EXISTENCE AND RELAXATION RESULTS FOR NONLINEAR EVOLUTION INCLUSIONS REVISITED

In this paper we confirm the validity of some recent results of Hu, Lakshmikantham, Papageorgiou [4] and Papageorgiou [13] concerning the existence and relaxation for nonlinear evolution inclusions. We fill a gap in the proofs of these results due to the use of incorrect Nagy's compactness embedding theorem.


Introduction
Recently Hu, Lakshmikantham and Papageorgiou [4] studied the properties of the solution set of nonlinear evolution inclusion driven by a time dependent maximal monotone coercive operators defined on an evolution triple (V,H,V').However, the main existence theorem of their paper (Theorem 3.1 in [4]) as well as the relaxation theorem (Theorem 3.2 in [4] have a gap in their proofs and therefore, in our opinion, these proofs are incorrect.The same remark concerns the proof of another result on existence of solutions to differential inclusion due to Papageorgiou (see Theorem 3.1 of [13]).Namely, in [4] and [13] the authors have exploited several times the result of Nagy (see Theorem 2 in [7]), which says that tr (the space where the solution of the inclusion is sought) is compactly embedded into C(O,T;H) (the space of continuous functions from the time interval [0, T] into H); see the notation in Section 2 below.Very recently, the author of this paper gave an example (see [6]) which shows that the above result of Nagy is false.For this rea- son, the proofs of existence and relaxation theorems cannot follow directly from arguments pre- sented in [4] and [13].
The purpose of this paper is to fill the gap due to the use of Nagy's incorrect result and to establish in this way the validity of some earlier results of [4] and [13].
As the referee noticed, it is important to observe that the proofs of Theorems 3.1 and 3.2 in [4] and Theorem 3.1 in [13] remain valid (even without assuming V to be a separable Hilbert space) provided x 0 E V, i.e., when the initial condition is more regular.In this situation, the result of Nagy [7] can be omitted.Indeed, in this case the solution set of the evolution inclusion is sequentially compact in C(O,T;w-V) (see [1] or [10]).Hence Corollary 4, p. 85 of Simon [15]  guarantees the compactness of this set in C(0, T; H).
In he light of the above observation, the present paper shows that it is not necessary to 144 S. MIGORSKI restrict x 0 in V, that is, we will prove that even if the initial datum is nonsmooth the solution set remains compact in C(0, T;H).We note that the new mathematical argument of this paper is based on the fact that the solution map (which assigns to the right-hand side the solution) of the associated evolution equation is sequentially continuous from ' endowed with its weak topology into C(O,T;H) (see Proposition 3.1).Nonetheless, for the sake of completeness and clarity, we will give (following [4] and [13]) the main steps of the proofs of two existence theorems and a re- laxation theorem established in [4] and [13] indicating places where the mentioned above result of Nagy has to be replaced by Proposition 3.1.Note that Theorem 3.1 has been already established for p q 2 in [12], while Theorem 3.2 can be found in a more general setting in [14].Another relevant work is [11], where a controlled evolution inclusion with control constraints was examin- ed.

Prehminaries
Let H be a separable Hilbert space and let V be a subspace of H carrying the structure of a separable reflexive Banach space, which embeds into H densely and continuously.Identifying H with its dual, we have the Gelfand triple (see e.g., [2, 5, 16]) V C H C V', where all embeddings are continuous and dense.Moreover, we assume in this paper that these embeddings are also com- pact.We denote by (-, the duality of V and its dual V' as well as the inner product on H, by I1" II, I" and I1" ]1 y' the norms in V, H and Y', respectively.Given a fixed real number T > 0 and 2 < p < +cx, we introduce the following spaces T'-LP(O,T;V), -LP(O,T;H), :'-Lq(O,T;H), i"-Lq(O,T;V'), (i/p+ 1/q 1) and {w E T'lw' E V'}, where the deri- vative is undcrstood in the sense of vector valued distributions.Clearly qr C T" C C T".The T pairing of V and V' and the duality between and :' are denoted by (If, v))-f (f(s), v(s))ds.0 Given a Banach space , the symbols w-, s-are always used to indicate the space equipped with the weak and the strong (norm) topology, respectively.
Let (,E,#) be a measure space, X be a separable Banach space.By ](c}(X) and (wlk(cl(X) we denote respectively the family of all nonempty, closed, (convex) subsets'of X and the family of all nonempty, (weak-) compact, (convex) subsets of X.A multifunction F defined X on with values in the space 2 of all nonempty subsets of X is called measurable if F-(E): {w : F(w)N E :/: } E, for every closed set E C X. F is called graph measurable if GrF: {(w,x) X:x F(w)} E %(X) (here %(X)is the family of all Borel subsets of X).We de- note by S (1 < p < cx) the set of all selectors of F that belong to LP(;X) i.e., LP(; X): f(w) F(w) # a.e.}.We know that S # 0 if and only if wHinf{ L_.The set S is said to be decomposable (see e.g., [9]) if A e %() (the Borel r-field of and fl, f2Sz imply XAfl +(I_-XA)f2S Let (Y, 7y), (Z, 7Z) be Hausdorff topological spaces.A multifunction G: Y---2 z is said to be "(ryrz) upper semicontinuous (u.s.c.

Existence and Relaxation Theorems
In this section we examine the continuity properties of the solution map of the Cauchy problem for the evolution equation associated with the following nonlinear inclusion" 2(t) + A(t,z(t))e F(t,x(t)) a.e.E (0, T), (3.1) (0)o" Then we will present concisely the proofs of existence and relaxation results by Hu, Lakshmikan- tham and Papageorgiou [4] and Papageorgiou [13].
F:[0, T] x H-PI(H is a multifunction such that H(F)I(1 holds, xF(t,x) is 1.s.c., H(F)I(3 holds.The hypotheses H(A), H(F)I H(F)2 for p q 2 coincide with the ones of [13] and [4].
It is well-known (see Theorem 4.2, p. 167 of Barbu [2] or Theorem 1.2, p. 162 of Lions [5]) that if H(A) holds, f ' and z 0 H, then evolution equation (3.2) admits a unique solution in .We consider below the solution map r::E'+ for (3.2) defined by r(f)= x, where z denotes the solution to (3.2).
This proves the claim.
T From these inequalities, we deduce that lim f )n(s)ds-0 which clearly implies that On--+0 strong 0 ly in LI(0, T).Therefore, we may assume, by taking a next subsequence, that n(t)--.O a.e.t e (0, T). (3.9) Using hypothesis H(A)(3)(4), for a.e. e (0, T), we get cOn(t) C ]] xn(t ]] P-b ]] X(t) ]] ]] Xn(t ]] p-1 I I xn(t)[I (a(t) + b I I x(t)II p 1) a(t)II x(t)II + c I I (t)II From the above inequality and (3.9), it follows that { [[ Xn(t II} is bounded for a.e. (0,T) and n >_ n 0. Thus we have shown that the sequence {Xn} belongs to a bounded set of L(O, T; V).Moreover, since {2n} lies in a bounded subset of c, and V C H compactly, we deduce by a ver- sion of the Arzelg-Asc01i theorem (compare Corollary 4, 8 of Simon [15]) that x--+x in

C(O,T;H).
As in the proof of Theorem 3.1 in [13], we show that x-r(f) is a solution to (3.2).From the uniqueness of solutions to (3.2), we infer that the whole sequence {x,} converges to x in both w-and C(0, T; H).This completes the proof of the proposition.
Theorem 3.1: (The convex case) If hypotheses H(A), H(F)I hold and x o E H, then (3.1) ad- mits a solution. Proof: Step 1.Every solution x E oz to (3.1) satisfies the following a priori estimates" (3.10) where M > 0 for i-1,2.
Step 4. One applies the Kakutani-Fan fixed point theorem to the multifunction % finding f* % such that f* E %(f*).Then x*: r(f*) solves (3.1) with F in place of F. The same esti- mate as in Step 1 and (3.12) implies (t,x*(t))-F(t,x*(t)) for a.e.t, which means that x* is a solution to (3.1).This completes the proof of the theorem.From Theorem 3.1 and (3.11), we obtain the following.Corollary 3.1: If hypotheses of Theorem 3.1 hold, then the solution set of (3.1) is a nonemp- ty, weakly compact subset of qr and a compact subset of C(0, T; H).
Theorem 3.2: (The nonconvex case) If hypotheses H(A), H(F)2 hold and x o H, then (3.1) admits a solution.
Step 1.As in the proof of Theorem 3.1, we get for the solutions a priori bounds (3.10) and (3.11).
Step 4. Applying the selection theorem of Fryszkowski [3], we get a continuous map r/: %--,% such that r/(f)E %(f) for all f E %.Then, by the Schauder-Tikhonov fixed point theorem, one finds f* % such that f*= rl(f*).As in the proof of Theorem 3.1, it is easy to check that x* r(f*) solves (3.1)This completes the proof.
Finally, we consider the following "convexified" version fo the Cauchy problem (3.1)" &(t)+ A(t,x(t)) e-d-dF(t,x(t)) a.e.t e (0, T), (0)-(3.14) Denote by S(Xo) the solution set of (3.1) and by Sc(xO) the solution set of (3.14).The relaxation theorem stated below shows that S(xo) is dense in Sc(Xo) for the C(O,T;H) topology.As regards the orientor field, we need now a hypothesis which is stronger than the ones considered previously.
From Corollary 3.1, it follows that S(Xo)C Sc(XO).It is sufficient to prove the Step 2 Let >0 and XSc(Xo).So (3.2) holds with fSF(.q z(.))"As in the proof of Theorem 3".1, one defines the set % and gets If(t) <_' (t) a.e.. Due to Theorem 4.1 of Papageor- 1 and a symmetric weak neighgiou [10] and Proposition 3.1, it is possible to find fl SF(.,x(.)) borhood U of the origin in ' such that f-fl E U fl% implies I I x-z I I I C(o,T;H)--, where Z 1 r(fl).
Step 4. From Step 3, again by Proposition 3.1, we deduce that z n -r(fn)---,r(f -"' in C(0, T; H) and next that F S(Xo).Furthermore, due to the inequalities vg(s)ds

Vi > 1
Zi+l(t)-zi(t)l <_-.0 we obtain I I I I c(o, T; H) exp I I o I I L1)" Since > 0 is arbitrary we conclude that