Chapter 5 Viability under carathéodory conditions

CHAPTER 5

Viability under Carath´ eodory conditions

In this chapter we extend to the case of Carath´eodory solutions the viability results in Chapter 3, referring to classical solutions. We notice that in this, a fortiori nonautonomous, case, due to some reasons explained below, we will confine ourselves to consider only cylindrical sets. After showing that an a.e. Nagumo-type tangency condition is necessary for Carath´eodory viability, we state and prove that the very same a.e. Nagumo-type tangency condition is also sufficient under some natural Carath´eodory-type extra-conditions combined with appropriate either compactness or Lipschitz conditions on the right hand-side. Finally, we focus our attention on the existence of noncontinuable or even global Carath´eodory solutions.

5.1. Necessary conditions for Carath´ eodory viability Let X be a real Banach space, K a nonempty subset in X, f : I × K → X a given function and let us consider the Cauchy problem ½ 0 u (t) = f (t, u(t)) (5.1.1) u(τ ) = ξ. Definition 5.1.1. By a Carath´eodory solution of (5.1.1) on [ τ, T ] we mean an absolutely continuous function u : [ τ, T ] → K which is a.e. differentiable on [ τ, T ], with u0 ∈ L1 (τ, T ; X) and satisfying the differential equation in (5.1.1) for a.a. t ∈ [ τ, T ] and u(τ ) = ξ. A Carath´eodory solution of (5.1.1) on the semi-open interval [ τ, T ) is defined similarly, with the mention that here, we have to impose that u0 ∈ L1loc ([ τ, T ); X). Remark 5.1.1. Since the function u in Definition 5.1.1 is a fortiori in W 1,1 (τ, T ; X) for each [ τ, T ] ⊆ I, if u is a Carath´eodory solution of (5.1.1) on [ τ, T ], then Z t u(t) = u(s) + u0 (θ) dθ s

for each τ ≤ s ≤ t ≤ T . 93

94

Viability under Carath´eodory conditions

Definition 5.1.2. The set I × K is Carath´eodory viable with respect to f if for each (τ, ξ) ∈ I × K there exist T ∈ I, T > τ , and a Carath´eodory solution u : [ τ, T ] → K of (5.1.1) We can now proceed to the main result in this section, i.e., a necessary condition for viability. Theorem 5.1.1. Let X be a Banach space. If K ⊆ X is separable and the set I × K is Carath´eodory viable with respect to the Carath´eodory function f : I × K → X, then there exists a negligible subset Z of I such that, for each (τ, ξ) in (I \ Z) × K, we have f (τ, ξ) ∈ FK (ξ). Proof. Since K is separable and f is a Carath´eodory function, we are in the hypotheses of Theorem 2.8.2. Therefore, there exists a negligible subset Z of I such that for each τ ∈ I \ Z and each continuous function u : [ τ, T ) → K, we have Z 1 τ +h lim kf (s, u(s)) − f (τ, u(τ ))k ds = 0. (5.1.2) h↓0 h τ Let (τ, ξ) ∈ (I \ Z) × K. In view of Definition 2.4.2, we have to prove 1 that lim dist (ξ + hf (τ, ξ); K) = 0. Since I × K is Carath´eodory viable h↓0 h with respect to f , there exist T ∈ I, T > τ , and an almost everywhere differentiable function u : [ τ, T ] → K satisfying u0 (s) = f (s, u(s)) for a.a. s ∈ [ τ, T ] and u(τ ) = ξ. In view of Remark 5.1.1, we have Z u(τ + h) − u(τ ) 1 τ +h = f (s, u(s)) ds. h h τ Thus 1 1 lim dist (ξ + hf (τ, ξ); K) ≤ lim ku(τ ) + hf (τ, ξ) − u(τ + h)k h↓0 h h↓0 h ° ° ° ° Z τ +h ° ° ° ° u(τ + h) − u(τ ) 1 ° = lim °f (τ, ξ) − ≤ lim ° f (τ, ξ) − f (s, u(s)) ds° ° ° ° ° h↓0 h↓0 h h τ Z 1 τ +h ≤ lim kf (τ, u(τ )) − f (s, u(s))k ds. h↓0 h τ Thanks to (5.1.2), f (τ, ξ) ∈ FK (ξ), which completes the proof. ¤ Corollary 5.1.1. Let X be a Banach space. If K ⊆ X is separable and I × K is Carath´eodory viable with respect to the Carath´eodory function f : I × K → X, then there exists a negligible subset Z of I such that, for each (τ, ξ) in (I \ Z) × K, we have f (τ, ξ) ∈ TK (ξ).

Sufficient conditions for Carath´eodory viability

95

Proof. The conclusion is a simple consequence of Theorem 5.1.1 and Remark 2.4.3. ¤ Theorem 5.1.2. Let X be a Banach space. If K ⊆ X is separable and locally closed and I × K is Carath´eodory viable with respect to the locally Carath´eodory function f : I × K → X, then there exists a negligible subset Z of I such that, for each (τ, ξ) in (I \ Z) × K, we have f (τ, ξ) ∈ FK (ξ). Proof. The proof is similar with the one of Theorem 5.1.1, with the mention that here, instead of Theorem 2.8.2, we have to use Theorem 2.8.5. ¤ The next example shows why, in this context, we cannot consider the fully general case of a noncylindrical domain C as in the case when we were looking for classical, i.e., C 1 solutions. At a first glace, it seems that we cannot do this because the usual reduction of the nonautonomous case u0 (t) = f (t, u(t)) to the autonomous one z 0 (t) = F (z(t)), with z = (t, u) and F (z) = (1, f (z)), cannot work, and this, due to the fact that whenever f is a Carath´eodory function, F may fail to be continuous. Surprisingly, this is not the only reason why we cannot extend the classical viability theory to noncylindrical domains in the Carath´eodory case, as we can see from the next “autonomous” example below. Example 5.1.1. Let C ⊆ R × R be the graph of the Cantor function g : [ 0, 1) → R which is continuous, strictly increasing and g 0 (t) = 0 a.e. for t ∈ [ 0, 1). See Gelbaum–Olmsted [100], 30, p. 105. Then the “autonomous” function f ≡ 0 satisfies the tangency condition (1, 0) ∈ TC (τ, g(τ )) for a.a. τ ∈ [ 0, 1), but the Cauchy problem u0 (t) = 0 and u(0) = 0 has no Carath´eodory solution. So, the situation is even worse than the one observed in Example 3.5.1, i.e., when under similar circumstances, we have concluded that the same Cauchy problem has no solution in the sense of Definition 3.1.1. This shows that, in the case of a noncylindrical set C, no matter how regular is the right hand side f : C → X, we cannot replace the “everywhere” tangency condition with an “almost everywhere” one in order to get Carath´eodory viability. 5.2. Sufficient conditions for Carath´ eodory viability The next class of functions will play a crucial role in the sequel. Definition 5.2.1. A compact-Carath´eodory function f : I × K → X is a locally Carath´eodory function satisfying (C5 ) for each fixed ξ ∈ K and each τ ∈ I there exist T > τ and ρ > 0 such that [ τ, T ] ⊆ I and, for each ε > 0, there exists a subset Hε

96

Viability under Carath´eodory conditions in [ τ, T ] with λ(Hε ) ≤ ε and such that the set {f (t, u) ; (t, u) ∈ ([ τ, T ] \ Hε ) × (D(ξ, ρ) ∩ K)}

is relatively compact in X. A Lipschitz-Carath´eodory function is a function satisfying (C1 ), (C4 ) in Definition 2.8.1 and (C6 ) for each ξ ∈ K, there exist ρ > 0 and L ∈ L1loc (I), such that kf (t, u) − f (t, v)k ≤ L(t)ku − vk for a.a. t ∈ I and for all u, v ∈ D(ξ, ρ) ∩ K. A locally β-compact-Carath´eodory function is a function satisfying (C1 ), (C2 ), (C4 ) in Definition 2.8.1 and (C7 ) for each ξ ∈ K, there exist ρ > 0 and a uniqueness Carath´eodory function ω : I × R+ → R+ in the sense of Definition 1.8.2 such that β(f (t, C)) ≤ ω(t, β(C)) a.e. for t ∈ I and for each C ⊆ D(ξ, ρ) ∩ K. An β-compact-Carath´eodory function is a function satisfying (C1 ), (C2 ), (C3 ) in Definition 2.8.1 and (C8 ) there exists a uniqueness Carath´eodory function ω : I × R+ → R+ in the sense of Definition 1.8.2 such that β(f (t, C)) ≤ ω(t, β(C)) a.e. for t ∈ I and for each bounded subset C in K. Remark 5.2.1. If X is finite dimensional, each locally Carath´eodory function is a β-compact-Carath´eodory function. Furthermore, compactCarath´eodory functions as well as Lipschitz-Carath´eodory functions are β-compact-Carath´eodory functions. Theorem 5.2.1. Let X be a separable Banach space, K a locally closed subset in X and f : I × K → X a locally β-compact-Carath´eodory function. Then I × K is Carath´eodory viable with respect to f if and only if there exists a negligible subset Z in I such that, for each (t, ξ) ∈ (I \ Z) × K, we have f (t, ξ) ∈ TK (ξ). (5.2.1) Theorem 5.2.2. Let X be a separable Banach space, K a locally closed subset in X and f : I × K → X a Lipschitz-Carath´eodory function. Then, a necessary and sufficient condition in order that I × K be Carath´eodory viable with respect to f is the tangency condition (5.2.1).

Existence of (ε, L)-approximate Carath´eodory solutions

97

Theorem 5.2.3. Let X be a separable Banach space, K a locally closed subset in X and f : I × K → X a compact-Carath´eodory function. Then, a necessary and sufficient condition in order that I × K be Carath´eodory viable with respect to f is the tangency condition (5.2.1). From Theorem 5.2.3, we easily deduce Theorem 5.2.4. Let X be finite dimensional, let K be a locally closed subset in X and let f : I ×K → X be a locally Carath´eodory function. Then, a necessary and sufficient condition in order that I × K be Carath´eodory viable with respect to f is the tangency condition (5.2.1). Finally, we have Theorem 5.2.5. Let X be a separable Banach space, K a nonempty and locally closed subset in X and let f : I×K → X be locally Carath´eodory. Let us assume that K is proximal and the norm k·k is Gˆ ateaux differentiable at each x ∈ X, x 6= 0. Then the following conditions are equivalent : (i) there exists a negligible subset Z of I such that for every (t, ξ) ∈ (I \ Z) × K, f (t, ξ) ∈ CK (ξ) ; (ii) there exists a negligible subset Z of I such that for every (t, ξ) ∈ (I \ Z) × K, f (t, ξ) ∈ TK (ξ) ; (iii) there exists a negligible subset Z of I such that for every (t, ξ) ∈ (I \ Z) × K, f (t, ξ) ∈ BK (ξ) ; (iv) the set I × K is Carath´eodory viable with respect to f . In general, if G : K ; X is such that CK (ξ) ⊆ G(ξ) ⊆ BK (ξ) for each ξ ∈ K, then each one of the conditions above is equivalent to (v) there exists a negligible subset Z of I such that for every (t, ξ) ∈ (I \ Z) × K, f (t, ξ) ∈ G(ξ). 5.3. Existence of (ε, L)-approximate Carath´ eodory solutions We begin with the following simple but useful result. Proposition 5.3.1. Let X be a real Banach space, K a nonempty and separable subset in X and f : I × K → X a Carath´eodory function. Then, the tangency condition (5.2.1) is equivalent to the condition (5.3.1) below: there exists a negligible subset Z of I such that, for every (t, ξ) ∈ (I \Z)×K, we have µ ¶ Z t+h 1 lim inf dist ξ + f (θ, ξ) dθ; K = 0. (5.3.1) h↓0 h t Problem 5.3.1. Prove Proposition 5.3.1.

98

Viability under Carath´eodory conditions

The main ingredient in the proof of the sufficiency of Theorem 5.2.1 is the next lemma which offers an existence result for “ε-approximate Carath´eodory solutions” of the Cauchy problem ½ 0 u (t) = f (t, u(t)) (5.3.2) u(τ ) = ξ. Lemma 5.3.1. Let X be a separable Banach space, K a nonempty, locally closed subset in X and f : I×K → X a locally Carath´eodory function satisfying the tangency condition (5.2.1). Let (τ, ξ) be arbitrarily fixed in I × K, let r > 0 be such that D(ξ, r) ∩ K is closed. Let Z = Z1 ∪ Z2 ∪ Z3 , where Z1 is the negligible set in (5.2.1), Z2 is the negligible set in (5.3.1), while Z3 is the negligible set in I such that for each t ∈ I \ Z3 , f (t, ·) is continuous on K. Then, there exist ρ ∈ (0, r ], T ∈ (τ, sup I ], θ0 ∈ I \ Z and M in L1 (τ, T ; R+ ), such that for each ε ∈ (0, 1) and each open set L of R, with Z ⊆ L and λ(L) < ε, there exist one family of nonempty and pairwise disjoint intervals: PT = {[ tm , sm ); m ∈ Γ}, with Γ finite or countable, and three functions, g ∈ L1 (τ, T ;X), r : [ τ, T ] → X Borel measurable and u : [ τ, T ] → X continuous, satisfying : S (i) m∈Γ [ tm , sm ) = [ τ, T ) and sm − tm ≤ ε for each m ∈ Γ ; (ii) if tm ∈ L then [ tm , sm ) ⊂ L ; (iii) u(tm ) ∈½D(ξ, ρ) ∩ K for each m ∈ Γ, u(T ) ∈ D(ξ, ρ) ∩ K ; f (s, u(tm )) a.e. on [ tm , sm ) if tm ∈ /L (iv) g(s) = f (θ0 , u(tm )) a.e. on [ tm , sm ) if tm ∈ L; (v) kg(t)k ≤ M(t) a.e. for t ∈ [ τ, T ] ; (vi) kr(t)k ≤ ε a.e. for t ∈ [ τ, T ] ; (vii) u(τ ) = ξ and, for each m ∈ Γ and each t ∈ [ tm , T ], u satisfies Z t Z t u(t) = u(tm ) + g(θ) dθ + r(θ) dθ. tm

tm

Before proceeding to the proof of Lemma 5.3.1, we introduce: Definition 5.3.1. Let ε > 0 and L an open set including the negligible set Z in Lemma 5.3.1. A quadruple (PT , g, r, u), satisfying (i)∼(vii) in Lemma 5.3.1, is called an (ε, L)-approximate Carath´eodory solution to the Cauchy problem (5.3.2) on the interval [ τ, T ]. We may now pass to the proof of Lemma 5.3.1. Proof. Let (τ, ξ) ∈ I × K be arbitrary and choose r > 0 such that D(ξ, r) ∩ K is closed and for which there exists `(·) ∈ L1loc (I) such that kf (t, u)k ≤ `(t)

(5.3.3)

Existence of (ε, L)-approximate Carath´eodory solutions

99

for a.a. t ∈ I and every u ∈ D(ξ, r) ∩ K. This is always possible because K is locally closed and f satisfies (C4 ) in Definition 2.8.1. Fix a θ0 ∈ I \ Z. Then v 7→ f (θ0 , v) is continuous on K. Taking a sufficiently small ρ ∈ (0, r ], we can find M > 0 satisfying kf (θ0 , v)k ≤ M

(5.3.4)

for each v ∈ D(ξ, ρ) ∩ K. Next, take T > τ such that [ τ, T ] ⊆ I and let us define M(t) = max{M, `(t)} a.e. for t ∈ [ τ, T ]. Clearly M ∈ L1 (τ, T ; R+ ), and therefore, diminishing T > τ , if necessary, we may assume that Z T T −τ + M(θ)dθ ≤ ρ. (5.3.5) τ

We first prove that the conclusion of Lemma 5.3.1 remains true if we replace T as above with a possibly smaller number µ ∈ (τ, T ] which, at this stage, is allowed to depend on ε ∈ (0, 1). This being done, by using the Brezis-Browder Ordering Principle, we will prove that we can take µ = T , independent of ε ∈ (0, 1). For ε ∈ (0, 1) take an open set L of R with Z ⊆ L and whose Lebesgue measure λ(L) < ε. Case 1. If τ ∈ L, since f satisfies the tangency condition (5.2.1) at (θ0 , ξ), it follows that there exist δ ∈ (0, ε) with [ τ, τ + δ) ⊆ L, and p ∈ X with kpk ≤ ε, and such that ξ + δf (θ0 , ξ) + δp ∈ K. Let us define g : [ τ, τ +δ ] → X, r : [ τ, τ +δ ] → X and u : [ τ, τ +δ ] → X by g(t) = f (θ0 , ξ), r(t) = p, and respectively by Z t Z t u(t) = ξ + g(θ) dθ + r(θ) dθ (5.3.6) τ

τ

for each t ∈ [ τ, τ + δ ]. By (5.3.4) and the definition of M, we deduce that the family Pτ +δ = {[ τ, τ + δ)} and the functions M, g, r and u satisfy (i)-(vii) with T substituted by τ + δ. Case 2. If τ ∈ / L, we have τ ∈ / Z and, in view of Proposition 5.3.1, there R τ +δ exist δ ∈ (0, ε) and p ∈ X, with kpk ≤ ε and ξ + τ f (θ, ξ) dθ + δp ∈ K. Setting g(θ) = f (θ, ξ) and r(θ) = p, for θ ∈ [ τ, τ + δ], and defining u by (5.3.6), we can easily see that, again, the family Pτ +δ = {[ τ, τ + δ)} and the functions M, g, r and u satisfy (i)-(vii) with T substituted by τ + δ. Next, we will show that there exists at least one quadruple (PT , g, r, u) satisfying (i)∼(vii) on [ τ, T ]. To this aim we shall use the Brezis-Browder Ordering Principle, as follows. Let U be the set of all quadruples (Pµ , g, r, u) with L fixed as above and µ ≤ T and satisfying (i)∼(vii) with µ instead of

100

Viability under Carath´eodory conditions

T . As we have already proved, this set is nonempty. On U we introduce a partial order as follows. We say that (Pµ1 , g1 , r1 , u1 ) ¹ (Pµ2 , g2 , r2 , u2 ), where Pµk = {[tkm , skm ); m ∈ Γk }, k = 1, 2, if (O1 ) µ1 ≤ µ2 and, if µ1 < µ2 , there exists i ∈ Γ2 such that µ1 = t2i ; (O2 ) for each m1 ∈ Γ1 there exists m2 ∈ Γ2 such that: t1m1 = t2m2 and s1m1 = s2m2 ; (O3 ) g1 (θ) = g2 (θ), r1 (θ) = r2 (θ) and u1 (θ) = u2 (θ) for θ ∈ [ τ, µ1 ]. Let us define the function N : U → R by N((Pµ , g, r, u)) = µ. It is clear that N is increasing on U. Let us take now an increasing sequence ((Pµj , gj , rj , uj ))j∈N in U and let us show that it is bounded from above in U. We define an upper bound as follows. First, set µ∗ = sup{µj ; j ∈ N}. If µ∗ = µj for some j ∈ N, (Pµj , gj , rj , uj ) is clearly an upper bound. If µj < µ∗ for each j ∈ N, we define Pµ∗ = {[ tjm , sjm ); j ∈ N, m ∈ Γj }. In the latter case, Pµ∗ can be written in the form Pµ∗ = {[ tm , sm ); m ∈ N}. We define g(t) = gj (t), r(t) = rj (t), u(t) = uj (t) for j ∈ N and every t ∈ [ τ, µj ]. Let us observe that (Pµ∗ , g, r, u), where Pµ∗ , g, r and u are defined as above, satisfies (i), (ii), the first condition in (iii), (iv), (v) and (vi) with T replaced with µ∗ . Notice that (vii) is also satisfied but only on [ τ, µ∗ ). Obviously we have u(tm ) ∈ D(ξ, ρ)∩K for each m ∈ N. Since g and r are a.e. defined on [ τ, µ∗ ], it remains to prove that u can be extended to [ τ, µ∗ ] and satisfies the second condition in (iii), i.e., u(µ∗ ) ∈ D(ξ, ρ) ∩ K and (vii) for t = µ∗ . To this aim, let us observe that, in view of (vii) on [ τ, µ∗), and of the fact that, by (v), we have kg(t)k ≤ M(t), for a.a. t ∈ [ τ, T ], with M integrable on [ τ, T ], it follows that u satisfies the Cauchy condition for the existence of the limit for t ↑ µ∗ . Consequently, there exists limt↑µ∗ u(t). Accordingly, u can be continuously extended at µ∗ by u(µ∗ ) = limt↑µ∗ u(t). Since u(tm ) ∈ D(ξ, ρ) ∩ K for m ∈ N, and D(ξ, ρ) ∩ K is closed, we easily see that u(µ∗ ) ∈ D(ξ, ρ) ∩ K and thus the last condition in (iii) is also satisfied. So, with u : [ τ, µ∗ ] → X, defined as above, we obviously have that (Pµ∗ , g, r, u) satisfies (i)∼(vi). It is also easy to see that (vii) holds for each m ∈ N and each t ∈ [ tm , µ∗ ). To check (vii)

Existence of (ε, L)-approximate Carath´eodory solutions

101

for t = µ∗ , we have to fix any m ∈ N, to take t = µj with µj > tm in (vii) and to pass to the limit for j tending to ∞ both sides in (vii). Thus (Pµ∗ , g, r, u) is an upper bound for ((Pµj , gj , rj , uj ))j∈N . So, the set U, endowed with the partial order ¹, and the function N satisfy the hypotheses of Brezis-Browder Ordering Principle. Accordingly, there exists at least one N-maximal element (Pν , gν , rν , uν ) in U, which means that if (Pν , gν , rν , uν ) ¹ (Pσ , gσ , rσ , uσ ) then ν = σ. Next, we show that ν = T , where T satisfies (5.3.5). To this aim let us assume by contradiction that ν < T and let ξν = uν (ν) which belongs to D(ξ, ρ) ∩ K. Throughout the function M is defined as in the beginning of the proof. In view of (i)∼ (vii), we have Z kξν − ξk ≤

τ

Z

ν

kgν (θ)k dθ +

τ

Z

ν

krν (θ)k dθ ≤ (ν − τ )ε +

ν

M(θ) dθ. τ

Recalling that ν < T and ε < 1, from (5.3.5) we get kξν − ξk < ρ.

(5.3.7)

There are two possibilities: either ν ∈ L or ν ∈ / L. If ν ∈ L, we act as in Case 1 above with ν instead of τ and with ξν instead of ξ. So, from the tangency condition (5.2.1) combined with (5.3.7), we infer that there exist δ ∈ (0, ε], with ν +δ ≤ T , [ ν, ν +δ) ⊆ L and p ∈ X, satisfying kpk ≤ ε, such that ξν + δf (θ0 , ξν ) + δp ∈ D(ξ, ρ) ∩ K. If ν ∈ / L, we act as in Case 2 above with ν instead of τ and with ξν instead of ξ. From Proposition 5.3.1 combined with (5.3.7), we infer that there exist δ ∈ (0, ε], with ν + δ ≤ T , and q ∈ X, satisfying kqk ≤ ε and Z ξν +

ν+δ

ν

f (θ, ξν ) dθ + δq ∈ D(ξ, ρ) ∩ K.

We define Pν+δ = Pν ∪ {[ ν, ν + δ)} and both gν+δ : [ τ, ν + δ ] → X and rν+δ : [ τ, ν + δ ] → X by ½ gν+δ (t) = ½ rν+δ (t) =

gν (t) if t ∈ [ τ, ν ] f (θ0 , ξν ) if t ∈ ( ν, ν + δ ] rν (t) if t ∈ [ τ, ν ] p if t ∈ ( ν, ν + δ ]

Viability under Carath´eodory conditions

102 if ν ∈ L, and

½ gν+δ (t) = ½ rν+δ (t) =

gν (t) if t ∈ [ τ, ν ] f (t, ξν ) if t ∈ ( ν, ν + δ ] rν (t) if t ∈ [ τ, ν ] q if t ∈ ( ν, ν + δ ]

if ν ∈ / L. Finally, we define uν+δ : [ τ, ν + δ ] → X by  uν (t) if t ∈ [ τ, ν ]  Z t Z t uν+δ (t) =  ξν + gν+δ (θ) dθ + rν+δ (θ) dθ if t ∈ (ν, ν + δ ]. ν

ν

Since uν+δ (ν + δ) ∈ K ∩ D(ξ, ρ), it follows that (Pν+δ , gν+δ , rν+δ , uν+δ ) satisfies (i)∼(vii) with ν + δ instead of T . So, (Pν+δ , gν+δ , rν+δ , uν+δ ) ∈ U and (Pν , gν , rν , uν ) ¹ (Pν+δ , gν+δ , rν+δ , uν+δ ) with ν < ν + δ. This contradiction can be avoided only if ν = T . The proof is therefore complete. ¤ 5.4. Convergence of (ε, L)-approximate Carath´ eodory solutions The proof of the sufficiency of Theorem 5.2.1 consists in showing the convergence of a suitably chosen sequence of (ε, L)-approximate Carath´eodory solutions. Proof. Let (τ, ξ) ∈ I × K and let r > 0 be such that D(ξ, r) ∩ K is closed. Let ρ ∈ (0, r ], T ∈ (τ, sup I ], θ0 ∈ I \ Z and M ∈ L1 (τ, T ; R+ ) given by Lemma 5.3.1, let (εn )n be a sequence in (0, 1) strictly decreasing to 0, let (Ln ) be a decreasing sequence of open subsets in R such that the negligible set Z, defined in Lemma 5.3.1, satisfy Z ⊆ Ln and λ(Ln ) < εn for every n ∈ N. Take L = ∩n≥1 Ln and a sequence of (εn , Ln )-approximate Carath´eodory solutions ((PnT , gn , rn , un ))n of (5.3.2) whose existence is ensured also by Lemma 5.3.1. Let us define σn : [ τ, T ] → [ τ, T ] by σn (t) = tnm for t ∈ [ tnm , snm ), σn (T ) = T , and [ Hn = [ tnm , snm ). tn m ∈Ln

In view of (ii)1 we have Hn ⊂ Ln and therefore λ(Hn ) < εn .

(5.4.1)

1Except otherwise specified, all references to (i), (ii),. . . , (vii) are to the correspond-

ing items in Lemma 5.3.1.

Convergence of (ε, L)-approximate Carath´eodory solutions

103

Set En = [ τ, T ] \ Hn , let t ∈ [ τ, T ], and let us define Hnt = [ τ, t ] ∩ Hn and Ent = [ τ, t ] ∩ En . Since by (vii) we have Z t Z t un (t) = ξ + gn (s) ds + rn (s) ds, (5.4.2) τ

τ

from Lemma 2.7.2, Remark 2.7.1 and (vi), we deduce β ({un (t); n ≥ k}) ¾¶ µ½Z t ¾¶ t ≤β gn (s) ds; n ≥ k +β rn (s) ds; n ≥ k τ τ Z t ≤ β ({gn (s); n ≥ k}) ds + (T − τ )εk τ Z β ({f (s, un (σn (s))); n ≥ k}) ds+ β ({gn (s); n ≥ k}) ds+(T −τ )εk µ½Z

Z ≤

Ekt

Hkt

Z

≤

Ekt

ω(s, β ({un (σn (s)); n ≥ k}))ds +

Z ≤

Ekt

Z Hk

M(s) ds + (T − τ )εk

ω(s, β({un (s); n ≥ k} + {un (σn (s)) − un (s); n ≥ k}))ds Z + Hk

M(s) ds + (T − τ )εk .

Let k = 1, 2, . . . and n ≥ k. From (5.4.2), we have (Z ) Z t t kun (σn (t)) − un (t)k ≤ sup M(θ)dθ+ krn (θ)kdθ ; t ∈ [ τ, T ] . σn (t)

Let us denote by (Z δk = sup

Z

t

)

t

M(θ) dθ+

σn (t)

σn (t)

σn (t)

krn (θ)k dθ ; t ∈ [ τ, T ], n ≥ k

.

By (vii), (v), (vi) and (i), we have lim kun (t) − un (σn (t))k ≤ lim δn = 0. n

n

(5.4.3)

Thus, β ({un (t); n ≥ k})

Z ≤

Ekt

ω(s, β({un (s); n ≥ k}) + β({un (σn (s)) − un (s); n ≥ k}))ds Z + Hk

M(s) ds + (T − τ )εk

Viability under Carath´eodory conditions

104 Z ≤ τ

Z

t

ω(s, β({un (s); n ≥ k}) + δk )ds +

Hk

M(s) ds + (T − τ )εk .

Denoting by xk (t) = β({un (t); n ≥ k}) + δk and Z γk = M(s) ds + (T − τ )εk + δk , Hk

we get

Z xk (t) ≤ γk +

t

τ

ω(s, xk (s)) ds

for k = 1, 2, . . . and t ∈ [ τ, T ]. As γn ↓ 0, we are in the hypotheses of Lemma 1.8.3, wherefrom we conclude that, diminishing T > τ if necessary, on a subsequence at least, we have limk xk (t) = 0, which means that limk β({un (t); n ≥ k}) = 0, uniformly for t ∈ [ τ, T ]. Now, Lemma 2.7.3 comes into play and shows that, for each t ∈ [ τ, T ], {un (t); n = 1, 2, . . . } is relatively compact. Since by (v), (vi) and (vii) in Lemma 5.3.1, it follows that {un ; n = 1, 2, . . . } is equicontinuous on [ τ, T ], from Theorem 1.3.6, we conclude that there exists u ∈ C([ τ, T ]; X) such that, on a subsequence at least, we have limn un (t) = u(t) uniformly for t ∈ [ τ, T ], where u ∈ C([ τ, T ]; X). To complete the proof, it remains to show that u is a solution of the Cauchy problem (5.3.2). To this aim, let us remark that by (iii), we have un (σn (t)) ∈ D(ξ, ρ) ∩ K and since D(ξ, ρ) ∩ K is closed, from (5.4.3), we deduce that u(t) ∈ D(ξ, ρ) ∩ K for every t ∈ [ τ, T ]. Again (5.4.3) yields lim gn (s) = lim f (s, un (σn (s))) = f (s, u(s)), n

n

for each s ∈ [ τ, T ] \ L. Using Lebesgue Dominated Convergence Theorem 1.2.3 in order to pass to the limit for n → ∞ both sides in (5.4.2), we conclude that Z t

u(t) = ξ +

f (s, u(s)) ds τ

for each t ∈ [ τ, T ]. This completes the proof.

¤

Problem 5.4.1. Give a direct proof to Theorem 5.2.2, avoiding the measure of noncompactness. Problem 5.4.2. Give a direct proof to Theorem 5.2.4, avoiding the measure of noncompactness. Problem 5.4.3. Give a direct proof to Theorem 5.2.3, avoiding the measure of noncompactness.

Noncontinuable Carath´eodory solutions

105

5.5. Noncontinuable Carath´ eodory solutions We next present some results concerning the existence of noncontinuable, or even global Carath´eodory solutions to u0 (t) = f (t, u(t)).

(5.5.1)

A Carath´eodory solution u : [ τ, T ) → K to (5.5.1) is called noncontinuable, if there is no other Carath´eodory solution v : [ τ, Te) → K of the same equation, with T < Te and satisfying u(t) = v(t) for all t ∈ [ τ, T ). The Carath´eodory solution u is called global if T = sup I. The next theorem follows from Brezis–Browder Theorem 2.1.1. Theorem 5.5.1. Let X be a Banach space, K ⊆ X a nonempty set and let f : I × K → X. Then, the following conditions are equivalent : (i) I × K is Carath´eodory viable with respect to f ; (ii) for each (τ, ξ) ∈ I × K there exists at least one noncontinuable Carath´eodory solution u : [τ, T ) → K of (5.5.1), satisfying the initial condition u(τ ) = ξ. Remark 5.5.1. Notice that in Theorem 5.5.1 we do not assume K to be locally closed or f to be locally Carath´eodory. Definition 5.5.1. A function f : I × K → X is called Carath´eodory positively sublinear, if there exist a, b ∈ L1loc (I), c ∈ L∞ loc (I) and a negligible subset Z of I such that kf (t, ξ)k ≤ a(t)kξk + b(t) for each (t, ξ) ∈ c K+ (f )

c (f ), K+

where

= {(t, ξ) ∈ (I \ Z) × K; kξk > c(t) and [ ξ, f (t, ξ) ]+ > 0} .

Theorem 5.5.2. Let X be a Banach space, K ⊆ X a nonempty and closed set and let f : I × K → X be a locally Carath´eodory function. If f is Carath´eodory positively sublinear and I × K is Carath´eodory viable with respect to f , then each Carath´eodory solution of (5.5.1) can be continued up to a global one, i.e., defined on [ τ, sup I). Proof. Since I × K is Carath´eodory viable with respect to f , by Theorem 5.5.1, it follows that for each (τ, ξ) ∈ I × K there exists at least one noncontinuable Carath´eodory solution, u : [ τ, T ) → K, of (5.5.1), satisfying u(τ ) = ξ. We will show that T = sup I. To this aim, let us assume the contrary, i.e., that T < sup I. In particular this means that T < +∞. As u0 (s) = f (s, u(s)) for a.a. s ∈ [ τ, T ), we deduce d+ (ku(·)k)(s) = [ u(s), f (s, u(s)) ]+ ds

106

Viability under Carath´eodory conditions

for a.a. s ∈ [ τ, T ). Let t ∈ [ τ, T ). Integrating over [ τ, t ] ⊆ [ τ, T ) this equality, we get Z t ku(t)k = kξk + [ u(s), f (s, u(s)) ]+ ds τ Z Z ≤ kξk + [ u(s), f (s, u(s)) ]+ ds + [ u(s), f (s, u(s)) ]+ ds, Et

Ht \Gt

where Et = {s ∈ [ τ, t ]; [ u(s), f (s, u(s)) ]+ > 0 and ku(s)k > c(s)}, Gt = {s ∈ [ τ, t ]; [ u(s), f (s, u(s)) ]+ ≤ 0}, Ht = {s ∈ [ τ, t ]; ku(s)k ≤ c(s)}. Taking into account that Ht ⊆ HT and that, by (ii) in Exercise 1.6.1, [ u, v ]+ ≤ kvk for each u, v ∈ X, we get Z Z ku(t)k ≤ kξk + [a(s)ku(s)k + b(s)] ds + kf (s, u(s))k ds Et

HT

for each t ∈ [ τ, T ). But f is a Carath´eodory function and therefore there exists ` ∈ L1loc (I) such that kf (s, u(s))k ≤ `(s) for a.a. s ∈ HT . See (C3 ) in Definition 2.8.1. Hence Z T Z T Z ku(t)k ≤ kξk + `(s) ds + b(s) ds + τ

τ

T

a(s)ku(s)k ds

τ

for each t ∈ [ τ, T ). By Gronwall Lemma 1.8.4, it follows that u is bounded on [ τ, T ). Using once again the fact that f is Carath´eodory, we deduce that f (·, u(·)) is bounded on [ τ, T ) by a function in L1 (τ, T ) and so, there exists limt↑T u(t) = u∗ . As K is closed and T < sup I, we get (T, u∗ ) ∈ I × K. Using this observation and recalling that I × K is Carath´eodory viable with respect to f , we conclude that u can be continued to the right of T . But this is absurd, because u is noncontinuable. This contradiction can be eliminated only if T = sup I, and this completes the proof. ¤