Tangent sets, viability for differential inclusions and applications

Nonlinear Analysis 71 (2009) e979–e990

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Tangent sets, viability for differential inclusions and applicationsI Ovidiu Cârjă a,b , Mihai Necula a , Ioan I. Vrabie a,b,∗ a

Faculty of Mathematics, ‘‘Al. I. Cuza’’ University, Iaşi 700506, Romania

b

‘‘Octav Mayer’’ Mathematics Institute, Romanian Academy, Iaşi 700506, Romania

article

info

Dedicated to Professor Dr. V. Lakshmikantham MSC: primary 34G20 34G25 47D06 47J35 secondary 35K55 35B05 93B05

abstract Let X be a real Banach space, let A : D(A) ⊆ X ; X be a given operator, K a nonempty and possible non-open subset in D(A), F : K ; X a given multi-function. In this lecture, we consider the differential inclusion, u0 (t ) ∈ Au(t ) + F (u(t )), with (a) A = 0, (b) A linear and (c) A nonlinear and, in each one of these cases, we give a short survey of the most important and very recent necessary and sufficient conditions for viability expressed in terms of tangent sets and A-quasi-tangent sets to K at a given point ξ ∈ K , concepts recently introduced by the authors. From a rather long list of applications, we confined ourselves only to: solutions in moving sets, a comparison result for a reaction–diffusion system, a comparison result for a nonlinear diffusion inclusion and a sufficient condition for null controllability. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Viability Tangency condition Compact semigroup Nonlinear diffusion equation Reaction–diffusion system Null controllability

1. Introduction Let X be a real Banach space, let B(X ) be the class of all nonempty and bounded subsets in X , let A : D(A) ⊆ X ; X be an m-dissipative operator, K a nonempty subset in D(A), F : K → B(X ) a given multi-function. We consider the Cauchy problem for the nonlinear quasi-autonomous perturbed differential inclusion



u0 (t ) ∈ Au(t ) + F (u(t )) u(0) = ξ

(1.1)

and we are interested in finding necessary and even necessary and sufficient conditions in order that K be C 0 -viable with respect to A + F , i.e. that for each ξ ∈ K there exists at least one C 0 -solution u : [0, T ] → K of (1.1). We recall that a subset K of a Banach space X is locally closed if for each ξ ∈ K there exists ρ > 0 such that K ∩ D(ξ , ρ) is closed. Here and thereafter, D(ξ , ρ) denotes the closed ball with center ξ and radius ρ . We begin by giving a short survey of the state of the art. A detailed source of information on this subject up to 1984 is Aubin, Cellina [1]. A more recent one is Cârjă, Necula, Vrabie [2]. Throughout, K is a locally closed subset in X .

I Supported by CNCSIS Grant A 1159/2006 (O. Cârjă and M. Necula) and the Project CEx05-DE11-36/05.10.2005 (I.I. Vrabie).

∗

Corresponding address: Faculty of Mathematics, ‘‘Al. I. Cuza’’ University, Bd Carol I, No 11, Iaşi, 700506, Romania. E-mail addresses: [email protected] (O. Cârjă), [email protected] (M. Necula), [email protected] (I.I. Vrabie).

0362-546X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2009.01.055

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The single-valued case (1) The case u0 (t ) = f (u(t )) with f : K → X continuous and X finite-dimensional was considered by Nagumo [3] who proved that, whenever f is continuous, a necessary and sufficient condition in order that K be viable with respect to f is the classical tangency condition f (ξ ) ∈ TK (ξ ),

for each ξ ∈ K ,

where TK (ξ ) = {η ∈ X ; lim infh↓0 1h dist(ξ + hη; K ) = 0} is the Bouligand [4], Severi [5] contingent cone to K at ξ . See also the pioneering work of Perron [6]. (2) The semilinear case, u0 (t ) = Au(t ) + f (u(t )), was first considered by Pavel [7] who showed that, if A : D(A) ⊆ X → X generates a compact C0 -semigroup, {S (t ) : X → X ; t ≥ 0}, and f is continuous, a sufficient condition for mild-viability is the mild tangency condition f (ξ ) ∈ UAK (ξ ),

for each ξ ∈ K ,

where UAK



(ξ ) = η ∈ X ; lim h ↓0

1 h



dist S (h)ξ +

h

Z

S (h − s)η ds; K





=0 .

0

Obviously, U0K (ξ ) ⊆ TK (ξ ). (3) The nonlinear single-valued perturbed case, u0 (t ) ∈ Au(t ) + f (u(t )), was first considered by Vrabie [8] who showed that, if A : D(A) ⊆ X ; X generates a compact semigroup of nonlinear contractions and the nonlinear perturbation f : K → X is continuous, a sufficient condition for C 0 -viability is the C 0 -tangency condition f (ξ ) ∈ e UAK (ξ ),

for each ξ ∈ K ,

where

  1 e UAK (ξ ) = η ∈ X ; lim dist (u(h, 0, ξ , η); K ) = 0 , h ↓0

h

u(·, 0, ξ , η) being the unique C 0 -solution of



v 0 (t ) ∈ Av(t ) + η v(0) = ξ .

We recall for easy reference Definition 1.1. Let A : D(A) ⊆ X ; X be m-dissipative and let f ∈ L1 (τ , T ; X ). A C 0 -solution1 of the equation u0 (t ) ∈ Au(t ) + f (t ) is a function u in C ([τ , T ]; X ) satisfying: for each τ < c < T and ε > 0 there exist (i) τ = t0 < t1 < · · · < c ≤ Rtn < T , tk − tk−1 ≤ ε for k = 1, 2, . . . , n; Pn t (ii) f1 , . . . , fn ∈ X with k=1 t k kf (t ) − fk k dt ≤ ε; k−1 (iii) v0 , . . . , vn ∈ X satisfying:

vk − vk−1 ∈ Avk + fk for k = 1, 2, . . . , n and such that tk − tk−1 ku(t ) − vk k ≤ ε for t ∈ [tk−1 , tk ), k = 1, 2, . . . , n. See Lakshmikantham, Leela [9]. So, whenever A is linear, e UAK (ξ ) = UAK (ξ ), i.e. it coincides with the tangency concept introduced by Pavel [7]. Therefore, in both cases, i.e. A linear or nonlinear, we will use the very same symbol UAK (ξ ). To summarize, the first step to overcome the main difficulty occurred in infinite-dimensional spaces, whenever A is unbounded, when the problem (1.1) has no differentiable solution, was done by Pavel [7] in the semilinear single-valued case, i.e. A linear and F single-valued and continuous, by introducing the mild tangency condition, a concept avoiding the use of the values of A at all points ξ ∈ K (which makes no sense whenever ξ ∈ K \ D(A)). The second step, in the fully nonlinear single-valued case, was done by Vrabie [8] who has shown that, if A generates a nonlinear compact semigroup of nonexpansive operators and F is single-valued and continuous on the locally closed subset K ⊆ D(A), a sufficient condition for K to be C 0 -viable with respect to A + F is the C 0 -tangency condition which, again, avoids the use of the values of A at all points ξ ∈ K . However, it should be noted that, in contrast with Nagumo [3], both Pavel [7] and Vrabie [8] use ‘‘lim’’ instead of ‘‘lim inf’’ in the tangency conditions. The multi-valued case (1) The case u0 (t ) ∈ F (u(t )), with F : K ; X u.s.c. with nonempty, closed convex and bounded values and X finitedimensional, was considered by Bebernes, Schuur [10] who proved that a necessary and sufficient condition in order that K be viable with respect to F is the extended classical tangency condition 1 This concept is also known under the name of DS-limit solution, or integral solution.

O. Cârjă et al. / Nonlinear Analysis 71 (2009) e979–e990

F (ξ ) ∩ TK (ξ ) 6= ∅,

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for each ξ ∈ K .

The tangency condition above can be equivalently expressed as: for each ξ ∈ K , F (ξ ) contains at least one tangent vector to K at ξ in the sense of Bouligand [4] and Severi [5]. In fact, the necessity part was proved earlier by Ważewski [11]. (2) The semilinear multi-valued case u0 (t ) ∈ Au(t ) + F (u(t )), with X reflexive, A : D(A) ⊆ X → X the infinitesimal generator of a C0 -semigroup and F : K ; X strongly–weakly u.s.c., locally bounded, with nonempty, closed and convex values, was considered by Pavel, Vrabie [12,13], who proved that a sufficient (but not necessary) condition in order that K be viable with respect to F is the extended mild tangency condition F (ξ ) ∩ UAK (ξ ) 6= ∅,

for each ξ ∈ K .

Subsequent developments, using a larger tangency concept, TKA (ξ ) ⊇ UAK (ξ ),



TKA (ξ ) =

η ∈ X ; lim inf h↓0

1 h



dist S (h)ξ +

h

Z

S (h − s)η ds; K



 =0 ,

0

can be found in Shi [14], Bothe [15] and the references therein. We refer also to Cârjă, Vrabie [16] who introduced an extended mild tangency concept involving the weak topology. For an extension to semilinear differential inclusions on graphs see Necula, Popescu, Vrabie [17]. (3) The fully nonlinear multi-valued case u0 (t ) ∈ Au(t ) + F (u(t )), with A : D(A) ⊆ X ; X the infinitesimal generator of a nonlinear semigroup of contractions and F : K ; X strongly–weakly u.s.c. with nonempty, closed convex and weakly compact values, was considered by Cârjă, Vrabie [18], by using an extended C 0 -tangency condition F (ξ ) ∩ TKA (ξ ) 6= ∅,

for each ξ ∈ K ,

i.e. that for each ξ ∈ K there exists η ∈ F (ξ ) such that lim inf h↓0

1 h

dist (u(h, 0, ξ , η); K ) = 0.

As far as the nonlinear l.s.c. perturbed multi-valued case is concerned, we mention the paper of Bressan, Staicu [19] where the condition above is assumed to hold for each ξ ∈ K and η ∈ F (ξ ), i.e. F (ξ ) ⊆ TKA (ξ ). The outline of the paper is as follows. In Section 2 we introduce the main set-tangency concepts, in Sections 3–5 we state the main viability results referring to the cases (a), (b) and (c). Section 6 contains some extensions to the quasiautonomous case, while Section 7 includes two results on noncontinuable and global solutions. In Sections 8–11 we collect some applications to: solutions on moving sets, a comparison result for a semilinear reaction–diffusion system, a comparison result for a nonlinear diffusion inclusion and a null controllability result for a class of nonlinear evolution inclusions. For detailed proofs, the interested reader is referred to Cârjă, Necula, Vrabie [2,20,21] and Necula, Popescu, Vrabie [22]. See also Necula, Popescu, Vrabie [17]. 2. Viability via tangent sets (1) The case u0 (t ) ∈ F (u(t )), with F : K ; X , considered by Bebernes, Schuur [10], was reconsidered in infinitedimensional Banach spaces by Cârjă, Necula, Vrabie [21] who showed that, whenever F is strongly–weakly u.s.c. with nonempty, convex and weakly compact values, a necessary and sufficient condition in order that K be viable is the classic set-tangency condition F (ξ ) ∈ TSK (ξ ) for each ξ ∈ K , TSK (ξ ) =



E ∈ B(X ); lim inf h↓0

1 h



dist(ξ + hE ; K ) = 0

being the set of all the so-called tangent sets to K at ξ . Here and thereafter, if E, G are two nonempty subsets in X , we denote by dist(E ; G) =

inf

(x,y)∈E ×G

kx − yk.

(2) The semilinear multi-valued case u0 (t ) ∈ Au(t ) + F (u(t )), with A : D(A) ⊆ X → X the infinitesimal generator of a C0 -semigroup and F : K ; X strongly–weakly u.s.c., locally bounded, with nonempty, convex and weakly compact values, was reconsidered again by Cârjă, Necula, Vrabie [21], who proved that a necessary and sufficient condition in order that K be viable with respect to F is the set A-quasi-tangency condition F (ξ ) ∈ QTSAK (ξ ),

for each ξ ∈ K ,

where QTSAK (ξ ) =

with FE = {f ∈



L1loc

E ∈ B(X ); lim inf h ↓0

1 h



dist S (h)ξ +

h

Z

S (h − s)FE ds; K



 =0 ,

0

(R; X ); f (s) ∈ E a.e. for s ∈ R}, is the class of all the so-called A-quasi-tangent sets to K at ξ .

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(3) The case u0 (t ) ∈ Au(t ) + F (u(t )) with A : D(A) ⊆ X ; X the infinitesimal generator of a nonlinear semigroup of contractions and F : K ; X strongly–weakly u.s.c. with nonempty, closed convex and weakly compact values, was reconsidered by Cârjă, Necula, Vrabie [20], who proved that a necessary and sufficient condition in order that K be C 0 -viable with respect to F is the set A-quasi-tangency condition A

g F (ξ ) ∈ Q TSK (ξ ),

for each ξ ∈ K ,

where A

g Q TSK (ξ ) =



E ∈ B(X ); lim inf h↓0

1 h



dist (u(h, 0, ξ , FE ); K ) = 0 ,

u(h, 0, ξ , FE ) = {u(h, 0, ξ , f ); f ∈ FE }, u(·, 0, ξ , f ) being the unique C 0 -solution, i.e. DS-limit (or integral) solution of



v 0 (t ) ∈ Av(t ) + f (t ) v(0) = ξ . A

g See Lakshmikantham, Leela [9]. Hence, if A is linear, Q TSK (ξ ) = QTSAK (ξ ). Therefore, in all that follows we will simply write A QTSK (ξ ) either for A linear or nonlinear. 3. The problem u0 (t ) ∈ F (u(t )) The main results in this section are important consequences of Cârjă, Necula, Vrabie [21]. Let X be a Banach space, K a nonempty subset in X , F : K ; X a given multi-function and let us consider the Cauchy problem



u0 (t ) ∈ F (u(t )) u(0) = ξ .

(3.1)

Definition 3.1. An exact solution of (3.1) on [0, T ] is an absolutely continuous function u : [0, T ] → K which is a.e. differentiable on [0, T ] with u0 ∈ L1 (0, T ; X ) and satisfies



u0 (t ) ∈ F (u(t )) u(0) = ξ .

at each point t ∈ [0, T ] at which u is differentiable

An exact solution of (3.1) on the semi-open interval [0, T ) is defined similarly, noticing that, in this case, we have to impose the weaker constraint u0 ∈ L1loc ([0, T ); X ). Definition 3.2. An almost exact solution of (3.1) on [0, T ] is an absolutely continuous function u : [0, T ] → K which is a.e. differentiable on [0, T ] with u0 ∈ L1 (0, T ; X ) and satisfies



u0 (t ) ∈ F (u(t )) u(0) = ξ .

a.e. for t ∈ [0, T ]

An almost exact solution of (3.1) on the semi-open interval [0, T ) is defined similarly, noticing that, in this case, we have to merely impose u0 ∈ L1loc ([0, T ); X ). Definition 3.3. The set K ⊆ X is (almost) exact viable with respect to F if for each ξ ∈ K , there exists T > 0 such that the Cauchy problem (3.1) has at least one (almost) exact solution u : [0, T ] → K . Definition 3.4. A multi-valued mapping F : K → B(X ) is called locally compact if for each ξ ∈ K there exists ρ > 0 such that F (K ∩ D(ξ , ρ)) is relatively compact. Theorem 3.1. Let X be a Banach space. If K ⊆ X is almost exact viable with respect to the multi-function F : K ; X , then for each point ξ ∈ K with F (ξ ) convex and weakly compact and at which F is u.s.c., we have F (ξ ) ∈ TSK (ξ ). Theorem 3.2. Let X be a Banach space, let K ⊆ X be a nonempty and locally closed set and let F : K ; X be a locally compact, u.s.c. multi-function with nonempty, closed and convex values. A necessary and sufficient condition in order that K be exact viable with respect to F is that F (ξ ) ∈ TSK (ξ ) for each ξ ∈ K .

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Theorem 3.3. Let X be a Banach space, let K ⊆ X be a nonempty and locally compact set and let F : K ; X be a strongly–weakly u.s.c. multi-function with nonempty, weakly compact and convex values. Then, a sufficient condition in order that K be exact viable with respect to F is that F (ξ ) ∈ TSK (ξ ) for each ξ ∈ K . 4. The semilinear problem u0 (t ) ∈ Au(t ) + F (u(t )) The results in this section are essentially due to Cârjă, Necula, Vrabie [21]. Let X be a real Banach space, let A : D(A) ⊆ X → X be the infinitesimal generator of a C0 -semigroup, {S (t ) : X → X ; t ≥ 0}, K a nonempty subset in X and F : K ; X a given multi-function. Definition 4.1. By a mild solution of the autonomous multi-valued semilinear Cauchy problem



u0 (t ) ∈ Au(t ) + F (u(t )) u(0) = ξ ,

(4.1)

on [0, T ], we mean a continuous function u : [0, T ] → K for which there exists f ∈ L1 (0, T ; X ) such that f (s) ∈ F (u(s)) a.e. for s ∈ [0, T ] and u(t ) = S (t )ξ +

t

Z

S (t − s)f (s) ds

for each t ∈ [0, T ].

(4.2)

0

Definition 4.2. The set K ⊆ X is mild viable with respect to A + F if for each ξ ∈ K , there exists T > 0 such that the Cauchy problem (4.1) has at least one mild solution u : [0, T ] → K . Theorem 4.1. Let X be a Banach space, A : D(A) ⊆ X → X the infinitesimal generator of a C0 -semigroup, {S (t ) : X → X ; t ≥ 0}, K a nonempty subset in X and F : K ; X a nonempty-valued multi-function. If K is mild viable with respect to A + F then, for each ξ ∈ K at which F is u.s.c. and F (ξ ) is convex and weakly compact, we have F (ξ ) ∈ QTSAK (ξ ). Theorem 4.2. Let X be a Banach space, A : D(A) ⊆ X → X the infinitesimal generator of a C0 -semigroup, {S (t ) : X → X ; t ≥ 0}, K a nonempty, locally closed subset in X and F : K ; X a nonempty, closed and convex valued, locally compact multifunction. Then K is mild viable with respect to A + F if and only if, for each ξ ∈ K , F (ξ ) ∩ TKA (ξ ) 6= ∅. Theorem 4.3. Let X be a Banach space, A : D(A) ⊆ X → X the infinitesimal generator of a compact C0 -semigroup, {S (t ) : X → X ; t ≥ 0}, K a nonempty, locally closed subset in X and F : K ; X a strongly–weakly u.s.c. multi-function with nonempty, weakly compact and convex values. If, for each ξ ∈ K , F (ξ ) ∈ QTSAK (ξ ), then K is mild viable with respect to A + F. Theorem 4.4. Let X be a Banach space, A : D(A) ⊆ X → X the infinitesimal generator of a C0 -semigroup, {S (t ) : X → X ; t ≥ 0}, K a nonempty, locally compact subset in X and F : K ; X a strongly–weakly u.s.c. multi-function with nonempty, weakly compact and convex values. If, for each ξ ∈ K , F (ξ ) ∈ QTSAK (ξ ), then K is mild viable with respect to A + F . 5. The fully nonlinear problem u0 (t ) ∈ Au(t ) + F (u(t )) The results in this section are from Cârjă, Necula, Vrabie [20]. We consider the Cauchy problem for the nonlinear perturbed differential inclusion



u0 (t ) ∈ Au(t ) + F (u(t )) u(τ ) = ξ .

(5.1)

Definition 5.1. A function u : [τ , T ] → K is a C 0 -solution of (5.1) if u(τ ) = ξ and there exists f ∈ L1 (τ , T ; X ) satisfying f (t ) ∈ F (u(t )) a.e. for t ∈ [τ , T ] and such that u is a DS-solution, or integral solution, on [τ , T ] of the equation u0 (t ) ∈ Au(t ) + f (t ) in the usual sense. In order to exhibit the dependence of the C 0 -solution u on τ , ξ and f , we denote it by u = u(·, τ , ξ , f ). Definition 5.2. The set K ⊆ D(A) is C 0 -viable with respect to A + F if for each ξ ∈ K there exist T > 0 and a C 0 -solution u : [0, T ] → K of (5.1) with τ = 0.

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Theorem 5.1. Let X be a Banach space, let A : D(A) ⊆ X ; X be an m-dissipative operator, K a nonempty subset in D(A) and F : K → B(X ) a weakly compact and convex valued, u.s.c. multi-function. Then a necessary condition in order that K be C 0 -viable with respect to A + F is that, for each ξ ∈ K , we have F (ξ ) ∈ QTSAK (ξ ). Definition 5.3. An m-dissipative operator A : D(A) ⊆ X ; X is called of complete continuous type if for each fixed (τ , ξ ) ∈ R × D(A), the graph of the C 0 -solution operator, f 7→ u(·, τ , ξ , f ), is weakly×strongly sequentially closed in L1 (τ , T ; X ) × C ([τ , T ]; X ). Remark 5.1. If A is linear and m-dissipative, then it is of complete continuous type. In addition, if A generates a compact C0 -semigroup, then f 7→ u(·, τ , ξ , f ) is weakly–strongly sequentially continuous from L1 (τ , T ; X ) to C ([τ , T ]; X ). Remark 5.2. If the dual of X is uniformly convex and A generates a compact semigroup, then A is of complete continuous type. In this case, the operator f 7→ u(·, τ , ξ , f ) is weakly–strongly sequentially continuous from L1 (τ , T ; X ) to C ([τ , T ]; X ). See Vrabie [23], Lemma 2.3.2, p. 48. Remark 5.3. For an example of an m-dissipative operator of complete continuous type, which is neither linear nor defined on a Banach space with uniformly convex dual, see Theorem 3.4 in Cârjă, Necula, Vrabie [20]. Theorem 5.2. Let X be a Banach space, let A : D(A) ⊆ X ; X be an m-dissipative operator of complete continuous type which generates a compact semigroup of contractions, let K be a nonempty, locally closed subset in D(A) and let F : K → B(X ) be a nonempty, weakly compact and convex valued strongly–weakly u.s.c. multi-function. Then a sufficient condition for K to be C 0 -viable with respect to A + F is that, for each ξ ∈ K , F (ξ ) ∈ QTSAK (ξ ).

(5.2)

If, in addition, F is u.s.c., then (5.2) is also necessary for K to be C 0 -viable with respect to A + F . If A is only m-dissipative and of complete continuous type but the semigroup generated by A is not compact, we can prove Theorem 5.3. Let X be a Banach space, let A : D(A) ⊆ X ; X be an m-dissipative operator of complete continuous type, let K be a nonempty, locally compact subset in D(A) and let F : K → B(X ) be a nonempty, weakly compact and convex valued strongly–weakly u.s.c. multi-function. Then a sufficient condition for K to be C 0 -viable with respect to A + F is (5.2). If, in addition, F is u.s.c., then (5.2) is also necessary for K to be C 0 -viable with respect to A + F . 6. The quasi-autonomous case Let Y be a real Banach space, B : D(B) ⊆ Y ; Y an m-dissipative operator, K a nonempty subset in R×D(B), G : K → B(Y ) a given multi-function. The goal of this section is to extend the necessary and sufficient conditions for C 0 -viability already proved in the autonomous case to the general frame of quasi-autonomous nonlinear evolution inclusions of the form



z 0 (t ) ∈ Bz (t ) + G(t , z (t )) z (τ ) = ζ .

(6.1)

We denote by X = R × Y . Endowed with the norm

k(t , z )k =

p

|t |2 + kz k2 ,

for each (t , z ) ∈ X , the former is a Banach space. In addition, if Y has uniformly convex dual, then X , endowed with the norm above, has uniformly convex dual too. Definition 6.1. By a C 0 -solution of the quasi-autonomous multi-valued nonlinear Cauchy problem (6.1), we mean a continuous function z : [τ , T ] → Y , with (t , z (t )) ∈ K for each t ∈ [τ , T ], and for which there exists g ∈ L1 (τ , T ; Y ) such that g (s) ∈ G(s, z (s)) a.e. for s ∈ [τ , T ] and z is a C 0 -solution of the evolution inclusion below



z 0 (t ) ∈ Bz (t ) + g (t ) z (τ ) = ζ .

(6.2)

Definition 6.2. The set K ⊆ R × D(B) is C 0 -viable with respect to B + G if for each (τ , ζ ) ∈ K , there exists T ∈ R, T > τ , such that the Cauchy problem (6.1) has at least one C 0 -solution z : [τ , T ] → Y .

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In order to introduce the tangency concept we are going to use in the sequel, let us first observe that the quasiautonomous Cauchy problem (6.1) can be equivalently rewritten as an autonomous one in the space X , by setting A = (0, B), u(s) = (t (s + τ ), z (s + τ )), F (u) = (1, G(u))2 and ξ = (τ , ζ ). Indeed, with the notations above, (6.1) can be equivalently rewritten as (5.1). It readily follows that A is m-dissipative. So, u is a C 0 -solution for the problem above if it is given by u(s) = (τ + s, z (τ + s, τ , ζ , g (τ + ·))), where g is a function as in Definition 6.1 and z is a C 0 -solution for the problem (6.2). One may easily see that K is C 0 -viable with respect to B + G in the sense of Definition 6.1 if and only if K is C 0 -viable with respect to A + F in the sense of Definition 5.2. Theorem 6.1. Let Y be a Banach space, B : D(B) ⊆ Y ; Y an m-dissipative operator, K a nonempty subset in R × D(B) and let G : K → B(Y ) be a weakly compact and convex valued, u.s.c. multi-function. If K is C 0 -viable with respect to B + G then, for each (τ , ζ ) ∈ K , we have

(1, G(τ , ζ )) ∈ QTSAK (τ , ζ ).

(6.3)

Remark 6.1. Sometimes, it would be useful to observe that K is the graph of the multi-function3 C : I ; Y , defined by C (t ) = {y ∈ Y ; (t , y) ∈ K } for each t ∈ I. If such identification is made and C has nonempty values, we have

(1, E ) ∈ QTSAK (τ , ζ ) if and only if lim inf h↓0

1 h

dist (z (τ + h, τ , ζ , E); C (τ + h)) = 0.

We have the following counterparts of Theorems 5.2 and 5.3. Theorem 6.2. Let Y be a Banach space, B : D(B) ⊆ Y ; Y an m-dissipative operator of complete continuous type which generates a compact semigroup of contractions, K a nonempty and locally closed subset in R × D(B) and let G : K → B(Y ) be a nonempty, weakly compact and convex valued strongly–weakly u.s.c. multi-function. Then a sufficient condition in order that K be C 0 -viable with respect to B + G is that, for each (τ , ζ ) ∈ K , (6.3) be satisfied. Theorem 6.3. Let Y be a Banach space, B : D(B) ⊆ Y ; Y an m-dissipative operator of complete continuous type, K a nonempty and locally compact subset in R×D(B) and let G : K → B(Y ) be a nonempty, weakly compact and convex valued strongly–weakly u.s.c. multi-function. Then a sufficient condition in order that K be C 0 -viable with respect to B + G is that, for each (τ , ζ ) ∈ K , (6.3) be satisfied. If, in addition, G is u.s.c., then (6.3) is also necessary for K to be C 0 -viable with respect to B + G. We emphasize that all the results in this section apply in the special case when B is linear and, of course, when B ≡ 0. 7. Noncontinuable C 0 -solutions Let B : D(B) ⊆ Y ; Y be an m-dissipative operator, let K ⊆ R × D(B) be nonempty and let G : K → B(Y ). Recall that, if x, y ∈ Y , we denote by [x, y]+ the right directional derivative of the norm calculated at x in the direction y. Definition 7.1. A multi-function G : K → B(Y ) is called positively sublinear if there exist three continuous functions a : R → R+ , b : R → R+ , and c : R → R+ such that

kg k ≤ a(t )kζ k + b(t ) c for each (t , ζ , g ) ∈ K+ (G), where c K+ (G) = {(t , ζ , g ) ∈ K × Y ; kζ k > c (t ), g ∈ G(t , ζ ), [ζ , g ]+ > 0} .

Remark 7.1. There are three notable specific cases in which G is positively sublinear: (i) when G is bounded on K ; (ii) when G has sublinear growth with respect to its last argument4 ; (iii) when g satisfies the ‘‘sign condition’’ [ζ , g ]+ ≤ 0 for each (t , ζ ) ∈ K and g ∈ G(t , ζ ).

2 Here (1, F (u)) = {(1, η); η ∈ F (u)}. 3 With possibly empty values. 4 This means that there exist two continuous functions a : R → R and b : R → R such that kg k ≤ a(t )kζ k + b(t ) for each (t , ζ ) ∈ K and g ∈ G(t , ζ ). + +

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Definition 7.2. The set K is Y -closed if for each sequence ((tn , ζn ))n in K and limn (tn , ζn ) = (t , ζ ), with t < Tk , where TK = sup{t ∈ R; there exists η ∈ Y , with (t , η) ∈ K },

(7.1)

it follows that (t , ζ ) ∈ K . A typical example of Y -closed set is K = I × C with I a nonempty and open to the right interval and C ⊆ Y nonempty and closed. Here we present some results concerning the existence of noncontinuable, or even global C 0 -solutions for the nonlinear evolution inclusion



z 0 (t ) ∈ Bz (t ) + G(t , z (t )) z (τ ) = ζ .

(7.2)

A C 0 -solution z : [τ , T ) → Y to (7.2) is called noncontinuable, if there is no other C 0 -solution w : [τ , e T ) → Y of the same equation, with T < e T and satisfying z (t ) = w(t ) for all t ∈ [τ , T ). The C 0 -solution z is called global if T = TK , which is given by (7.1). The next theorem follows from the Brezis–Browder Ordering Principle [24]. Theorem 7.1. Let Y be a Banach space, B : D(B) ⊆ Y ; Y an m-dissipative operator, let K ⊆ R × D(B) be nonempty and let G : K → B(Y ). Then the following conditions are equivalent: (i) K is C 0 -viable with respect to B + G; (ii) for each (τ , ζ ) ∈ K , (7.2) has at least one noncontinuable C 0 -solution z : [τ , T ) → X . The next result concerns the existence of global solutions. Theorem 7.2. Let Y be a Banach space, B : D(B) ⊆ Y ; Y an m-dissipative operator, let K ⊆ R × D(B) be nonempty and let G : K → B(Y ). If K is Y -closed, G is positively sublinear and maps bounded subsets in K into bounded subsets in Y and K is C 0 -viable with respect to B + G, then each C 0 -solution of (7.2) can be continued up to a global one, i.e., defined on [τ , TK ), where TK is given by (7.1). 8. Solutions in ‘‘moving sets’’ The results in this section are adapted from Necula, Popescu, Vrabie [17]. Let X be a real Banach space, let C ⊂ X be a closed convex cone with C ∩ (−C ) = {0}, let ‘‘’’ be the partial order on X defined by C , i.e., x  y if and only if y − x ∈ C . Let a : I → X be a continuous function, and let K : I ; X be defined by K (t ) := {x ∈ X ; a(t )  x} for each t ∈ I. Let be the graph of K and F : ; X be a given multi-function. We are interested in finding sufficient conditions in order that be viable with respect to (1, F ), i.e., in order that, for each (τ , ξ ) ∈ I × X with a(τ )  ξ , to exist at least one solution u : [τ , T ] → X of the problem u0 (t ) ∈ F (t , u(t )) u(τ ) = ξ a(t )  u(t ) for each t ∈ [τ , T ].

(

(8.1)

We notice that the first necessary and sufficient condition for the viability of in the specific case X = R and C = [0, ∞), is due to Perron [6].

with respect to a single-valued function,

1 ,1

Theorem 8.1. Let a ∈ W loc (I ; X ), let and F be as above and let us assume that F is a nonempty, convex, closed valued compact u.s.c. multi-function. Then, a necessary and sufficient condition in order that be viable with respect to F is that

−a0 (τ ) + F (τ , a(τ ) + ξ ) ∈ TSC (ξ )

for each (τ , ξ ) ∈ I × ∂ C .

1 ,1

Theorem 8.2. Let a ∈ W loc (I ; X ), let and F be as above and let us assume that C is locally compact and F is nonempty, convex, weakly compact valued multi-function strongly–weakly u.s.c. Then, a sufficient condition in order that be viable with respect to F is that

−a0 (τ ) + F (τ , a(τ ) + ξ ) ∈ TSC (ξ ) for each (τ , ξ ) ∈ I × ∂ C . Theorems 8.1 and 8.2 follow from Theorems 3.2 and respectively 3.3, with the help of the simple lemma below. 1 ,1

Lemma 8.1. Let a ∈ W loc (I ; X ) and let be as above. Let τ ∈ I be a point of differentiability from the right for a, let ξ ∈ C and let E ∈ B(X ). Then, the following conditions are equivalent: (i) −a0 (τ ) + E ∈ TSC (ξ ) (ii) E ∈ TS (τ , a(τ ) + ξ ). Moreover, (iii) dist (−a0 (τ ) + E ; C ) = 0 implies both (i) and (ii).

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e987

9. A comparison result for a reaction–diffusion system The next application is from Burlică [25]. Let Ω ⊆ Rn , n = 1, 2, . . ., be a bounded domain with C 2 boundary Γ , let δi ≥ 0, i = 1, 2, a > 0, r > 0, let fi : R+ × R → R+ and gi : R+ × R → R− be four functions with f1 , g1 l.s.c., f2 , g2 u.s.c. and let us consider the following general predator–pray system

 u ∈ δ1 ∆u − au + [f1 (u, v), f2 (u, v)]   t vt ∈ δ2 ∆v + r v + [g1 (u, v), g2 (u, v)]  u = v = 0 u(τ , x) = ξ (x) v(τ , x) = η(x)

in (τ , +∞) × Ω in (τ , +∞) × Ω on (τ , +∞) × Γ in Ω .

(9.1)

Here, ξ , η ∈ L2 (Ω ), ξ (x) ≥ 0 and η(x) ≥ 0 a.e. for x ∈ Ω . Let e f : R+ × R → R+ and e g : R+ × R → R− be two continuous functions such that



f1 (u, v) ≤ min{f2 (u, v),e f (u, v)} g2 (u, v) ≥ max{g1 (u, v),e g (u, v)}

(9.2)

for each (u, v) ∈ R × R. Let us consider also the comparison predator–pray system

 e  ut = δ1 ∆u − au + f (u, v) vt = δ 2 ∆v + r v + e g (u, v) u = v = 0   u(0, x) = u0 (x), v(0, x) = v0 (x)

in (0, +∞) × Ω in (0, +∞) × Ω on (0, +∞) × Γ , in Ω

(9.3)

and let (e u,e v) : R+ × Ω → R+ × R+ be a mild solution of (9.3). The problem is to find a sufficient condition in order that for each ξ , η ∈ L2 (Ω ), with 0 ≤ ξ (x) ≤ e u(τ , x) e v(τ , x) ≤ η(x)



(9.4)

a.e. for x ∈ Ω , to exist one solution, (u, v) : R+ × Ω → R+ × R+ , of the predator–pray system (9.1), such that, for each t ∈ [τ , ∞) 0 ≤ u(t , x) ≤ e u(t , x) e v(t , x) ≤ v(t , x)



(9.5)

a.e. for x ∈ Ω . Let C ⊆ R × L2 (Ω ) × L2 (Ω ) be defined by C = (t , u, v) ∈ R+ × L2 (Ω ) × L2 (Ω ); (t , u, v) satisfy (9.7) below



0 ≤ u(x) ≤ e u(t , x) e v(t , x) ≤ v(x)



(9.6)



(9.7)

a.e. for x ∈ Ω . We intend to show that, for each (τ , ξ , η) ∈ C, the problem (9.1) has at least one solution (u, v) : [τ , ∞) → L2 (Ω ) × L2 (Ω ) with (t , u(t ), v(t )) ∈ C for each t ∈ [τ , ∞). To this aim, let us assume that there exist ci ≥ 0, i = 1, . . . , 5, such that



|e f (u, v)| ≤ c1 |u| + c2 |e g (u, v)| ≤ c3 |u| + c4 |v| + c5

(9.8)

for each (u, v) ∈ R+ × R+ . Theorem 9.1. Let fi : R × R → R+ and gi : R × R → R− , i = 1, 2 be as above and let e f : R × R → R+ and e g : R × R → R− be continuous and such that, for each (u0 , v0 ) ∈ R+ × R+ , u 7→ e f (u, v0 ) and v 7→ e g (u0 , v) are nondecreasing, u 7→ e g (u, v0 ) and v 7→ e f (u0 , v) are nonincreasing and satisfy (9.2) and (9.8). Let (u0 , v0 ) ∈ L2 (Ω ) × L2 (Ω ) with u0 (x) ≥ 0 and v0 (x) ≥ 0 a.e. for x ∈ Ω and let (e u,e v) : R+ → L2 (Ω ) × L2 (Ω ) be a global mild solution of (9.3) with e u ≥ 0 for each t ≥ 0 and a.e. for x ∈ Ω . Let C be defined by (9.6). Then, for each (τ , ξ , η) ∈ C, the problem (9.1) has at least one global mild solution (u, v) : [τ , ∞) → L2 (Ω ) × L2 (Ω ) satisfying for each τ < δ < T : (i) u, v ∈ C ([τ , T ]; L2 (Ω )) ∩ L2 (δ, T ; H 2 (Ω )) ∩ W 1,1 (δ, T ; H01 (Ω )); (ii) for each t ∈ [τ , ∞), we have (t , u(t ), v(t )) ∈ C. Idea of proof. First we show that the locally closed set C in R × L2 (Ω ) × L2 (Ω ) is mild viable with respect to (1, δ1 ∆ − aI + f , δ2 ∆ + rI + g ), and second that every mild solution (t , u, v) : [τ , T ) → R × L2 (Ω ) × L2 (Ω ), satisfying (t , u(t ), v(t )) ∈ C for each t ∈ [τ , T ), can be extended to a global one obeying the very same constraints. For related results obtained via combinations of monotonicity and viability (flow-invariance) techniques see Ladde, Lakshmikantham, Vatsala [26]. The case of a fully nonlinear reaction–diffusion system was very recently considered by Roşu [27].

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10. A comparison result for a nonlinear diffusion inclusion The application here included is from Cârjă, Necula, Vrabie [20]. Let us consider the Cauchy problem for the nonlinear diffusion equation zt ∈ ∆ϕ(z ) + [g1 (t , x, z ), g2 (t , x, z )] z=0 z (τ , x) = η(x)

(

in Qτ ,T on Στ ,T in Ω ,

(10.1)

where Qτ ,T = (τ , T ) × Ω , Στ ,T = (τ , T ) × Γ , ∆ϕ is the nonlinear diffusion operator, gi : R × Ω × R → R+ for i = 1, 2. We assume that g1 is l.s.c. and g2 is u.s.c. on R × Ω × R and 0 ≤ g1 (t , x, y) ≤ g2 (t , x, y) for each (t , x, y) ∈ R × Ω × R. Definition 10.1. By a C 0 -solution of the problem (10.1), on [τ , T ], we mean a continuous function z : [τ , T ) → L1 (Ω ), for which there exists g ∈ L1 (0, T ; L1 (Ω )) with g (t , x) ∈ [g1 (t , x, z (t , x)), g2 (t , x, z (t , x))] a.e. for t ∈ [τ , T ] and x ∈ Ω , such that z is a C 0 -solution of the problem



z 0 (t ) = Bz (t ) + g (t ) z (τ ) = ζ

in the usual sense, where B = ∆ϕ is the nonlinear diffusion operator with homogeneous boundary conditions in L1 (Ω ), i.e.



D(B) = {u ∈ L1 (Ω ); ϕ(u) ∈ W 1,1 (Ω ), ∆ϕ(u) ∈ L1 (Ω )} Bu = ∆ϕ(u) for u ∈ D(B).

By a C 0 -solution of the problem (10.1), on [τ , e T ), τ < e T ≤ ∞, we mean a function z ∈ C ([τ , e T ); L1 (Ω )) such that for each 0 e τ < T < T , z is a C -solution of (10.1) on [τ , T ] in the sense stated before. Next let us consider another function h : R × Ω × R → R+ which is continuous, bounded, nondecreasing with respect to its last argument and such that g1 (t , x, y) ≤ h(t , x, y) for each (t , x, y) ∈ R × Ω × R. Our aim is to show that, under fairly general assumptions, for each z0 ∈ L1 (Ω ), with z0 (x) ≥ 0 a.e. for x ∈ Ω , and for every global C 0 -solution e z : R+ × Ω → R+ of the nonlinear diffusion equation zt = ∆ϕ(z ) + h(t , x, z ) z=0 z (0, x) = z0 (x)

(

in Q0,∞ on Σ0,∞ in Ω ,

(10.2)

the nonlinear diffusion equation (10.1) has at least one global C 0 -solution satisfying 0 ≤ z (t , x) ≤ e z (t , x), for each t ∈ [τ , ∞) and a.e. for x ∈ Ω , if 0 ≤ ζ (x) ≤ e z (τ , x) a.e. for x ∈ Ω . The main result concerning this problem is: Theorem 10.1. Let ϕ : R → R be continuous on R and C 1 on R \ {0}, with ϕ(0) = 0, and for which there exist C > 0 and a > 0 if n ≤ 2 and a > (n − 2)/n if n ≥ 3 such that ϕ 0 (r ) ≥ C |r |a−1 for each r ∈ R \ {0}. Let gi , h : R × Ω × R → R+ , i = 1, 2, with g1 l.s.c., g2 u.s.c. and h continuous, bounded and nondecreasing with respect of its last argument. Let us assume also that 0 ≤ g1 (t , x, y) ≤ min{g2 (t , x, y), h(t , x, y)}, for each (t , x, y) ∈ R × Ω × R. Let z0 ∈ L1 (Ω ), z0 (x) ≥ 0 a.e. for x ∈ Ω , and let C be defined by (10.3), where e z : R+ → L1 (Ω ) 1 e is a global C0 -solution of (10.2). Then, for each (τ , ζ ) ∈ R × L (Ω ) with 0 ≤ ζ (x) ≤ z (τ , x), a.e. for x ∈ Ω , there exists at least one global C 0 -solution z : [τ , ∞) → L1 (Ω ) of (10.1) satisfying 0 ≤ z ( t , x) ≤ e z (t , x) for each t ≥ τ and a.e. for x ∈ Ω . Idea of proof. Let C ⊆ R × L1 (Ω ) be the infinite tube defined by C = (t , z ) ∈ R+ × L1 (Ω ); 0 ≤ z (x) ≤ e z (t , x) a.e. for x ∈ Ω .





(10.3)

First we show that C is C 0 -viable with respect to (1, B + G), and second we prove that each C 0 -solution u : [τ , T ) → L1 (Ω ) of (10.1) whose graph lies in C can be continued to a C 0 -solution defined on R+ and whose graph is in C too. For a systematic study of comparison results by using monotonicity along with viability arguments, we refer to Ladde, Lakshmikantham, Vatsala [26].

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e989

11. A sufficient condition for null controllability The main result of this section is a multi-valued extension of Theorem 12.1 in Cârjă, Necula, Vrabie [20]. Let X be a Banach space, a ∈ R, B : D(B) ⊆ X ; X such that B − aI is an m-dissipative operator, H : X → B(X ) a given multi-function, ξ ∈ D(B), and c (·) a measurable control taking values in D(0, 1). The problem is to find a control c (·) in order to reach the origin starting from the initial point ξ in some time T , by C 0 -solutions of the state equation



u0 (t ) ∈ Bu(t ) + H (u(t )) + c (t ) u(0) = ξ .

(11.1)

Let G : X ; X be defined by G(x) = ax + H (x) + D(0, 1). We can reformulate the above problem as: for a given ξ ∈ D(A), find T > 0 and a C 0 -solution of multi-valued fully nonlinear Cauchy problem



u0 (t ) ∈ (B − aI )u(t ) + G(u(t )) u(0) = ξ ,

(11.2)

that satisfies u(T ) = 0. Theorem 11.1. Let X be reflexive and let B : D(B) ⊆ X ; X be such that, for some a ∈ R, B − aI is an m-dissipative operator of complete continuous type and which is the infinitesimal generator of a compact semigroup of contractions, {S (t ) : D(B) → D(B); t ≥ 0}. Let H : X → B(X ) be a convex valued and strongly–weakly u.s.c. multi-function such that for some L > 0 we have sup kyk ≤ Lkxk,

(11.3)

y∈H (x)

for every x ∈ X . Assume 0 ∈ D(B) and 0 ∈ B0. Then, for each ξ ∈ D(B) with ξ 6= 0 there exists a C 0 -solution u : [0, ∞) → X of (11.2) which satisfies

ku(t )k ≤ kξ k − t + (L + a)

t

Z

ku(s)kds

(11.4)

0

for every t ≥ 0 for which u(t ) 6= 0. Idea of proof. We consider the space R × X , the operator A = (0, B − aI ), that generates a compact semigroup of contractions (1, S (t )) on R × D(B), the locally closed set K = {(λ, x) ∈ R+ × D(B) \ {0}; kxk ≤ λ}, and the multi-function F : R × X ; R × X defined by F (t , x) = ((L + a)kxk − 1, G(x)), for every (t , x) ∈ R × X . Clearly F is convex, weakly compact valued and strongly–weakly u.s.c. A routine calculation shows that

((L + a)kξ k − 1, G(ξ )) ∈ QTSAK (λ, ξ ), for every (λ, ξ ) ∈ K . Therefore, from Theorems 5.2 and 7.1, we deduce that for each ξ ∈ D(A), ξ 6= 0, there exist T > 0 and a noncontinuable mild solution (z , u) : [0, T ) → R × X of the Cauchy problem z 0 (t ) = (L + a)ku(t )k − 1 u0 (t ) ∈ (B − aI )u(t ) + G(u(t )) z (0) = kξ k and u(0) = ξ ,

(

(11.5)

which satisfies (z (t ), u(t )) ∈ K for every t ∈ [0, T ). This means that (11.4) is satisfied for every t ∈ [0, T ). Now, let us observe that u, as a solution of (11.2), can be continued to R+ simply because G has sublinear growth. So, u(T ) exists, even though the solution (z , u) of (11.5) is defined merely on [0, T ). Clearly u(T ) must be 0 since otherwise, (z , u) can be continued to the right of T , thereby contradicting the fact that (z , u) is noncontinuable.  Corollary 11.1. Under the hypothesis of Theorem 11.1: (i) if L + a ≤ 0, for any ξ ∈ D(B), ξ 6= 0, there exist a control c (·) and a C 0 -solution of (11.1) that reaches the origin of X in some time T ≤ kξ k and satisfies ku(t )k ≤ kξ k − t for every 0 ≤ t ≤ T ; (ii) if L + a > 0, for every ξ ∈ D(B) satisfying 0 < kξ k < 1/(L + a), there exist a control c (·) and a C 0 -solution of (11.1) that reaches the origin of X in some time T ≤ L+1 a log 1−(L+1a)kξ k , and satisfies

 ku(t )k ≤ e(L+a)t kξ k −

1 L+a

 +

1 L+a

for every 0 ≤ t ≤ T .

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Example 11.1. Let Ω ⊆ Rn , n = 1, 2, . . ., be a bounded domain with C 2 boundary Γ , let β : R ; R be a maximal monotone operator with β(0) = 0 and let us consider the Cauchy problem for the nonlinear diffusion equation ut ∈ ∆u + [h1 (u), h2 (u)] + c (t , x) −uν ∈ β(u) u(0, x) = ξ (x)

(

in Q0,T on Σ0,T in Ω ,

(11.6)

where Q0,T = (0, T )× Ω , Σ0,T = (0, T )× Γ , uν is the exterior normal to Γ , hi : R → R+ for i = 1, 2 and c : [0, T ] → L2 (Ω ) is a control. From Theorem 11.1 combined with both Example 1.5.3, p. 17, and Remark 2.2.6, p. 44, in Vrabie [23], we deduce Theorem 11.2. Let Ω ⊆ Rn , n = 1, 2, . . ., be a bounded domain with C 2 boundary Γ and let β : R → R be a maximal monotone operator with β(0) = 0. Let h1 , h2 : R → R+ with h1 l.s.c. and h2 u.s.c. Further we assume that 0 ≤ h1 (u) ≤ h2 (u) for each u ∈ R and also that there exists L > 0 such that |hi (u)| ≤ L|u| for i = 1, 2 and each u ∈ R. Then, for each ξ ∈ L2 (Ω ), kξ kL2 (Ω ) ≤ 1/L there exist T > 0 and a control c : [0, T ] → L2 (Ω ), kc (t )kL2 (Ω ) ≤ 1 a.e. for t ∈ [0, T ] and such that the corresponding solution of (11.6) satisfies u(T , x) = 0 a.e. for x ∈ Ω . Acknowledgement The third author expresses his warmest thanks to the organizers for their kind invitation to deliver this one-hour lecture at WCNA-2008. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

J.P. Aubin, A. Cellina, Differential Inclusions, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1984. O. Cârjă, M. Necula, I.I. Vrabie, Viability, Invariance and Applications, in: North-Holland Mathematics Studies, vol. 207, Elsevier, 2007. M. Nagumo, Über die Lage der Integralkurven gewöhnlicher Differentialgleichungen, Proc. Phys.-Math. Soc. Japan 24 (1942) 551–559. H. Bouligand, Sur les surfaces dépourvues de points hyperlimités, Ann. Soc. Polon. Math. 9 (1930) 32–41. F. Severi, Su alcune questioni di topologia infinitesimale, Ann. Soc. Polon. Math. 9 (1931) 97–108. O. Perron, Ein neuer Existenzbeweis für Integrale der Differentialgleichung, y0 = f (x, y), Math. Ann. 76 (1915) 471–484. N.H. Pavel, Invariant sets for a class of semi-linear equations of evolution, Nonlinear Anal. 1 (1976–1977) 187–196. I.I. Vrabie, Compactness methods and flow-invariance for perturbed nonlinear semigroups, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 27 (1981) 117–125. V. Lakshmikantham, S. Leela, Nonlinear Differential Equations in Abstract Spaces, in: International Series in Nonlinear Mathematics, vol. 2, Pergamon Press, 1981. W. Bebernes, I.D. Schuur, The Ważewski topological method for contingent equations, Ann. Mat. Pura Appl. 87 (1970) 271–278. T. Ważewski, Sur un condition équivalent à l’équation au contingent, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 9 (1961) 865–867. N.H. Pavel, I.I. Vrabie, Équations d’évolution multivoques dans des espaces de Banach, C. R. Acad. Sci. Paris Sér. A–B 287 (1978) A315–A317. N.H. Pavel, I.I. Vrabie, Semilinear evolution equations with multivalued right-hand side in Banach spaces, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 25 (1979) 137–157. S.Z. Shi, Viability theorems for a class of differential operator inclusions, J. Differential Equations 79 (1989) 232–257. D. Bothe, Multivalued differential equations on graphs, Nonlinear Anal. 18 (1992) 245–252. O. Cârjă, I.I. Vrabie, Some new viability results for semilinear differential inclusions, NoDEA Nonlinear Differential Equations Appl. 4 (1997) 401–424. M. Necula, M. Popescu, I.I. Vrabie, Evolution equations on locally closed graphs and applications, in: Proceedings of WCNA-2008, Nonlinear Anal. (submitted for publication). O. Cârjă, I.I. Vrabie, Viability results for nonlinear perturbed differential inclusions, Panamer. Math. J. 9 (1) (1999) 63–74. A. Bressan, V. Staicu, On nonconvex perturbations of maximal monotone differential inclusions, Set-Valued Anal. 2 (1994) 415–437. O. Cârjă, M. Necula, I.I. Vrabie, Necessary and sufficient conditions for viability for nonlinear evolution inclusions, Set-Valued Anal. 16 (2008) 701–731. O. Cârjă, M. Necula, I.I. Vrabie, Necessary and sufficient conditions for viability for semilinear differential inclusions, Trans. Amer. Math. Soc. 361 (2009) 343–390. M. Necula, M. Popescu, I.I. Vrabie, Viability for differential inclusions on graphs, Set-Valued Anal. 16 (2008) 961–981. I.I. Vrabie, Compactness Methods for Nonlinear Evolutions, second Ed., in: Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman, 1995. H. Brezis, F.E. Browder, A general principle on ordered sets in nonlinear functional analysis, Adv. Math. 21 (1976) 355–364. M. Burlică, Viability for semi-multi-valued reaction–diffusion systems, AIP Conference Proceedings Volume 1048 (2008) 126–129. G.S. Ladde, V. Lakshmikantham, A.S. Vatsala, Monotone Iterative Techniques for Nonlinear Differential Equations, in: Monographs, Advanced Texts and Surveys in Pure and Applied Mathematics, vol. 27, Pitman, 1985. D. Roşu, Viability for nonlinear multi-valued reaction–diffusion systems, AIP Conference Proceedings Volume 1048 (2008) 458–461.