- Email: [email protected]
Nagumo viability theorem. Revisited
Nonlinear Analysis 64 (2006) 2043 – 2052 www.elsevier.com/locate/na
Nagumo viability theorem. Revisited夡 Ioan I. Vrabie Faculty of Mathematics, “Al. I. Cuza” University of Ia¸si & “O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania Received 10 April 2005; accepted 26 July 2005
Abstract We consider the nonlinear ordinary differential equation u (t) = f (t, u(t)) + h(t, u(t)), where X is a real Banach space, I is a nonempty and open interval, K a nonempty and locally closed subset in X, f : I × K → X a compact function, and h : I × K → X continuous on I × K and locally Lipschitz with respect to its last argument. We prove that a necessary and sufficient condition in order that for each (, ) ∈ I × K there exists T > such that the equation above has at least one solution u : [, T ] → K is the tangency condition below lim inf s↓0
1 d( + s[f (, ) + h(, )]; K) = 0 s
for each (, ) ∈ I × K. As an application, we deduce the existence of positive solutions for a class of pseudoparabolic semilinear equations. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: Primary 34G20; 47J35; secondary 35K70; 80A20 Keywords: Viable set; Tangency condition; b-compact function; Lipschitz function; Pseudoparabolic equation; Slow fluids
1. Introduction Let X be a real Banach space, I a nonempty and open interval, K a nonempty, locally closed subset in X, f : I × K → X a b-compact function and h : I × K → X a continuous 夡
Supported by CERES Grant Nr 4-194/2004. E-mail address: [email protected].
0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.07.037
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function on I × K, which is locally Lipschitz with respect to its last argument. The goal of this paper is to prove a viability result referring to the nonlinear differential equation u = f (t, u) + h(t, u).
(1.1)
Although essentially referring to ordinary differential equations, it turns out that our main result, i.e., Theorem 1.1, proves useful in obtaining the existence of positive solutions to some classes of partial differential equations as well. See Section 3. We begin by recalling that a subset K in X is locally closed if for each ∈ K there exists > 0 such that D(, ) ∩ K is closed.1 Obviously, each subset in X which is either open or closed is locally closed. Moreover, each subset K in X which is closed relative to some open subset D, i.e., for which there exists a closed subset C ⊆ X such that K = C ∩ D, is locally closed in X (and conversely, whenever X is finite dimensional). A function f : I × K → X is called b-compact if it is continuous and for each [, T ] ⊂ I , ∈ X and > 0 with D(, ) ∩ K closed, f ([, T ] × (D(, ) ∩ K)) is relatively compact in X. The function f is called compact if it carries bounded subsets in I × K into relatively compact subsets in X. Clearly, each compact function is b-compact, and whenever I = R and K = X, the converse statement is also true. However, we notice that, if either I does not coincide with R or K does not coincide with X, there exist b-compact functions which are not compact. We say that h : I × K → X is locally Lipschitz with respect to its last argument if for each (, ) ∈ I × K there exists T > , r > 0 and L = L, > 0 such that [, T ] ⊂ I and h(t, u) − h(t, v) Lu − v
(1.2)
for each (t, u), (t, v) ∈ [, T ] × (D(, ) ∩ K). By a solution of (1.1) we mean a C1 -function u : [, T ] → K with [, T ] ⊂ I and satisfying u (t) = f (t, u(t)) + h(t, u(t)) for each t ∈ [, T ], and u() = . We also recall that K is viable with respect to f + h if for each (, ) ∈ I × K there exists T > , such that Eq. (1.1) has at least one solution u : [, T ] → K satisfying u() = . The first necessary and sufficient condition for viability goes back to Nagumo [8] who analyzed the case when K is a closed, or merely locally closed subset in R n and h ≡ 0. It is interesting to notice that Nagumo’s result (or variants of it) has been rediscovered independently, in the late sixties and early seventies, by Yorke [12,13], Crandall [4] and Hartman [7], among others. Yorke [12] considered subsets K in R n which are closed relative to an open subset D in R n . But, as we already pointed out, each locally closed subset is closed relative to some open subset and conversely, all his results hold true for locally closed sets as well. Crandall [4] considers the case K ⊆ D is locally closed and f : I × D → R n is continuous. He shows that a sufficient condition for K to be viable (forward invariant in his terminology) with respect to f is (1.3). Hartman [7] proves essentially the same result for D open, K closed relative to D and f : I ×D → R n continuous, and shows, in addition, that (1.3) (with h ≡ 0) is necessary for the viability of K with respect to f. Brezis [2] analyzes the case when D is open in an arbitrary Banach space X, K ⊆ D is relatively closed, f ≡ 0 and h locally Lipschitz, and proves that (1.3) with “lim” instead of “lim inf” (and f ≡ 0) is necessary and sufficient for K to be local invariant, “flow invariant” in his terminology, with respect to h. For more details on this subject, we refer the reader to Cârj˘a–Vrabie [3]. 1 As usual, D(, ) denotes the closed ball with center and radius .
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The main result in this paper, i.e. Theorem 1.1, extends the main results in [8,12,4,7], as well as the existence, i.e. viability, part in [2]. We notice that the case K open and f, h Carathédory functions with f “compact” and h locally Lipschitz has been considered by Frigon–O’Reagan [6]. Theorem 1.1. Let f : I × K → X be a b-compact function, and let h : I × K → X be continuous on I × K, and locally Lipschitz with respect to its last argument. Then K is viable with respect to f + h if and only if for each (, ) ∈ I × K, we have lim inf s↓0
1 d( + s[f (, ) + h(, )]; K) = 0.2 s
(1.3)
We conclude this section with a result concerning the existence of global solutions. A function g : I × K → X is called positively sublinear if there exist a norm · on X, a > 0, b ∈ R, and c > 0 such that g(t, )a + b c (g(t, ·)), where for each t ∈ I and ∈ K+ c (g(t, ·) = { ∈ K; > c and [, g(t, )]+ > 0}. K+
We note that here [, ]+ is the right directional derivative of the norm · calculated at in the direction , i.e. [, ]+ = lim s↓0
1 ( + s − ). s
See Definition 3.5.2 [9, p. 95]. Using similar arguments as those in the proof of Theorem 6.5.2 in [3], we can prove: Theorem 1.2. Assume that all the hypotheses of Theorem 1.1 are satisfied and, in addition, that K is closed and f + h is positively sublinear. Then, each solution of (1.1) can be continued up to a global one, i.e. defined on [0, +∞). 2. Proof of Theorem 1.1
Proof (Necessity). This follows from the simple result below, in which neither K is assumed to be locally closed, nor f or/and h to be continuous and, in addition, “lim inf” is replaced by “lim”. Theorem 2.1. If K is viable with respect to f + h : I × K → X, then, for each (, ) ∈ I × K, we have lim s↓0
1 d( + s[f (, ) + h(, )]; K) = 0. s
2 Here and thereafter, d(; K) denotes the distance between the point ∈ X and the subset K in X.
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See Theorem 3.2.1 in [3]. Sufficiency. The first step is concerned with the existence of approximate solutions. Let (, ) ∈ I × K be arbitrary and let us choose > 0, M > 0 and T ∈ I , T > , such that D(, ) ∩ K be closed, f (t, x) + h(t, x) M
(2.1)
for every t ∈ [, T ] and x ∈ D(, ) ∩ K, and (T − )(M + 1) .
(2.2)
The existence of these three numbers is ensured by the fact that K is locally closed (from where the existence of > 0), by the continuity of f + h which implies its boundedness on [, T ] × (D(, ) ∩ K), provided > 0 is small enough. So, we deduce the existence of M > 0 satisfying (2.1), and of T ∈ I , T > , sufficiently close to , in order to have (2.2). Diminishing T > and/or > 0, if necessary, we may assume that h satisfies (1.2) on [, T ] × (D(, ) ∩ K). Lemma 2.1. For each ε ∈ (0, 1) and > 0, M > 0 and T ∈ I , as above, there exist three functions: : [, T ] → [, T ], nondecreasing, g : [, T ] → X, Riemann integrable and u : [, T ] → X, continuous, such that: (i) (ii) (iii) (iv)
t − ε (t)t for each t ∈ [, T ]; g(t) ε for each t ∈ [, T ]; u((t)) ∈ D(, ) ∩ K for each t ∈ [, T ] and u(T ) ∈ D(, ) ∩ K; u satisfies t t t u(t) = + f (s, u((s))) ds + h(s, u((s))) ds + g(s) ds
for each t ∈ [, T ]. The proof of Lemma 2.1 will be omitted because it follows exactly the same lines as those in the proof of sufficiency of both Theorem 3.1.1 in [3], and Theorem 7.7.2 [11, p. 289], of course, with the help of the simple Proposition 2.1. Proposition 2.1. Assume that, for some ∈ K, t → f (t, ) + h(t, ) is continuous from the right at ∈ I . Then f + h satisfies (1.3) at (, ) if and only if +s +s 1 lim inf d + f (, ) d + h(, ) d; K = 0. s↓0 s For the sake of simplicity, in all that follows, we will say that a triple (, g, u), satisfying (i), (ii), (iii) and (iv) in Lemma 2.1, is an ε-approximate solution to the Cauchy problem u (t) = f (t, u(t)) + h(t, u(t)), (2.3) u() = on the interval [, T ].
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In the second step we will prove the convergence of a suitably chosen sequence of approximate solutions. First, we state a variant of the well-known Gronwall’s Lemma whose proof follows exactly the same lines as those of Lemma 10.2.1 [10, p. 232]. Lemma 2.2 (Gronwall). Let x, k : [, T ) → R+ be measurable and m 0. Let us assume that s → k(s)x(s) is locally integrable on [, T ) and t x(t) m + k(s)x(s) ds a
for every t ∈ [, T ). Then x(t) me
t a
k(s) ds
for every t ∈ [, T ). So, let us consider a sequence (εn )n∈N in (0, 1), decreasing to 0, and let ((n , gn , un ))n∈N be a sequence of εn -approximate solutions of (2.3) defined on [, T ]. We will show first that (un (n ))n∈N has at least one Cauchy subsequence in the sup norm. We have un (n (t)) − um (m (t)) un (n (t)) − un (t) + un (t) − um (t) + um (t) − um (m (t)).
(2.4)
At this point let us observe that, for each n ∈ N and each t ∈ [, T ], we have t un (t) − un (n (t)) f (s, un (n (s))) ds n (t) t t + h(s, un (n (s))) ds + gn (s) ds. n (t)
n (t)
From (2.1) and (iii), we have f (s, un (n (s))) + h(s, un (n (s))) M for each n ∈ N and each s ∈ [, T ]. In view of (i) and (ii), we deduce that for each ε > 0, there exists n0 (ε) ∈ N , such that un (t) − un (n (t)) (M + εn )(t − n (t)) (M + 1)εn ε for each n ∈ N, nn0 (ε), and each t ∈ [, T ]. Next, we have un (t) − um (t) t t f (s, u ( (s))) ds − f (s, u ( (s))) ds n n m m t + h(s, un (n (s))) − h(s, um (m (s))) ds t + gn (s) − gm (s) ds
(2.5)
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for every n, m ∈ N and every t ∈ [, T ]. Since h satisfies (1.2) on [, T ] × (D(, ) ∩ K), we get t h(s, un (n (s))) − h(s, um (m (s))) ds t L un (n (s)) − um (m (s)) ds.
Accordingly un (t) − um (t) t t f (s, um (m (s))) ds supt∈[,T ] f (s, un (n (s))) ds − t +L un (n (s)) − um (m (s)) ds T gn (s) − gm (s) ds +
(2.6)
for every n, m ∈ N and every t ∈ [, T ]. From (ii), we deduce that, given ε > 0, there exists n1 (ε) > 0 such that, for each n, m ∈ N, with n, m n1 (ε), we have T gn (s) − gm (s) ds ε. (2.7)
Next, let n ∈ N∗ and let us denote by Fn : [, T ] → X the function defined by t f (s, un (n (s))) ds Fn (t) =
for each t ∈ [, T ]. As f ([, T ] × (K ∩ D(, ))) is relatively compact, it follows that, for each t ∈ [, T ], the set {Fn (t) ; n ∈ N∗ } is relatively compact in X. See for instance Lemma A.1.3 in [10, p. 295]. Moreover, from (2.1), we get f (s, un (n (s))) M for each n ∈ N and each s ∈ [, T ]. Therefore Fn (t) − Fn (s) M|t − s| for each n ∈ N and each t, s ∈ [, T ]. Thus {Fn ; n ∈ N} is equicontinuous on [, T ]. In view of Arzelà–Ascoli’s Theorem A.2.1 in [10, p. 296], there exists F ∈ C([, T ]; X) such that, on a subsequence at least, lim Fn (t) = F (t),
n→∞
uniformly for t ∈ [, T ]. So, for the very same ε > 0, there exists n2 (ε) ∈ N such that, for each n, m ∈ N, with n, m n2 (ε), we have t t ε. sup f (s, u ( (s))) ds − f (s, u ( (s))) ds n n m m t∈[,T ]
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Denote by n(ε) = max{n0 (ε), n1 (ε), n2 (ε)}. From (2.6), (2.7) and the preceding inequality, we deduce un (t) − um (t) 2ε + L
t
un (n (s)) − um (m (s)) ds
for all n, m ∈ N, n, m n(ε) and t ∈ [, T ]. In view of (2.4), (2.5) and the last inequality, we deduce un (n (t)) − um (m (t)) 4ε + L
t
un (n (s)) − um (m (s)) ds
for all n, m ∈ N, n, m n(ε) and t ∈ [, T ]. Gronwall’s Lemma 2.2 implies un (n (t)) − um (n (t)) 4εeL(T −) for all n, m ∈ N, n, m n(ε) and t ∈ [, T ]. But this inequality shows that (un (n ))n∈N is a Cauchy sequence in the sup norm. So, there exists a function u : [, T ] → X such that lim un (n (s)) = u(s) n
uniformly for s ∈ [, T ]. Taking into account (iii) and of the fact that D(, ) ∩ K is closed, we deduce that u(t) ∈ D(, ) ∩ K for every t ∈ [, T ]. Moreover, (2.5), along with limn εn = 0, implies lim un (s) = u(s) n
uniformly for s ∈ [, T ]. Using (i) and (ii), we conclude that both lim n (t) = t
n→∞
and
lim gn (t) = 0
n→∞
uniformly on [, T ]. Finally, passing to the limit for n → +∞ both sides in the approximate equations in (iv), i.e., in t un (t) = + f (s, un (n (s))) ds t t + h(s, un (n (s))) ds + gn (s) ds,
we deduce that u(t) = +
t
f (s, u(s)) ds +
t
h(s, u(s)) ds
for every t ∈ [, T ], which achieves the proof of the theorem.
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3. Existence of positive solutions to some pseudoparabolic equations Let be a bounded domain in R3 , with smooth boundary and let us consider the following semilinear pseudoparabolic partial differential equation: ut = ut + u + g(u), (t, x) ∈ QT , (3.1) u = 0, (t, x) ∈ T , u(0, x) = (x), x ∈ . Here and thereafter QT = (0, T ) × and T = (0, T ) × . Theorem 3.1. Let g : R → R be continuous and let us assume that u + g(u)0
(3.2)
for each u ∈ R+ . Then, for each ∈ H 2 () ∩ H01 () with − 0 a.e. on , there exists T > 0 such that problem (3.1) has at least one solution u ∈ C 1 ([0, T ]; H 2 () ∩ H01 ()) satisfying both u(t)−u(t)0 and u(t)0 for each t ∈ [0, T ] and a.e. on . If, in addition, g is positively sublinear, then each solution of (3.1) can be continued up to a global one. Proof. Clearly, (3.1) is equivalent to (u − u)t = u + g(u), (t, x) ∈ QT , u = 0, (t, x) ∈ T , u(0, x) = (x), x ∈ .
(3.3)
Let A : D(A) ⊆ L2 () → L2 () be defined by D(A) = H01 () ∩ H 2 (), Au = u for each u ∈ D(A), let J = (I − A)−1 and let us denote by u = J v. Since AJ = J − I , (3.3) can be rewritten as an abstract differential equation in the space L2 (), i.e. v = f (v) + h(v), (3.4) v(0) = , where f : L2 () → L2 () is defined by f (v)(x) = g(J v)(x) for each v ∈ L2 () and a.e. for x ∈ , h = J − I and = (I − A). Since the operator J is continuous from L2 () to H 2 () ∩ H01 (), and, in our specific case, i.e. n = 3, H 2 () is compactly imbedded in C() (see (iii) in Theorem 1.5.4 in [10, p. 18]), it follows that f is well-defined and b-compact (in fact compact). Moreover, J is obviously Lipschitz, being linear continuous. At this point, let us recall that = (I − A)0 a.e. on . Let K = {v ∈ L2 (); v 0 a.e. on }.
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In view of (3.2), we deduce that f, h and K satisfy the tangency condition (1.3). Indeed, to prove that for each ∈ K lim inf s↓0
1 d( + s(J − ) + sf (J ); K) = 0, s
it suffices to show that, for each s ∈ (0, 1), we have + s(J − ) + sf (J )0
(3.5)
a.e. on . But this is certainly the case, because 0 a.e. on , along with the maximum principle for elliptic equations, implies both (1 − s) 0 and J 0 a.e. on . By virtue of (3.2), it follows that J + f (J ) 0 and thus (3.5) holds. So, we are in the hypotheses of Theorem 1.1. Therefore K is viable with respect to f + h, and this implies that there exists at least one solution, v : [0, T ] → L2 (), of (3.4), with v(t) ∈ K for each t ∈ [0, T ]. But u(t) = J v(t), and consequently u(t) − u(t) 0 for each t ∈ [0, T ] and a.e. on . Using again the maximum principle for elliptic equations, we conclude that u(t)0 for each t ∈ [0, T ] and a.e. on . Since v ∈ C 1 ([0, T ]; L2 ()), and u = J v, with J linear continuous from L2 () to H 2 () ∩ H01 (), we conclude that u ∈ C 1 ([0, T ]; H 2 () ∩ H01 ()), and the proof is complete. We conclude the paper by noticing that this problem appears in the study of some special kind of fluids. For some other mathematical models which can be treated with similar arguments see [1,5] and the references therein.
Acknowledgements The author takes this opportunity to thank Professor J.I. Díaz for his useful remarks on a preliminary version of the paper.
References [1] S.N.Antontsev, J.I. Díaz, S.I. Shmarev, Energy methods for free boundary problems:Applications to nonlinear PDSs and fluid mechanics, Progress in Nonlinear Differential Equations and Their Applications, vol. 48, Birkhäuser, Basel, Boston, Berlin, 2002. [2] H. Brezis, On a characterization of flow-invariant sets, Comm. Pure Appl. Math. 23 (1970) 261–263. [3] O. Cârj˘a, I.I. Vrabie, Differential equations on closed sets, in: A. Cañada, P. Drabek, A. Fonda (Eds.), Handbook of Differential Equations, vol. 2, Ordinary Differential Equations, 2005, pp. 147–238. [4] M.G. Crandall, A generalization of Peano’s existence theorem and flow-invariance, Proc. Am. Math. Soc. 36 (1972) 151–155. [5] J.I. Díaz, R. Quintanilla, Spatial and continuous dependence estimates in linear viscoelasticity, J. Math. Anal. Appl. 273 (2002) 1–16. [6] M. Frigon, D. O’Reagan, Existence results for initial value problems in Banach spaces, Differential Equations Dynamical Syst. 2 (1994) 41–48. [7] P. Hartman, On invariant sets and on a theorem of Wa˙zewski, Proc. Am. Math. Soc. 32 (1972) 511–520. [8] M. Nagumo, Über die Lage der Integralkurven gewönlicher Differentialgleichungen, Proc. Phys. Math. Soc. Jpn. 24 (1942) 551–559.
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[9] I.I. Vrabie, Compactness methods for nonlinear evolutions, second ed., Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 75, Longman, 1995. [10] I.I. Vrabie, C0 -semigroups and applications, North-Holland Mathematics Studies, vol. 191, Elsevier, Amsterdam, 2003. [11] I.I. Vrabie, Differential Equations. An Introduction to Basic Concepts, Results and Applications, World Scientific, Singapore, 2004. [12] J.A. Yorke, Invariance for ordinary differential equations, Math. Syst. Theory 1 (1967) 353–372. [13] J.A. Yorke, Invariance for ordinary differential equations: correction, Math. Syst. Theory 2 (1968) 381.
Nagumo viability theorem. Revisited夡 Ioan I. Vrabie Faculty of Mathematics, “Al. I. Cuza” University of Ia¸si & “O. Mayer” Mathematics Institute of the Romanian Academy, Ia¸si, Romania Received 10 April 2005; accepted 26 July 2005
Abstract We consider the nonlinear ordinary differential equation u (t) = f (t, u(t)) + h(t, u(t)), where X is a real Banach space, I is a nonempty and open interval, K a nonempty and locally closed subset in X, f : I × K → X a compact function, and h : I × K → X continuous on I × K and locally Lipschitz with respect to its last argument. We prove that a necessary and sufficient condition in order that for each (, ) ∈ I × K there exists T > such that the equation above has at least one solution u : [, T ] → K is the tangency condition below lim inf s↓0
1 d( + s[f (, ) + h(, )]; K) = 0 s
for each (, ) ∈ I × K. As an application, we deduce the existence of positive solutions for a class of pseudoparabolic semilinear equations. 䉷 2005 Elsevier Ltd. All rights reserved. MSC: Primary 34G20; 47J35; secondary 35K70; 80A20 Keywords: Viable set; Tangency condition; b-compact function; Lipschitz function; Pseudoparabolic equation; Slow fluids
1. Introduction Let X be a real Banach space, I a nonempty and open interval, K a nonempty, locally closed subset in X, f : I × K → X a b-compact function and h : I × K → X a continuous 夡
Supported by CERES Grant Nr 4-194/2004. E-mail address: [email protected].
0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.07.037
2044
I. I. Vrabie / Nonlinear Analysis 64 (2006) 2043 – 2052
function on I × K, which is locally Lipschitz with respect to its last argument. The goal of this paper is to prove a viability result referring to the nonlinear differential equation u = f (t, u) + h(t, u).
(1.1)
Although essentially referring to ordinary differential equations, it turns out that our main result, i.e., Theorem 1.1, proves useful in obtaining the existence of positive solutions to some classes of partial differential equations as well. See Section 3. We begin by recalling that a subset K in X is locally closed if for each ∈ K there exists > 0 such that D(, ) ∩ K is closed.1 Obviously, each subset in X which is either open or closed is locally closed. Moreover, each subset K in X which is closed relative to some open subset D, i.e., for which there exists a closed subset C ⊆ X such that K = C ∩ D, is locally closed in X (and conversely, whenever X is finite dimensional). A function f : I × K → X is called b-compact if it is continuous and for each [, T ] ⊂ I , ∈ X and > 0 with D(, ) ∩ K closed, f ([, T ] × (D(, ) ∩ K)) is relatively compact in X. The function f is called compact if it carries bounded subsets in I × K into relatively compact subsets in X. Clearly, each compact function is b-compact, and whenever I = R and K = X, the converse statement is also true. However, we notice that, if either I does not coincide with R or K does not coincide with X, there exist b-compact functions which are not compact. We say that h : I × K → X is locally Lipschitz with respect to its last argument if for each (, ) ∈ I × K there exists T > , r > 0 and L = L, > 0 such that [, T ] ⊂ I and h(t, u) − h(t, v) Lu − v
(1.2)
for each (t, u), (t, v) ∈ [, T ] × (D(, ) ∩ K). By a solution of (1.1) we mean a C1 -function u : [, T ] → K with [, T ] ⊂ I and satisfying u (t) = f (t, u(t)) + h(t, u(t)) for each t ∈ [, T ], and u() = . We also recall that K is viable with respect to f + h if for each (, ) ∈ I × K there exists T > , such that Eq. (1.1) has at least one solution u : [, T ] → K satisfying u() = . The first necessary and sufficient condition for viability goes back to Nagumo [8] who analyzed the case when K is a closed, or merely locally closed subset in R n and h ≡ 0. It is interesting to notice that Nagumo’s result (or variants of it) has been rediscovered independently, in the late sixties and early seventies, by Yorke [12,13], Crandall [4] and Hartman [7], among others. Yorke [12] considered subsets K in R n which are closed relative to an open subset D in R n . But, as we already pointed out, each locally closed subset is closed relative to some open subset and conversely, all his results hold true for locally closed sets as well. Crandall [4] considers the case K ⊆ D is locally closed and f : I × D → R n is continuous. He shows that a sufficient condition for K to be viable (forward invariant in his terminology) with respect to f is (1.3). Hartman [7] proves essentially the same result for D open, K closed relative to D and f : I ×D → R n continuous, and shows, in addition, that (1.3) (with h ≡ 0) is necessary for the viability of K with respect to f. Brezis [2] analyzes the case when D is open in an arbitrary Banach space X, K ⊆ D is relatively closed, f ≡ 0 and h locally Lipschitz, and proves that (1.3) with “lim” instead of “lim inf” (and f ≡ 0) is necessary and sufficient for K to be local invariant, “flow invariant” in his terminology, with respect to h. For more details on this subject, we refer the reader to Cârj˘a–Vrabie [3]. 1 As usual, D(, ) denotes the closed ball with center and radius .
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The main result in this paper, i.e. Theorem 1.1, extends the main results in [8,12,4,7], as well as the existence, i.e. viability, part in [2]. We notice that the case K open and f, h Carathédory functions with f “compact” and h locally Lipschitz has been considered by Frigon–O’Reagan [6]. Theorem 1.1. Let f : I × K → X be a b-compact function, and let h : I × K → X be continuous on I × K, and locally Lipschitz with respect to its last argument. Then K is viable with respect to f + h if and only if for each (, ) ∈ I × K, we have lim inf s↓0
1 d( + s[f (, ) + h(, )]; K) = 0.2 s
(1.3)
We conclude this section with a result concerning the existence of global solutions. A function g : I × K → X is called positively sublinear if there exist a norm · on X, a > 0, b ∈ R, and c > 0 such that g(t, )a + b c (g(t, ·)), where for each t ∈ I and ∈ K+ c (g(t, ·) = { ∈ K; > c and [, g(t, )]+ > 0}. K+
We note that here [, ]+ is the right directional derivative of the norm · calculated at in the direction , i.e. [, ]+ = lim s↓0
1 ( + s − ). s
See Definition 3.5.2 [9, p. 95]. Using similar arguments as those in the proof of Theorem 6.5.2 in [3], we can prove: Theorem 1.2. Assume that all the hypotheses of Theorem 1.1 are satisfied and, in addition, that K is closed and f + h is positively sublinear. Then, each solution of (1.1) can be continued up to a global one, i.e. defined on [0, +∞). 2. Proof of Theorem 1.1
Proof (Necessity). This follows from the simple result below, in which neither K is assumed to be locally closed, nor f or/and h to be continuous and, in addition, “lim inf” is replaced by “lim”. Theorem 2.1. If K is viable with respect to f + h : I × K → X, then, for each (, ) ∈ I × K, we have lim s↓0
1 d( + s[f (, ) + h(, )]; K) = 0. s
2 Here and thereafter, d(; K) denotes the distance between the point ∈ X and the subset K in X.
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See Theorem 3.2.1 in [3]. Sufficiency. The first step is concerned with the existence of approximate solutions. Let (, ) ∈ I × K be arbitrary and let us choose > 0, M > 0 and T ∈ I , T > , such that D(, ) ∩ K be closed, f (t, x) + h(t, x) M
(2.1)
for every t ∈ [, T ] and x ∈ D(, ) ∩ K, and (T − )(M + 1) .
(2.2)
The existence of these three numbers is ensured by the fact that K is locally closed (from where the existence of > 0), by the continuity of f + h which implies its boundedness on [, T ] × (D(, ) ∩ K), provided > 0 is small enough. So, we deduce the existence of M > 0 satisfying (2.1), and of T ∈ I , T > , sufficiently close to , in order to have (2.2). Diminishing T > and/or > 0, if necessary, we may assume that h satisfies (1.2) on [, T ] × (D(, ) ∩ K). Lemma 2.1. For each ε ∈ (0, 1) and > 0, M > 0 and T ∈ I , as above, there exist three functions: : [, T ] → [, T ], nondecreasing, g : [, T ] → X, Riemann integrable and u : [, T ] → X, continuous, such that: (i) (ii) (iii) (iv)
t − ε (t)t for each t ∈ [, T ]; g(t) ε for each t ∈ [, T ]; u((t)) ∈ D(, ) ∩ K for each t ∈ [, T ] and u(T ) ∈ D(, ) ∩ K; u satisfies t t t u(t) = + f (s, u((s))) ds + h(s, u((s))) ds + g(s) ds
for each t ∈ [, T ]. The proof of Lemma 2.1 will be omitted because it follows exactly the same lines as those in the proof of sufficiency of both Theorem 3.1.1 in [3], and Theorem 7.7.2 [11, p. 289], of course, with the help of the simple Proposition 2.1. Proposition 2.1. Assume that, for some ∈ K, t → f (t, ) + h(t, ) is continuous from the right at ∈ I . Then f + h satisfies (1.3) at (, ) if and only if +s +s 1 lim inf d + f (, ) d + h(, ) d; K = 0. s↓0 s For the sake of simplicity, in all that follows, we will say that a triple (, g, u), satisfying (i), (ii), (iii) and (iv) in Lemma 2.1, is an ε-approximate solution to the Cauchy problem u (t) = f (t, u(t)) + h(t, u(t)), (2.3) u() = on the interval [, T ].
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In the second step we will prove the convergence of a suitably chosen sequence of approximate solutions. First, we state a variant of the well-known Gronwall’s Lemma whose proof follows exactly the same lines as those of Lemma 10.2.1 [10, p. 232]. Lemma 2.2 (Gronwall). Let x, k : [, T ) → R+ be measurable and m 0. Let us assume that s → k(s)x(s) is locally integrable on [, T ) and t x(t) m + k(s)x(s) ds a
for every t ∈ [, T ). Then x(t) me
t a
k(s) ds
for every t ∈ [, T ). So, let us consider a sequence (εn )n∈N in (0, 1), decreasing to 0, and let ((n , gn , un ))n∈N be a sequence of εn -approximate solutions of (2.3) defined on [, T ]. We will show first that (un (n ))n∈N has at least one Cauchy subsequence in the sup norm. We have un (n (t)) − um (m (t)) un (n (t)) − un (t) + un (t) − um (t) + um (t) − um (m (t)).
(2.4)
At this point let us observe that, for each n ∈ N and each t ∈ [, T ], we have t un (t) − un (n (t)) f (s, un (n (s))) ds n (t) t t + h(s, un (n (s))) ds + gn (s) ds. n (t)
n (t)
From (2.1) and (iii), we have f (s, un (n (s))) + h(s, un (n (s))) M for each n ∈ N and each s ∈ [, T ]. In view of (i) and (ii), we deduce that for each ε > 0, there exists n0 (ε) ∈ N , such that un (t) − un (n (t)) (M + εn )(t − n (t)) (M + 1)εn ε for each n ∈ N, nn0 (ε), and each t ∈ [, T ]. Next, we have un (t) − um (t) t t f (s, u ( (s))) ds − f (s, u ( (s))) ds n n m m t + h(s, un (n (s))) − h(s, um (m (s))) ds t + gn (s) − gm (s) ds
(2.5)
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for every n, m ∈ N and every t ∈ [, T ]. Since h satisfies (1.2) on [, T ] × (D(, ) ∩ K), we get t h(s, un (n (s))) − h(s, um (m (s))) ds t L un (n (s)) − um (m (s)) ds.
Accordingly un (t) − um (t) t t f (s, um (m (s))) ds supt∈[,T ] f (s, un (n (s))) ds − t +L un (n (s)) − um (m (s)) ds T gn (s) − gm (s) ds +
(2.6)
for every n, m ∈ N and every t ∈ [, T ]. From (ii), we deduce that, given ε > 0, there exists n1 (ε) > 0 such that, for each n, m ∈ N, with n, m n1 (ε), we have T gn (s) − gm (s) ds ε. (2.7)
Next, let n ∈ N∗ and let us denote by Fn : [, T ] → X the function defined by t f (s, un (n (s))) ds Fn (t) =
for each t ∈ [, T ]. As f ([, T ] × (K ∩ D(, ))) is relatively compact, it follows that, for each t ∈ [, T ], the set {Fn (t) ; n ∈ N∗ } is relatively compact in X. See for instance Lemma A.1.3 in [10, p. 295]. Moreover, from (2.1), we get f (s, un (n (s))) M for each n ∈ N and each s ∈ [, T ]. Therefore Fn (t) − Fn (s) M|t − s| for each n ∈ N and each t, s ∈ [, T ]. Thus {Fn ; n ∈ N} is equicontinuous on [, T ]. In view of Arzelà–Ascoli’s Theorem A.2.1 in [10, p. 296], there exists F ∈ C([, T ]; X) such that, on a subsequence at least, lim Fn (t) = F (t),
n→∞
uniformly for t ∈ [, T ]. So, for the very same ε > 0, there exists n2 (ε) ∈ N such that, for each n, m ∈ N, with n, m n2 (ε), we have t t ε. sup f (s, u ( (s))) ds − f (s, u ( (s))) ds n n m m t∈[,T ]
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Denote by n(ε) = max{n0 (ε), n1 (ε), n2 (ε)}. From (2.6), (2.7) and the preceding inequality, we deduce un (t) − um (t) 2ε + L
t
un (n (s)) − um (m (s)) ds
for all n, m ∈ N, n, m n(ε) and t ∈ [, T ]. In view of (2.4), (2.5) and the last inequality, we deduce un (n (t)) − um (m (t)) 4ε + L
t
un (n (s)) − um (m (s)) ds
for all n, m ∈ N, n, m n(ε) and t ∈ [, T ]. Gronwall’s Lemma 2.2 implies un (n (t)) − um (n (t)) 4εeL(T −) for all n, m ∈ N, n, m n(ε) and t ∈ [, T ]. But this inequality shows that (un (n ))n∈N is a Cauchy sequence in the sup norm. So, there exists a function u : [, T ] → X such that lim un (n (s)) = u(s) n
uniformly for s ∈ [, T ]. Taking into account (iii) and of the fact that D(, ) ∩ K is closed, we deduce that u(t) ∈ D(, ) ∩ K for every t ∈ [, T ]. Moreover, (2.5), along with limn εn = 0, implies lim un (s) = u(s) n
uniformly for s ∈ [, T ]. Using (i) and (ii), we conclude that both lim n (t) = t
n→∞
and
lim gn (t) = 0
n→∞
uniformly on [, T ]. Finally, passing to the limit for n → +∞ both sides in the approximate equations in (iv), i.e., in t un (t) = + f (s, un (n (s))) ds t t + h(s, un (n (s))) ds + gn (s) ds,
we deduce that u(t) = +
t
f (s, u(s)) ds +
t
h(s, u(s)) ds
for every t ∈ [, T ], which achieves the proof of the theorem.
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3. Existence of positive solutions to some pseudoparabolic equations Let be a bounded domain in R3 , with smooth boundary and let us consider the following semilinear pseudoparabolic partial differential equation: ut = ut + u + g(u), (t, x) ∈ QT , (3.1) u = 0, (t, x) ∈ T , u(0, x) = (x), x ∈ . Here and thereafter QT = (0, T ) × and T = (0, T ) × . Theorem 3.1. Let g : R → R be continuous and let us assume that u + g(u)0
(3.2)
for each u ∈ R+ . Then, for each ∈ H 2 () ∩ H01 () with − 0 a.e. on , there exists T > 0 such that problem (3.1) has at least one solution u ∈ C 1 ([0, T ]; H 2 () ∩ H01 ()) satisfying both u(t)−u(t)0 and u(t)0 for each t ∈ [0, T ] and a.e. on . If, in addition, g is positively sublinear, then each solution of (3.1) can be continued up to a global one. Proof. Clearly, (3.1) is equivalent to (u − u)t = u + g(u), (t, x) ∈ QT , u = 0, (t, x) ∈ T , u(0, x) = (x), x ∈ .
(3.3)
Let A : D(A) ⊆ L2 () → L2 () be defined by D(A) = H01 () ∩ H 2 (), Au = u for each u ∈ D(A), let J = (I − A)−1 and let us denote by u = J v. Since AJ = J − I , (3.3) can be rewritten as an abstract differential equation in the space L2 (), i.e. v = f (v) + h(v), (3.4) v(0) = , where f : L2 () → L2 () is defined by f (v)(x) = g(J v)(x) for each v ∈ L2 () and a.e. for x ∈ , h = J − I and = (I − A). Since the operator J is continuous from L2 () to H 2 () ∩ H01 (), and, in our specific case, i.e. n = 3, H 2 () is compactly imbedded in C() (see (iii) in Theorem 1.5.4 in [10, p. 18]), it follows that f is well-defined and b-compact (in fact compact). Moreover, J is obviously Lipschitz, being linear continuous. At this point, let us recall that = (I − A)0 a.e. on . Let K = {v ∈ L2 (); v 0 a.e. on }.
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In view of (3.2), we deduce that f, h and K satisfy the tangency condition (1.3). Indeed, to prove that for each ∈ K lim inf s↓0
1 d( + s(J − ) + sf (J ); K) = 0, s
it suffices to show that, for each s ∈ (0, 1), we have + s(J − ) + sf (J )0
(3.5)
a.e. on . But this is certainly the case, because 0 a.e. on , along with the maximum principle for elliptic equations, implies both (1 − s) 0 and J 0 a.e. on . By virtue of (3.2), it follows that J + f (J ) 0 and thus (3.5) holds. So, we are in the hypotheses of Theorem 1.1. Therefore K is viable with respect to f + h, and this implies that there exists at least one solution, v : [0, T ] → L2 (), of (3.4), with v(t) ∈ K for each t ∈ [0, T ]. But u(t) = J v(t), and consequently u(t) − u(t) 0 for each t ∈ [0, T ] and a.e. on . Using again the maximum principle for elliptic equations, we conclude that u(t)0 for each t ∈ [0, T ] and a.e. on . Since v ∈ C 1 ([0, T ]; L2 ()), and u = J v, with J linear continuous from L2 () to H 2 () ∩ H01 (), we conclude that u ∈ C 1 ([0, T ]; H 2 () ∩ H01 ()), and the proof is complete. We conclude the paper by noticing that this problem appears in the study of some special kind of fluids. For some other mathematical models which can be treated with similar arguments see [1,5] and the references therein.
Acknowledgements The author takes this opportunity to thank Professor J.I. Díaz for his useful remarks on a preliminary version of the paper.
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