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Nonlinear functional variational inequalities governed by time-dependent subdifferentials
Nonlinear Analysrs, Theory, Methods Printed in Great Britain.
& Applications.
NONLINEAR GOVERNED
Vol. 17, No. 9, pp. 863.883,
1991. 0
0362-546X/91 $3.00+ .OO 1991 Pergamon Press plc
FUNCTIONAL VARIATIONAL INEQUALITIES BY TIME-DEPENDENT SUBDIFFERENTIALS
NOBUYUKI KENMOCHI Department of Mathematics, Faculty of Education, Chiba University, Yayoi-cho, Chiba, 260 Japan
and TETSUYA KOYAMA Department of Mathematics, Hiroshima Institute of Technology, Miyake, Saiki-ku, Hiroshima, 731-51 Japan (Received Key words and phrases:
1 August
1990; received for publication
20 February
Functional variational inequalities, time-dependent
1991)
subdifferentials,
existence
and uniqueness.
INTRODUCTION THIS PAPER is concerned
equation
(CR uo)
of the following
with the abstract Cauchy problem (CP; u,) for a functional form in a real Hilbert space H:
u’(t) +
@&W)
3
G(u)(t),
differential
Oct
on I-Q, 01
u = ug
where u’(t) = (d/dt)u(t), {@JtZo is a family of proper lower semicontinuous (1.s.c.) and convex functionals I# on H, dp,’ denotes the subdifferential of p’, u,, is a given function in C([-r,, 01; H) and G is a mapping from a subset of C([-T,, T]; H) into L2(0, T; H). We regard G as an abstract model of functionals with nonlinear memory. The aim of this paper is to establish a local existence theorem under some boundedness and demicontinuity condition of G, and prove the global existence and uniqueness of solutions under further restrictions on G. Functional differential equations of the same type have been studied by many authors [l-5] in order to solve nonlinear integral equations of the Volterra type. Also, for related works we quote here some results from [6], [7] and [8]. For the existence and uniqueness of solutions to (CP; ZQ), one of the simple sufficient conditions on G might be of the form:
for any U, u E C([-r,, T]; H) and 0 < t 5 T, where y E L’(0, T). However, this is too restrictive to apply (0; u,) to some physical problems such as heat conduction with timedelay. Recently, Mitidieri and Vrabie [5] gave an existence result for (CP; u,) under general conditions on G, in case I$ is time-independent. We shall generalize this result to the case of time-dependent pf, making use of results in [9]. Our abstract results are applied, for instance, 863
N. KENMOCHIand T. KOYAMA
864
to the following
functional
variational
u, - (lu, lp%,
inequality
with memory: in Q = (0, T) x (0, l),
2 G(u)(t, X)
in Q,
I.42 g(l, x)
(u, - (~GI~-~~,),
in Q,
- G(u))(u - 8) = 0
u(t, x) = 240 where G is a functional
in (--TV, 01 x [O, 11,
given by
f G(u)(t, 4 =
f
fo tt, x, s, e, i -70
4, a,
xl) d.s +
J-1(t, xv s, N, xl, m, .a%(& s -70
xl cl%
and g and u0 are given functions on Q and (-r,, 0] x [0, 11, respectively. Also, in a final section of this paper, we will discuss some applications to the same type of functional variational inequalities involving the so-called hysteresis operators (cf. [lo]) as a nonlinear memory term G(u). Notations. For a (real) Banach space V, 1-Iv means the norm C([a, b]; V), LP(a, 6; V), etc., mean the usual function spaces.
of
V, and
the notations
Throughout this paper, we assume that H is a real Hilbert space, with inner product (* , a) and norm 1. (, and use the notion of subdifferentials of convex functionals on H with some basic properties of subdifferentials without any explanation; we refer them to [l I]. 1. MAIN
RESULTS
Let r0 2 0 and r, > 0 be given two numbers, and let (pf; -tO 5 t 5 To] be a family of proper 1.s.c. and convex functionals on H. Also, let G be an operator from D(G) := U,,
(CR
uo>
u’(t) +
@%40) 3 G(u)(t),
0
u(t) = u,(f),
< t < T(s T,),
-50 5 t 5 0,
where &D’ denotes the subdifferential of 9’ and u,, is a given function in C([-r,, function u E C([-s,, T]; H) is called a solution of (CP; zq,) on [0, T], if u E W1*2(0, T; H), G(u) E L’(O, T; H),
G(u)(t)
- u’(t) E %f(W))
and u(t) = 4(f)
for --to 5 t 5 0.
for a.e. t E [0, T]
01; H). A
Nonlinear functional variational inequalities
865
We note here that if r, = 0, then in the above formulation of (CP; u,), condition u(t) = u,(t) for -rO I t I 0 is replaced by the usual condition u(0) = u0 for a given u,, E H. Our problem (CP; u,) is discussed under the following assumptions (Al), (~2) and (Gl)-(G3). (~1). For each r > 0, there are functions (Y,E W’*‘(O, &) andp, E W”‘( 0, T,) for which the following condition holds.
Assumption
For each s, t E [0, T,] with s 5 t and each z E D($) with [z( 5 r, there is Z E D($),
Condition.
such that le - ZI 5 la,(t) - 44I(f
+ l~“(Z)Y2),
and V’(a - V”(Z)5 IA(t) - P&)l(l Assumption in H.
+ lV”(Z)l).
(~72). For each t E [0, T,] and R > 0, the set (z E H; &z> 5 R, 1.~15 R] is compact
Assumption (Gl). For every T E (0, To], X(T) contains the set X,,(t) := (u E W’*2(-t,,, T; H); p”‘(u) E L”(-t,, T) and there exists u* E L2(-T,,. T; H) such that u*(t) E &&u(t)) for a.e.
TII.
t E I-Q,
(G2). (Boundedness.) There are a constant v E (0, l), a nondecreasing I: R: + R, and a function 1, E L’(0, T,) such that
Assumption
\G(u)(t)12 5 vlu*(t)12 + &&+ro,r~;wj, for
a.e. t E [0, T], whenever u*(t) E &&u(t)) for a.e. t E [-zo,
0 < Ts
To,
~~%4~~-~-r,,r~r lu*l,+a,l;~j) u E X,(T)
and
function
+ [o(t)
U* E L2(-ro, T; H)
with
T].
Assumption (G3). (Continuity.) For every T E (0, To], G is demicontinuous as a mapping from X,(T) into L’(-ro, T; H) in the sense that if U, EX~(T), u, + u in C([-r,, T]; H) (as n -+ co), (uh) is bounded in L2(--ro, T; H), (q~“‘(u,)) is bounded in P(-rO, T) and there is a bounded sequence {u,*) in L2(--ro, T; H) with u;(t) E &&u,(t)) for a.e. t E [-r,,, T], then G(u,) + G(u) weakly in L2(0, T; H).
The main results in this paper are formulated
as follows.
1.1. Suppose that conditions (pl), (~2), (Gl)-(G3) hold. Let u. be any function in W’,2(-r,, 0; H) such that $“(u,) E L”(-T,, 0), u,(O) E II and there is u$ E L2(-ro, 0; H) satisfying z@(t) E ?Qr(uo(t)) for a.e. t E [-ro, 01. Then there exists a number T E (0, To] for which (CP; u,) has at least one solution on [O, T].
THEOREM
Concerning following.
global existence, i.e. existence on the whole interval
[0, To], we prove the
866
N. KENMOCHIand T. KOYAMA
THEOREM 1.2. In addition to the assumptions of theorem 1.1, suppose that the functions CX, and p, in condition (cpl) are independent of r, i.e.
(Y,=: CY
and
P, =:P
for any r > 0, and that in condition (G2), I(.) is of the form U19 r2 9<3) = c,crf + r2 + r: + 1) for a nonnegative constant C, . Then (CP; uO) has a solution on [0, r,]. 1.1. The above theorems include the case of r0 = 0. In this case, it is enough to assume for the initial value u0 E H that u0 E D(cp’).
Remark
These theorems will be proved in Sections 4 and 5. 2. SOME
EXAMPLES
In order to illustrate our main theorems, we give some examples of functional differential equations and variational inequalities of parabolic type in one-dimensional space. Example 2.1. We consider the following problem with flux condition of Signorini type with nonlinear memory:
Ut - %X = G(u)(t, x)
in Q := (0, T) x (0, l),
u(t, 0) L h(t)
for 0 5 t 5 T,
u*(t, 0) 5 0
for 0 < t < T,
u,(t, O)(u(t, 0) - h(t)) = 0
for 0 < t < T,
u(t, 1) = 0
for 0 zz t 5 T,
u = f&J
in [-ro,O]
(2.1)
x [0, 11.
Here we are given h E W’92(-ro, &) and u. E W1.2(-ro,0; W1V2(0, 1)) with uo(t,O) 1 h(t) and u,(t, 1) = 0 for any t E [-To, 01. The functional G is defined in the following way. Let f = f(t, x, s, Cl, r2, Cl, C2):[0, To] x [O, 11 x [-ro, To] x R x R x R x R + R be a function such that (fl) fis measurable in (t, x, s) for fixed (ri, (t2, cl, c2) and is continuous in ({i , r2, cl, c2) for a.e. fixed (t, x, s); (f2) bV,x9~~~l~52~ Cl.C2)i 5 po(t,x,s) + ~~(15~1, kll) + ~~(k~i, lC11)(k21 + lC21) where pLoE L2((0, To) x (0, 1) x (-to, T,)) and iu,: R, x R, -, R, , k = 1,2 are nondecreasing func-
tions with respect to all variables. Now, put X(T)
= W1P2(-.ro, T; L2(0, 1)) 0 L”(-to,
T; W’.2(0, 1))
and define G: U.
_I%, x9 s, w, 4, u,(t, xl, a, -To
4, ~,cc x)) dJ
(2.2)
867
Nonlinear functional variational inequalities
for u E X(T), 0 < T I To. Next consider functionals q~’on H = L’(O, 1) given by 4(z) =
the family
[pi; -r,
tlzxlE~(o,l, if +cO
5 t I T,) of proper
1.s.c. convex
z E K(t), (2.3)
otherwise,
where K(t) = {z E WlT2(0, 1); z(O) 1 h(t),z(l) = 0). Then we have the following facts: (a) Z* E a&z) if and only if z E K(t), z* E L2(0, l), z* = -z,, a.e. in (0, I), z,(O) I 0 and z,(O)(Z(O) - h(t)) = 0. Hence 8s’ is single-valued. (b) Conditions (91) and (~2) are satisfied. Proof
of(b).
Clearly
(~2) holds.
To verify (pl),
for any z E K(s), we take
Z(x) = z(x) - (1 - x)(h(.s) - h(t)) E K(t). We then see by elementary
computations a,(t)
that (~1) holds with
= P,(t) = const.
* lh’(z)l dr, 0
which is independent (c) Conditions Proof
of(c).
of r.
(Gl)-(G3) Trivially,
I&([-
(Gl) is satisfied.
Next, by (f2) and (2.3), we have
C0h0(0l~~((0,1) X(-so,To)) + Pl(I&~[-sO,t] X[O,l]) 3 I&([-~,,*, x [0,1])12 (2.4) 14C(]-ro,t] x [0,1])~21~~l~-~-~~,t;~~~o,1))~ + L12(14c([-r,,t]x[0,1])~
IG(u)(t)l &0,1) 5
where Co is a positive
are satisfied.
constant.
so,t]X[O,l])
5
Cl
Since
+ IUlc([-~o,tl;LZ~o,l))) (2.5) lulC(I-~O,tl;~*(o,l))~l~Xl~-m(-~O,t;~~(o,l))
(2.4) yields (G2). for any ueX(T) and O< TI To, where C, is a positive constant, Finally, to verify (G3), let u, E X(T), n = 1,2, . . ., and u E X(T) such that U, + u in weakly in L2(-ro, T; L2(0, l)), (u,,,] is bounded in u,,t --+ ut C([-- 70, 7’1; L’(O, I)), T; L2(0, 1)). Then, by the compactness L2(-ro, T; L’(O, 1)) and ]u,,,) is bounded in L m(-70, theorem, we have in L2(-ro, T; L2!0, l)), %I*, + UX and u, + u in C([-7o, Tl x 10, 11) L2(-7,, T, L2(0, 1)). On account of (a), problem (CP; u,) is a weak formulation (f2) the function ,~r is of the form
so that (fl) implies G(u,) + G(u) in
of (2.1). Moreover,
if in condition
~~(15~1~ kll) = c0nst.U + l&l + 15h1 and if ,u2 is a nonnegative constant, then G, given by (2.2), satisfies 1.2 for global existence of solution.
the assumption
of theorem
N. KENMOCHI and T. KOYAMA
868
Example
2.2. We consider
u, - (Iu,]~-~u,),
an obstacle
problem
of the following
form: O
in Q = (0, T) x (0, l),
2 G(u)(l,x)
24 2 g(t, x)
in Q
(u, - (]u,I~-~U,
- G(u))(u
- g) = 0
in
Q,
(2.6)
u(t,x) = 0
for 0 5 t 5 T,
U = Ug
in [-rO, 0] x [0, 11.
x = 0, 1,
Here, p 2 2, g E W1r2(-q,, To; W’,“(O, 1)) and u0 E W1,2(-r,, 0; WISp(O, 1)) such that g(t, x) = 0 for -q, 5 t 5 To, x = 0, 1, u0 2 g in [-rO, 0] x [0, l] and u,(t, x) = 0 for -rO 5 t I To, x = 0, 1. Assume that G is the functional defined by
f fo(t, x, s, u(t,x), MS,x))ds +
G(u)(t, X) =
.i -70
t fi(t 9x, s, u(t,x)94% X))&(&x) ds .i-nl (2.7)
for any u E X(T), X(T)
and fk (fl)’
0 < T 5 To. Here
= W’+t,,,
T: L2(0, 1)) fl L”(-r,,
T; W1yp(O, l)),
O<
= fk(t,x, s, r, [), k = 1,2, satisfy conditions. fk is measurable in (t, x, s) for fixed ([, [) and is continuous
Ts
To,
in (r, 4) for a.e. fixed
(t, X, s);
Ifi(t~x~~~r,i)I (iiio(t,s) +iiildtIJcl), for a.e. (t, x, s) and for all r and [, where ,u, E L2((0, To) x (0, 1) x (-r,,, To)), ,D,, E L2((0, &) x functions with respect to any T,)), and pl, PI : R, x R, + R, are nondecreasing variable. Also, the family 1~‘; -rO I t I To] of proper 1.s.c. convex functionals on H = L2(0, 1) is given by (-Q,
w-9 otherwise, where K(t)
= (z E W,‘sp(O, 1); z 2 g(t, .) on [0, 11).
In this case it is easy to see that z* E @‘(z) if and only if z E K(t), z* E L2(0, l), 1 Izxlp-2zx~,
dx 2
i ,O ”
\ .0
1 z*u 10
dx
for any nonnegative
1 z*cz
-
g(t, *)I dx =
1l Izxlp-2zx(zx - g,(t,
! ,O
.)) dx.
q E W,‘,p(O, l),
(2.9)
Nonlinear functional variational inequalities
869
Therefore (CP; z+,) is a weak formulation of (2.6), and we see that conditions (eel), (~72), (Cl) and (G2) are similarly checked as in the first example. Condition (G3) is seen from the fact that G(u,) --t G(u) weakly in L2(0, T; L2(0, l)), if u,, u E X(T), (u,,,) is bounded in T; Lp(O, l)), u, -+ u in C([-s,,, T] x [0, 11) and u,,, + U, weakly in L2( -rO, T, Lp(O, 1)). L”(-r0, Moreover, the operator G satisfies the condition for global existence of a solution in theorem 1.2, in case ,D, and jir are of the forms ~i(l
Ill) = const.0
,f&(j
ltlp’2 + lClp’2)T
+
const.(l
+ l@+“‘”
+ lp-2’2).
in this case we have, by (2.5),
/G(W)/
&O,l) 5
for any u E X(T),
Co[Iruo(t)lZzcc-,,To)x(O,l)) + liio@)l~~cTo,To) + IU*IP,m(-s,,r;rp~o,1)) + 11
0 < T I To and a.e. t E [0, T], where Co is a positive 3. AUXILIARY
ESTIMATES
In this section, let u. be the same function as in theorem f E L2(0, T; H) with 0 < T s &, we consider the problem u’(t) + @‘(W))
constant.
sf(t),
1.1. Then,
for each
O
-ro I t 5 0.
i u(t) = uo(t),
given
By some results in [9] (or [12]), this problem is uniquely solvable under condition (~1); in fact, there is a unique solution u of (3.1) such that u E W”2(0, T; H), u(0) = u,(O) and f(t) - u’(t) E &p’(u(t)) for a.e. t E [0, T]. In what follows, we assume all the assumptions of theorem 1.l and denote by U, the solution of (3.1). LEMMA 3.1. There
are constants
A, > 0 and A; > 0 such that (3.2)
and
lq(t)l foranyOSs5
ts
T,O<
TI
-(
l.qM + -4; + (t - s)
lf(412d7
(3.3)
T,andfEL2(0,T;H).
Proof. Let h be a function in W1,2(0, To; H) t E [0, To]. Then, for any 0 5 s 5 t IS T, +(qt)
.\s
- h(t)12 I guy(s)
- h(.s)12 +
such
that
h’(t) + @‘(h(t))
’ If(r)/ by(~) - MT)/ dT ss
and hence
If we take Al = 2b31ccIo,Tol;Hj and A; = Al + 1, then (3.2) and (3.3) hold.
H
3 0 for
a.e.
870
N. KENMOCHI and T. KOYAMA
COROLLARY3.1. There are constants
A2 > 0 and A$ > 0 depending
\&~~o,T~;H~
5 A, + i
and \+Ic(to,rl;~j
;\A7)\
d7
5 Ai + Z-.‘: If( /
(3.4)
d7
for any 0 < T 5 T, and f E L’(O, T; H). In fact, by lemma 3.1, we can take JuO(0)I + A, and lu,(O)j COROLLARY3.2. There are constants
only on (uo(0)( such that
(3.5)
+ A’, as AZ and A; respectively.
A, > 0 and A; > 0 such that t , s If(r)1 d7 + 1 i >
(3.6)
and
b’@~W)Ic: &q(t)) + A; lq(s)l + (t - s) for any 0 5 s 5 t 5 T, whenever Proof. Condition
((~1) implies
Ji
’If(
dr + 1 >
0 < T 5 T, and f E L2(0, T; H). (by [9; lemma
1.5.11) that there is a constant
0 > 0 satisfying
p’(z) + 0121 + 0 2 0 for any 0 I t 5 T, and z E H. Combining obtain the corollary. n For simplicity,
(3.7)
this inequality
(3.8)
with (3.2) and (3.3) in lemma
3.1, we
given R > 0 and 0 < T s: T,, we put KR(O, T) = tf E L2(0, T; H); If b<,,,T;u) 5 RI.
LEMMA 3.2. There are a constant R > 0 and 6 E (0, l] such that
A, > 0 and a function
r,rR,6 E rf(0,
T,) depending
only on
for any 6 E (0, 11, 0 5 E s t, 0 < T 5 T, and f E KR(O, T). Proof.
Let R > 0, 0 < T % T, and f E KR(O, T). Then,
5 r, := IulC~~O,Tl;H~ GivenO<
by corollary
A, + T,“2R.
1,weput '1RJC7) = f
/4(7)12 + ku7>l9
3.1, u := uf satisfies
Nonlinear functional variational inequalities
871
where (Y,and /3, are the functions with r = rl + 1 in condition (91). Now we recall the energy inequality for the solution u (cf. [9]): h4~N
- ce49)
+
(u’, u’ - f) dr
ft
rt
JS
JS
(3.9) for any 0 I s 5 t 5 T, where mi, i = 1,2, 3,4, are positive constants depending only on the constant cr of (3.8). Hence, for each given 0 < 6 I 1, &u(t))
” Id -f[‘dr
- q?@(s)) +
1s 16
‘1~’ -j-f12dr+ .i5
-
t IfI
Iu’ -j-l
I’s
dr
i’: 1; la:12 + iP:$a’(u)J
+ A,lul + A)dr
(3.10)
for any 0 I s 5 t 5 T, where A4 = 3mf + 3rni + m3 + m4. Here use the inequalities “f
!
IfI Id - fl dr I $1 - E)
(j-l2 dr + &
1’ 1~’ - j-I2 dr s
.s
5 +(l - E) ’ lf12dt ss
+ +(l + 2.5) t Id -f 12dt is
for 0 I E I 5. Then, from (3.10) we obtain the required inequality. Remark
3.1.
n
The energy inequality (3.9) is obtained by letting I 10 in the energy inequality
5 ’ tlail l~~~(u~>l(l~~(~~>l”’ + ml141’2+ md+ IP~l(ld(ud+ m3bXI + mJ1dr ss for
the solution ux of the approximate problem u:(t) + dq&(u,(t)) = f(t), 0 < t < T, where 89; is the Yosida-approximation of a$. See [9; Section 2.81 for the details.
u,(O) = u,(O),
4. LOCAL
EXISTENCE
OF SOLUTIONS
We use the same notation as in the previous section. The key of the proof of theorem 1.1 is the following lemma. LEMMA
4.1. There are T E (0, To] and R > 0 such that
G(q)
E
KR@ T)
for any f E KR(O, T).
(4.1)
N. KENMOCHI and T. KOYAMA
872
Proof. According to the result [9; theorem 2.6.11, for each R > 0 there is a constant M, > 0, depending only on R, such that l&[O,Wf)
5 MI7
l~jlL~@J.;H) 5 Mi?,
t6+"("f)tLm(o,~ 5
(4.2)
MR,
for every 0 < T 5 To and f E KR(0, T). Applying lemma 3.2 for s = 0 and t = T, we see that ‘T vT(q(T))
+
(+ -
6 -
E)
o
i
b;-f12dr 'T
I
$(I
-
E)R~
+
(hfR
+ x‘t4(kfR +
1))
i0
)lR,Sdt
+
(4.3)
ul'(Uo(o)).
By (3.7) of corollary 3.2, v~(+(T))
1 l~~(q(T))l
(4.4)
- A;]\uo(O)] + TR* + 11.
From (4.3) and (4.4) it follows that (4 - 6 - E) ~TIUj- fj’dt I0 T 5
+(I
-
& +
A;T)R*
+ A;(lu,(O)I
+ 1) + rii,
~R,?I~~
+
ul"(Uo(o)),
(4.5)
i0
where tiR = MR + A4(MR + 1). Besides, by condition (G2) ‘T iG(u f )I2 L'(O,T;H) 5
vb;
-
f I&O,T;H)
+
WJo
+ MRr
Jo + MI?, Jo +
MR
+
R)
j,dr
+
(4.6)
.0 1
where Jo := I~OIC([-,,OI;H)+
lv)(~)@o)ILy-To,o) + I~o*lL+~O,o;H)
for a function u,*(t) E &p’(uo(t)) for a.e. t E [-TV, 01. Therefore, we derive from (4.5) and (4.6) that for small 6 > 0 and E > 0, (f - 6 -
~)IG(~~)l&,T;H) T
5
;(I
-
&
+A;T)R*
+ vA;(lu,(O)l
+ 1) + d&
qR,Sdf.
+
"(t -
6 -
~)~"(~o(o))
0
s T
+
($ -
6 -
E)T/(J~ + it'fR,Jo
+ hfR, Jo + b’fR + R) + (3 - 6 - E)
I,
dr
0
so
that
+
TI(Jo + MR, Jo + MR, Jo + MR + R) +
I,dr + q”(uo(0)). 0
(4.7)
873
Nonlinear functional variational inequalities
Since 0 < Y < 1, it follows that v(1 - E + A; T’)/(l 6 > 0, E > 0 and T’ > 0. For such 6, E and T’, put 1 - 2v, := Clearly
- 26 - 2~) < 1 for sufficiently
small
~(1 - E + A;T’) .
1 - 26 - 2E
v1 > 0. Next, choose R > 0 sufficiently
~;(l~o(o)I + 1) + 1 - 26 - 2E
large so that p”(uo(0))
I v,R2
and then T E (0, T’] so that 2li&
T
1 - 26 - 2E s 0 Then,
T
rIR,&dt + Tl(Jo + MRr Jo + MR, Jo + MR + R) +
for such a couple
lo dr I v,R’. 0
{R, T), inequality
(4.7) implies
(4.1).
W
Proof of theorem 1.1. Let R > 0 and T E (0, To] be the numbers define a mapping S: KR(O, T) + KR(O, T) by putting Sf := G(uf)
found
in lemma
4.1, and
for every f E KR(O, T).
Then S is continuous in KR(O, T) with respect to the weak topology of L’(O, T; H). In fact, if f, E KR(O, T) and f, + f weakly in L2(0, T, H) as n + 00, then by the uniform estimate (4.2) in the proof of lemma 4.1 and by condition (~2) we see that u?l := z.4fn+ U := 24 f
u:, + u’
Tl; HI,
in C([-T~,
weakly in L2(-
T,
,
T; H)
and is bounded
W’(u n)]
in L”(-to,
T).
Therefore condition (G3) implies that G(u,) + G(u) weakly in tinuous in KR(O, T). Since KR(O, T) is a nonempty, weakly L2(0, T; H), it follows from the usual fixed point theorem that KR (0, T). It is clear that for any fixed point f of S in KR(O, on[O,T]. W 5. GLOBAL
EXISTENCE
L2(0, T; H) so S is weakly concompact and convex subset of S has at least one fixed point in T), uf is a solution of (CP; u,)
OF SOLUTIONS
In this section, suppose always that all of the assumptions use the same notations as in Section 3.
of theorem
1.2 are satisfied,
and
LEMMA 5.1.
Let u be any solution of (CP; u,) on [0, T] with 0 < T I To. Then there are A, > 0, A, > 0 and a function qs E L2(0, To) depending only on 6 E (0, l/2) such
constants that
l4wt))l +
1 - (t - s)A, - (t - Wh(l
+ [jr&)d7)
(t - ~)Iv%&-~s,~~
+ H,(s)
v + (t - s)A,
- ;]
[:kWi)l’dr
I 5
s
~cd~)~vW))~
ds + 4
+
cs
10(5) dr
1
(5.1)
N. KENMOCHI and T. KOYAMA
874
T, where
for any 0 I s I t I X(9
= I&-q#f)
+ I~*l~~(-,,,;lY)
+ Ie+“@)IP(-TO,s) + 19
and u*
Proof.
=
First we have, by lemma
I-70
9 01,
uo*
on
G(u) - u’
on (0, T].
3.2,
+ (+ - 6)
&u(t))
I +
’ IG(u)12d7 rs
+
‘q,#(u)l s
dr + A4
‘qa(lul is
+ 1)dt
+ cp”(u(s))
for any 0 5 s I t 5 T, where ~~(7) = (1/~5)la’(r)l~ + 1/3’(7)1 for functions p(7) := /3,(t) in condition (~1). Also by condition (G2),
(y(r) := q(7)
(5.2) and
is
for any 0 I s I t I
T. Besides,
by lemma
3.1 and its corollaries,
f A4 (t -
.rs
~(luj
+ 1)dr
I A4 (t - s) ’ IG(u)j2d7 i ‘I’s
+ lu(s)l + A; + 1
‘qsd7, s
(5.4)
wA14&-T0,r];ff) + l~*l3-,,,t;H)I 5
(t - s)k,
’ IG(u)12 dr + t 1~’ - Wd2d~ is (1 s
+ t&,,sl;~~
+ b4*1:2(-,o,s;~j + 1 > (5.5)
and ul’(W))
2
Id@u))l - (t -
4k2 (1;
kWl”d7)
for any 0 I .s I t I T, where k, (LC~), k2 are positive Substituting estimates (5.5) into (5.3), we obtain
(1 - (t - s)k,)
constants
-
k2(149)1
independent
’ IG(~)1~dr 1s
5 (v + (t - s)k,)
’ Id - G(u)12 dt + (t - s)k,b”“(u)b(s,t) s
+
1)
(5.6)
of S, t, T and U.
Nonlinear
and furthermore,
multiplying
functional
875
by (* - 6)/[v
+ (t - s)kJ,
we get
11 v + - (t - 2)/k, MI1 .i f ,G(u),2d 5 s
6
1 -2
5
’ Id G(u)l’dr
6
(5.2)
+ ~k,/~‘~‘(~)]~~~~,~~
+ &
‘!,,dr S
To + Id%41L-(-70,s) + 1. ,:Iq , + I~*IZ~(-,,,;H)
+ l&tTherefore, from A, > 0. n
inequalities
both sides of this inequality
( > ( >i -21
variational
with
(5.4)-(5.7)
we derive
(5.1)
for
some
constants
(5.7) A, > 0 and
Proof of theorem 1.2. Let u be a function from [0, T’) with 0 < T’ I TOinto H, and suppose that u is a solution of (CP; ZQ,) on [0, T] for any 0 < T < T’. Then we show that p”‘(u) E L”(0, T’) and G(U) E L’(O, T’; H). In fact, we do it in the following two cases. Case 1. If l&u(t))1
is ’ b ounded
as t t T’, then G(u) E L’(O, T’; H).
Proof. Since 0 < v < 1, we see that the coefficient of j: 1G(u) 1’ dr in (5.1) is positive if 6 > 0 is sufficiently small, and s and t are sufficiently close to T’. Therefore, (5.1) implies that G(u) E L2(0, T’; H). Case 2. If lfp’(u(t))( is unbounded Proof. By assumption, 6 such that
1
there are a sequence
(t,) with t, t T’ as n + ~0 and a positive
number
I~‘“W,NI = Id”(U)IL-(-T0, fn) 9 1 - (T’ - t&4,
(--> 2
and
6
as t t T’, then G(u) E L2(0, T’; H).
v + (T’ - t&l,
-(T’-t,)A,(I
+ i:O’dr)
-;>O
T'
1 > (T’ - t,J&
+ rll
vs dr
for n = 1,2, . . . . Then by (5.1), for a fixed m we see that ” (G(u))~ dr s tfn
is bounded
as n + 00,
which shows that G(u) E L2(0, T’; H). As is seen above, G(u) E L2(0, T’; H) in any case, hence it follows from the abstract result in [9, theorem 2.1.11 that u E W1S2(0, T’; H) and q”‘(u) E L”(0, T’). Thus u is a solution of (CP; q,) on [0, T’], so that by the local existence result the solution u can be extended beyond time T’, if T’ < TO. This fact completes the proof of theorem 1.2. n
N. KENMOCHIand T. KOYAMA
876
6. UNIQUENESS
OF SOLUTIONS
In such a nonlinear problem as (CP; u,) it might be not easy to give sufficient conditions on the memory term G(u) for uniqueness of solutions. In this section we give a result about uniqueness. We assume that (PI), (~2) and (Gl)-(G3) hold. 6.1. Suppose that there are a Banach space I/ with norm 1.(v and a seminorm [.I on Vsuch that D((o’) C Vfor any t E [0, T,], Vis a dense subspace of Hand [s] + 1.( is a norm on Y equivalent to 1.1V. Also, suppose that there are constants .eO> 0 and p L 1 such that
THEOREM
(z* - z*, 2 - Z) L EO[Z - ZIP for any t E (0, T,], z, z E D(p’), z* E i@‘(z) L > 0 there are constants C, with 0 I
(6.1)
and z* E a@(z). Further, suppose that C, < E,,, CL 2 0 and a nonnegative
for each function
yr. = yL(t) E L’(0, To) such that (G(u)(t)
-
G(Q)(t),
u(t)
-
a(t)) t
5 CL[U(f) - ii(t)y
+ CL
h(d .i -70
-
a(Wdr
+ y,Wb
- &&,,tl;~~
for a.e. t E [0, T], whenever 0 < T s T,, u, ti E X(T) and u*, ii* E L’(--r,, u*(t) E @‘(u(t)) and a*(t) E lhp’(ti(t)) for a.e. t E [--to, T] and l4W~+,,T;H)
+ le+“(~)It-(-70,T)
MV’+“,r;H)
+
+ Iu*lL+TO,r;H)
Proof.
Let u and ii be two solutions
5 L,
[0, T], 0 < T s To, provided
(6.3) that u0
of (CP; u,) on [0, T], and put OstsT
o(t) := lu - &([--T0,11;H)7 u*(t) :=
T, H) such that
l~,‘%Qhy-*o,T) + ldLq-70,T;H)5 L
then (0; u,) has at most one solution on any interval satisfies the same properties as in theorem 1.1.
(6.2)
u,*(t) i .G(u)(t)
for -
u’(t)
-7
<
t 5 0,
for 0 < t < T,
and n*(t) :=
for -r
u,*(t) L G(a)(t)
- ii’(t)
< t 5 0,
for 0 < t < T,
for a.e. t E [--to, T]. Note that o(t) = Ju - a)ccfo,rl;Hi) for all t E (0, T] and u*(t) E &&u(t)) and and ii*(t) E &&a(t)) for a.e. t E [--t,,, T]. Besides, note that a(O) = 0, 0 is nondecreasing 0 E WlY2(0, T), since 0 5 a’(t) 5 /u’(t) - ii’(t)1 for a.e. t E [0, T]. Now choose
a number a(s) = 0
which is obtained
as follows.
L > 0 so that (6.3) holds.
Then we see that
for small s > 0,
Let s be any number
in (0, T] and choose s’ E [0, s] to satisfy
a(s) = lu(s’) - ii@‘)].
Nonlinear
functional
variational
877
inequalities
We observe that ;$
[u(r)
-
ii(r)l2
+ (U*(T)
-
ii*(~),
u(t)
-
ii(r))
= (G(u(s))
-
G(ii(r)),
u(t)
-
ii(z))
(6.4)
for a.e. r E [0, T]. Integrating (6.4) over [0, s’], we get +]u(.s’) - ii(s’)j2 +
” (u* - a”, u - ii) ds = -” (G(u) - G(ii), u - ii) dr. I0 i0
(6.5)
From (6.5), with conditions (6.1) and (6.2), it follows that ns’ 5’ [U - iilp dr I +o(s)2 + (Eo - c, - Cis’) YL(r)o(r)2 dr i0 i0 s
s yL
I
(s)(T(s)~
dr 5 a(~)~
3 0
yr. (7)
dr.
I 0
Here, if s is sufficiently close to 0, then (co - C, - CLs’) > 0 and
and hence c = 0 near s = 0. Thus u(r) = ii(t) for small t > 0. Consequently, by repeating the same argument as above. n
u = ii on [0, 7’1
Example 6.1. Let us consider the same type of problem as (2.1) in example 2.1. Here, suppose that G is given by formula (2.7), and in addition to (fl)’ and (f2)’ with p = 2, the functions fk = fk(t, x, s, l, c) satisfy the following condition.
For each number A4 > 0 there is pM E L2(0, T,) such that
Condition.
Ifk(f, x, s, r, 0 - fk@, x, s, r; L)I 5 PM(f)~k - 41 + I< - 01
(6.6)
- -
for a.e. (t, x, s) and all 5, 5, c, c E R with j?j], Jr], Ifj, ]fl I M. Then problem (2.1) has at most one solution by theorem 6.2 for p = 2, I/ = W1,2(0, 1) and In fact, inequality (6.1) clearly holds, and (6.2) is seen as follows. Let L > 0 be bl = ldL~(O,l). any number, and choose a number M > 0 so that ~~~~~~~~~~~~~~~~~~~ I M for any u E X(T) satisfying (6.3). In this case, for any two functions U, ii E X(T) satisfying (6.3), we have (G(u)(t) - G(a)(t), u(t) - a(t)) 1 “I =
(h(t,
!.I0
1’
=:
r,(t)
s,
u(t,
4,
NJ,
x)1
-
fo(t, x, s, a(t, -4, w, x)))(u(t, x) - o(t, 4) d.9dx
f
(fi (t, x, s, u(t, x), 4s, x))Q,
+
x
x,
-70
ii 0
-70
(u(t,
x)
+
-
I,(f).
zqt,
x))
d.s
dx
x) - fi (t, x, s, w, x)9 m, X))%(h x))
N. KENMOCHI and T. KOYAMA
878
From
(6.6) we see that 1
t 1, Cl) 5 ,&4(f)
5
Moreover,
&,(~)(T
(IW,
4
-
u(t, x)l
+
lu(s,
4
-
ii(s, x)lW,
XI -
@Cl, XII dxb
<1-7”
Glu
-
(6.7)
~I&-,0,t);L2(o,1))~
from (6.6) again,
t
1 (Ifi ct,x, s, u(tvXL f4s,4) - “6tt, x, s, act,4, w, ml I%(%41 I,(f) 5 1 -70i 0
Now we note that &,,(t)
5 const. k(t)ln
since
(6.8)
- 4&l-a,t1;L2~o,1n9
”I “I lu% ! -70
,) d.s 5 const. *)I Lmco,
5
!
const.((T
(I%(% *)lL%,1)+ IMh .h(O,I,>ds -To + ro)L1’2 + (T + ro)1’2L).
Besides, &,2(f)
5 :
where kL is a positive constant depending inequality of the form (6.2) holds.
only on L. We see from estimates
(6.7)-(6.9)
that an
Example 6.2. Consider the same problem as (2.6) with p = 2 in example 2.2; suppose that p = 2, G is given by (2.7) and condition (6.6) is satisfied as well. Moreover, suppose that f, (t, x, s, r, c) is independent of t, x, s and r, i.e. f, = fi (0. Then problem (2.6) has at most one solution by theorem 6.2, in the case wherep = 2, V = W,‘V2(0, 1) and [z] = (z~[~z~~,~). In fact, let L > 0 be any number and choose A4 > 0 in the same way as in example 6. I. We then have
879
Nonlinear functional variational inequalities
for any functions
U, ti E X(T)
satisfying
(G(u)(t)
- G(D)(t), u(t) - a(t)) 5 Zr(t) + &2(f),
where Zr(t) is the same function 1 &
(6.3)
as in (6.7) and
t
=
u-i
110
(4.c
x))u,(.%
4
-
fi
(a.
x)h%(s,
x)))(u,
ct,
x)Mt,
XI
-
iict,
4)
ds
d-x
-70 “1
t
ii 0
-70
=--
(fl
with the primitivefr
offi Il;Wl
(MS,
4)
-
fl
(ii
satisfyingyr(0) 5 VI&(0
6,
4
-
Rx
(f,
x))
ds
dx
= 0. Therefore,
for any positive
- ~XWlt~(o,l) + CV&
- 4&-,,,M2~0,1))
where CM is a constant depending only on v and M. From estimates that an inequality of the form (6.2) holds in the case of p = 2. 7. VARIATIONAL
INEQUALITIES
WITH
number
v > 0,
for I, (t) and I2 (t) we see
HYSTERESIS
Let us consider variational inequalities of the forms (2.1) and (2.6) with nonlinear source terms G described by hysteresis functionals. Let f, , fdE C(R) be monotone nondecreasing and Lipschitz continuous functions such that and put
Also, let w. E R such that {(c, Now, we define a hysteron operator X(=X(*; wo)): C([O, T]) + C([O, T]), 0 < T < m, associated with f,, fd and w, as follows. Firstly we define X on the set of all piecewise linear functions PLC([O, T]) := (5 E C([O, T]); there exists a partition 0 = to < t, < ... < t, = T of [0, T] such that r is linear on each [ti_, , ti], i = 1,2, . . . , n) by the formula minffd(W)), x(r)(t)
=
if t = 0,
~~~1~
maxffuW-N
minffd(t(thmaxIf,WN, W3(L1)JL
for any r E PLC([O, 1,2 ,..., n.
if t e [ti-l,
T]) with 0 = to < t, < ... < t, = T such that
til,
i=
r is linear
1,2 ,..., on [ti_l,
n. t;],
i=
LEMMA 7.1. For any rr, & E PLC([O,
holds,
and hence IWrr)
holds,
T]),
where L is a common
- WZ)IC([O,T])
Lipschitz
constant
5 Zk, of
-
&([O,T])~
fd and f, .
(7.2)
880
N.
KENMOCHI
T.
and
KOYAMA
Proof. In order to get (7.1), it is sufficient to show the following fact. If ?ji, c2 E C([O, T]) are monotone functions, and if w~,~, w~,~ E R satisfy (ri(O), wi,J E M, i = 1, 2, then the functions i= 1,2, wi(t) := mWfd(ti(t)), mdf,(ti(t)), wi,0119 satisfy the inequality wi(t)
- ~~(0
5 max]w,,,
- w2,0,
fa(rl(t))
-
fo(r2(f)),fd(rl(f))
-
To show this, we note for each i = 1,2, that (i) if Pi := (C(t), wi(t)) E M” := M\(D U A), then Wi(t) = wi.0, (ii) if Pi E D\A, then fd(ri(f)) = wi(t),
(2) if P2 E M\D
w2(f)
‘fd(
and P, E A, then w,(t) Wl
(3) if P2 E M\D
-
(t)
-
and P, E M\A,
%
Cl)
= 5
(7.3) is proved
f,(r,
then wi(t)
in any case, whence
0))
(7.3)
~i,~, ~i,~,
fd(t;2(f)k
and thus
(t))
fa (Cl
WI(t) - %(f) therefore
-
fd(r2(f))l.
-
f,(&
(t));
I w~,~, wz(t) 2 w2,0 and thus 5
w1.0
-
w,,,;
(7.1) and (7.2) are obtained.
W
By (7.2), the operator X(=X(.; w,)): PLC([O, T]) + C([O, T]) has a unique extension, denoted by X(=X( *, wo)) again, from C([O, T]) into itself, and it is a Lipschitz continuous operator in C([O, T]) with Lipschitz constant L. Now let us define a mapping G: U,, Tc ToX(T) -+ Uo, Ts =,L2(0, T, L2(0, 1)) in terms of the hysteron X, where X(T) = C([O, T] x [0, l]), as given in the following example. Example 7.1. (i) Let w. E L2(0, 1) such that (([, w,(x)); r E R] fl A4 # 6 for a.e. x E (0, l), and f. E L2(Qo), Q. := (0, T,) x (0, 1). Then, for 0 < T I To, we put G(u)(t, x) := fo (t, xW(u(
* , xl;
wo (x))(t)
(7.4) for a.e. (t, x) E (0, T) x (0, 1) The definition (7.4) of G makes fo E Lm(Qo),then
and any u E X(T).
sense as an operator
iGO4 - G(&-~~o,T~x (o,l))5
Ifoirx&U
for any U, u E X(T).
from X(T)
-
into L2(0, T; L’(O, 1)). If
&([o,TI
x [O,II)
(7.5)
Nonlinear functional variational inequalities
7.1. (ii) Let W, E
Example
881
R such that ((<, w,); c E RJ f~A4 # 4, and f0 E L2(Q,). Then we
define G(u)(t, x) : = A, (f, xWO4 * 1xo); wJ(t) (7.6) for a.e. (t, x) E (0, T) x (0,1)
and any u E X(T),
where x0 is a prescribed point in (0, 1). This definition (7.6) makes sense and (7.5) holds true just as in the case of (7.4). In the remainder of this section we discuss problems (2.1) and (2.6) with r, = 0 and the operator G given by (7.4) or (7.6). (a) Let G be the operator given by (7.4) with& E L”(Q,,), and let (~‘1 be the family of proper 1.s.c. convex functionals on L2(0, 1) given by (2.3) [resp. (2.8)]. Let u0 E D(p’). Then (CP; u,), corresponding to problem (2.1) [resp. (2.6)], has one and only one solution on [O, &I. PROPOSITION 7.1.
PROPOSITION 7.1. (b) Let G be the operator given by (7.6) with_&, E L”(Q,), and let {v’) and u. be as in (a). Then (CP; u,), corresponding to problem (2.1) [resp. (2.6)] has one and only one solution on [0, To].
Proof. In any case, it is easy to verify conditions (Gl)-(G3), and the conditions for global existence of solutions in theorem 1.2. Therefore, the existence of a solution to (CP; uo) on [0, T,] is a direct consequence of our abstract theorems in Section 1. Next, we will show the uniqueness of solutions of (CP; u,) only in the case corresponding to problem (2.6) with G defined by (7.4), since the proof of uniqueness in any other case is quite similar. Let ui and u2 be any two solutions of (CP; u,) on [0, To]. Then, using the characterization (2.9) of a@, we see that for sufficiently large q E R, 1 @I,,@)
i
-
W,Wb,
(f)
-
~2(0l~-?~,
(t)
-
~2
(t))
k
0 1 =
k,(f)
r
Jo 5 -
WMtNiu,W
-
s
@)fl% I~l,xu)lq-2%,x
:
+
=-
-
.i
; 14,x(f)lq-2~1,x
(f)
~2(01~-~#41(0
-
Web,
~2(olq-2041
-
U2(olq-2(U2(t
U)
-
g(f))
-
@2(t)
-
g(t))), dx
-
g(f)))
1 - g(f))l, dx
r1
I IqxwlP-2~l,x
= -44
WI%(t)
o
-
1)
l l~I,xw-2 50
%,,(0b,
-
~2(olq-2Mo
(r)
-
-
~2wlq-2@l,,(o
UZ(f)Nx
dx
-
u2,xw
dx
dx
N. KBNMOCHI
882
and T. KOYAMA
and similarly
Therefore
5
I(W,)(0
-
C(UZ)(~))~L-((O,T)X(O,~))IU~(~) -
~2(~)1&;,,~>
for a.e. f E [0, T] and any 0 < T 5 To; note here that G(u,), G(u~) E Lm(Qo), since u,, u2 E C([O, &] x [0, I]). From this with the Lipschitz continuity (7.5) of G it follows that
andanyO<
for a.e. t e [0, T]
TI
To,
where L’ = Ifolt~cQ,,L. Hence
l%(f) - M)lLQ(O,I)--= TJ+Q - %lC~~O,Tl x [O, 11) for all t E (0, T]
and
O
Letting 4 + W, we have I%(0 -
~2WlC([O,I~)
for all t E [0, T] Accordingly,
5
TL’l% and
%I.((o,Tlx[o,lj)
0<
Ts
To.
u, = uz for near f = 0, and it results consequently that u, = u2 on [0, q].
It
Remark 7.1. For detailed studies on various hysteresis functionals we quote [IO] and [14], and for some related works, [lo, 13-181 on evolution equations involving hysteresis functionals in boundary conditions or source terms as systems of feedback control. Also, in the forthcoming paper [19], existence and uniqueness questions for more general variational inequalities with hysteresis in multi-dimensional spaces will be discussed.
Nonlinear functional variational inequalities
883
REFERENCES 1. AGXOVICI S., Abstract integral equations of Volterra type, Lincei Rend. Sci. Fis. Mat. Nat. 58, 868-879 (1975). 2. AIZICOVICI S., Time-dependent Volterra integrodifferential equations, J. Integral Eqns 10, 45-60 (1985). 3. AIZICOVICI S., Asymptotic properties of solutions of time-dependent Volterra integral equations, J. Math. Anal. Appl. 131, 421-440 (1988). 4. ATTOUCH H. & DAMLAMIANA., A nonlinear Volterra equation in variable domain, MRC Tech. Summary Rep. No. 2028, Madison, Wisconsin (1979). 5. MITID~ERI E. & VRABIE I., A class of strongly nonlinear functional differential equations, Ann. Muter. pura uppl. 151, 125-147 (1988). 6. CRANDALL M. G. & NOHEL J., An abstract functional differential equation and a related nonlinear Volterra equation, Israel J. Math. 29, 313-328 (1978). 7. FITZGIBBON W., Representation and approximation of solutions to semilinear Volterra equations with delay, J. diff. Eqns 32, 233-249 (1979). 8. MITIDIERI E. & VRABIE I., Existence for nonlinear functional differential equations, Hiroshima Math. J. 17, 627-649 (1987). 9. KENMOCHI N., Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fat. Educ. Chiba Univ. 30, l-87 (1981). 10. VISINTIN A., A model for hysteresis of distributed systems, Ann. Muter. pura appl. 131, 203-231 (1982). 11. BR~ZIS H., Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, in Math. Studies, Vol. 5. North-Holland, Amsterdam (1973). 12. OTANI M., Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauch problems, J. dzff. Eqns 46, 268-299 (1982). 13. HILPERT M., On uniqueness for evolution problems with hysteresis, in Mathematical Models for Phase Change Problems (Edited by J. F. RODRIGUES), Int. Ser. Numel. Math. 88, 377-388. Birkhauser, Base1 (1989). 14. VISINTIN A., On the Preisach model for hysteresis, Nonlinear Analysis 8, 977-996 (1984). 15. FRIEDMAN A. & HOFFMANN K.-H., Control of free boundary problems with hysteresis, SIAM J. Control Optim. 26, 42-55 (1988). 16. FRIEDMAN A. & LI-SHANG JIANG, Periodic solutions for a thermostat control problem, Commurzspartial dif$ Eqns 13, 515-550 (1988). 17. HOFFMANN K.-H. & KEMMOCHI N., Two-phase Stefan problems with feedback controls, in Mathematical Models for Phase Change Problems (Edited by J. F. RODRIGUES), Int. Ser. Numel. Math. 88, 239-260. Birkhauser, Base1 (1989). 18. VISINTIN A., Evolution problems with hysteresis in source term, SIAM J. Math. Anal. 17, 1113-1138 (1986). 19. KOYAMA T., KENMOCHI N. & VISINTIN A., On a class of variational inequalities with memory terms (in preparation).
& Applications.
NONLINEAR GOVERNED
Vol. 17, No. 9, pp. 863.883,
1991. 0
0362-546X/91 $3.00+ .OO 1991 Pergamon Press plc
FUNCTIONAL VARIATIONAL INEQUALITIES BY TIME-DEPENDENT SUBDIFFERENTIALS
NOBUYUKI KENMOCHI Department of Mathematics, Faculty of Education, Chiba University, Yayoi-cho, Chiba, 260 Japan
and TETSUYA KOYAMA Department of Mathematics, Hiroshima Institute of Technology, Miyake, Saiki-ku, Hiroshima, 731-51 Japan (Received Key words and phrases:
1 August
1990; received for publication
20 February
Functional variational inequalities, time-dependent
1991)
subdifferentials,
existence
and uniqueness.
INTRODUCTION THIS PAPER is concerned
equation
(CR uo)
of the following
with the abstract Cauchy problem (CP; u,) for a functional form in a real Hilbert space H:
u’(t) +
@&W)
3
G(u)(t),
differential
Oct
on I-Q, 01
u = ug
where u’(t) = (d/dt)u(t), {@JtZo is a family of proper lower semicontinuous (1.s.c.) and convex functionals I# on H, dp,’ denotes the subdifferential of p’, u,, is a given function in C([-r,, 01; H) and G is a mapping from a subset of C([-T,, T]; H) into L2(0, T; H). We regard G as an abstract model of functionals with nonlinear memory. The aim of this paper is to establish a local existence theorem under some boundedness and demicontinuity condition of G, and prove the global existence and uniqueness of solutions under further restrictions on G. Functional differential equations of the same type have been studied by many authors [l-5] in order to solve nonlinear integral equations of the Volterra type. Also, for related works we quote here some results from [6], [7] and [8]. For the existence and uniqueness of solutions to (CP; ZQ), one of the simple sufficient conditions on G might be of the form:
for any U, u E C([-r,, T]; H) and 0 < t 5 T, where y E L’(0, T). However, this is too restrictive to apply (0; u,) to some physical problems such as heat conduction with timedelay. Recently, Mitidieri and Vrabie [5] gave an existence result for (CP; u,) under general conditions on G, in case I$ is time-independent. We shall generalize this result to the case of time-dependent pf, making use of results in [9]. Our abstract results are applied, for instance, 863
N. KENMOCHIand T. KOYAMA
864
to the following
functional
variational
u, - (lu, lp%,
inequality
with memory: in Q = (0, T) x (0, l),
2 G(u)(t, X)
in Q,
I.42 g(l, x)
(u, - (~GI~-~~,),
in Q,
- G(u))(u - 8) = 0
u(t, x) = 240 where G is a functional
in (--TV, 01 x [O, 11,
given by
f G(u)(t, 4 =
f
fo tt, x, s, e, i -70
4, a,
xl) d.s +
J-1(t, xv s, N, xl, m, .a%(& s -70
xl cl%
and g and u0 are given functions on Q and (-r,, 0] x [0, 11, respectively. Also, in a final section of this paper, we will discuss some applications to the same type of functional variational inequalities involving the so-called hysteresis operators (cf. [lo]) as a nonlinear memory term G(u). Notations. For a (real) Banach space V, 1-Iv means the norm C([a, b]; V), LP(a, 6; V), etc., mean the usual function spaces.
of
V, and
the notations
Throughout this paper, we assume that H is a real Hilbert space, with inner product (* , a) and norm 1. (, and use the notion of subdifferentials of convex functionals on H with some basic properties of subdifferentials without any explanation; we refer them to [l I]. 1. MAIN
RESULTS
Let r0 2 0 and r, > 0 be given two numbers, and let (pf; -tO 5 t 5 To] be a family of proper 1.s.c. and convex functionals on H. Also, let G be an operator from D(G) := U,,
(CR
uo>
u’(t) +
@%40) 3 G(u)(t),
0
u(t) = u,(f),
< t < T(s T,),
-50 5 t 5 0,
where &D’ denotes the subdifferential of 9’ and u,, is a given function in C([-r,, function u E C([-s,, T]; H) is called a solution of (CP; zq,) on [0, T], if u E W1*2(0, T; H), G(u) E L’(O, T; H),
G(u)(t)
- u’(t) E %f(W))
and u(t) = 4(f)
for --to 5 t 5 0.
for a.e. t E [0, T]
01; H). A
Nonlinear functional variational inequalities
865
We note here that if r, = 0, then in the above formulation of (CP; u,), condition u(t) = u,(t) for -rO I t I 0 is replaced by the usual condition u(0) = u0 for a given u,, E H. Our problem (CP; u,) is discussed under the following assumptions (Al), (~2) and (Gl)-(G3). (~1). For each r > 0, there are functions (Y,E W’*‘(O, &) andp, E W”‘( 0, T,) for which the following condition holds.
Assumption
For each s, t E [0, T,] with s 5 t and each z E D($) with [z( 5 r, there is Z E D($),
Condition.
such that le - ZI 5 la,(t) - 44I(f
+ l~“(Z)Y2),
and V’(a - V”(Z)5 IA(t) - P&)l(l Assumption in H.
+ lV”(Z)l).
(~72). For each t E [0, T,] and R > 0, the set (z E H; &z> 5 R, 1.~15 R] is compact
Assumption (Gl). For every T E (0, To], X(T) contains the set X,,(t) := (u E W’*2(-t,,, T; H); p”‘(u) E L”(-t,, T) and there exists u* E L2(-T,,. T; H) such that u*(t) E &&u(t)) for a.e.
TII.
t E I-Q,
(G2). (Boundedness.) There are a constant v E (0, l), a nondecreasing I: R: + R, and a function 1, E L’(0, T,) such that
Assumption
\G(u)(t)12 5 vlu*(t)12 + &&+ro,r~;wj, for
a.e. t E [0, T], whenever u*(t) E &&u(t)) for a.e. t E [-zo,
0 < Ts
To,
~~%4~~-~-r,,r~r lu*l,+a,l;~j) u E X,(T)
and
function
+ [o(t)
U* E L2(-ro, T; H)
with
T].
Assumption (G3). (Continuity.) For every T E (0, To], G is demicontinuous as a mapping from X,(T) into L’(-ro, T; H) in the sense that if U, EX~(T), u, + u in C([-r,, T]; H) (as n -+ co), (uh) is bounded in L2(--ro, T; H), (q~“‘(u,)) is bounded in P(-rO, T) and there is a bounded sequence {u,*) in L2(--ro, T; H) with u;(t) E &&u,(t)) for a.e. t E [-r,,, T], then G(u,) + G(u) weakly in L2(0, T; H).
The main results in this paper are formulated
as follows.
1.1. Suppose that conditions (pl), (~2), (Gl)-(G3) hold. Let u. be any function in W’,2(-r,, 0; H) such that $“(u,) E L”(-T,, 0), u,(O) E II and there is u$ E L2(-ro, 0; H) satisfying z@(t) E ?Qr(uo(t)) for a.e. t E [-ro, 01. Then there exists a number T E (0, To] for which (CP; u,) has at least one solution on [O, T].
THEOREM
Concerning following.
global existence, i.e. existence on the whole interval
[0, To], we prove the
866
N. KENMOCHIand T. KOYAMA
THEOREM 1.2. In addition to the assumptions of theorem 1.1, suppose that the functions CX, and p, in condition (cpl) are independent of r, i.e.
(Y,=: CY
and
P, =:P
for any r > 0, and that in condition (G2), I(.) is of the form U19 r2 9<3) = c,crf + r2 + r: + 1) for a nonnegative constant C, . Then (CP; uO) has a solution on [0, r,]. 1.1. The above theorems include the case of r0 = 0. In this case, it is enough to assume for the initial value u0 E H that u0 E D(cp’).
Remark
These theorems will be proved in Sections 4 and 5. 2. SOME
EXAMPLES
In order to illustrate our main theorems, we give some examples of functional differential equations and variational inequalities of parabolic type in one-dimensional space. Example 2.1. We consider the following problem with flux condition of Signorini type with nonlinear memory:
Ut - %X = G(u)(t, x)
in Q := (0, T) x (0, l),
u(t, 0) L h(t)
for 0 5 t 5 T,
u*(t, 0) 5 0
for 0 < t < T,
u,(t, O)(u(t, 0) - h(t)) = 0
for 0 < t < T,
u(t, 1) = 0
for 0 zz t 5 T,
u = f&J
in [-ro,O]
(2.1)
x [0, 11.
Here we are given h E W’92(-ro, &) and u. E W1.2(-ro,0; W1V2(0, 1)) with uo(t,O) 1 h(t) and u,(t, 1) = 0 for any t E [-To, 01. The functional G is defined in the following way. Let f = f(t, x, s, Cl, r2, Cl, C2):[0, To] x [O, 11 x [-ro, To] x R x R x R x R + R be a function such that (fl) fis measurable in (t, x, s) for fixed (ri, (t2, cl, c2) and is continuous in ({i , r2, cl, c2) for a.e. fixed (t, x, s); (f2) bV,x9~~~l~52~ Cl.C2)i 5 po(t,x,s) + ~~(15~1, kll) + ~~(k~i, lC11)(k21 + lC21) where pLoE L2((0, To) x (0, 1) x (-to, T,)) and iu,: R, x R, -, R, , k = 1,2 are nondecreasing func-
tions with respect to all variables. Now, put X(T)
= W1P2(-.ro, T; L2(0, 1)) 0 L”(-to,
T; W’.2(0, 1))
and define G: U.
_I%, x9 s, w, 4, u,(t, xl, a, -To
4, ~,cc x)) dJ
(2.2)
867
Nonlinear functional variational inequalities
for u E X(T), 0 < T I To. Next consider functionals q~’on H = L’(O, 1) given by 4(z) =
the family
[pi; -r,
tlzxlE~(o,l, if +cO
5 t I T,) of proper
1.s.c. convex
z E K(t), (2.3)
otherwise,
where K(t) = {z E WlT2(0, 1); z(O) 1 h(t),z(l) = 0). Then we have the following facts: (a) Z* E a&z) if and only if z E K(t), z* E L2(0, l), z* = -z,, a.e. in (0, I), z,(O) I 0 and z,(O)(Z(O) - h(t)) = 0. Hence 8s’ is single-valued. (b) Conditions (91) and (~2) are satisfied. Proof
of(b).
Clearly
(~2) holds.
To verify (pl),
for any z E K(s), we take
Z(x) = z(x) - (1 - x)(h(.s) - h(t)) E K(t). We then see by elementary
computations a,(t)
that (~1) holds with
= P,(t) = const.
* lh’(z)l dr, 0
which is independent (c) Conditions Proof
of(c).
of r.
(Gl)-(G3) Trivially,
I&([-
(Gl) is satisfied.
Next, by (f2) and (2.3), we have
C0h0(0l~~((0,1) X(-so,To)) + Pl(I&~[-sO,t] X[O,l]) 3 I&([-~,,*, x [0,1])12 (2.4) 14C(]-ro,t] x [0,1])~21~~l~-~-~~,t;~~~o,1))~ + L12(14c([-r,,t]x[0,1])~
IG(u)(t)l &0,1) 5
where Co is a positive
are satisfied.
constant.
so,t]X[O,l])
5
Cl
Since
+ IUlc([-~o,tl;LZ~o,l))) (2.5) lulC(I-~O,tl;~*(o,l))~l~Xl~-m(-~O,t;~~(o,l))
(2.4) yields (G2). for any ueX(T) and O< TI To, where C, is a positive constant, Finally, to verify (G3), let u, E X(T), n = 1,2, . . ., and u E X(T) such that U, + u in weakly in L2(-ro, T; L2(0, l)), (u,,,] is bounded in u,,t --+ ut C([-- 70, 7’1; L’(O, I)), T; L2(0, 1)). Then, by the compactness L2(-ro, T; L’(O, 1)) and ]u,,,) is bounded in L m(-70, theorem, we have in L2(-ro, T; L2!0, l)), %I*, + UX and u, + u in C([-7o, Tl x 10, 11) L2(-7,, T, L2(0, 1)). On account of (a), problem (CP; u,) is a weak formulation (f2) the function ,~r is of the form
so that (fl) implies G(u,) + G(u) in
of (2.1). Moreover,
if in condition
~~(15~1~ kll) = c0nst.U + l&l + 15h1 and if ,u2 is a nonnegative constant, then G, given by (2.2), satisfies 1.2 for global existence of solution.
the assumption
of theorem
N. KENMOCHI and T. KOYAMA
868
Example
2.2. We consider
u, - (Iu,]~-~u,),
an obstacle
problem
of the following
form: O
in Q = (0, T) x (0, l),
2 G(u)(l,x)
24 2 g(t, x)
in Q
(u, - (]u,I~-~U,
- G(u))(u
- g) = 0
in
Q,
(2.6)
u(t,x) = 0
for 0 5 t 5 T,
U = Ug
in [-rO, 0] x [0, 11.
x = 0, 1,
Here, p 2 2, g E W1r2(-q,, To; W’,“(O, 1)) and u0 E W1,2(-r,, 0; WISp(O, 1)) such that g(t, x) = 0 for -q, 5 t 5 To, x = 0, 1, u0 2 g in [-rO, 0] x [0, l] and u,(t, x) = 0 for -rO 5 t I To, x = 0, 1. Assume that G is the functional defined by
f fo(t, x, s, u(t,x), MS,x))ds +
G(u)(t, X) =
.i -70
t fi(t 9x, s, u(t,x)94% X))&(&x) ds .i-nl (2.7)
for any u E X(T), X(T)
and fk (fl)’
0 < T 5 To. Here
= W’+t,,,
T: L2(0, 1)) fl L”(-r,,
T; W1yp(O, l)),
O<
= fk(t,x, s, r, [), k = 1,2, satisfy conditions. fk is measurable in (t, x, s) for fixed ([, [) and is continuous
Ts
To,
in (r, 4) for a.e. fixed
(t, X, s);
Ifi(t~x~~~r,i)I (iiio(t,s) +iiildtIJcl), for a.e. (t, x, s) and for all r and [, where ,u, E L2((0, To) x (0, 1) x (-r,,, To)), ,D,, E L2((0, &) x functions with respect to any T,)), and pl, PI : R, x R, + R, are nondecreasing variable. Also, the family 1~‘; -rO I t I To] of proper 1.s.c. convex functionals on H = L2(0, 1) is given by (-Q,
w-9 otherwise, where K(t)
= (z E W,‘sp(O, 1); z 2 g(t, .) on [0, 11).
In this case it is easy to see that z* E @‘(z) if and only if z E K(t), z* E L2(0, l), 1 Izxlp-2zx~,
dx 2
i ,O ”
\ .0
1 z*u 10
dx
for any nonnegative
1 z*cz
-
g(t, *)I dx =
1l Izxlp-2zx(zx - g,(t,
! ,O
.)) dx.
q E W,‘,p(O, l),
(2.9)
Nonlinear functional variational inequalities
869
Therefore (CP; z+,) is a weak formulation of (2.6), and we see that conditions (eel), (~72), (Cl) and (G2) are similarly checked as in the first example. Condition (G3) is seen from the fact that G(u,) --t G(u) weakly in L2(0, T; L2(0, l)), if u,, u E X(T), (u,,,) is bounded in T; Lp(O, l)), u, -+ u in C([-s,,, T] x [0, 11) and u,,, + U, weakly in L2( -rO, T, Lp(O, 1)). L”(-r0, Moreover, the operator G satisfies the condition for global existence of a solution in theorem 1.2, in case ,D, and jir are of the forms ~i(l
Ill) = const.0
,f&(j
ltlp’2 + lClp’2)T
+
const.(l
+ l@+“‘”
+ lp-2’2).
in this case we have, by (2.5),
/G(W)/
&O,l) 5
for any u E X(T),
Co[Iruo(t)lZzcc-,,To)x(O,l)) + liio@)l~~cTo,To) + IU*IP,m(-s,,r;rp~o,1)) + 11
0 < T I To and a.e. t E [0, T], where Co is a positive 3. AUXILIARY
ESTIMATES
In this section, let u. be the same function as in theorem f E L2(0, T; H) with 0 < T s &, we consider the problem u’(t) + @‘(W))
constant.
sf(t),
1.1. Then,
for each
O
-ro I t 5 0.
i u(t) = uo(t),
given
By some results in [9] (or [12]), this problem is uniquely solvable under condition (~1); in fact, there is a unique solution u of (3.1) such that u E W”2(0, T; H), u(0) = u,(O) and f(t) - u’(t) E &p’(u(t)) for a.e. t E [0, T]. In what follows, we assume all the assumptions of theorem 1.l and denote by U, the solution of (3.1). LEMMA 3.1. There
are constants
A, > 0 and A; > 0 such that (3.2)
and
lq(t)l foranyOSs5
ts
T,O<
TI
-(
l.qM + -4; + (t - s)
lf(412d7
(3.3)
T,andfEL2(0,T;H).
Proof. Let h be a function in W1,2(0, To; H) t E [0, To]. Then, for any 0 5 s 5 t IS T, +(qt)
.\s
- h(t)12 I guy(s)
- h(.s)12 +
such
that
h’(t) + @‘(h(t))
’ If(r)/ by(~) - MT)/ dT ss
and hence
If we take Al = 2b31ccIo,Tol;Hj and A; = Al + 1, then (3.2) and (3.3) hold.
H
3 0 for
a.e.
870
N. KENMOCHI and T. KOYAMA
COROLLARY3.1. There are constants
A2 > 0 and A$ > 0 depending
\&~~o,T~;H~
5 A, + i
and \+Ic(to,rl;~j
;\A7)\
d7
5 Ai + Z-.‘: If( /
(3.4)
d7
for any 0 < T 5 T, and f E L’(O, T; H). In fact, by lemma 3.1, we can take JuO(0)I + A, and lu,(O)j COROLLARY3.2. There are constants
only on (uo(0)( such that
(3.5)
+ A’, as AZ and A; respectively.
A, > 0 and A; > 0 such that t , s If(r)1 d7 + 1 i >
(3.6)
and
b’@~W)Ic: &q(t)) + A; lq(s)l + (t - s) for any 0 5 s 5 t 5 T, whenever Proof. Condition
((~1) implies
Ji
’If(
dr + 1 >
0 < T 5 T, and f E L2(0, T; H). (by [9; lemma
1.5.11) that there is a constant
0 > 0 satisfying
p’(z) + 0121 + 0 2 0 for any 0 I t 5 T, and z E H. Combining obtain the corollary. n For simplicity,
(3.7)
this inequality
(3.8)
with (3.2) and (3.3) in lemma
3.1, we
given R > 0 and 0 < T s: T,, we put KR(O, T) = tf E L2(0, T; H); If b<,,,T;u) 5 RI.
LEMMA 3.2. There are a constant R > 0 and 6 E (0, l] such that
A, > 0 and a function
r,rR,6 E rf(0,
T,) depending
only on
for any 6 E (0, 11, 0 5 E s t, 0 < T 5 T, and f E KR(O, T). Proof.
Let R > 0, 0 < T % T, and f E KR(O, T). Then,
5 r, := IulC~~O,Tl;H~ GivenO<
by corollary
A, + T,“2R.
1,weput '1RJC7) = f
/4(7)12 + ku7>l9
3.1, u := uf satisfies
Nonlinear functional variational inequalities
871
where (Y,and /3, are the functions with r = rl + 1 in condition (91). Now we recall the energy inequality for the solution u (cf. [9]): h4~N
- ce49)
+
(u’, u’ - f) dr
ft
rt
JS
JS
(3.9) for any 0 I s 5 t 5 T, where mi, i = 1,2, 3,4, are positive constants depending only on the constant cr of (3.8). Hence, for each given 0 < 6 I 1, &u(t))
” Id -f[‘dr
- q?@(s)) +
1s 16
‘1~’ -j-f12dr+ .i5
-
t IfI
Iu’ -j-l
I’s
dr
i’: 1; la:12 + iP:$a’(u)J
+ A,lul + A)dr
(3.10)
for any 0 I s 5 t 5 T, where A4 = 3mf + 3rni + m3 + m4. Here use the inequalities “f
!
IfI Id - fl dr I $1 - E)
(j-l2 dr + &
1’ 1~’ - j-I2 dr s
.s
5 +(l - E) ’ lf12dt ss
+ +(l + 2.5) t Id -f 12dt is
for 0 I E I 5. Then, from (3.10) we obtain the required inequality. Remark
3.1.
n
The energy inequality (3.9) is obtained by letting I 10 in the energy inequality
5 ’ tlail l~~~(u~>l(l~~(~~>l”’ + ml141’2+ md+ IP~l(ld(ud+ m3bXI + mJ1dr ss for
the solution ux of the approximate problem u:(t) + dq&(u,(t)) = f(t), 0 < t < T, where 89; is the Yosida-approximation of a$. See [9; Section 2.81 for the details.
u,(O) = u,(O),
4. LOCAL
EXISTENCE
OF SOLUTIONS
We use the same notation as in the previous section. The key of the proof of theorem 1.1 is the following lemma. LEMMA
4.1. There are T E (0, To] and R > 0 such that
G(q)
E
KR@ T)
for any f E KR(O, T).
(4.1)
N. KENMOCHI and T. KOYAMA
872
Proof. According to the result [9; theorem 2.6.11, for each R > 0 there is a constant M, > 0, depending only on R, such that l&[O,Wf)
5 MI7
l~jlL~@J.;H) 5 Mi?,
t6+"("f)tLm(o,~ 5
(4.2)
MR,
for every 0 < T 5 To and f E KR(0, T). Applying lemma 3.2 for s = 0 and t = T, we see that ‘T vT(q(T))
+
(+ -
6 -
E)
o
i
b;-f12dr 'T
I
$(I
-
E)R~
+
(hfR
+ x‘t4(kfR +
1))
i0
)lR,Sdt
+
(4.3)
ul'(Uo(o)).
By (3.7) of corollary 3.2, v~(+(T))
1 l~~(q(T))l
(4.4)
- A;]\uo(O)] + TR* + 11.
From (4.3) and (4.4) it follows that (4 - 6 - E) ~TIUj- fj’dt I0 T 5
+(I
-
& +
A;T)R*
+ A;(lu,(O)I
+ 1) + rii,
~R,?I~~
+
ul"(Uo(o)),
(4.5)
i0
where tiR = MR + A4(MR + 1). Besides, by condition (G2) ‘T iG(u f )I2 L'(O,T;H) 5
vb;
-
f I&O,T;H)
+
WJo
+ MRr
Jo + MI?, Jo +
MR
+
R)
j,dr
+
(4.6)
.0 1
where Jo := I~OIC([-,,OI;H)+
lv)(~)@o)ILy-To,o) + I~o*lL+~O,o;H)
for a function u,*(t) E &p’(uo(t)) for a.e. t E [-TV, 01. Therefore, we derive from (4.5) and (4.6) that for small 6 > 0 and E > 0, (f - 6 -
~)IG(~~)l&,T;H) T
5
;(I
-
&
+A;T)R*
+ vA;(lu,(O)l
+ 1) + d&
qR,Sdf.
+
"(t -
6 -
~)~"(~o(o))
0
s T
+
($ -
6 -
E)T/(J~ + it'fR,Jo
+ hfR, Jo + b’fR + R) + (3 - 6 - E)
I,
dr
0
so
that
+
TI(Jo + MR, Jo + MR, Jo + MR + R) +
I,dr + q”(uo(0)). 0
(4.7)
873
Nonlinear functional variational inequalities
Since 0 < Y < 1, it follows that v(1 - E + A; T’)/(l 6 > 0, E > 0 and T’ > 0. For such 6, E and T’, put 1 - 2v, := Clearly
- 26 - 2~) < 1 for sufficiently
small
~(1 - E + A;T’) .
1 - 26 - 2E
v1 > 0. Next, choose R > 0 sufficiently
~;(l~o(o)I + 1) + 1 - 26 - 2E
large so that p”(uo(0))
I v,R2
and then T E (0, T’] so that 2li&
T
1 - 26 - 2E s 0 Then,
T
rIR,&dt + Tl(Jo + MRr Jo + MR, Jo + MR + R) +
for such a couple
lo dr I v,R’. 0
{R, T), inequality
(4.7) implies
(4.1).
W
Proof of theorem 1.1. Let R > 0 and T E (0, To] be the numbers define a mapping S: KR(O, T) + KR(O, T) by putting Sf := G(uf)
found
in lemma
4.1, and
for every f E KR(O, T).
Then S is continuous in KR(O, T) with respect to the weak topology of L’(O, T; H). In fact, if f, E KR(O, T) and f, + f weakly in L2(0, T, H) as n + 00, then by the uniform estimate (4.2) in the proof of lemma 4.1 and by condition (~2) we see that u?l := z.4fn+ U := 24 f
u:, + u’
Tl; HI,
in C([-T~,
weakly in L2(-
T,
,
T; H)
and is bounded
W’(u n)]
in L”(-to,
T).
Therefore condition (G3) implies that G(u,) + G(u) weakly in tinuous in KR(O, T). Since KR(O, T) is a nonempty, weakly L2(0, T; H), it follows from the usual fixed point theorem that KR (0, T). It is clear that for any fixed point f of S in KR(O, on[O,T]. W 5. GLOBAL
EXISTENCE
L2(0, T; H) so S is weakly concompact and convex subset of S has at least one fixed point in T), uf is a solution of (CP; u,)
OF SOLUTIONS
In this section, suppose always that all of the assumptions use the same notations as in Section 3.
of theorem
1.2 are satisfied,
and
LEMMA 5.1.
Let u be any solution of (CP; u,) on [0, T] with 0 < T I To. Then there are A, > 0, A, > 0 and a function qs E L2(0, To) depending only on 6 E (0, l/2) such
constants that
l4wt))l +
1 - (t - s)A, - (t - Wh(l
+ [jr&)d7)
(t - ~)Iv%&-~s,~~
+ H,(s)
v + (t - s)A,
- ;]
[:kWi)l’dr
I 5
s
~cd~)~vW))~
ds + 4
+
cs
10(5) dr
1
(5.1)
N. KENMOCHI and T. KOYAMA
874
T, where
for any 0 I s I t I X(9
= I&-q#f)
+ I~*l~~(-,,,;lY)
+ Ie+“@)IP(-TO,s) + 19
and u*
Proof.
=
First we have, by lemma
I-70
9 01,
uo*
on
G(u) - u’
on (0, T].
3.2,
+ (+ - 6)
&u(t))
I +
’ IG(u)12d7 rs
+
‘q,#(u)l s
dr + A4
‘qa(lul is
+ 1)dt
+ cp”(u(s))
for any 0 5 s I t 5 T, where ~~(7) = (1/~5)la’(r)l~ + 1/3’(7)1 for functions p(7) := /3,(t) in condition (~1). Also by condition (G2),
(y(r) := q(7)
(5.2) and
is
for any 0 I s I t I
T. Besides,
by lemma
3.1 and its corollaries,
f A4 (t -
.rs
~(luj
+ 1)dr
I A4 (t - s) ’ IG(u)j2d7 i ‘I’s
+ lu(s)l + A; + 1
‘qsd7, s
(5.4)
wA14&-T0,r];ff) + l~*l3-,,,t;H)I 5
(t - s)k,
’ IG(u)12 dr + t 1~’ - Wd2d~ is (1 s
+ t&,,sl;~~
+ b4*1:2(-,o,s;~j + 1 > (5.5)
and ul’(W))
2
Id@u))l - (t -
4k2 (1;
kWl”d7)
for any 0 I .s I t I T, where k, (LC~), k2 are positive Substituting estimates (5.5) into (5.3), we obtain
(1 - (t - s)k,)
constants
-
k2(149)1
independent
’ IG(~)1~dr 1s
5 (v + (t - s)k,)
’ Id - G(u)12 dt + (t - s)k,b”“(u)b(s,t) s
+
1)
(5.6)
of S, t, T and U.
Nonlinear
and furthermore,
multiplying
functional
875
by (* - 6)/[v
+ (t - s)kJ,
we get
11 v + - (t - 2)/k, MI1 .i f ,G(u),2d 5 s
6
1 -2
5
’ Id G(u)l’dr
6
(5.2)
+ ~k,/~‘~‘(~)]~~~~,~~
+ &
‘!,,dr S
To + Id%41L-(-70,s) + 1. ,:Iq , + I~*IZ~(-,,,;H)
+ l&tTherefore, from A, > 0. n
inequalities
both sides of this inequality
( > ( >i -21
variational
with
(5.4)-(5.7)
we derive
(5.1)
for
some
constants
(5.7) A, > 0 and
Proof of theorem 1.2. Let u be a function from [0, T’) with 0 < T’ I TOinto H, and suppose that u is a solution of (CP; ZQ,) on [0, T] for any 0 < T < T’. Then we show that p”‘(u) E L”(0, T’) and G(U) E L’(O, T’; H). In fact, we do it in the following two cases. Case 1. If l&u(t))1
is ’ b ounded
as t t T’, then G(u) E L’(O, T’; H).
Proof. Since 0 < v < 1, we see that the coefficient of j: 1G(u) 1’ dr in (5.1) is positive if 6 > 0 is sufficiently small, and s and t are sufficiently close to T’. Therefore, (5.1) implies that G(u) E L2(0, T’; H). Case 2. If lfp’(u(t))( is unbounded Proof. By assumption, 6 such that
1
there are a sequence
(t,) with t, t T’ as n + ~0 and a positive
number
I~‘“W,NI = Id”(U)IL-(-T0, fn) 9 1 - (T’ - t&4,
(--> 2
and
6
as t t T’, then G(u) E L2(0, T’; H).
v + (T’ - t&l,
-(T’-t,)A,(I
+ i:O’dr)
-;>O
T'
1 > (T’ - t,J&
+ rll
vs dr
for n = 1,2, . . . . Then by (5.1), for a fixed m we see that ” (G(u))~ dr s tfn
is bounded
as n + 00,
which shows that G(u) E L2(0, T’; H). As is seen above, G(u) E L2(0, T’; H) in any case, hence it follows from the abstract result in [9, theorem 2.1.11 that u E W1S2(0, T’; H) and q”‘(u) E L”(0, T’). Thus u is a solution of (CP; q,) on [0, T’], so that by the local existence result the solution u can be extended beyond time T’, if T’ < TO. This fact completes the proof of theorem 1.2. n
N. KENMOCHIand T. KOYAMA
876
6. UNIQUENESS
OF SOLUTIONS
In such a nonlinear problem as (CP; u,) it might be not easy to give sufficient conditions on the memory term G(u) for uniqueness of solutions. In this section we give a result about uniqueness. We assume that (PI), (~2) and (Gl)-(G3) hold. 6.1. Suppose that there are a Banach space I/ with norm 1.(v and a seminorm [.I on Vsuch that D((o’) C Vfor any t E [0, T,], Vis a dense subspace of Hand [s] + 1.( is a norm on Y equivalent to 1.1V. Also, suppose that there are constants .eO> 0 and p L 1 such that
THEOREM
(z* - z*, 2 - Z) L EO[Z - ZIP for any t E (0, T,], z, z E D(p’), z* E i@‘(z) L > 0 there are constants C, with 0 I
(6.1)
and z* E a@(z). Further, suppose that C, < E,,, CL 2 0 and a nonnegative
for each function
yr. = yL(t) E L’(0, To) such that (G(u)(t)
-
G(Q)(t),
u(t)
-
a(t)) t
5 CL[U(f) - ii(t)y
+ CL
h(d .i -70
-
a(Wdr
+ y,Wb
- &&,,tl;~~
for a.e. t E [0, T], whenever 0 < T s T,, u, ti E X(T) and u*, ii* E L’(--r,, u*(t) E @‘(u(t)) and a*(t) E lhp’(ti(t)) for a.e. t E [--to, T] and l4W~+,,T;H)
+ le+“(~)It-(-70,T)
MV’+“,r;H)
+
+ Iu*lL+TO,r;H)
Proof.
Let u and ii be two solutions
5 L,
[0, T], 0 < T s To, provided
(6.3) that u0
of (CP; u,) on [0, T], and put OstsT
o(t) := lu - &([--T0,11;H)7 u*(t) :=
T, H) such that
l~,‘%Qhy-*o,T) + ldLq-70,T;H)5 L
then (0; u,) has at most one solution on any interval satisfies the same properties as in theorem 1.1.
(6.2)
u,*(t) i .G(u)(t)
for -
u’(t)
-7
<
t 5 0,
for 0 < t < T,
and n*(t) :=
for -r
u,*(t) L G(a)(t)
- ii’(t)
< t 5 0,
for 0 < t < T,
for a.e. t E [--to, T]. Note that o(t) = Ju - a)ccfo,rl;Hi) for all t E (0, T] and u*(t) E &&u(t)) and and ii*(t) E &&a(t)) for a.e. t E [--t,,, T]. Besides, note that a(O) = 0, 0 is nondecreasing 0 E WlY2(0, T), since 0 5 a’(t) 5 /u’(t) - ii’(t)1 for a.e. t E [0, T]. Now choose
a number a(s) = 0
which is obtained
as follows.
L > 0 so that (6.3) holds.
Then we see that
for small s > 0,
Let s be any number
in (0, T] and choose s’ E [0, s] to satisfy
a(s) = lu(s’) - ii@‘)].
Nonlinear
functional
variational
877
inequalities
We observe that ;$
[u(r)
-
ii(r)l2
+ (U*(T)
-
ii*(~),
u(t)
-
ii(r))
= (G(u(s))
-
G(ii(r)),
u(t)
-
ii(z))
(6.4)
for a.e. r E [0, T]. Integrating (6.4) over [0, s’], we get +]u(.s’) - ii(s’)j2 +
” (u* - a”, u - ii) ds = -” (G(u) - G(ii), u - ii) dr. I0 i0
(6.5)
From (6.5), with conditions (6.1) and (6.2), it follows that ns’ 5’ [U - iilp dr I +o(s)2 + (Eo - c, - Cis’) YL(r)o(r)2 dr i0 i0 s
s yL
I
(s)(T(s)~
dr 5 a(~)~
3 0
yr. (7)
dr.
I 0
Here, if s is sufficiently close to 0, then (co - C, - CLs’) > 0 and
and hence c = 0 near s = 0. Thus u(r) = ii(t) for small t > 0. Consequently, by repeating the same argument as above. n
u = ii on [0, 7’1
Example 6.1. Let us consider the same type of problem as (2.1) in example 2.1. Here, suppose that G is given by formula (2.7), and in addition to (fl)’ and (f2)’ with p = 2, the functions fk = fk(t, x, s, l, c) satisfy the following condition.
For each number A4 > 0 there is pM E L2(0, T,) such that
Condition.
Ifk(f, x, s, r, 0 - fk@, x, s, r; L)I 5 PM(f)~k - 41 + I< - 01
(6.6)
- -
for a.e. (t, x, s) and all 5, 5, c, c E R with j?j], Jr], Ifj, ]fl I M. Then problem (2.1) has at most one solution by theorem 6.2 for p = 2, I/ = W1,2(0, 1) and In fact, inequality (6.1) clearly holds, and (6.2) is seen as follows. Let L > 0 be bl = ldL~(O,l). any number, and choose a number M > 0 so that ~~~~~~~~~~~~~~~~~~~ I M for any u E X(T) satisfying (6.3). In this case, for any two functions U, ii E X(T) satisfying (6.3), we have (G(u)(t) - G(a)(t), u(t) - a(t)) 1 “I =
(h(t,
!.I0
1’
=:
r,(t)
s,
u(t,
4,
NJ,
x)1
-
fo(t, x, s, a(t, -4, w, x)))(u(t, x) - o(t, 4) d.9dx
f
(fi (t, x, s, u(t, x), 4s, x))Q,
+
x
x,
-70
ii 0
-70
(u(t,
x)
+
-
I,(f).
zqt,
x))
d.s
dx
x) - fi (t, x, s, w, x)9 m, X))%(h x))
N. KENMOCHI and T. KOYAMA
878
From
(6.6) we see that 1
t 1, Cl) 5 ,&4(f)
5
Moreover,
&,(~)(T
(IW,
4
-
u(t, x)l
+
lu(s,
4
-
ii(s, x)lW,
XI -
@Cl, XII dxb
<1-7”
Glu
-
(6.7)
~I&-,0,t);L2(o,1))~
from (6.6) again,
t
1 (Ifi ct,x, s, u(tvXL f4s,4) - “6tt, x, s, act,4, w, ml I%(%41 I,(f) 5 1 -70i 0
Now we note that &,,(t)
5 const. k(t)ln
since
(6.8)
- 4&l-a,t1;L2~o,1n9
”I “I lu% ! -70
,) d.s 5 const. *)I Lmco,
5
!
const.((T
(I%(% *)lL%,1)+ IMh .h(O,I,>ds -To + ro)L1’2 + (T + ro)1’2L).
Besides, &,2(f)
5 :
where kL is a positive constant depending inequality of the form (6.2) holds.
only on L. We see from estimates
(6.7)-(6.9)
that an
Example 6.2. Consider the same problem as (2.6) with p = 2 in example 2.2; suppose that p = 2, G is given by (2.7) and condition (6.6) is satisfied as well. Moreover, suppose that f, (t, x, s, r, c) is independent of t, x, s and r, i.e. f, = fi (0. Then problem (2.6) has at most one solution by theorem 6.2, in the case wherep = 2, V = W,‘V2(0, 1) and [z] = (z~[~z~~,~). In fact, let L > 0 be any number and choose A4 > 0 in the same way as in example 6. I. We then have
879
Nonlinear functional variational inequalities
for any functions
U, ti E X(T)
satisfying
(G(u)(t)
- G(D)(t), u(t) - a(t)) 5 Zr(t) + &2(f),
where Zr(t) is the same function 1 &
(6.3)
as in (6.7) and
t
=
u-i
110
(4.c
x))u,(.%
4
-
fi
(a.
x)h%(s,
x)))(u,
ct,
x)Mt,
XI
-
iict,
4)
ds
d-x
-70 “1
t
ii 0
-70
=--
(fl
with the primitivefr
offi Il;Wl
(MS,
4)
-
fl
(ii
satisfyingyr(0) 5 VI&(0
6,
4
-
Rx
(f,
x))
ds
dx
= 0. Therefore,
for any positive
- ~XWlt~(o,l) + CV&
- 4&-,,,M2~0,1))
where CM is a constant depending only on v and M. From estimates that an inequality of the form (6.2) holds in the case of p = 2. 7. VARIATIONAL
INEQUALITIES
WITH
number
v > 0,
for I, (t) and I2 (t) we see
HYSTERESIS
Let us consider variational inequalities of the forms (2.1) and (2.6) with nonlinear source terms G described by hysteresis functionals. Let f, , fdE C(R) be monotone nondecreasing and Lipschitz continuous functions such that and put
Also, let w. E R such that {(c, Now, we define a hysteron operator X(=X(*; wo)): C([O, T]) + C([O, T]), 0 < T < m, associated with f,, fd and w, as follows. Firstly we define X on the set of all piecewise linear functions PLC([O, T]) := (5 E C([O, T]); there exists a partition 0 = to < t, < ... < t, = T of [0, T] such that r is linear on each [ti_, , ti], i = 1,2, . . . , n) by the formula minffd(W)), x(r)(t)
=
if t = 0,
~~~1~
maxffuW-N
minffd(t(thmaxIf,WN, W3(L1)JL
for any r E PLC([O, 1,2 ,..., n.
if t e [ti-l,
T]) with 0 = to < t, < ... < t, = T such that
til,
i=
r is linear
1,2 ,..., on [ti_l,
n. t;],
i=
LEMMA 7.1. For any rr, & E PLC([O,
holds,
and hence IWrr)
holds,
T]),
where L is a common
- WZ)IC([O,T])
Lipschitz
constant
5 Zk, of
-
&([O,T])~
fd and f, .
(7.2)
880
N.
KENMOCHI
T.
and
KOYAMA
Proof. In order to get (7.1), it is sufficient to show the following fact. If ?ji, c2 E C([O, T]) are monotone functions, and if w~,~, w~,~ E R satisfy (ri(O), wi,J E M, i = 1, 2, then the functions i= 1,2, wi(t) := mWfd(ti(t)), mdf,(ti(t)), wi,0119 satisfy the inequality wi(t)
- ~~(0
5 max]w,,,
- w2,0,
fa(rl(t))
-
fo(r2(f)),fd(rl(f))
-
To show this, we note for each i = 1,2, that (i) if Pi := (C(t), wi(t)) E M” := M\(D U A), then Wi(t) = wi.0, (ii) if Pi E D\A, then fd(ri(f)) = wi(t),
(2) if P2 E M\D
w2(f)
‘fd(
and P, E A, then w,(t) Wl
(3) if P2 E M\D
-
(t)
-
and P, E M\A,
%
Cl)
= 5
(7.3) is proved
f,(r,
then wi(t)
in any case, whence
0))
(7.3)
~i,~, ~i,~,
fd(t;2(f)k
and thus
(t))
fa (Cl
WI(t) - %(f) therefore
-
fd(r2(f))l.
-
f,(&
(t));
I w~,~, wz(t) 2 w2,0 and thus 5
w1.0
-
w,,,;
(7.1) and (7.2) are obtained.
W
By (7.2), the operator X(=X(.; w,)): PLC([O, T]) + C([O, T]) has a unique extension, denoted by X(=X( *, wo)) again, from C([O, T]) into itself, and it is a Lipschitz continuous operator in C([O, T]) with Lipschitz constant L. Now let us define a mapping G: U,, Tc ToX(T) -+ Uo, Ts =,L2(0, T, L2(0, 1)) in terms of the hysteron X, where X(T) = C([O, T] x [0, l]), as given in the following example. Example 7.1. (i) Let w. E L2(0, 1) such that (([, w,(x)); r E R] fl A4 # 6 for a.e. x E (0, l), and f. E L2(Qo), Q. := (0, T,) x (0, 1). Then, for 0 < T I To, we put G(u)(t, x) := fo (t, xW(u(
* , xl;
wo (x))(t)
(7.4) for a.e. (t, x) E (0, T) x (0, 1) The definition (7.4) of G makes fo E Lm(Qo),then
and any u E X(T).
sense as an operator
iGO4 - G(&-~~o,T~x (o,l))5
Ifoirx&U
for any U, u E X(T).
from X(T)
-
into L2(0, T; L’(O, 1)). If
&([o,TI
x [O,II)
(7.5)
Nonlinear functional variational inequalities
7.1. (ii) Let W, E
Example
881
R such that ((<, w,); c E RJ f~A4 # 4, and f0 E L2(Q,). Then we
define G(u)(t, x) : = A, (f, xWO4 * 1xo); wJ(t) (7.6) for a.e. (t, x) E (0, T) x (0,1)
and any u E X(T),
where x0 is a prescribed point in (0, 1). This definition (7.6) makes sense and (7.5) holds true just as in the case of (7.4). In the remainder of this section we discuss problems (2.1) and (2.6) with r, = 0 and the operator G given by (7.4) or (7.6). (a) Let G be the operator given by (7.4) with& E L”(Q,,), and let (~‘1 be the family of proper 1.s.c. convex functionals on L2(0, 1) given by (2.3) [resp. (2.8)]. Let u0 E D(p’). Then (CP; u,), corresponding to problem (2.1) [resp. (2.6)], has one and only one solution on [O, &I. PROPOSITION 7.1.
PROPOSITION 7.1. (b) Let G be the operator given by (7.6) with_&, E L”(Q,), and let {v’) and u. be as in (a). Then (CP; u,), corresponding to problem (2.1) [resp. (2.6)] has one and only one solution on [0, To].
Proof. In any case, it is easy to verify conditions (Gl)-(G3), and the conditions for global existence of solutions in theorem 1.2. Therefore, the existence of a solution to (CP; uo) on [0, T,] is a direct consequence of our abstract theorems in Section 1. Next, we will show the uniqueness of solutions of (CP; u,) only in the case corresponding to problem (2.6) with G defined by (7.4), since the proof of uniqueness in any other case is quite similar. Let ui and u2 be any two solutions of (CP; u,) on [0, To]. Then, using the characterization (2.9) of a@, we see that for sufficiently large q E R, 1 @I,,@)
i
-
W,Wb,
(f)
-
~2(0l~-?~,
(t)
-
~2
(t))
k
0 1 =
k,(f)
r
Jo 5 -
WMtNiu,W
-
s
@)fl% I~l,xu)lq-2%,x
:
+
=-
-
.i
; 14,x(f)lq-2~1,x
(f)
~2(01~-~#41(0
-
Web,
~2(olq-2041
-
U2(olq-2(U2(t
U)
-
g(f))
-
@2(t)
-
g(t))), dx
-
g(f)))
1 - g(f))l, dx
r1
I IqxwlP-2~l,x
= -44
WI%(t)
o
-
1)
l l~I,xw-2 50
%,,(0b,
-
~2(olq-2Mo
(r)
-
-
~2wlq-2@l,,(o
UZ(f)Nx
dx
-
u2,xw
dx
dx
N. KBNMOCHI
882
and T. KOYAMA
and similarly
Therefore
5
I(W,)(0
-
C(UZ)(~))~L-((O,T)X(O,~))IU~(~) -
~2(~)1&;,,~>
for a.e. f E [0, T] and any 0 < T 5 To; note here that G(u,), G(u~) E Lm(Qo), since u,, u2 E C([O, &] x [0, I]). From this with the Lipschitz continuity (7.5) of G it follows that
andanyO<
for a.e. t e [0, T]
TI
To,
where L’ = Ifolt~cQ,,L. Hence
l%(f) - M)lLQ(O,I)--= TJ+Q - %lC~~O,Tl x [O, 11) for all t E (0, T]
and
O
Letting 4 + W, we have I%(0 -
~2WlC([O,I~)
for all t E [0, T] Accordingly,
5
TL’l% and
%I.((o,Tlx[o,lj)
0<
Ts
To.
u, = uz for near f = 0, and it results consequently that u, = u2 on [0, q].
It
Remark 7.1. For detailed studies on various hysteresis functionals we quote [IO] and [14], and for some related works, [lo, 13-181 on evolution equations involving hysteresis functionals in boundary conditions or source terms as systems of feedback control. Also, in the forthcoming paper [19], existence and uniqueness questions for more general variational inequalities with hysteresis in multi-dimensional spaces will be discussed.
Nonlinear functional variational inequalities
883
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