FOR NONLINEAR EVOLUTION INCLUSIONS1

YOKOHAMA MATHEMATICAL JOURNAL VOL. 43, 1995

EXISTENCE AND STRONG RELAXATION THEOREMS FOR NONLINEAR EVOLUTION INCLUSIONS1 By NIKOLAOS S. PAPAGEORGIOU and NASEER SHAHZAD (Received February 26, 1995;

Revised August 30, 1995)

Abstract. In this paper we study nonlinear evolution inclusions defined on a Gelfand triple of spaces. First we prove an existence and compactness result for the set of solutions of the “convex” problems. Then we look at extremal solutions and show that under reasonable hypotheses such solutions exist. Moreover if the orientor field (multivalued perturbation term) is h-Lipschitz in the state-variable, we show that the set of extremal solutions is dense in the solution set of the convexified problem (“strong relaxation theorem”). We also show that the solution set is compact in $C(T, H)$ if and only if the orientor field is convex-valued. Finally we present two examples of parabolic distributed parameter systems which illustrate the applicability of our abstract results.

1. Introduction One of the central results in the theory of evolution inclusions (i.e. evolution equations with multivalued terms) is the “relaxation theorem”. It says that if the orientor field (i.e. multivalued perturbation term) is h-Lipschitz in the state variable, then the solution set of the original problem is dense in that (i.e. the system obtained by replacing the orientor of the convexified field by its closed convex hull). Such a result is important in control theory in connection with the ”bang-bang principle”. Roughly speaking it tells us that we can produce a control system with essentially the same reachable sets, by economizing in the set of admissible controls. In this paper we prove a strong version of the relaxation theorem, in which the approximating trajectories are ”extremal solutions” (i.e. solutions moving through the extreme points of the orientor field). First we prove a general existence result for the convexified problem under very general conditions on the orientor field. Then we establish the existence of extremal trajectories, under the stronger hypothesis that the orientor field is h-continuous in . Subsequently by strengthening the hypothesis further to h-Lipschitzness, we show that the extremal trajectories are $pro_{\backslash }blem$

$x$

1991 Mathematics Subject Classification: $34G20$ , 35R70 Key words and phrases: evolution triple, compact embedding, evolution inclusion, extremal -set, parabolic distributed parameter system. solution, relaxation, Choquet function, lThis work was done while the authors were visiting the Florida Institute of Technology. $G_{\delta}$

2Revised version.

74

N. S. PAPAGEORGIOU AND N. SHAHZAD

dense in those of the convexified problem. A two-dimensional counterexample due to Plis [11], shows that simple h-continuity in the x-variable is not enough to guarantee the validity of the relaxation theorem. Finally we show that the solution set is compact in $C(T, H)$ if and only if the orientor field is convexvalued. In the last section we present examples of parabolic distributed parameter systems.

2. Preliminaries Let be a measurable space and $X$ a separable Banach space. Throughout this paper we will be using the following notations: $(\Omega,\Sigma)$

$P_{f(c)}(X)=$

and

$P_{(\omega)k(c)}=$

{

{ $A\subseteq X:nonempty$ , closed, (convex)} : nonempty, (weakly-) compact, (convex)}.

$A\subseteq X$

A multifunction (set-valued function) is said to be measurable if for every $x\in X,$ $\omega\rightarrow d(x, F(\omega))=\inf\{\Vert x-y\Vert:y\in F(\omega)\}$ is measurable. Now let be a $\sigma- finite$ measure on , we . By denote the set of selectors of which belong in the Lebesgue-Bochner space , i.e. $S_{F}^{p}=\{f\in L^{p}(\Omega, X):f(\omega)\in F(\omega)\mu-a.e.\}$ . In general this set may be empty. However, if satisfies $GrF=\{(\omega,x)\in\Omega\times X$ : $x\in F(\omega)\}\in\Sigma\times B(X)$ with $B(X)$ being the Borel of $X$ (i.e. is graph measurable), then if and only if . Note that $forP_{f}(X)$ -valued multifunctions measurability implies graph measurability, while the converse is true if is -complete. Also remark that is decomposable, i.e. for every , we have with being the characteristic function of a set . On $P_{f}(X)$ we can define a generalized metric, known in the literature as the ”Hausdorff metric” by setting, for $A,$ $C\in P_{f}(X),$ $h(A,C)=\max\{\sup_{a\in A}d(a,C),$ $F:\Omega\rightarrow P_{f}(X)$

$(\Omega,\Sigma)$

$\mu(\cdot)$

$S_{F}^{p},$

$ 1\leq p\leq\infty$

$F(\cdot)$

$L^{p}(\Omega,X)$

$F:\Omega\rightarrow 2^{X}\backslash \{\phi\}$

$\sigma- field$

$F(\cdot)$

$\omega\rightarrow\inf\{\Vert x\Vert:x\in F(\omega)\}\in U(\Omega)$

$ S_{F}^{p}\neq\phi$

$\Sigma$

$S_{F}^{p}$

$\mu$

$(A,f_{1},f_{2})\in\Sigma\times S_{F}^{p}\times S_{F}^{\rho}$

$x_{4}f_{1}+x_{A^{c}}f_{2}\in S_{F}^{P}$

$ A\in\Sigma$

$x_{A}$

$\sup_{c\in C}$

$d(c,A)\}$

. It is well-known that

$F:X\rightarrow P_{f}(X)$

is complete. A multifunction is said to be h-continuous (resp. h-Lipschitz), if it is continuous $(P_{f}(X),h)$

into $(P_{f}(X),h)$ . be Hausdorff topological spaces and

(resp. Lipschitz) from

$X$

Let . We say that is upper-semicontinuous (usc) (resp. closed), if for every closed $G^{-}(C)=\{y\in Y:G(y)\cap C\neq\phi\}$ is closed (resp. $GrG=\{(y,z)\in Y\times Z:z\in G(y)$ ) is closed). For a $P_{f}(Z)$ -valued multifunction upper-semicontinuity implies closedness, while the converse is true if is compact in Z. We say that (lsc), if for every is lower-semicontinuous closed $G^{+}(C)=\{y\in Y$ : is closed. For details we refer to DeBlasi-Myjak [2] and KleinThompson [8]. $Y,$

$Z$

$G:Y\rightarrow 2^{z}\backslash \{\phi\}$

$G(\cdot)$

$C\subseteq Z$

$\overline{G(Y)}$

$G(\cdot)$

$C\subseteq Z$

$G(y)\subseteq C\}$

75

NONLINEAR EVOLUTION INCLUSIONS

Now let $H$ be a separable Hilbert space with norm denoted by . . Let $X$ be a separable reflexive Banach space which embeds continuously and densely into $H$ . Identifying $H$ with its dual (pivot space), we have with all embeddings being continuous and dense. Such a triple of spaces is known in the literature as “evolution triple” or ”Gelfand triple”. We will also assume that the embeddings are compact. This is often the case in applications where evolution ) we will denote the (resp. triples are generated by Sobolev spaces. By $X\rightarrow H\rightarrow X^{*}$

$\Vert\cdot\Vert$

$\Vert\cdot\Vert_{*}$

we will denote the duality brackets for Also by ) and by (., ) the inner-product of $H$ . The two are compatible in the pair (X, and define the sense that . Let $T=[0, b]$ and $W_{pq}(T)=\{x\in U(T,X):\dot{x}\in L^{q}(T,X^{*})\}$ The derivative involved in this definition is understood in the sense of vector-valued distributions. Equipped with the

norm of

$X$

(resp. of

$X^{*}$

).

$\langle\cdot,\cdot\rangle$

$X^{*}$

$\cdot$

$10$ and a quasinorm on $X$ such

$(ii)x\rightarrow A(t,x)$

$|\mu(t,x)\Vert_{*}\leq a_{1}(t)+c_{1}\Vert x\Vert^{p-1}a.e$

(iv)

$\langle A(t,x),x\rangle\geq c[x]^{p}a.e$

$[x]+\lambda|x|\geq\gamma\Vert x\Vert$

for some

$\lambda,\gamma>0$

$a_{1}\in L^{q}(T),$

$x$

$[\cdot]$

.

is a multifunction such that (i) is measurable, (ii) $GrF(t,.)$ is sequentially closed in (here denotes the Hilbert space $H$ furnished with the weak topology), (iii) $|F(t,x)|=\sup\{|v|:v\in F(t,x)\}\leq a_{2}(t)+c_{2}|x|a.e$ . with $a_{2}\in L^{2}(T),c_{2}>0$ }. $\underline{H(F)}:F:T\times H\rightarrow P_{fc}(H)$

$t\rightarrow F(t,x)$

$H\times H_{\omega}$

$H_{\omega}$

Theorem 1. If hypotheses $H(A),$ is nonempty and compact.

$H(F)$

hold and

$x_{0}\in H$

then

$P_{c}(x_{0})$

$\subseteq C(T,H)$

Proof. We start by deriving some a priori bounds for the elements in . So let . By definition we have $x(t)+A(t,x(t))=f(t)$ , $x(O)=x_{0}$ with . Multiply this equation with and recall that there , to get: exists such that $P_{c}(x_{0})$

$x\in P_{c}(x_{0})$

$a.e.$

$f\in S_{F(\cdot,x())}^{2}$

$\beta_{1}>0$

$x$

$\Vert x\Vert_{*}\leq\beta_{1}|x|$

$\langle x(t),x(t)\rangle+\langle A(t,x(t)),x(t)\rangle=\langle f(t),x(t)\rangle$

$a.e$

.

$\Rightarrow$

$\frac{1}{2}\frac{1}{dt}|x(t)|^{2}+c[x(t)]^{p}\leq\Vert f(t)\Vert_{*}\Vert x(t)\Vert\leq\hat{\beta}_{1}|f(t)|\cdot[x(t)]+\lambda\hat{\beta}_{1}|f(t)|\cdot|x(t)|$

with

$\hat{\beta}_{1}=\frac{\beta_{1}}{\gamma}$

.

From Cauchy’s inequality with

$\epsilon>0$

$a.e$

.

we get

$\hat{\beta}_{1}|f(t)|[x(t)]\leq\hat{\beta}_{1}\frac{\epsilon^{p}}{p}[x(t)]^{\rho}+\hat{\beta}_{1}\frac{1}{\epsilon^{q}q}|f(t)|$

In addition from classical Cauchy’s inequality we have $\lambda\hat{\beta}_{1}|f(t)||x(t)|\leq\frac{1}{2}\lambda\hat{\beta}_{1}|f(t)|^{2}+\frac{1}{2}\lambda\hat{\beta}_{I}|x(t)|^{2}$

So after integration over

$[0, t]$

,

.

we get

$\frac{1}{2}|x(t)|^{2}+c\int_{0}^{t}[x(s)]^{\rho}ds\leq\frac{1}{2}|x_{0}|^{2}+\hat{\beta}_{1}\frac{\epsilon^{p}}{p}\int_{0}^{t}[x(s)]^{p}ds+\hat{\beta}_{1}\frac{1}{\epsilon^{q}q}\int_{0}^{t}|f(s)|^{q}ds$

$+\frac{1}{2}\lambda\hat{\beta}_{1}\int_{0}^{t}|f(s)|^{2}ds+\frac{1}{2}\lambda\hat{\beta}_{1}\int_{0}^{t}|x(s)|^{2}ds$

Let

$\epsilon>0$

we have

be such that

$\hat{\beta}_{1}\frac{\epsilon^{p}}{p}=\frac{c}{2}$

. Also note that since

.

$2\leq p$

(hence

$10$

which we have $\frac{1}{2}|x(t)|^{2}+\frac{c}{2}\int_{0}^{t}[x(s)]^{\rho}ds\leq\frac{1}{2}|x_{0}|^{2}+\gamma_{1}\int_{0}^{t}|f(s)|^{2}ds+\gamma_{1}\int_{0}^{t}|x(s)|^{2}ds$

Because of hypothesis $H(F)$ (iii) we have we can find some such that

.

$|f(s)|^{2}\leq 2a_{2}(s)^{2}+2c_{2}|x(s)|^{2}a.e.$

.

So

$\gamma_{2}>0$

$\frac{1}{2}|x(t)|^{2}+\frac{c}{2}\int_{0}^{t}[x(s)]^{p}ds\leq\gamma_{2}+\gamma_{2}\int_{0}^{t}|x(s)|^{2}ds$

Through Gronwall’s lemma, we see that there exists $|x(t)|\leq M_{I}$

for every

$t\in T$

and every

From (2) and (3) above we get the existence of $\int_{0}^{b}[x(s)]^{p}ds\leq M_{2}$

From (3) and (4) above and since $M_{3}>0$

$x\in P_{c}(x_{0})$

.

(3)

such that

$x\in P_{c}(x_{0})$

$\gamma\Vert x\Vert\leq[x]+\lambda|x|$

(2)

such that

$M_{1}>0$

$M_{2}>0$

for every

.

.

(4)

and from

$H(A)$

(iii)

we get an

such that

$\Vert x\Vert L^{\rho}(T.X)\leq M_{3}$

and

$\Vert x\Vert L^{q}(T,X)\leq M_{3}$

for every

$x\in P_{c}(x_{0})$

.

(5)

Thus without any loss of generality we may assume that $|F(t,x)|\leq\alpha_{2}(t)+$ $c_{2}M_{1}=\psi(t)a.e$ . with $\psi\in L^{2}(T)$ . Otherwise replace $F(t,x)$ by $F(t,r_{M_{1}}(x))$ where is the -radial retraction on $H$ . Let $K=\{g\in L^{2}(T,H):|g(t)|\leq\psi(t)a.e.\}$ . $r_{M_{1}}(\cdot)$

$M_{1}$

Furnished with the relative weak-L2 $(T,H)$ topology $K$ becomes a compact metrizable space. In what follows this is the topology considered on $K$ . Let $R:K\rightarrow P_{fc}(K)$ be the multifunction defined by , where $p:L^{2}(T,H)\rightarrow C(T,H)$ is the map which to each $g\in L^{2}(T,H)$ assigns the unique solution of $\dot{x}(t)+A(t,x(t))=g(t)a.e.,$ $x(O)=x_{0}.We$ claim that GrR is sequentially closed in . To this end let $[g_{n},f_{n}]\in GrRn\geq 1$ and assume that in . From the same a priori estimation as in the beginning of this proof, we get that is bounded in $W_{pq}(T)$ , thus relative compact in $L^{p}(T,H)$ . So by passing to a subsequence if necessary we may assume that $p(g_{n})(t)\rightarrow p(g)(t)a.e$ . in $H$ . Using theorem 3.1 of [9] and hypothesis $H(F)(ii)$ we get $f(t)\in F(t,p(g)(t))a.e.$ . So $[g,f]\in GrR$ and hence GrR is sequentially closed in . Since $K$ (with the relative weak-L2 $(T,H)$ topology) is compact metrizable, we can apply the Kakutani-Ky Fan fixed point theorem (see for example Klein-Thompson [8]) and get $f\in R(f)$ . Evidently . Now we will show that is compact in $C(T,H)$ . To this end let $R(g)=S_{F\langle\cdot,p1g)\langle\cdot))}^{2}$

$K\times K$

$\rightarrow[g,f]$

$[g_{n},f_{n}]$

$K\times K$

$\{p(g_{n})\}_{n\geq 1}$

$K\times K$

$ p(f)\in P_{c}(x_{0})\neq\emptyset$

$P_{c}(x_{0})$

N. S. PAPAGEORGIOU AND N. SHAHZAD

78 $\{x_{l}\}_{n\geq 1}\subseteq S_{c}(x_{0})$

compact in $W_{\rho q}(T)$

. We

know that

, Note that

$x_{n}=p(f_{n}),$

is bounded in

. So we may assume that

$L^{2}(T,H)$

$n\geq 1$

So

$x\in W_{0}$

$W_{pq}(T)$

in and

, hence

relatively

in is convex closed and thus $x(O)=x_{0}$ . Also let such $x_{n}\rightarrow x$

$x_{\iota}\in W_{0}=\{y\in W_{pq}(T):y(O)=x_{0}\}$

(hence weakly closed).

that

$\{x_{\iota\iota}\}_{n\geq 1}$

$L^{2}(T,H)$

and

$x_{l}\rightarrow^{\omega}x$

$W_{0}$

$f_{n}\in S_{F\langle\cdot.x_{1}\langle\cdot))}^{2}$

. We may assume that

in

$f_{n}\rightarrow^{\omega}f$

.

$L^{2}(T,H)$

Let

$\hat{A}(x)(\cdot)=A(\cdot,x(\cdot))$

.

be the Nemitsky operator corresponding to $A(t,x)$ , Moreover let be the duality brackets for the pair

$(L^{\rho}(T,X),L^{q}(T,X^{*}))$

.

We have

$\hat{A}:L^{\rho}(T,X)\rightarrow L^{q}(T,X^{*})$

i.e.

$((\cdot,\cdot))$

$((x_{n},x_{n}-x))+((\hat{A}(x_{n}),x_{n}-x))=((f_{n},x_{n}-x))$

From the integration by parts formula for functions in p. 422) we have

.

(6)

$W_{\rho q}(T)$

(see

Zeidler [17], (7)

$((x_{n},x_{n}-x))=\frac{1}{2}|x_{n}(b)-x(b)|^{2}+((x,x_{n}-x))$

Replacing (7) in (6) we get that $((\hat{A}(x_{n}),x_{n}-x))\leq((f_{n},x_{n}-x))-((x,x_{n}-x))$

Also

$((f_{n},x_{n}-x))=(f_{n},x_{n}-x)_{L^{2}\langle T,H)}$

by

we denote the inner

$(\cdot,\cdot)_{L^{2}(T.H)}$

. So we get $\xi_{n}(t)=\langle A(t,x_{n}(t))-A(t,x(t)),x_{n}(t)-x(t)\rangle\geq 0$ . From Fatou’s

product in the Hilbert space let

(here

.

$L^{2}(T,H))$

$\varlimsup((\hat{A}(x_{n}),x_{n}-x))\leq 0$

lemma

. Now

we have

$0\leq\int_{0}^{b}\varliminf\xi_{n}(t)dt\leq\varlimsup\int_{0}^{b}\xi_{n}(t)dt=\varlimsup[((\hat{A}(x_{n}),x_{n}-x))-((\hat{A}(x),x_{n}-x))]\leq 0$

hence

$\xi_{n}\rightarrow 0$

in

$L^{1}(T)$

and so we may assume that

$\xi_{n}(t)\rightarrow 0$

a.e. . We have

$ c_{3}[\Vert x_{n}(t)\Vert^{\rho}+||x(t)\Vert^{\rho}]-c_{4}-\Vert x_{n}(t)\Vert[a_{1}(t)+c_{1}\Vert x(t)\Vert^{\rho- 1}]-\Vert x(t)\Vert[a_{1}(t)+c_{1}\Vert x_{n}(t)\Vert^{\rho- 1}]a.e\xi_{n}(t)\geq c[[x_{n}(t)]^{\rho}+[x(t)]^{\rho}-||x(t)\Vert[a_{1}(t)+c_{1}||x(t)I|^{\rho- 1}]-||x(t)\Vert[a_{1}(t)+c_{1}\Vert x_{n}(t)\Vert^{\rho- 1}]\geq$

. for some $c_{3},c_{4}>0$ . From the above inequality we deduce that is bounded in . So we can apply corollary 4, p. 85 of Simon [14] and conclude that is relatively compact in $C(T,H)$ . Thus in $C(T,H)$ . and $x=p(f),$ , i.e. $C(T,H)$ . So is compact in Q.E.D. . $\{x_{n}\}_{n\geq 1}$

$L^{\infty}(T, H)$

$\{x_{l}\}_{n\geq 1}$

$x_{l}\rightarrow x$

$f\in S_{F\langle\cdot,x(\cdot))}^{2}$

$x\in P_{\iota}(x_{0})$

$P_{c}(x_{0})$

A careful reading of the above proof reveals that the following result holds:

Corollary 2.

If hypothesis

$H(A)$

holds and

which to each $g\in L^{2}(T,H)$ $x(t)+A(t,x(t))=g(t)a.e.,$ $x(O)=x_{0}$ , is compact. $p:L^{2}(T,H)\rightarrow C(T,H)$

then the solution map assigns the unique solution of

$x_{0}\in H$

,

79

NONLINEAR EVOLUTION INCLUSIONS

4. Extremal solutions In conjunction with (1) we also consider the following problem: $x(t)+A(t,x(t)\in extF(t,x(t)a.e$

(8)

$x(0)=x_{0}$

we denote the extreme points of the orientor field $F(t, x)$ . we will denote the solution set of (8). In this section we establish the

Here by

$extF(t, x)$

By . nonemptiness of In what follows let $P_{e}(x_{0})$

$P_{e}(x_{0})$

equipped with norm

$L_{\omega}^{1}(T,H)$

denote the Lebesgue-Bochner space

$|\triangleright\Vert_{\omega}=\sup[\Vert\int_{s}^{t}x(\tau)d\tau\Vert:0\leq s\leq t\leq b]$

will need the following simple auxiliary result. Lemma 3. and

(the

$\{f_{n},f\}_{n\geq 1}\subseteq L^{2}(T,H),$ $f_{n}^{II}4f$

$L^{2}(T,H)$

weak norm”). We then

$0$ . (i)

$t\rightarrow F(t,x)$

Theorem 4.

If hypotheses

$H(A),$

$x\rightarrow F(t,x)$

$H(F)_{1}$

is h-continuous, (iii)

hold and

$x_{0}\in H$

,

then

$|F(t,x)|\leq$

$P_{e}(x_{0})$

is

nonempty.

Proof. As in the proof of Theorem 1, we may assume without any loss of generality that $|F(t,x)|\leq\psi(t)$ $a.e$ . with $\psi\in L^{2}(T)$ . Again set $K=$ $\{g\in L^{2}(T,H):|g(t)|\leq\psi(t) a.e.\}$ and let From corollary 2 we . Let know that is compact in $C$ $(T, H)$ . Hence so is $R:V\rightarrow P_{akc}(L^{1}(T,H))$ be defined by . Invoking theorem 1.1 of a continuous map such that Tolstonogov [15] we get $V_{0}=\overline{p(K)}^{C(T,H)}$

$V=\overline{conv}V_{0}$

$V_{0}$

$R(y)=S_{F\langle\cdot,y\langle\cdot))}^{1}$

$r:V\rightarrow L_{\omega}^{1}(T,H)$

N. S. PAPAGEORGIOU AND N. SHAHZAD

80

. From Benamara [1] we know that . Let $u=p\circ r:V\rightarrow V$ . Using Lemma 3 and Corollary 2 we

$r(y)\in extR(y)=extS_{F(.v(\cdot))}^{1}$

ext

$S_{F(\cdot,v())}^{1}=S_{extF\langle.y())}^{1}$

see that

$u(\cdot)$

Evidently

for every

$y\in V$

is compact. Thus via Schauder’s fixed point theorem we get $x=u(x)$ . Q.E.D. .

$ x\in P_{e}(x_{0})\neq\emptyset$

5. Strong relaxation theorem In this section we stablish the density $ofP_{e}(x_{0})$ in for the $C(T, H)-$ norm and in addition that is a -subset of . To prove this last property we will have to use the Choquet function associated with the orientor field $F(t, x)$ . More specifically let be a sequence which is dense in the $H$ unit sphere of . Following DeBlasi-Pianigiani [3], [4] we define : by $\overline{e}$

$P_{c}(x_{0})$

$P_{e}(x_{0})$

$P_{e}(x_{0})$

$G_{\delta}$

$\{z_{k}\}_{k\geq 1}$

$\gamma_{F}$

$T\times H\times H\rightarrow R\cup\{+\infty\}$

$\gamma_{F}(t,x,v)=\left\{\begin{array}{ll}\sum_{k\geq 1}2^{-k}(z_{k},v)^{2} & ifv\in F(t,x)\\+\infty & otherwise\end{array}\right.$

Let $Aff(H)=$ {set of all continuous affine functions $a:H\rightarrow R$ }. Let $=\inf$ { $a(v):a\in aff(H)$ and $a(z)\geq\gamma_{F}(t,x,v)$ for every $z\in F(t,x)$ } (as usual we adopt the convention that ). Then the Choquet function : is defined by $\delta_{F}(t,x,v)=\hat{\gamma}_{F}(t,x,v)-\gamma_{F}(t,x,v)$ . The next proposition summarizes the properties of (see DeBlasi-Pianigiani [3], [4] and Papageorgiou [10]). $\hat{\gamma}_{F}(t,x,v)$

$\inf\emptyset=-\infty$

$\delta_{F}$

$T\times H\times H\rightarrow R\cup\{-\infty\}$

$\delta_{F}(\cdot,\cdot,\cdot)$

Proposition 5.

If hypothesis

holds, then

is is $usc,$ is concave and strictly concave on $F(t,x,),$ $(d)0\leq\delta_{F}(t,x,v)\leq 4a_{2}(t)^{2}+4c_{2}^{2}|x|^{2}$ for every $F(t,x),$ $(e)\delta_{F}(t,x,v)=0$ if and only if $v\in extF(t,x)$ . measurable,

$H(F)_{1}$

$(b)(x,v)\rightarrow\delta_{F}(t,x,v)$

$(a)(t,x,v)\rightarrow\delta_{F}(t,x,v)$

$(c)v\rightarrow\delta_{F}(t,x,v)$

$ v\in$

As we already mentioned in the introduction, simple h-continuity of does not suffice to have the desired relaxation theorem. So we impose the following stronger conditions on $F(t,x)$ : $H(F)_{2}$ : is a multifunction such that $F(\cdot,\cdot)$

$F:T\times H\rightarrow P_{\omega \mathfrak{t}c}(H)$

(i)

$t\rightarrow F(t,x)$

is measurable, (ii)

$h(F(t,x),F(t,y))\leq k(t)|x-y|a.e$

for every

$x$

with , (iii) $|F(t,x)|\leq a_{2}(t)+c_{2}|x|a.e$ with $a_{2}\in L^{2}(T),c_{2}>0$ . Then our stronger relaxation theorem reads as follows: $k\in L^{1}(T)$

Theorem 6. If hypotheses dense -subset of $G_{\delta}$

$H(A),H(F)_{2}$

$P_{c}(x_{0})\subseteq C(T, H)$

.

hold and

$x_{0}\in H$

, then

$P_{e}(x_{0})$

is a

NONLINEAR EVOLUTION INCLUSIONS

Proof. Let

$\lambda>0$

and define

$ D_{\lambda}=\{x\in S_{c}(x_{0}):\int_{0}^{b}\delta_{F}(t,x(t),f(t))dt0$

Theorem 8.

If

$\lambda(V_{\epsilon}^{\mathfrak{c}})0$ and as above) we can produce hence closed with . Let and l.s.c.. Evidently and $t_{0}\in T_{1}\cap T_{2}t_{0}0$

$T_{1}\subseteq T$

$\lambda(\cdot)$

$t\in T_{1}$

$T\times H\rightarrow P_{f}(H)$

$F_{0}$

$\Delta\subseteq T$

$\Delta$

$x,y:\Delta\rightarrow H$

$\epsilon>0$

$\Delta$

$\lambda(T\backslash C_{\epsilon})\leq\epsilon$

$C_{\epsilon}\subseteq T$

$F_{0}|C_{\epsilon}\times H$

$F_{0}$

$V=C_{\epsilon/2}$

$\lambda(C_{\epsilon/2}\backslash T_{2})$

$T_{2}\subseteq C_{\epsilon/2}\subseteq T$

$\lambda(T_{1}\backslash T_{2})