, Nonlinear Functional Analysis and Its Applications, Vol. II, Springer, New York, 1990.
www.elsevier.com/locate/na
Evolution hemivariational inequalities with hysteresis Leszek Gasi$nski Institute of Computer Science, Jagiellonian University, ul. Nawojki 11, Cracow 30072, Poland Accepted 6 February 2004
Abstract In this paper we study evolution hemivariational inequalities containing a hysteresis operator. For such problems we establish an existence result by reducing the order of the equation and then by the use of the time-discretization procedure. ? 2004 Elsevier Ltd. All rights reserved. MSC: 47J20; 49J40 Keywords: Hemivariational inequality; Clarke subdi4erential; Hysteresis operator; Time-discretization method; Gronwall inequality
1. Introduction Theory of variational inequalities provides us with an appropriate mathematical model to describe many physical problems. This theory was started in 1960s by Baiocchi, H. Brezis, G. Duvaut, G. Ficher, D. Kinderlehrer, J.L. Lions, G. Stampacchia and many others. In 1980s, Panagiotopoulos introduced the so-called hemivariational inequalities (see [13,14]), using the notion of Clarke’s subdi4erential (see [3]), which is deAned for locally Lipschitz functions. Hemivariational inequalities are an e4ective tool to treat problems with nonmonotonicity and multivaluedness. For a great amount of examples of applications we refer to Naniewicz and Panagiotopoulos [10] and Panagiotopoulos [12–14]. Parallel to the theory of hemivariational inequalities, the theory of hysteresis has been developed. The latter is based on the notion of the hysteresis operator, introduced
This paper was partially supported by the KBN Grants no 2 P03A 003 25 and no 7 T07A 047 18. E-mail address: [email protected] (L. Gasi$nski).
0362-546X/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2004.02.016
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L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
by Krasnoselskii and Pokrovskii [8] and developed e.g. by Brokate and Sprekels [2], Visintin [15] and others. In this paper we study evolution hemivariational inequality with a memory hysteresis operator which occurs in the source term. The unknown function u may represent the displacement and w describe a force term characterized by continuous hysteresis cycles. The existence and uniqueness results for the corresponding semilinear parabolic and hyperbolic equations with hysteresis in the lower term can be found in Visintin [15, Chapter X.1]. Similar approach to the one exploited in this paper can be done when the hysteresis operator depends on the time derivative of the unknown function. We mention that in other problems with hysteresis arising in continuous mechanics, hysteresis operators may also appear in constitutive laws. For such situations we refer to Brokate and Sprekels [2] and Visintin [15]. For the evolution hemivariational inequalities similar to the one presented in this paper (but not containing hysteresis operator), we refer to the papers of Gasi$nski [4,5] and Gasi$nski and Sma lka [6,7]. In all these papers the proofs of the existence theorems are based on the surjectivity result for coercive pseudomonotone operators. In our case, because of the presence of the hysteresis operator, this method does not work any more. The method of the proof in the present paper is based on the “reduction of the order” of the inequality and then on the time discretization method and the surjectivity result for a stationary problem (cf. [2, Chapter 3.3]; [15,9]).
2. Preliminaries For the convenience of the reader we recall some basic deAnitions and notions from the theory of pseudomonotone operators, the theory of Clarke’s subdi4erential and the theory of hysteresis operators. Let X be a Banach space and X its topological dual. By · X we will denote a norm in X and by ·; ·X the duality brackets for the pair (X; X ). Let X and Y be two Banach spaces. If the embedding X ⊆ Y is continuous, then by cYX we denote the continuity constant, i.e. such a constant that for all x ∈ X , we have that xY 6 cYX xX . A single valued operator A : X → X is said to be pseudomonotone, if (a) A is bounded; (b) if x n → x weakly in X and lim supAx n ; x n − xX 6 0, then for all y ∈ X , we have that Ax; x − yX 6 lim inf Ax n ; x n − yX .
A multivalued operator A : X → 2X is said to be pseudomonotone, if (a) for all x ∈ X , set Ax is nonempty, convex and weakly compact in X ; (b) for every Y , Anite-dimensional subspace of X , operator A|Y is upper semicontinuous into X furnished with the weak topology (i.e. if U ⊆ X is weakly open, then the set {x ∈ Y : Ax ⊆ U } is open in Y );
L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
325
(c) if x n → x weakly in X , x∗n ∈ Ax n for n ¿ 1 and lim supx∗n ; x n − xX 6 0, then for every y ∈ X , there exists x∗ (y) ∈ Ax such that x∗ (y); x − yX 6 lim inf x∗n ; x n − yX . If A is bounded (i.e. maps bounded sets into bounded sets) and satisAes condition (c), then it satisAes condition (b) too. An operator A : X → 2X is said to be generalized pseudomonotone, if x n → x weakly in X, x∗n → x∗ weakly in X , x∗n ∈ Ax n for n ¿ 1 and lim supx∗n ; x n − xX 6 0, imply that x∗ ∈ Ax and x∗n ; x n X → x∗ ; xX . We recall some basic facts concerning pseudomonotone operators (for the proofs see [10, Theorems 2.2–2.4, 2.6]).
Theorem 2.1. If X is a re4exive Banach space, A : X → 2X is a bounded generalized pseudomonotone operator and for each x ∈ X , Ax is a nonempty, closed and convex subset of X , then A is pseudomonotone.
Theorem 2.2. If X is a re4exive Banach space and A : X → 2X is a maximal monotone operator, which is everywhere de9ned, then A is pseudomonotone.
Theorem 2.3. If X is a re4exive Banach space and A; B : X → 2X are two pseudomonotone operators, then A + B is pseudomonotone.
Theorem 2.4. If X is a re4exive Banach space, A : X → 2X is a pseudomonotone operator, which is coercive in the following sense: there exists a function C : R+ → R with limr→+∞ mC(r) = +∞, such that for all x ∈ X and x∗ ∈ X , we have x∗ ; xX ¿ C(xX )xX , then A is surjective. In analogy with the directional derivative of a convex function, we deAne the generalized directional derivative of a locally Lipschitz function ’ : X → R at x ∈ X in the direction h ∈ X , by ’(x + th) − ’(x ) df ’0 (x; h) = lim sup : t x →x t0
The function X h → ’0 (x; h) ∈ R is sublinear, continuous and by the Hahn–Banach theorem it is the support function of a nonempty, convex and w∗ -compact subset of X , deAned by df
@’(x) = {x∗ ∈ X : x∗ ; hX 6 ’0 (x; h) for all h ∈ X }:
The multifunction X x → @’(x) ∈ 2X is called generalized or Clarke’s subdi:erential of ’ at x. Now we recall the basic facts and notions from the theory of hysteresis operators. Let T ¿ 0. A function v : [0; T ] → R is called piecewise monotone if there exists a partition 0 6 t0 ¡ t1 ¡ · · · ¡ tn = T of interval [0; T ] (called monotonicity partition), such that v|[ti−1 ;ti ] is monotone, for all i = 1; : : : ; n. Such a partition is called standard monotonicity partition if the number of subintervals n is minimal. By Cpm ([0; T ]), we denote the space of all continuous and piecewise monotone functions on [0; T ].
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Functional H : Cpm ([0; T ]) → R is called rate independent if and only if H[v ◦ ’] = H[v];
(2.1)
for all v ∈ Cpm ([0; T ]) and all continuous increasing functions ’ : [0; T ] → R satisfying ’(0) = 0 and ’(T ) = T . It can be easily seen that only the local extremal values of v have an inNuence on H[v]. By S we denote the set of all Anite strings of real numbers and by SA , the set of all alternating strings, i.e. df
SA = {(s0 ; : : : ; sn ) ∈ S: (si+1 − si )(si − si−1 ) ¡ 0; 1 6 i 6 n − 1}
(2.2)
We deAne the restriction operator A : Cpm ([0; T ]) → SA , as follows df
A (v) = (v(t0 ); v(t1 ); : : : ; v(tn ));
(2.3)
for all v ∈ Cpm ([0; T ]), where 0 = t0 ¡ t1 ¡ · · · ¡ tn = T is the standard monotonicity partition of v. By A : SA → Cpm ([0; T ]), we denote the so-called prolongation operator, which maps any string s = (s0 ; s1 ; : : : ; sn ) into the linear interpolate function v : [0; T ] → R of the points (iT=n; si ) for i = 0; 1 : : : ; n. Using these operators, we obtain : SA → R and rate independent the bijective correspondence between functionals H functionals H : Cpm ([0; T ]) → R, as follows ◦ A ; H=H
= H ◦ A : H
(2.4)
An operator W : Cpm ([0; T ]) → C([0; T ]) is said to be a hysteresis operator on Cpm ([0; T ]) if there exists a rate independent functional H called a generating functional of W, such that W[v](t) = H[vt ] where
df
vt ( ) =
∀t ∈ [0; T ]; ∀v ∈ Cpm ([0; T ]);
v( )
for 0 6 6 t;
v(t)
for t ¡ 6 T:
(2.5)
(2.6)
: SA → S is called a hysteresis operator on SA , if An operator W 0 ); H(s 0 ; s1 ); : : : ; H(s)) = (H(s W(s)
∀s = (s0 ; s1 ; : : : ; sn ) ∈ SA ;
(2.7)
where H=H◦ A is called a generating functional of W and H is a rate independent functional on Cpm ([0; T ]). are called 9nal value mappings and The unique generating functionals H and H f , respectively. Due to (2.4), we also have a bijective are denoted by Wf and W deAned correspondence between hysteresis operators W deAned on Cpm ([0; T ]) and W and also Wf for both on SA . We will use the same notation W for both W and W f . Finally, we can extend the hysteresis operators deAned on Cpm ([0; T ]) Wf and W to the set of all continuous functions C([0; T ]), by the use of the density of the embedding Cpm ([0; T ]) ⊆ C([0; T ]). For more details on hysteresis operators we refer to Brokate and Sprekels [2, Chapter 2], Krasnoselskii and Pokrovskii [8] and Visintin [15, Chapter I].
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3. Formulation of the problem Let ∈ RN be an open, bounded set with Lipschitz boundary @ and T ¿ 0. Let H = L2 ( ). By V we denote a Hilbert space such that the embedding V ⊆ H 1 ( ) is dense and continuous. In our evolution case we will need the following spaces: H = L2 (0; T ; H ) = L2 ((0; T ) × ); V = L2 (0; T ; V ); Y = H 1 (0; T ; H ) ∩ L∞ (0; T ; V ): We consider the following evolution hemivariational inequality: Find u ∈ C([0; T ]; V ) with u ∈ Y and % ∈ H; such that u (t) + w(t) + A(u (t)) + B(u(t)) + %(t) = f(t) in V u(0) = 0 ; u (0) = 1 in (HVI) w(t; x) = W[u(·; x); x](t) for a:a: (t; x) ∈ (0; T ) × %(t; x) ∈ @j(x; g(u(t; x); u (t; x))) for a:a: (t; x) ∈ (0; T ) × : Our hypotheses on the right-hand side f, initial conditions 0 , 1 , the hysteresis operator W, the operators A and B, the nonsmooth potential function j and the function g are the following: H (f; ) f ∈ H, 0 ; 1 ∈ V . H (W) W[·; x] is a family of hysteresis operators (indexed by x ∈ ), such that (i) for all x ∈ , operator W[·; x] : C([0; T ]) → C([0; T ]) is continuous; (ii) the parameterized Anal value mapping x → Wf (s; x) ∈ R is measurable for all n ∈ N and all s = (s0 ; s1 ; : : : ; sn ) ∈ S; we also have |Wf (s; x)| 6 aW (x) + cW max |sk |; 06k6n
∀x ∈ ; s ∈ S; n ∈ N;
where aW ∈ H and cW ¿ 0; (iii) Wf ( 1 (·); ·) ∈ H . H (g) g : R × R → R is a function, such that (i) g is continuous; (ii) for all ; , ∈ R, we have |g( ; ,)| 6 -g | | + .g |,|, where -g ; .g ¿ 0. H (j) j :
× R → R is a function, such that
(i) for all ∈ R, the function x → j(x; ) ∈ R is measurable; (ii) for almost all x ∈ , the function R → j(x; ) ∈ R is locally Lipschitz;
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(iii) for almost all x ∈ , all ∈ R and all / ∈ @j(x; ), we have that |/| 6 cj (1 + | |), where cj ¿ 0. H (A) A : V → V is a linear operator, such that (i) for all v; w ∈ V , we have that Av; wV 6 -A vV wV , with some -A ¿ 0. (ii) for all v; w ∈ V , we have Av; wV = Aw; vV ; (iii) for all v ∈ V , we have Av; vV ¿ .A v2V − 0A v2H , with some .A ¿ 2(cHV )2 cj .g and 0A ¿ 0. H (B) B : V → V is a linear operator, such that (i) for all v; w ∈ V , we have Bv; wV 6 -B vV wV , with some -B ¿ 0; (ii) for all v ∈ V , we have Bv; vV ¿ 0. Now we can formulate our main result. Theorem 3.1. If hypotheses H (W), H (A), H (B), H (j), H (g), H (f; ) hold, then the problem (HVI) admits a solution. 4. Auxiliary results Now we will give some existence result for an elliptic hemivariational inequality, which will be useful in the sequel. Theorem 4.1. Let AQ : V → V be a pseudomonotone operator, which is weakly coercive in the following sense: there exist two constants .A ¿ 0 and 0A ¿ 0 such that Q vV ¿ .A v2V − 0A vV . Let fQ ∈ V and z ∈ V . Finally, let for all v ∈ V , we have Av; g and j satisfy hypotheses H (g) and H (j), respectively. Then for all h ¿ 0 such that h(h-g + .g ) ¡ .A =2(cHV )2 cj the following problem Find u ∈ V and %Q ∈ H; such that Q + h%Q = fQ in V (EHVI) Au %(x) Q ∈ @j(x; g((z + hu)(x); u(x))) for a:a: x ∈ admits a solution. Remark 4.2. Compare the above theorem with the result of Naniewicz and Panagiotopoulos [10, Theorem 4.25, p. 120]. In our formulation we use a stronger coercivity condition but we do not assume any sign condition (cf. [10, (4.3.27), p. 120]). Proof of Theorem 4.1. First let us introduce a multivalued operator G : V → 2H , deAned for all v ∈ V , by df
Gv = {/ ∈ H : /(x) ∈ @j(x; g((z + hv)(x); v(x))) for a:a: x ∈ }:
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Using hypotheses H (j)(iii) and H (g)(ii) for any v ∈ V and / ∈ Gv, we have 2 2 2cj2 (1 + |g((z + hv)(x); v(x))|2 ) d x /H = |/(x)| d x 6 6 2cj2
(1 + 2-g2 |z(x)|2 + 2(-g2 h2 + .g2 )|v(x)|2 ) d x = c1 + c2 v2H ;
df
(4.1)
df
where c1 = 2cj2 (| | + 2-g2 z2H ) and c2 = 4cj2 (-g2 h2 + .g2 ). Using operator G, we can rewrite the problem (EHVI) as follows: Find u ∈ V; such that Q Q + hGu f: Au
(4.2)
In order to obtain the existence of a solution for (4.2), we examine the range of the multivalued mapping AQ + hG. First, let us note that G has nonempty, weakly compact and convex values. Next we will check that operator G is generalized pseudomonotone. To this purpose, let us choose two sequences {vn }n¿1 ⊆ V and {/n }n¿1 ⊆ H , such that w−V
w−V
vn → v ∈ V , /n → / ∈ V , /n ∈ Gvn for n ¿ 1 and lim sup/n ; vn − vV 6 0. From the H compactness of the embedding V ⊆ H , we have that vn →v. So, from estimate (4.1), w−H Q Using the convergence passing to a subsequence if necessary, we get that /n → /. theorem of Aubin–Cellina (cf. e.g. [1, Theorem 7.2.2, p. 273]), we can easily show that /Q ∈ Gu. From the uniqueness of the weak limit in V , we know that / = /Q and so also that / ∈ Gv. Finally, we also have /n ; vn V = (/n ; vn )H → (/; v)H = /; vV : Thus G is generalized pseudomonotone. From Theorem 2.1, we get that G is pseudomonotone and from Theorem 2.3, we have that AQ + hG is also pseudomonotone. To verify the coercivity of AQ + hG, let us choose any v ∈ V and / ∈ Gv. Using the coercivity of A and (4.1), we have Q + h/; vV ¿ .A v2V − 0A vV + h(/; v)H Av √ √ ¿ .A v2V − 0A vV − h( c1 + c2 vH )vH √ ¿ (.A − 2hcj (cHV )2 (h-g + .g ))v2V − (0A + h c1 cHV )vV : Q From the choice of h ¿ 0, we get that operator A+hG is coercive. Now using Theorem 2.4, we also see that operator AQ + hG is surjective and so there exists u ∈ V , such that Q Q + hGu f. Au 5. The proof of the main theorem Proof of Theorem 3.1. First let us introduce the following operator: t df (Ku)(t) = u(s) ds + 0 : 0
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Using the operator K, we can rewrite the problem (HVI) as the following problem (P): Find u ∈ Y; and % ∈ H such that u (t) + w(t) + A(u(t)) + B(z(t)) + %(t) = f(t) u(0) =
1
in
in V ;
(5.1)
;
w(t; x) = W[z(·; x); x](t)
(5.2) for a:a: (t; x) ∈ (0; T ) × ;
%(t; x) ∈ @j(x; g(z(t; x); u(t; x)))
(5.3)
∀t ∈ [0; T ]; for a:a: x ∈ ;
(5.4)
z = Ku:
(5.5)
Now, u is a solution of (P) if and only if Ku is a solution of (HVI). For problem (P), we will use time-discretization method (cf. [2, Chapter 3.3; 15,9]). For any m ∈ N, let h = T=m. We deAne 1 ih fhi (x) = f(t; x) dt ∀i = 1; : : : ; m; (5.6) h (i−1)h uh0 (x) =
1 (x);
(5.7)
zh0 (x) =
0 (x):
(5.8)
We formulate the following semilinear problem (Ph ): Find uhi ∈ V and whi ; %hi ∈ H (for i = 1; : : : ; m); such that uhi − uhi−1 + whi + Auhi + Bzhi + %hi = fhi h whi (x) = Wf (zh0 (x); zh1 (x); : : : ; zhi (x); x) %hi (x) ∈ @j(x; g(zhi (x); uhi (x)))
in V ;
(5.9)
for a:a: x ∈ ;
(5.10)
for a:a: x ∈ ;
(5.11)
zhi = zhi−1 + huhi :
(5.12)
In order to solve problem (Ph ), let us assume that we have already solve it for k = 1; : : : ; i − 1, so we already know functions uh0 ; uh1 ; : : : ; uhi−1 ∈ V (and thus also zh0 ; zh1 ; : : : ; zhi−1 ∈ V ) and let us solve it for k = i. We rewrite problem (Ph ) as follows: Find uhi ∈ V and whi ; %hi ∈ H; such that hAuhi + h2 Buhi + uhi + hwhi + h%hi = hfhi + uhi−1 − hBzhi−1
in V ;
(5.13)
whi (x) = Wf (zh0 (x); zh1 (x); : : : ; zhi−1 (x); (zhi−1 + huhi )(x); x) for a:a: x ∈ ; %hi (x) ∈ @j(x; g((zhi−1 + huhi )(x); uhi (x)))
(5.14) for a:a: x ∈ :
(5.15)
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Let us introduce the following operators and functions: df AQ = A1 + A2 ; df
A1 v = hAv + h2 Bv + v; df
(A2 v)(x) ≡ b(x; v(x)) = hWf (zh0 (x); zh1 (x); : : : ; zhi−1 (x); (zhi−1 + hv)(x); x); df fQ = hfhi + uhi−1 − hBzhi−1 ∈ V :
Now we can rewrite problem (5.13)–(5.15) as follows: Find uhi ∈ V and %hi ∈ H; such that Q hi + h%hi = fQ Au
in V ;
(5.16)
%hi (x) ∈ @j(x; g((zhi−1 + huhi )(x); uhi (x)))
for a:a: x ∈ :
(5.17)
To solve problem (5.16)–(5.17), we will apply Theorem 4.1. First, we assume that h 6 1=0A (or h is arbitrary if 0A = 0), so that A1 is linear and from hypotheses H (A)(iii) and H (B)(ii), it is coercive in the following sense: A1 v; vV ¿ h.A v2V
∀v ∈ V
(5.18)
and thus A1 is maximal monotone. As A1 is everywhere deAned, using Theorem 2.2, we get that A1 is pseudomonotone. From hypotheses H (W), we see that b : × R → R is a Caratheodory function satisfying the following growth condition: |b(x; )| 6 haW (x) + h2 cW | |
∀(x; ) ∈
× R;
(5.19)
df
with aW ∈ H , namely aW (x) = |aW (x)| + cW max06k6i−1 |zhk (x)|. Thus A2 is a continuous and bounded Nemytski operator from H to H (see e.g. [16, Proposition 26.7, p. 563]) and so also a pseudomonotone operator from V to V . From Theorem 2.3, we get that AQ is pseudomonotone. From (5.18) and (5.19), for any v ∈ V , we get Q vV ¿ h.A v2V − A2 vV vV Av; ¿ h.A v2V − (haW V + h2 cW vV )vV ¿ h(.A − h(cHV )2 cW )v2V − hcHV aW H vV ; so if h ¡ .A =((cHV )2 cW ), then we have that AQ is coercive in the sense of Theorem 4.1, with .A = h(.A − h(cHV )2 cW ). Because of hypothesis H (A)(iii), we can choose h ¿ 0 so small, that .A =2(cHV )2 cj = h(.A − h(cHV )2 cW )=2(cHV )2 cj ¿ h.g . Thus, decreasing again h ¿ 0, if necessary, we may assume that h(h-g + .g ) ¡ .A =(2(cHV )2 cj ). Applying Theorem 4.1, we obtain uhi ∈ V , a solution of (5.16)–(5.17).
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Next we obtain some a priori estimate for the solutions of (5.9)–(5.12). We multiply (5.9) by uhi − uhi−1 and sum it from i = 1 to l (with 1 6 l 6 m). Then we estimate separately each term. First, we have
l l uhi − uhi−1 i 1 i i−1 = uh − uhi−1 2H ; uh − uh h h i=1
i=1
H
1 ¿ T ¿
l
2 (uhi H
−
i=1
uhi−1 H )
1 l 2 1 uh H − 1 2H : 2T T
(5.20)
From hypothesis H (W)(ii) and (5.10), we get |whi (x)| 6 a (x) + cW h
i
|uhk (x)|
for a:a: x ∈ ;
(5.21)
k=1 df
with a ∈ H , namely a (x) = |aW (x)| + cW | 0 (x)|. Using the Cauchy–Schwarz inequality, the Young inequality and estimate (5.21), we have
1=2 l
1=2 l l 1 i i−1 i−1 2 i i i 2 (wh ; uh − uh )H ¿ − h wh H uh − uh H h i=1
i=1
¿−
¿−
i=1
l
l
i=1
i=1
h i 2 8 i wh H − uh − uhi−1 2H 48 h l i 2 Th2 cW T a 2H − uhk 2H 28 28 i=1 k=1
l
−
8 i uh − uhi−1 2H : h i=1
Because of l
1 l 2 1 1 i uh − uhi−1 2H uh H − 1 2H + 2 2 2 i=1
l
=
1 (uhi 2H − uhi−1 2H + uhi − uhi−1 2H ) 2 i=1
l (uhi − uhi−1 ; uhi )H = i=1
(5.22)
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333
and the Young inequality, we get l
1 i 1 l 2 uh − uhi−1 2H uh H + 2 2 i=1
6
l
l
i=1
i=1
8 i h i 2 1 1 2H + uh H + uh − uhi−1 2H : 48 h 2
Using hypotheses H (A) and the last inequality, we have l i=1
Auhi ; uhi − uhi−1 V l
=
1 (Auhi ; uhi V − Auhi−1 ; uhi−1 V + A(uhi − uhi−1 ); uhi − uhi−1 V ) 2 i=1
l
1 1 1 A(uhi − uhi−1 ); uhi − uhi−1 V = Auhl ; uhl V − Auh0 ; uh0 V + 2 2 2 i=1
¿
l 1 .A i .A l 2 uh V + uh − uhi−1 2V − A 1 ; 2 2 2
1 V
i=1
l 0A l 2 0A i − uh H − uh − uhi−1 2H 2 2 i=1
l .A l 2 .A i -A ¿ uh V + uh − uhi−1 2V − 1 2V 2 2 2 i=1
−
l
l
i=1
i=1
0A 0A h i 2 0A 8 i uh H − uh − uhi−1 2H : 1 2H − 2 48 h
(5.23)
Next, from (5.12), hypotheses H (B) and the Young inequality, we have l i=1
Bzhi ; uhi − uhi−1 V
=Bzh1 ; uh1 −
1 V
+
l i=2
(Bzhi ; uhi V − Bzhi ; uhi−1 V )
l l (Bzhi ; uhi V − Bzhi−1 ; uhi−1 V ) − h Buhi ; uhi−1 V + Bzh1 ; uh1 − = i=2
i=2
1 V
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=Bzhl ; uhl V − Bzh1 ;
1 V
−h
l i=2
=B 0 ; uhl −
1 V
+h
l i=1
Buhi ; uhi−1 V
Buhi ; uhl − uhi−1 V
¿− -B 0 V (uhl V + 1 V ) − h-B
l i=1
uhi V (uhl V + uhi−1 V )
-B -B 0 2V − -B 8uhl 2V − ( 0 2V + 1 2V ) 2 48 l l h-B 1 i 2 l 2 uh V + 8uh V − (uhi 2V + uhi−1 2V ) −h-B 48 2
¿−
i=1
i=1
l−1
h h h-B (1 + 48) i 2 + ¿− uh V − -B 8(1 + T ) + 48 2 48 −-B
1 1 + 48 2
uhl 2V
i=1
0 2V −
-B (1 + h) 1 2V : 2
(5.24)
As zhi 2H 6 2 0 2H + 2Th
i
uhk 2H
∀i = 1; : : : ; m;
k=1
so using the Cauchy–Schwartz inequality, the Young inequality, (5.11), hypotheses H (j)(iii) and H (g)(ii), we have l i=1
(%hi ; uhi − uhi−1 )H ¿ −
¿−
l
l
i=1
i=1
h i 2 8 i %h H − uh − uhi−1 2H 48 h l cj2 h (1 + 2-g zhi 2H + 2.g uhi 2H ) 28 i=1
l
−
8 i uh − uhi−1 2H h i=1
¿−
l i 2cj2 -g Th2 cj2 T uhk 2H (1 + 4-g 0 2H ) − 28 8 i=1 k=1
−
l l cj2 .g h 8 i uhi 2H − uh − uhi−1 2H : 8 h i=1
i=1
(5.25)
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335
Using the Cauchy–Schwartz inequality, the Young inequality and (5.6), we obtain l
(fhi ; uhi
−
i=1
l
8 i 1 uh − uhi−1 2H : 6 f2H + 48 h
uhi−1 )H
(5.26)
i=1
Putting together all the above estimates (5.20), (5.22)–(5.26), we get .A hc4 h 2 c6 − 8c3 − − hc5 − uhl 2V 2 8 8 +
l l uhi − uhi−1 2H .A i uh − uhi−1 2V + (1 − 38 − 0A 8) 2 h i=1
i=1
hc6 c10 c8 + hc9 + h -B + + 6 c7 + 8 8 8
l−1 i l−1 h 2 c6 k 2 × uh V uhi 2V + 8 i=1
(5.27)
i=1 k=1
df -B df 1 V 2 2 2 , c6 = 2 T (cH ) (cW + df df c7 = 12 (-A + -B ) 1 2V + 02A 1 2H + -2B 0 2V , c8 = 14 (2T a 2H + -B 0 2V + df df B + 4-g 0 2H ) + f2H ), c9 = 1 2V , c10 = 14 (0A (cHV )2 + -B ) + cj2 .g (cHV )2 . df
df
with c3 = -B (1 + T ), c4 = 14 (-B + (cHV )2 (0A + 4cj2 .g )), c5 = 4cj2 -g ), 2cj2 T (1
2 First, let us choose 0 ¡ 8 ¡ min{1=(3+0A ); .A =(2c3 )}. Then, we see that there exists h0 ¿ 0 small enough, such that for all 0 ¡ h ¡ h0 , the constant in front of uhl 2V is positive. Applying Theorem 6.1, we obtain max uhi 2V 6 const
16i6m
for 0 ¡ h 6 h0 :
(5.28)
From (5.27), we have m i=1
uhi − uhi−1 2V 6 const
m uhi − uhi−1 2H 6 const h
for 0 ¡ h 6 h0 ;
for 0 ¡ h 6 h0 :
(5.29)
(5.30)
i=1
As zhi V 6 0 V + h
i
k=1
max zhi V 6 const
16i6m
uhk V , so from (5.28), we have for 0 ¡ h 6 h0 :
(5.31)
From (5.12) and (5.28), we see that zhi − zhi−1 V 6 const 16i6m h max
for 0 ¡ h 6 h0
(5.32)
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L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
and m i=1
zhi − zhi−1 V 6 const
for 0 ¡ h 6 h0 :
(5.33)
From (5.11), hypotheses H (j)(iii), H (g)(ii), (5.28) and (5.31), we have max %hi H 6 const
16i6m
for 0 ¡ h 6 h0
(5.34)
and from (5.21) and (5.28), we obtain max whi H 6 const
16i6m
for 0 ¡ h 6 h0 :
(5.35)
For s ∈ (0; 1] and i = 1; : : : ; m, let us deAne the following interpolates: df
uh ((i − 1 + s)h; x) = (1 − s)uhi−1 (x) + suhi (x); df
uh ((i − 1 + s)h; x) = uhi (x); df
zh ((i − 1 + s)h; x) = (1 − s)zhi−1 (x) + szhi (x); df
zh ((i − 1 + s)h; x) = zhi (x); df
wh ((i − 1 + s)h; x) = whi (x); df
%h ((i − 1 + s)h; x) = %hi (x); df
fh ((i − 1 + s)h; x) = fhi (x): Using these deAnitions, we can rewrite the problem (Ph ), as follows: uh (t) + wh (t) + Auh (t) + Bzh (t) + %h (t) = fh (t) wh (t; x) = W[zh (·; x); x](t)
in V ∀t ∈ (0; T );
for a:a: (t; x) ∈ (0; T ) × ;
%h (t; x) ∈ @j(x; g(zh (t; x); uh (t; x)))
for a:a: (t; x) ∈ (0; T ) × :
(5.36) (5.37) (5.38)
From (5.28)–(5.34), we know that all uh L2 (0; T ;H ) ;
uh L∞ (0; T ;V ) ;
zh L∞ (0; T ;V ) ;
wh L∞ (0; T ;H ) ;
uh L∞ (0; T ;V ) ;
zh W 1; ∞ (0; T ;V ) ;
%h L∞ (0; T ;H )
are bounded, for 0 ¡ h ¡ h0 . Thus, we can choose subsequences and limit functions, such that uh → u
weakly in H 1 (0; T ; H ) and weakly∗ in L∞ (0; T ; V );
(5.39)
uh → uQ
weakly∗ in L∞ (0; T ; V );
(5.40)
zh → z
weakly in H 1 (0; T ; V ) and weakly∗ in W 1; ∞ (0; T ; V );
(5.41)
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zh → zQ
337
weakly∗ in L∞ (0; T ; V );
(5.42)
weakly∗ in L∞ (0; T ; H );
wh → wQ
(5.43)
weakly∗ in L∞ (0; T ; H ):
%h → %Q
(5.44)
From (5.29), we deduce that uh − uh 2L2 (0; T ;H ) =
m i=1
ih
(i−1)h
i−
t 2 i uh − uhi−1 2H dt h
m h i = u − uhi−1 2H 6 hconst 3 h i=1
and so from (5.39) and (5.40), we see that u = u. Q In the same manner, we obtain that z = z. Q As zh = uh , so also Ku = z and thus (5.5) holds. Passing to the limit in (5.36) we obtain (5.1). From (5.39) and the fact that uh (0) = 1 , we conclude that (5.2) holds. From the compactness of the embedding Y ⊆ L2 ( ; C([0; T ])) (see e.g. [2, p. 129; 15, p. 265]] and from (5.41), we get zh → z
in L2 ( ; C([0; T ]))
(5.45)
and consequently zh (·; x) → z(·; x)
uniformly in [0; T ] for a:a x ∈ :
(5.46)
Let us deAne df
wh∗ (t; x) = W[zh (·; x); x](t); df
w∗ (t; x) = W[z(·; x); x](t); df
:h (x) =
2 sup |wh∗ (t; x) − w∗ (t; x)| ;
06t6T
2 2 df ;h (x) = 2 aW (x) + cW sup |zh (t; x)| + 2 aW (x) + cW sup |z(t; x)| ; df
;(x) = 4 aW (x) + cW
06t6T
06t6T
2 sup |z(t; x)| :
06t6T
From H (W), we obtain sup |wh∗ (t; x)| 6 aW (x) + cW sup |zh (t; x)|
06t6T
06t6T
sup |w∗ (t; x)| 6 aW (x) + cW sup |z(t; x)|
06t6T
06t6T
for a:a: x ∈ ; for a:a: x ∈
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and thus |:h (x)| 6 |;h (x)|
for a:a: x ∈
∀0 ¡ h 6 h0 :
(5.47)
From (5.46), we see that ;h (x) → ;(x)
for a:a: x ∈
and from (5.45), we also have ;h (x) d x → ;(x) d x
(5.48) as h → 0+ :
(5.49)
From hypothesis H (W)(i) and (5.46), we have wh∗ (·; x) → w∗ (·; x);
uniformly in [0; T ] for a:a: x ∈
and thus :h (x) → 0
for a:a: x ∈ :
(5.50)
Properties (5.47)–(5.50) allow us to use the generalized majorized convergence theorem (see [16, Appendix, Theorem 19a, p. 1015]) and obtain that lim+ :h (x) d x = lim+ :h (x) d x; h→0
h→0
what in the connection with (5.50), gives that :h (x) d x → 0; thus wh∗ → w∗
in L2 ( ; C([0; T ])):
As, for t = ih, with i = 1; : : : ; m, we have wh (t; x) = wh∗ (t; x) = W[uh (·; x); x](t); so wh is the piecewise constant interpolate of wh∗ and sup |wh (t; x)| 6 aW (x) + cW sup |zh (t; x)|; 06t6T
06t6T
wh → w ∗ ;
for a:a: x ∈
uniformly in [0; T ] for a:a: x ∈ :
Arguing in the same way as before, we obtain that wh → w ∗
in L2 ( ; C([0; T ]))
and from (5.43) we have that wQ = w∗ , so (5.3) holds. Finally from (5.40), (5.42), (5.44) and hypothesis H (g)(i), we have that g(zh (t; x); uh (t; x)) → g(z(t; Q x); u(t; Q x)) %h → %Q
for a:a: (t; x) ∈ (0; T ) ×
weakly in L1 ((0; T ) × );
so from (5.38), using the so-called convergence theorem of Aubin and Cellina (cf. e.g. [1, Theorem 7.2.2, p. 273]), we get (5.4).
L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
339
Remark 5.1. Note that if we assume the dependence of W on the time derivative u of the unknown function u (and not on the function u itself) then the argumentation will be similar (in place of hypothesis H (W)(iii), we will need that Wf ( 0 (·); ·) ∈ H ). Remark 5.2. Note that if the function g depends only on the Arst variable (i.e. there is no dependence of @j on the derivative u ), then the additional assumption on the coercivity constant .A in hypothesis H (A)(iii) is no more needed (i.e. it is enough to assume that BW ¿ 0). Remark 5.3. It seems to be interesting to have a similar result to this of Theorem 3.1, but with a noncoercive operator A (see [7] for analogous hemivariational inequality) and in particular for wave-type hemivariational inequality with hysteresis (i.e. with A = 0; see [6]). Remark 5.4. Another interesting generalization of problem (HVI) would be one with a nonlinear operator A. Appendix We formulate and proof a generalized discrete version of the Gronwall inequality (see e.g. [11] for a continuous version). Theorem 6.1. Let {ak }k¿1 ⊆ R be a sequence such that ak ¿ 0 for all k ¿ 1 and there exist numbers A; B; C; h ¿ 0, such that ak 6 A + hB
k−1
ai + h 2 C
i=1
k−1 i
aj
∀k ¿ 1:
(A.1)
i=1 j=1
Then ak 6 Aeh(k−1)B+h
2
(k−1)2 C
∀k ¿ 1:
(A.2)
Proof. Case 1: A = 1. Let us denote the right-hand side of (A.1) by Rk . For any k ¿ 2, we have ak 6 Rk = 1 + hB
k−1
ai + h 2 C
k−1 i
i=1
= Rk−1 + hBak−1 + h2 C
aj
i=1 j=1 k−1
aj
j=1
6 Rk−1 + hBRk−1 + h2 (k − 1)C max aj j=1;:::; k−1
6 Rk−1 + hBRk−1 + h2 (k − 1)CRk−1 = (1 + hB + h2 (k − 1)C)Rk−1 :
340
L. Gasi+nski / Nonlinear Analysis 57 (2004) 323 – 340
As R1 = 1, we obtain ak 6 (1 + hB + h2 (k − 1)C)k−1 =[(1 + hB + h2 (k − 1)C)1=(hB+h
2
(k−1)C) (k−1)(hB+h2 (k−1)C)
]
Exploiting the estimate (1 + x)1=x 6 e for all x ¿ 0, we obtain ak 6 eh(k−1)B+h
2
(k−1)2 C
∀k ¿ 1
and this Anishes the proof in Case 1. Case 2: A ¿ 0. We divide (A.1) by A and obtain that k−1
k−1
i=1
i=1 j=1
i
ai aj ak + h2 C 6 1 + hB A A A
∀k ¿ 1:
Let us deAne bk =ak =A for k ¿ 1. Applying Case 1 to the sequence {bk }k¿1 , we obtain that bk 6 eh(k−1)B+h
2
(k−1)2 C
∀k ¿ 1;
so (A.2) holds. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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