INVARIANT SETS FOR NONLINEAR EVOLUTION EQUATIONS, CAUCHY PROBLEMS AND PERIODIC PROBLEMS

In the case of K≠D(A)¯, we study Cauchy problems 
and periodic problems for nonlinear evolution equation u(t)∈K, u′(t)


Introduction
Let E be a Banach space and let A ⊂ E × E be an m-accretive operator.Let K be a closed subset of E, let T > 0, and let f : [0,T] × K → E. In the case of K = D(A), many researchers have studied initial value problems or periodic problems for nonlinear evolution equation u(t) ∈ K, u (t) + Au(t) f t,u(t) for 0 ≤ t ≤ T; (1.1) see [3,5,13,14,16,17,18,21,22,23,24,25,26,27,28].Recently, in the case when K = D(A) and f is of Carathéodory type, Bothe [7] showed the existence of solutions of the initial value problem with u(0) = x ∈ K ∩ D(A) for (1.1) under a tangential condition: where S f (t,x) (•)x is the solution of w(0) = x and w (s) + Aw(s) f (t,x) for s ≥ 0. Bothe [8] also showed the existence of T-periodic solutions for (1.1) under a subtangential condition: K-invariance of resolvent operators and Nagumo-type condition lim s→+0 1 s inf z∈K x + s f (t,x) − z = 0 for a.e.t ∈ (0,T) and for every x ∈ K. (1.3) of the form ∂u ∂t (t,x) + Au(t,x) = g t,x,u(t,x) in R × Ω, where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, g : R × Ω × R → R is a continuous mapping, A is a nonlinear elliptic operator, B is a boundary operator, and c is a real number.
For semilinear cases, Amann [2] considered initial value problems and periodic problems for (1.1) in the case when K = D(A) and f is not necessarily of Carathéodory type with respect to the topology of E. The results in [2] can be applied to derive the existence of T-periodic solutions of the problem ∂u ∂t (t,x) + Lu(t,x) = g t,x,u(t,x),∇u(t,x) in R × Ω, where L is a second-order linear elliptic operator, B is a first-order boundary operator, and g : R × Ω × R × R N → R is a continuous function.We can see that in problem (1.5), function g cannot be of Carathéodory type in L 2 (Ω).To deal with this kind of problems, it was assumed in [2] that f (t,•) is defined on a subspace V , which is endowed with a stronger topology than that of E and f (t,•) : V → E is continuous with respect to this topology.Under these conditions, the existence of solutions of the problems were established in [2] imposing a subtangential condition: K-invariance of evolution operators and Nagumo type condition.Our purpose in this paper is to establish existence results which can cover problems of the form (1.5) with L replaced by nonlinear elliptic operators.That is, in the case of K = D(A), we give existence results for solutions of initial value problems and periodic problems for (1.1) under a tangential or subtangential condition in the case when H is a Hilbert space, V is a subspace of H, and f : [0,T] × V → H is a mapping, which is not necessarily of Carathéodory type with respect to the topology of H.
The organization of this paper is the following.Section 2 is devoted to some preliminaries and notations.We state our main results in Section 3 and we prove them in Section 4. Finally, we study an example to which our results are applicable.

Preliminaries and notations
Throughout this paper, we denote by N, R, and R + the set of positive integers, the set of real numbers, and the set of nonnegative real numbers, respectively.For a subset X of a normed linear space, we denote by ∂X the boundary of X.
Let (H, •, • ) be a Hilbert space.We denote by | • | the norm defined by |x| 2 = x,x for x ∈ H.We also denote by B H (x,r) the closed ball in H with center x ∈ H and radius r > 0. Let K be a closed, convex subset of H and let P be the metric projection from H onto K, that is, for each x ∈ H, Px is the unique point in K with |x − Px| = d H (x,K), where d H (x,K) = min y∈K |x − y|.We know that y − Px,x − Px ≤ 0 for all x ∈ H and y ∈ K.We define a tangential cone T K (x) for K at x ∈ K by Let A be a maximal monotone subset of H × H.For each λ > 0, we define a resolvent and a Yosida approximation by J λ = (I + λA) −1 and A λ = (I − J λ )/λ, respectively.We denote by {S(t) : t ≥ 0} the semigroup generated by the negative of A; see [4,10,19].We say the semigroup for every a ≤ t ≤ b, and for every (y,z) ∈ A and s, t with a ≤ s ≤ t ≤ b.It is known that the initial value problem (2.2) has a unique integral solution; see [4,6].We remark that for each x ∈ D(A), S(•)x is the integral solution of u(0) = x, u (t) + Au(t) 0 for t ≥ 0. (2.4) For each x ∈ D(A) and z ∈ H, we denote by S z (•)x the integral solution of and we define T A K by We remark that in the case of K ⊂ D(A), T A K coincides with the one in [7].Let (V , • ) be a reflexive Banach space which is continuously imbedded into H.We identify V with a subspace of H. Let ω,ε ≥ 0, let p > 1, and let A be a maximal monotone subset of for every (x 1 , y 1 ),(x 2 , y 2 ) ∈ A. In this case, if u, v are the integral solutions of (2.2) corresponding to (x,g),(y,h) ∈ D(A) × L 1 (a,b;H), respectively, then To prove our results, we need the following propositions and theorems.The first one is a property of the Dini derivative.For a proof, see [ (2.9) The next one is a fixed-point theorem, which can be derived from the Leray-Schauder degree theory [11,20].
Theorem 2.2.Let X be a bounded, closed, convex subset of a normed linear space E with nonempty interior.Let H be a continuous mapping from [0,1] × X into a compact subset of E such that Then H(0,•) has a fixed point in X.
The next proposition shows a sufficient condition that the negative of a maximal monotone operator generates a compact semigroup; see [18,Lemma 2].Proposition 2.3.Let (V , • ) be a reflexive Banach space which is compactly imbedded into a Hilbert space (H, •, • ) and let A be a maximal monotone subset of H × H which satisfies D(A) ⊂ V and with a constant p > 1.Then the negative of A generates a compact semigroup.
The following two theorems are concerning properties of integral solutions; see [4,  Theorem 2.4.Let H be a Hilbert space and let A be a maximal monotone subset of H × H. Let u 0 ∈ D(A), let T > 0, and let g ∈ L 1 (0,T;H).Let λ > 0 and let u λ be the solution of the initial value problem u λ (0) = u 0 , u λ (t) + A λ u λ (t) = g(t) for almost every 0 < t < T. (2.11) N. Hirano and N. Shioji 187 Then {u λ } converges to some u ∈ C(0,T;H) as λ → +0 with respect to the topology of C(0,T;H), and the limit function u is the integral solution of the initial value problem

Main results
We begin this section with hypotheses and notations which we will use in our results.The following are the hypotheses for our general framework: for every (x 1 , y 1 ),(x 2 , y 2 ) ∈ A, where 1 < p < ∞ and ω > 0 are constants; is the closure of D(A) with respect to the topology of H, and the metric projection P from H onto K with respect to the metric in H satisfies (i) for almost every t ∈ (0,T) and for every x ∈ K ∩ V .
Each one of the following hypotheses guarantees the boundedness of solutions of (1.1).We remark that if , a 1 ∈ L pq/(p−qα) (0,T;R + ), and a 2 ∈ L q (0,T;R + ), where q is the constant with 1/ p + 1/q = 1; for every x ∈ D(A) and for almost every and for every x ∈ D(A) and for almost every t ∈ [0,T].
Each one of the following hypotheses is a tangential or subtangential condition which guarantees K-invariance of solutions for (1.1).In applications to elliptic-parabolic problems, K-invariance of the semigroup in (T2) corresponds to the comparison principle for parabolic equations, and K-invariance of the resolvents in (T3) corresponds to the comparison principle for elliptic equations; see examples in [2] and this paper: [0,T] and for every x ∈ K ∩ V ; (T3) J λ K ⊂ K for every λ > 0, and f (t,x) ∈ T K (x) for almost every t ∈ [0,T] and for every x ∈ K ∩ V .Now, we state our viability theorem.
In the case of K = H, we have the following corollaries as direct consequences of Theorems 3.1 and 3.2 with assumption (T3); see also [26].
Corollary 3.5.Assume that the hypotheses of Theorem 3.2 hold.Assume in addition that f is t-independent, a 1 , a 2 are nonnegative constants, and b 1 , b 2 are nonnegative constants in the case of (B4).Then there exists x ∈ K ∩ D(A) with Ax f (x).

Proof of theorems
Throughout this section, we assume (H1), (H2), (H3), and (H4) and | • | ≤ • without loss of generality.We consider that space C(0,T;H) ∩ L p (0,T;V ) is endowed with a norm First, we give the proof of Theorem 3.2.The reason is that we want to give the proof of Theorem 3.2 precisely since its proof is more complicated than that of Theorem 3.1.
The following Lemmas 4.1, 4.2, 4.4, and 4.5 are obtained by similar arguments as those in [18].Lemma 4.1.For each ε > 0 and g ∈ L 1 (0,T;H), there exists a unique T-periodic, integral solution of Proof.Let ε > 0 and let g ∈ L 1 (0,T;H).We define a mapping U : We define In particular, for each bounded subset From (2.7), we know and we obtain ωTM p−1 ≤ 2 p C. Hence, it is easy to see that the conclusion holds.
N. Hirano and N. Shioji 193 Remark 4.8.Using any (z,w) ∈ A instead of (x δ , y δ ), by the same proof, we can show that if one of the conditions of (B1), (B2), and (B4) holds, then there exists R 1 > 0 such that for every T-periodic, integral solution u of u (t) + Au(t) f (t,Pu(t)) for 0 ≤ t ≤ T satisfying u(t) ∈ K 1 for all t ∈ [0,T], there holds u L p (0,T;V ) < R 1 .
We fix R 1 as in the previous lemma, and we define a subset X δ of C(0,T;H) ∩ L p (0, T;V ) by in the case of (B1), and by in each case of (B2) or (B4), where R 2 is a positive constant satisfying We remark that we can choose such R 2 by Lemma 4.2.
Hence, we obtain the conclusion.
Let v be any element of X δ and set u = Q δ (ε, f (•,Pv)).From (2.7), we have and hence By the previous lemma, there exists Then we also have Hence, we obtain the conclusion.
Next, we will show that the mapping u → Q δ (ε, f (•,Pu)) has no fixed point on ∂X δ for every ε > 0. The following play an important role to show this property.
Lemma 4.11.The following hold: Hence, by z ∈ T A K (Px), we obtain The reason why we define an approximate equation by (4.1) can be found in the proof of the next lemma.Lemma 4.12.Assume one of the conditions of (B1), (B2), and (B4), and assume also one of the conditions of (T1), (T2), and (T3).Then for each ε > 0 and u ∈ X δ with u Since one of the conditions of (B1), (B2), and (B4) is assumed, by Lemma 4.7 and the definition of X δ , we have u L p (0,T;V ) < R 1 and u − x δ C(0,T;H) < R 2 in the case of (B2) and (B4), respectively.Thus it is enough to show u(t) ∈ ∂K δ for all t ∈ [0,T].First, we consider the case of (T3).Let g ∈ C(0,T;H).Let λ > 0 and let v be the C 1 (0,T;H)-solution of the initial value problem Let t ∈ [0,T) and let s > 0 with t + s ≤ T. Since we have for every t ∈ [0,T).By Proposition 2.1, we have which implies, by Theorem 2.4, Lemma 4.11(ii), and u(0 Let P be the metric projection from H onto K δ/2 .Since x δ ∈ K δ/2 and Pu(τ) is in the line segment between u(τ) and Pu(τ) for all τ ∈ [0,T], we have which implies u(t) ∈ ∂K δ for all t ∈ [0,T].Next, we consider the cases of (T1) and (T2).Let x ∈ D(A), let g ∈ C 1 (0,T;H), and let v be the integral solution of N. Hirano and N. Shioji 197 By Theorem 2.5, v is everywhere differentiable from the right on [0, T) and there exists y ∈ L ∞ (0,T;H) such that Then we get for every t ∈ [0,T).By Proposition 2.1 and Lemma 4.11(ii), (iii), and (iv), we have which implies (4.32).By the same argument as above, we have u(t) ∈ ∂K δ for all t ∈ [0,T].
Next, we give the proof of Theorem 3.1.We show the following proposition concerning the existence of local solutions for the initial value problem.Proposition 4.14.Assume (H1), (H2), (H3), and (H4) and one of the conditions of (T1), (T2), and (T3).Then for each x ∈ K ∩ D(A), there exists T 0 ∈ (0,T] and an integral solution u of For each T 0 ∈ (0,T], we define a mapping G T0 : L 1 (0,T 0 ;H) → C(0,T 0 ;H) ∩ L p (0,T 0 ;V ) by G T0 g = u for g ∈ L 1 (0,T 0 ;H), where u ∈ C(0,T 0 ;H) ∩ L p (0,T 0 ;V ) is the unique integral solution of the initial value problem For each T 0 ∈ (0,T], we also define a subset X T0 of C(0,T 0 ;H) ∩ L p (0,T 0 ;V ) by X T0 = u ∈ C 0,T 0 ;H ∩ L p 0,T 0 ;V : By similar arguments as those in the case of the periodic problem, it is easy to see that the mapping v → G T0 ( f (•,Pv)) is compact and continuous from X T0 into C(0,T 0 ;H) ∩ L p (0,T 0 ;V ) for each T 0 ∈ (0,T].It is also easy to see that if T 0 > 0 is sufficiently small, then G T0 ( f (•,Pv)) ∈ X T0 for all v ∈ X T0 .Fix such T 0 ∈ (0,T].By Schauder's fixed point theorem, there exists u ∈ X T0 with G T0 ( f (•,Pu)) = u, that is, u is an integral solution of the problem u(0) = x and u (t) + Au(t) f (t,Pu(t)) for 0 ≤ t ≤ T 0 .By similar lines as those in the proof of Lemma 4.12, we can show that t → |u(t) − Pu(t)| 2 is decreasing on [0,T 0 ].Hence, u(t) ∈ K for all t ∈ [0,T 0 ] and u is an integral solution of (4.43).
Remark 4.15.Intuitively, (T3) seems to imply (T1) in the case of K ⊂ D(A), and (T2) seems to imply (T1).But it seems to be difficult to give a proof even after we obtain the proposition above.
So, there exists z ∈ K such that |u(t) − z| → 0 as t → T * − 0, and hence we can think u(T * ) = z and u ∈ C(0,T * ;H).It is easy to see that u is an integral solution of (4.43) on [0, T * ].We know T * = T. Indeed, if T * < T, we can derive a contradiction by similar lines as those in Proposition 4.14.Therefore, u is a desired solution.
.20) N. Hirano and N. Shioji 195(ii) Let x ∈ H and let z ∈ T K (Px).From 0 ≤ x − P(x + sz) 2 = x + sz − P(x + sz) 2 − 2s z,x + sz − P(x + sz) + s 2 |z| 2 (4.21)for every s > 0, we have z,x − Px ≤ 0. (iii) Assume K ⊂ D(A) and S(t)K ⊂ K for every t ≥ 0. Let (x, y) ∈ A. Then we have ).Then there exists a T-periodic, integral solution u of is continuous and compact by (H4), Lemmas 4.4, and 4.5.By Lemma 4.10 and Theorem 2.2, for each n ∈ N, there exists u n ∈ X δ such thatu n = Q δ (1/n, f (•,Pu n )).Then u n is a T-periodic, integral solution of u n (t) + 1/n(u n (t) − x δ ) + Au n (t) f (t,Pu n (t)) for 0 ≤ t ≤ T. By Lemmas 4.2 and 4.7, ∈ A, s,t with 0 ≤ s ≤ t ≤ T, and n ∈ N, we obtain ∈ A and s, t with 0 ≤ s ≤ t ≤ T, which implies that u is an integral solution of (4.38).By Lemma 4.13, for each n ∈ N, there exists a T-periodic, integral solution u n of (4.38) satisfying u n (t) ∈ K 1/n for all t ∈ [0,T].We know that {u n } is bounded in L p (0,T;V ) by Remark 4.8, and {u n } is bounded in C(0,T;H) by Remark 4.3.We also know that {u n } is relatively compact by Remark 4.6.So we may assume that {u n } converges strongly to some u ∈ C(0,T;H) ∩ L p (0,T;V ).It is easy to see that u is T-periodic and u(t) ∈ K for all t ∈ [0,T].By similar lines as those in the proof of Lemma 4.13, there holds (4.40) for every (x, y) ∈ A and s, t with 0 ≤ s ≤ t ≤ T. Since Pu(t) = u(t) for all t ∈ [0,T], u is a desired solution.Proof of Corollary 3.5.From Theorem 3.2, for each n ∈ N, there exists a 1/2 n -periodic, integral solution u n ofu n (t) ∈ K, u n (t) + Au n (t) f u n (t) for 0 ≤ t ≤ 1. (4.41)By Remarks 4.3 and 4.8, {u n } is bounded in C(0,1;H) ∩ L p (0,1;V ).So {u n } is relatively compact in C(0,1;H) ∩ L p (0,1;V ) by Remark 4.6.Hence, there exists a constant function u(t) ≡ x ∈ K ∩ V , which is a cluster point of {u n }.Since f (x) − w,x − z =