Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 3, 569–584

Compactness and Existence Results for Ordinary Differential Equations in Banach Spaces

J. Appell, M. Väth and A. Vignoli

Dedicated to Prof. P. P. Zabreĭko on the occasion of his 60th birthday

Abstract. We prove that the Picard-Lindelöf operator

Hx(t) = ∫_{t_{0}}^{t} f(s, x(s)) ds

with a vector function f is continuous and compact (condensing) in C, if f satisfies only a mild boundedness condition, and if f(s, ·) is continuous and compact (resp. condensing). This generalizes recent results of the second author and immediately leads to existence theorems for local weak solutions of the initial value problem for ordinary differential equations in Banach spaces.

Keywords: Ordinary differential equations in Banach spaces, nonlinear Volterra integral operators, Picard-Lindelöf operators, compactness, condensing operators, measures of non-compactness

AMS subject classification: Primary 34 A 12, 34 G 20, 47 H 30, secondary 45 N 10, 45 P 05, 46 G 10

1. Introduction

We consider the Cauchy problem

\begin{matrix}x′ & = f(t, x) \\ x(t_{0}) & = u \end{matrix} \}  (1)

in a Banach space U. By a solution of problem (1) we mean a solution of the integral equation

x(t) = u + ∫_{t_{0}}^{t} f(s, x(s)) ds,

where the integral is understood in the sense of Bochner. The existence of solutions has been studied by many authors, e.g. by Krasnoselskiĭ and Krein [12], Ambrosetti

J. Appell: Dept. Math. Univ., Am Hubland, D-97074 Würzburg M. Väth: Dept. Math. Univ., Am Hubland, D-97074 Würzburg A. Vignoli: Univ. of Rome “Tor Vergata”, Dept. Math., Via della Ricerca Sci., I-00133 Roma

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

[2], and Sadovskii [17] (see also [6: §2.1 and §8.1], [1: Section 4.1], and the references therein). The usual assumption for such results is that f be uniformly continuous. In the article [19] we have replaced this condition by a condition on “uniform measurability”. However, Mönch [14] and Mönch and von Harten [15] have found a related result where f is assumed to be only continuous. Their main ideas carry over to the case when f just satisfies a Carathéodory condition.

We employ these ideas in the proof of the basic Proposition 1.1 below. In Section 2 we extend that proposition to uncountable sets and to subsets of C. Finally, in Section 3 we apply our results to prove that the integral operator

Hx(t) = ∫_{t_{0}}^{t} f(s, x(s)) ds  (2)

is compact (or condensing) under the natural assumption that f(s, ·) is compact (or condensing); only the Carathéodory condition and a very mild boundedness condition are required additionally. This result is apparently new and generalizes [19]. It immediately leads to existence results for problem (1).

2. Integration and compactness of sequences

Given a Banach space U and a subset M ⊆ U, we denote by χ(M), χ_{i}(M), α(M) the Hausdorff, inner Hausdorff, and the Kuratowski measures of non-compactness of the set M, respectively (see, e.g., [1]). Recall that χ(M) (resp. χ_{i}(M)) is the infimum of all ε > 0 such that one can find a finite ε-net for M in U (resp. in M). We also write χ_{U}(M) instead of χ(M) to emphasize the dependence on U. Similarly, α(M) is the infimum of all δ > 0 such that M can be covered by finitely many sets with diameter less than δ. Evidently,

χ(M) ≤ χ_{U_{0}}(M) ≤ χ_{i}(M) ≤ α(M) ≤ 2χ(M)  (3)

for each closed subspace U_{0} ⊆ U with M ⊆ U_{0}. In contrast to the other measures of non-compactness, χ_{i} is neither monotone nor invariant under passing to the convex hull. Nevertheless, it is very natural and useful for the following considerations.

If M ⊆ U is separable, then there exists always a countable dense subset M_{0} ⊆ M, and so χ(M_{0}) = χ(M). In general, one can only say the following (similar to [1: Theorem 1.4.5]):

Lemma 1.1. For any set M in a Banach space U there exists a countable subset M_{0} ⊆ M such that α(M_{0}) ≥ χ_{i}(M).

Proof. Without loss of generality, let χ_{i}(M) > 0. Then there exists a sequence μ_{n} with 0 < μ_{n} ↑ χ_{i}(M). Fix n for the moment and define a sequence x_{k} ∈ M inductively as follows. If x_{1}, …, x_{k−1} are already defined, choose x_{k} ∈ M such that |x_{k} − x_{j}| > μ_{n} for all j < k. Such an x_{k} exists, since otherwise {x_{1}, …, x_{k−1}} is a finite μ_{n}-net for M, contradicting μ_{n} < χ_{i}(M). Now let M_{n} be the set of all x_{k}, and M_{0} = ⋃ M_{n}. If M_{0} is covered by finitely many sets D_{1}, …, D_{N}, we find for each n some set D_{k} which contains infinitely many points of M_{n}. In particular, diam D_{k} > μ_{n}

Considering the Hausdorff measure χ, we lost the factor 2 in non-separable spaces in Lemma 1.1. But in general one can not do better, as can be seen from the following example which slightly simplifies the example in [1: Theorem 1.4.5]:

Example 1.1. Let I be an uncountable set, and let U consist of all bounded functions x : I → ℝ with at most countable support. Endowed with the sup-norm, U is a Banach space, since the support of a sequence x_{n} ∈ U is countable, too. Assume that M = {x ∈ U : 0 ≤ x(t) ≤ 1} has a finite ε-net N ⊆ U. Since N has at most countable support S, but I is uncountable, there is a function x ∈ M with x(t) = 1 for some t ∉ S. Hence ε ≥ dist (x, N) ≥ 1, i.e. χ(M) ≥ 1. On the other hand, for each countable M_{0} ⊆ M we have χ(M_{0}) ≤ \frac{1}{2}. Indeed, define x_{0} ∈ M by x_{0}(t) = \frac{1}{2} on the support of M_{0}, and x_{0}(t) = 0 otherwise. Then \{x_{0}\} is a finite \frac{1}{2}-net for M_{0}.

The following lemma is a part of [14: Propostion 1.4] (see also [15: Proposition 2]):

Lemma 1.2. Let U be a separable Banach space. Then there exist finite-dimension-al subspaces U_{1} ⊆ U_{2} ⊆ … such that for any countable bounded M ⊆ U, M = {u_{1}, u_{2}, …}, the equality

χ(M) = lim_{k→∞} lim sup_{n→∞} dist (u_{n}, U_{k})  (4)

holds.

The idea to use Lemma 1.2 for the proof of the following proposition is apparently due to [14, 15]. However, the proof of [15: Proposition 3] (see also [14]) contains a minor mistake in the application of Fatou’s lemma (lim sup instead of lim inf), which will be avoided in the following proof by applying Lebesgue’s dominated convergence theorem instead. Moreover, we consider measurable functions instead of continuous functions and a much weaker boundedness condition.

Proposition 1.1. Let U be a separable Banach space, T be some measure space, and x_{n} : T → U be (strongly) (Bochner) measurable. Then t ↦ χ({x_{1}(t), x_{2}(t), …}) is measurable. Moreover, if x_{n} is bounded in L_{1} and of equicontinuous norm, i.e.

sup_{n} ∫_{T} ∥x_{n}(t)∥ dt < ∞,  (5)

and

lim_{k→∞} sup_{n} ∫_{D_{k}} ∥x_{n}(t)∥ dt = 0  (6)

for each sequence D_{k} ↓ of measurable sets with ⋂ D_{k} = ∅, then

χ ({∫_{T} x_{1}(t) dt, ∫_{T} x_{2}(t) dt, …}) ≤ ∫_{T} χ({x_{1}(t), x_{2}(t), … }) dt.  (7)

Proof. The set T_{∞} of all t ∈ T such that {x_{1}(t), x_{2}(t), …} is unbounded can be written as

T_{∞} = ⋂_{k=1}^{∞} ⋃_{n=1}^{∞}{t : ∥x_{n}(t)∥ ≥ k}.

In particular, T_{∞} ∩ E belongs to the Lebesgue extension of T for each set E of finite measure (see [9: Section III.6/Theorem 10]). Hence, to prove the measurability of g(t) = χ({x_{1}(t), x_{2}(t), …}) (or (7)), we may assume that T_{∞} = ∅. Similarly, we may assume that T is σ-finite (recall that measurability of g is equivalent to measurability on each σ-finite subset, and that integrable functions have σ-finite support).

We show now that for each integrable simple function x and each subspace V ⊆ U

dist (∫_{T} x(t) dt, V ) ≤ ∫_{T} dist (x(t), V) dt.  (8)

Indeed, if x = ∑_{k=1}^{K} u_{k}χ_{E_{k}} with pairwise disjoint sets E_{k} of finite measure μ_{k}, then (8) is equivalent to

dist (∑_{k=1}^{K} μ_{k}u_{k}, V ) ≤ ∑_{k=1}^{K} μ_{k}dist (u_{k}, V).

But this inequality is true, since dist (·, V) is subadditive and positively homogeneous.

For each n there is a sequence of integrable simple functions x_{n,j} such that x_{n,j} → x_{n}, as j → ∞, almost everywhere on T. We may additionally assume that ∥x_{n,j}(t)∥ ≤ ∥x_{n}(t)∥ (see, e.g., [18: Lemma 4.1.1]). Choosing U_{k} as in Lemma 1.2, we have that each function g_{n,k,j}(t) = dist (x_{n,j}(t), U_{k}) is simple. Since, for almost all t ∈ T,

g_{n,k,j}(t) → g_{n,k}(t) = dist (x_{n}(t), U_{k}) (j → ∞),

we may conclude that each g_{n,k} is measurable. By (4), also

g(t) = χ({x_{1}(t), x_{2}(t), …}) = lim_{k→∞} lim sup_{n→∞} g_{n,k}(t)

is measurable.

To prove estimate (7), we first assume additionally that ∥x_{n}(t)∥ ≤ y(t) for some integrable function y. By (8),

dist (∫_{T} x_{n,j}(t) dt, U_{k}) ≤ ∫_{T} g_{n,k,j}(t) dt.

Since the integrands are dominated by y, we may apply Lebesgue’s dominated convergence theorem to both sides of this inequality and find for j → ∞ that

dist (∫_{T} x_{n}(t) dt, U_{k}) ≤ ∫_{T} g_{n,k}(t) dt.

In particular, for each n and k,

sup_{m≥n} dist (∫_{T} x_{m}(t) dt, U_{k}) ≤ ∫_{T} sup_{m≥n} g_{m,k}(t) dt.

The integrand to the right is dominated by y and converges, as n → ∞, almost everywhere on T to lim sup_{n→∞} g_{n,k}(t). Applying Lebesgue’s dominated convergence theorem two times (once for m and once for k), we arrive at

lim_{k→∞} lim sup_{n→∞} dist (∫_{T} x_{n}(t) dt, U_{k}) ≤ ∫_{T} g(t) dt.

In view of (4), this is the desired inequality (8).

To drop the assumption ∥x_{n}(t)∥ ≤ y(t), we show first that (6) implies

lim_{δ→0} sup_{mes D<δ} sup_{n} ∫_{D} |x_{n}(t)| dt = 0.  (9)

If this were false, there exist some ε > 0 and sequences E_{k} with mes E_{k} < k^{−2} and n_{k} ∈ ℕ such that ∫_{E_{k}} |x_{n_{k}}(t)| dt ≥ ε. Putting D_{k} = ⋃_{j>k} E_{j}, we have mes D_{k} ≤ ∑_{j>k} j^{−2} → 0, and get a contradiction to (6).

Now observe that the support S_{n} of x_{n} is σ-finite, and so is S = ⋃ S_{n}. Thus, there exist sets T_{k} ↑ of finite measure with ⋃ T_{k} = S. Let D_{k}(n) denote the set of all t ∈ T with |x_{n}(t)| ≥ k. Then

sup_{n} mes D_{k}(n) = sup_{n} ∫_{D_{k}(n)} dt ≤ k^{−1} sup_{n} ∫_{T} ∥x_{n}(t)∥ dt → 0  (10)

as k → ∞. Now define for each k a sequence x_{k,n}(t) = x_{n}(t) for t ∈ T_{k} \ D_{k}(n), and put x_{k,n}(t) = 0 for all other t ∈ T. Then

∥∫_{T} x_{k,n}(t) dt − ∫_{T} x_{n}(t) dt∥ ≤ ∫_{S\T_{k}} ∥x_{n}(t)∥ dt + ∫_{D_{k}(n)} ∥x_{n}(t)∥ dt.

Both integrals tend to 0, uniformly in n, as k → ∞: For the first integral apply (6) with D_{k} = S \ T_{k}, for the second integral use (9) and (10). Since χ is continuous with respect to the Hausdorff distance [1], we may conclude that

χ ({∫_{T} x_{k,1}(t) dt, ∫_{T} x_{k,2}(t) dt, …}) → χ ({∫_{T} x_{1}(t) dt, ∫_{T} x_{2}(t) dt, …})

as k → ∞. But, for fixed k, the sequence x_{k,n} is dominated by the integrable function kχ_{T_{k}}, and so, by our previous proof,

χ ({∫_{T} x_{k,1}(t) dt, ∫_{T} x_{k,2}(t) dt, …}) ≤ ∫_{T} χ({x_{k,1}(t), x_{k,2}(t), …}) dt.

Since evidently χ({x_{k,1}(t), x_{k,2}(t), …}) ≤ χ({x_{1}(t), x_{2}(t), …}), a combination of these formulas gives (7)

If T is a σ-finite measure space, condition (5) may be dropped, but the corresponding extension of the proof is rather technical. For our application Proposition 1.1 suffices: If T is σ-finite and without atoms, i.e. each set may be divided into finitely many sets of arbitrarily small measure, then (5) is a consequence of (6).

Indeed, let T_{1} ⊆ T_{2} ⊆ … be sets of finite measure with ⋃ T_{k} = T. Then (6) with D_{k} = T \ T_{k} implies that for some set E = T_{k} of finite measure we have

sup_{n} ∫_{T \E} ∥x_{n}(t)∥ dt < ∞.

Now it remains to apply (9) by dividing E into a finite number of sets of measure less than δ.

2. Integration and compactness of sets

For a set M of functions x : T → U we introduce the shortcuts

M(t) := {x(t) : x ∈ M} (t ∈ T)

and

∫_{T} M(t) dt := {∫_{T} x(t) dt : x ∈ M} .

Proposition 2.1. Let U be a Banach space, T be a measure space, and M be a set of measurable functions x : T → U. Assume that there is a measurable function q such that

χ_{i}(Q) ≤ q(t) (for each countable Q ⊆ M(t)). (11)

Moreover, assume that M is bounded in L_{1} and of absolutely continuous norm,

sup_{x∈M} ∫_{T} ∥x(t)∥ dt = 0  (12)

and

lim_{k→∞} sup_{x∈M} ∫_{D_{k}} ∥x(t)∥ dt = 0  (13)

for each sequence D_{k} ↓ of measurable sets with ⋂ D_{k} = ∅. Then

χ_{i} (∫_{T} M(t) dt) ≤ 2 ∫_{T} q(t) dt.  (14)

If M is countable or U is separable, we even have

χ (∫_{T} M(t) dt) ≤ ∫_{T} q(t) dt.

For separable U we may even additionally replace (11) by

χ(Q) ≤ q(t)  (for each countable Q ⊆ M(t)),  (15)

and for countable M we may additionally replace (11) by

χ_{i}(M(t)) ≤ q(t).  (16)

If T is σ-finite and atomic free, then (12) is a consequence of (13).

Proof. Recall that each integrable function is essentially separably valued (see [9: Section III.6/Theorem 10]). By Proposition 1.1, (7) and (11) we have, for each countable M_{0} ⊆ M,

χ_{U_{0}} (∫_{T} M_{0}(t) dt) ≤ ∫_{T} χ_{U_{0}}(M_{0}(t)) dt ≤ ∫_{T} q(t) dt,

where U_{0} is the closed linear hull of the essential range of the functions in M_{0}. If M is countable, we may choose M_{0} = M, and are done. In general, we have by (3) in particular

α (∫_{T} M_{0}(t) dt) ≤ 2 ∫_{T} q(t) dt.

Now (14) follows by Lemma 1.1. If U is separable, we may choose U_{0} = U and observe that the set ∫_{T} M(t) dt contains a countable subset with the same Hausdorff measure of non-compactness (recall the remark before Lemma 1.1). The final statement is proved analogously to the countable case A sufficient condition for (11) is of course

α(M(t)) ≤ q(t).

Be aware that (16) is in general not sufficient for (11), since χ_{i} is not monotone.

We give now an example which shows that the factor 2 in (14) is best possible in general, even if we replace χ_{i} by χ in (14) and even when T = [0, 1], if we assume the continuum hypothesis.

Example 2.1. We assume that there is a linear ordering ≤ on T = [0, 1] such that the predecessor set P (t) = {s ∈ T : s ≤ t} is at most countable for all t ∈ T. (If we assume the continuum hypothesis, we even find a well-ordering with this property, namely the order inherited by a bijection of the first uncountable ordinal ℵ_{1} onto T). Define U as in Example 1.1 for I = T. For s ∈ T we define an element u_{s} ∈ U as the characteristic function of the set P (s). Moreover, we define x_{s} : T → U by

x_{s}(t) = \{ \begin{matrix}u_{s} & if s ≤ t \\ 0 & otherwise \end{matrix}

and put M = {x_{s} : s ∈ T } ∪ {2x_{s} : s ∈ T }. On the one hand,

M(t) = {x(t) : x ∈ M} = {0} ∪ {u_{s} : s ≤ t} ∪ {2u_{s} : s ≤ t}.

Hence the set {u_{t}} ⊆ M(t) provides a finite 1-net for M(t), i.e. χ_{i}(M(t)) ≤ 1. On the other hand, each function x_{s} takes almost everywhere the value u_{s} (except for the countable set {t ∈ T : t < s}), and so

∫_{T} M(t) dt = {∫_{T} x(t) dt : x ∈ M} = {u_{s} : s ∈ T } ∪ {2u_{s} : s ∈ T }.

If N ⊆ U is a finite ε-net for ∫_{T} M(t) dt, observe that N has at most countable support. Since I is uncountable, there is some s ∈ T such that all y ∈ N satisfy y(s) = 0. For this s we have dist (2u_{s}, N) ≥ 2; consequently, χ(∫_{T} M(t) dt) ≥ 2 ≥ 2χ_{i}(M(t)).

A result which is similar to Proposition 1.1 (and Proposition 2.1 for countable M) can be found in [11]. There one can also find quite sophisticated examples showing that the results considered there are sharp in a certain sense. In particular, [11: Example 4.4] shows that one may not replace (11) by (15) even for countable M without loosing the factor 2 (in that example one even has T = [0, 1] and M consists of uniformly bounded countinuous functions).

With an argument of [14, 15] we may enlarge the class of spaces U for which (11) may be replaced by the weaker assumption (15):

Definition 2.1. We say that a Banach space U has the retraction property, if for each separable subspace U_{0} ⊆ U there exists a non-expanding mapping P : U → U (i.e. ∥P u − P v∥ ≤ ∥u − v∥) with separable range and P u = u on U_{0}.

The retraction property is important for us in view of the following observation:

Lemma 2.1. Let U have the retraction property, and let U_{0} ⊆ U be a separable subspace. Then there exists a separable closed subspace U_{1} ⊆ U with U_{0} ⊆ U_{1} such that χ_{U_{1}}(M) = χ(M) for each M ⊆ U_{0}.

Proof. Choose P as in Definition 2.1, and let U_{1} denote the closed linear hull of P (U_{0}). If N ⊆ U is a finite ε-net for M ⊆ U_{0}, then P (N) ⊆ U_{1} is a finite ε-net for P (M) = M, and so χ_{U_{1}}(M) ≤ χ(M)

Of course, any separable Banach space U has the retraction property. Moreover, any Hilbert space has the retraction property: Choose P as the orthogonal projection of U onto \overline{U}_{0}. More general, if we assume the axiom of choice, any weakly compactly generated Banach space has the retraction property:

Recall that a Banach space U is called weakly compactly generated, if there is some weakly compact set K ⊆ U whose linear span is dense in U. All separable spaces and all reflexive spaces are weakly compactly generated. For weakly compactly generated spaces U, there exists even a projection P of U onto a separable subspace U_{1} ⊇ U_{0} with ∥P ∥ = 1 [8: Chapter 5, §2/ Theorem 3] (however, the proof makes essential use of the axiom of choice).

Corollary 2.1. If U has the retraction property, then we may replace (11) by (15) in Proposition 2.1.

Proof. The proof is almost the same as for separable spaces: Just replace U_{0} by the space U_{1} ⊇ U_{0} of Lemma 2.1

We now consider T = I with a (not necessarily compact) interval I on the extended real line. By C(I, U) we denote the set of all continuous functions x : I → U which have a continuous extension to the compact closure \overline{I} of I; we equip C(I, U) with the usual sup-norm.

We need an extension of the classical Arzelà-Ascoli criterion. For the Kuratowski measure of non-compactness similar results as the following lemma have been proved by several authors (see, e.g., [2, 10, 16]). However, if we consider the Kuratowski measure of non-compactness, we loose again the factor 2 in our applications.

Lemma 2.2. For any M ⊆ C(I, U) we have

χ(M) ≥ χ( ⋃_{x∈M} x(I)) ≥ sup_{t∈I} χ(M(t))  (17)
χ_{i}(M) ≥ sup_{t∈I} χ_{i}(M(t))  (18)
χ_{i}(M) ≥ χ_{i}( ⋃_{x∈M} x(I))  (19)
α(M) ≥ sup_{t∈I} α(M(t)).  (20)

Moreover, if M has an equicontinuous extension to the closure \overline{I} of I in the extended real line, then

χ(M) = sup_{t∈I} χ(M(t)) = max_{t∈\overline{I}} χ(M(t)) = χ( ⋃_{x∈M} x(I)).  (21)

Proof. The estimate (20) is a special case of a result proved in [16]. The second estimate (17) follows from the monotonicity of χ, and the first has been proved in [19]; however the proof is analogous to that of (19).

Let c > χ_{i}(M), and N ⊆ M be a finite c-net for M. Then N(t) ⊆ M(t) is a finite c-net for M(t), and (18) follows. Since \overline{I} is compact, x(I) ⊆ x(\overline{I}) is precompact for each x ∈ N. Given ε > 0, we thus find a finite ε-net for each set x(I) in x(I). The union of these nets evidently is a finite (c + ε)-net for ⋃_{x∈M} x(I). Hence, (19) is established.

To prove (21), let ε > 0 and c > sup_{t} χ_{i}(M(t)) be given. Choose a finite partition t_{1} < … < t_{n} of \overline{I} such that ∥x(t) − x(s)∥ < ε for all t, s in the same section of this partition and all x ∈ M. For each k = 1, …, n there exists a finite c-net N_{k} for M(t_{k}). Let N be the set of all functions which are linear in each of the sections and which pass at t_{k} through a point of N_{k}. Evidently, N is a finite (c + 2ε)-net for M. Hence

χ(M) ≤ sup_{t∈\overline{I}} χ(M(t)).

The function t ↦ χ(M(t)) is continuous, since χ and M both are continuous with respect to the Hausdorff distance (recall that M is equicontinuous on \overline{I}). Hence the supremum is even a maximum on the compact set \overline{I}, and we have

χ(M) ≤ sup_{t∈I} χ(M(t)) ≤ ⋃_{x∈M} x(I).

The converse estimate follows by (17)

We say that a sequence of measurable functions x_{n} converges to a function x in measure on sets of finite measure, if the restrictions x_{n}|_{E} converge to x|_{E} in measure for each set E of finite measure. If the underlying measure space is σ-finite, this notion of convergence is generated by a metric. Observe that in particular each a.e.-convergent sequence of measurable functions also converges with respect to this metric.

With Proposition 2.1 and Lemma 2.2 we can prove now:

Theorem 2.1. Let U be a Banach space, I a (not necessarily compact) interval, and M a set of measurable functions x : I → U, equipped with the above metric. Let t_{0} ∈ \overline{I}, and let J denote the integration operator

Jx(t) = ∫_{t_{0}}^{t} x(s) ds.  (22)

Assume that M satisfies

lim_{k→∞} sup_{x∈M} ∫_{D_{k}} ∥x(t)∥ dt = 0  (23)

for each sequence D_{k} ↓ of measurable sets in I with ⋂ D_{k} = ∅. Then J : M → C(I, U) is defined and continuous. The range J(M) has an equicontinuous extension to \overline{I}. Moreover, if there is a measurable function q : I → [0, ∞] with

χ_{i}(Q) ≤ q(t)  (for each countable Q ⊆ M(t)),  (24)

then

χ(J(M)) ≤ 2 ∫_{I} q(t) dt.  (25)

If M is countable or U is separable, we may drop the factor 2 in (25). If M is countable, we may additionally replace (24) by χ_{i}(M(t)) ≤ q(t). If U has the retraction property, we may always replace χ_{i} in (24) by χ.

Proof. Exhausting \overline{I} by closed and bounded intervals I_{k}, we find, for each ε > 0, some k with

sup_{x∈M} ∫_{I\I_{k}} ∥x(t)∥ dt < ε.

Since, analogously to (9),

lim_{mes D→0} sup_{x∈M} ∫_{D} ∥x(t)∥ dt = 0,

we may conclude that J(M) is defined and has an equicontinuous extension to \overline{I}. Moreover, M is bounded in L_{1}. If x_{n} ∈ M converges to x ∈ M in measure on sets of finite measure, then we have for all t ∈ I that

∥Jx(t) − Jx_{n}(t)∥ ≤ ∫_{I} ∥x(s) − x_{n}(s)∥ ds → 0 (n → ∞),

where the last expression converges to 0 by Vitali’s convergence theorem (see, e.g., [9: Section III.6/Theorem 15] or [18: Theorem 3.3.3]). To prove (25), apply Proposition 2.1 for T = [t_{0}, t] resp. T = [t, t_{0}] to find that

χ (∫_{t_{0}}^{t} M(s) ds) ≤ 2 ∫_{T} q(s) ds ≤ 2 ∫_{I} q(s) ds

holds for all t ∈ I. Now apply (21). The proof of the last statements is analogous

3. Compactness and existence of solutions

Let I be an interval of the extended real line, and t_{0} ∈ \overline{I}. We consider the Picard-Lindelöf operator

Hx(t) = ∫_{t_{0}}^{t} f(s, x(s)) ds.  (26)

Let f be defined on a subset of I ×U with a Banach space U and take values in a Banach space V. The operator (26) may be written as the composition H = JF of the linear integration operator (22) studied in the last section and of the nonlinear superposition operator

F x(s) = f(s, x(s)).  (27)

Theorem 3.1. Let f : I × U → V, and let B be a set of measurable functions x : I → U with the property that (27) is a.e. defined on I and measurable for each x ∈ B. Moreover, assume that

lim_{k→∞} sup_{x∈B} ∫_{D_{k}} ∥f(s, x(s))∥ ds = 0  (D_{k} ↓ ∅).  (28)

Then (26) maps B into a subset of C(I, V) which has an equicontinuous extension to \overline{I}. If almost all f(t, ·) are continuous on their set D(t) of definition, this mapping is continuous (where B is equipped with the metric of convergence in measure on sets of finite measure). Moreover, if B ⊆ C(I, U) and

χ_{i}(f({t} × D_{0})) ≤ q(t, γ(D_{0}))  (for each countable D_{0} ⊆ D(t))  (29)

holds for almost all t ∈ I where γ ∈ {α, χ, χ_{i}} and q is measurable in the first argument and non-decreasing in the second, then

χ(H(B_{0})) ≤ ∫_{I} q(t, γ(B_{0})) dt  (for each countable B_{0} ⊆ B).  (30)

If we have either γ ≠ χ_{i} or even

α(f({t} × D_{0})) ≤ q(t, γ(D_{0}))  (for each countable D_{0} ⊆ D(t)),  (31)

then also

χ(H(B_{0})) ≤ 2 ∫_{I} q(t, γ(B_{0})) dt  (B_{0} ⊆ B).  (32)

If V has the retraction property, we may replace χ_{i} in (29) and α in (31) by χ; if V is even separable, we may additionally drop the factor 2 in (32).

Proof. The first statement follows immediately from Theorem 2.1 and the decomposition H = JF. For the continuity, it suffices to prove that if a sequence x_{n} ∈ B converges in measure on sets of finite measure to x ∈ B, then there is a subsequence such that Hx_{n_{k}} → Hx uniformly on I. However, since I is σ-finite, there is a subsequence with x_{n_{k}} → x a.e. on I (see, e.g., [18: Lemma 2.2.3]). Then F x_{n_{k}} → F x a.e. on I, and it remains to apply Theorem 2.1.

We now prove (30) and (32) for γ ∈ {α, χ}. For almost all t ∈ I we may argue as follows: For each countable Q ⊆ f({t} × B_{0}(t)) we have Q = f({t} × B_{1}(t)) for some countable B_{1} ⊆ B_{0}. Since Lemma 2.2 implies that γ(B_{1}(t)) ≤ γ(B_{1}) ≤ γ(B_{0}), we find

χ_{i}(Q) = χ_{i}(f({t} × B_{1}(t))) ≤ q(t, γ(B_{1}(t))) ≤ q(t, γ(B_{0})).

Now apply Theorem 2.1. To prove (30) for γ = χ_{i} we argue analogously, using the estimate

χ_{i}(Q) ≤ α(Q) ≤ α(f({t} × B_{0}(t))) ≤ q(t, γ(B_{0}(t))) ≤ q(t, γ(B_{0})).

Replacing everywhere χ_{i} and α by χ, we get the result if V has the retraction property

The question under which conditions (27) maps measurable functions into measurable functions is a delicate problem, even in the scalar case (see [3: Chapter 1]). Product-measurability of f is neither necessary nor sufficient.

In the most important case that the sets D(t) are independent of t, i.e. if f : I×D → V for some D ⊆ U, it is sufficient that f satisfies the Carathéodory condition, i.e. that f(·, u) is measurable for all u ∈ D and that f(s, ·) is continuous for almost all s ∈ I.

Indeed, if x : I → U is measurable with x(I) ⊆ D, there exists a sequence of measurable functions x_{n} : I → U which converges a.e. on I to x and satisfies x_{n}(I) ⊆ D (consider, e.g., the proof of [9: Section III.6./Theorem 10] to see this). Evidently, F x_{n} is measurable and converges a.e. on I to F x.

Corollary 3.1. Let f : I × U → V satisfy the Carathéodory condition and the boundedness condition (28) for B = C(I, D). If f(t, ·) is compact, then the operator (26) defines a compact and continuous mapping from C(I, D) into C(I, V).

Corollary 3.1 is somewhat surprising, since under its assumptions, f(I×D) is usually not contained in a compact set.

In connection with Darbo’s fixed point theorem [5], Theorem 3.1 immediately gives an existence result for the Cauchy problem (1). However, since we have a better estimate (30) for countable subsets, we may gain the factor 2, if we use the following fixed point theorem of Daher [4] instead:

Theorem 3.2. Let K be a non-empty, bounded, closed and convex subset of a Banach space, and let A : K → K be continuous. Assume that for all countable C ⊆ K we have χ(A(C)) < χ(C), if C is not precompact. Then A has a fixed point in K.

Theorem 3.2 is a special case of [14: Theorem 2.1] (see also [7: Theorem 18.2]).

Corollary 3.2. Let V = U, and let D ⊆ U contain some closed ball with radius r > 0 around the initial value u. Assume that f : I × D → V satisfies the Carathéodory condition and the boundedness condition (28) with B = {x ∈ C(I, U) : x(I) ⊆ D}. Moreover, assume that there is a non-degenerate interval I_{0} ⊆ I with t_{0} ∈ \overline{I}_{0} and ε > 0 such that

∫_{I_{0}} q(t, λ) dt < λ  (0 < λ ≤ ε)  (33)

for some function q measurable in the first argument and non-decreasing in the second such that for almost all t ∈ I

χ_{i}(f({t} × D_{0})) ≤ q(t, χ(D_{0}))  (for each countable D_{0} ⊆ D).  (34)

Then there exists a local solution of the Cauchy problem (1).

More precisely, if ε ≤ r, and B_{ε} denotes the set of all functions x ∈ C(I, U) which satisfy ∥x(t) − u∥ ≤ ε, then there exists a non-degenerate interval J_{0} ⊆ I_{0} such that t_{0} ∈ \overline{J}_{0},

sup_{t,s∈J_{0}} sup_{x∈B_{ε}} ∥Hx(t) − Hx(s)∥ ≤ ε,  (35)

and problem (1) has a solution on J_{0} in B_{ε}.

Proof. Estimate (35) is a consequence of the equicontinuity of H(B). We consider B_{ε} as a subset of X = C(J_{0}, U). By Theorem 3.1, the mapping Ax = Hx + u is defined and continuous from B_{ε} into X, even into B_{ε} by (35) and Hx(t_{0}) = 0, which satisfies

χ(A(B_{0})) = χ(H(B_{0})) ≤ ∫_{J_{0}} q(t, χ(B_{0})) dt (for each countable B_{0} ⊆ B_{ε}).

Hence, in view of (33) and Theorem 3.2, A : B_{ε} → B_{ε} has a fixed point x ∈ B_{ε}

If U has the retraction property, we may even replace χ_{i} in (34) by χ.

The crucial condition (33) holds in particular for some I_{0} (and each ε > 0) if we have an estimate of the form

α(f(t, D_{0})) ≤ q(t)α(D_{0}) (D_{0} ⊆ D)

with an integrable function q (consider q̃(t, u) = q(t)u). For D being the ball with center u and radius ε > 0, this is the case, if almost all f(t, ·) can be written in the form

f(t, u) = L(t, u) + K(t, u),

where L(t, ·) satisfies a Lipschitz condition

∥L(t, u_{1}) − L(t, u_{2})∥ ≤ q(t)∥u_{1} − u_{2}∥ (∥u_{i} − u∥ ≤ ε),

and K(t, ·) is a compact mapping. This generalizes the classical result of Krasnoselskiĭ and Krein [12] where it is assumed that K(I × D) be precompact.

Observe that Corollary 3.2 also covers the terminal value problem, since the cases t_{0} = ±∞ are not excluded. We remark, however, that Corollary 3.2 does not cover all cases that are contained in [15]. For example, for I = [0, 1] and t_{0} = 0, the Nagumo function q(t, λ) = \frac{λ}{t} does not satisfy (33).

Example 3.1. Consider the nonlinear Barbashin equation

\frac{∂x(t, s)}{∂t} = c(t, s, x(t, s)) + ∫_{a}^{b} k(t, s, σ, x(t, σ)) dσ (t ∈ [0, ∞), s ∈ [a, b])  (36)

under the initial value condition x(0, s) = φ(s). Assume that c(t, ·, ·), k(t, ·, ·, ·), and φ are continuous and c is even Lipschitz continuous with respect to the last argument (for simplicity with a global Lipschitz constant). We consider x as a function from [0, ∞) into X = C([a, b]) by putting x(t)(s) = x(t, s). To prove that (36) has a local solution (for t near 0), it suffices to prove that the initial value problem in the space X,

\begin{matrix}\frac{dx(t)}{dt} & = C(t, x(t)) + K(t, x(t)) \\ x(0) & = φ \end{matrix} \}  (37)

has a local solution. Here, C and K are defined by

C(t, u)(s) = c(t, s, u(s))

and

K(t, u)(s) = ∫_{a}^{b} k(t, s, σ, u(σ)) dσ.

Observe that C(t, ·) satisfies a Lipschitz condition, and K(t, ·) is continuous and compact. If c and k do not depend continuously on the first parameter t, one can not expect a classical solution of (37). The results in the cited literature do not apply for this case. However, if C(·, u) and K(·, u) are at least measurable for u ∈ C([a, b]), it makes sense to confine ourselves to a weak solution of (37), i.e. to a solution of the integrated form of (37). Corollary 3.2 implies that such a local solution exists if, e.g., c and k are uniformly bounded.

Let us finally state a parameter-dependent version of Theorem 3.1 which we will need in a forthcoming paper:

Let f : D ⊆ I × U × Λ → V with I, U, V as before, and Λ being a non-empty set. Consider the operator

H(x, λ)(t) = ∫_{t_{0}}^{t} f(s, x(s), λ) ds  (38)

and the corresponding superposition operator

F (x, λ)(t) = f(t, x(t), λ).  (39)

By D(t) we denote the set of all u such that (t, u, λ) ∈ D for all λ ∈ Λ.

Theorem 3.3. Let f : D ⊆ I×U ×Λ → V, and B be a set of measurable functions x satisfying x(t) ∈ D(t) for almost all t ∈ I. Let (39) be measurable for all (x, λ) ∈ B ×Λ. Moreover, assume that

lim_{k→∞} sup_{(x,λ)∈B×Λ} ∫_{D_{k}} ∥f(s, x(s), λ)∥ ds = 0 (D_{k} ↓ ∅).

Then (38) maps B × Λ into a subset of C(I, V) which has an equicontinuous extension to \overline{I}. Moreover, if B ⊆ C(I, U) and

α(f({t} × D_{0} × Λ_{0})) ≤ q(t, γ(D_{0})) (each countable D_{0} ⊆ D(t), Λ_{0} ⊆ Λ)  (40)

holds for almost all t ∈ I, where γ ∈ {α, χ} and q is measurable in the first argument and nondecreasing in the second, then

χ(H(B_{0} × Λ)) ≤ 2 ∫_{I} q(t, γ(B_{0})) dt  (for each B_{0} ⊆ B)  (41)

and for each countable Λ_{0} ⊆ Λ even

χ(H(B_{0} × Λ_{0})) ≤ ∫_{I} q(t, γ(B_{0})) dt (for each countable B_{0} ⊆ B).

If V has the retraction property, we may replace α by χ in (40); if V is even separable, we may additionally drop the factor 2 in (41).

Compactness and Existence Results 583

Proof. Observe that H(B × Λ) = JF (B × Λ). By Theorem 2.1, H(B × Λ) thus is equicontinuous. For the second part, apply Theorem 2.1 to the set F (B_{0} × Λ). If Q ⊆ F (B_{0} ×Λ)(t) is countable, we find countable B_{1} ⊆ B_{0}, Λ_{0} ⊆ Λ with Q ⊆ F (B_{1} ×Λ_{0})(t). Since, by Lemma 2.2, we have γ(B_{1}(t)) ≤ γ(B_{1}) ≤ γ(B_{0}), we may conclude that, for almost all t ∈ I,

χ_{i}(Q) ≤ α({f(t, x(t), λ) : (x, λ) ∈ B_{1} × Λ_{0}}) ≤ 2q(t, γ(B_{1}(t))) ≤ 2q(t, γ(B_{0})),

as required. If V has the retraction property, we may everywhere replace χ_{i} and α by χ

If (40) holds even for arbitrary D_{0} ⊆ D(t) (not only for countable), we may also allow γ = χ_{i}, since also in this case

α({f(t, x(t), λ) : (x, λ) ∈ B_{1} × Λ_{0}}) ≤ 2q(t, γ(B_{0}(t))) ≤ 2q(t, γ(B_{0})).

Immediately after this text was finished, the article [13] appeared which partly overlaps with our results: Our Proposition 1.1 is the same as [13: Corollary 3.1] (but with a different proof) and our Proposition 2.1/Corollary 2.1 is only slightly more general than [13: Theorem 3.12].

References

[1] Akhmerov, R. R., Kamenskiĭ, M. I., Potapov, A. S., Rodkina, A. E., and B. N. Sadovskiĭ: Measures of Noncompactness and Condensing Operators. Basel - Boston - Berlin: Birkhäuser 1992.

[2] Ambrosetti, A.: Un teorema di esistenza per le equazioni differenziali negli spazi di Banach. Rend. Sem. Mat. Univ. Padova 39 (1967), 349 – 361.

[3] Appell, J. and P. P. Zabreĭko: Nonlinear Superposition Operators. Cambridge: Univ. Press 1990.

[4] Daher, S. J.: On a fixed point principle of Sadovskii. Nonlin. Anal. 2 (1978), 643 – 645.

[5] Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rend. Sem. Mat. Univ. Padova 24 (1955), 84 – 92.

[6] Deimling, K.: Ordinary Differential Equations in Banach Spaces. Lect. Notes Math. 596 (1977).

[7] Deimling, K.: Nonlinear Functional Analysis. Berlin - Heidelberg: Springer 1985.

[8] Diestel, J.: Geometry of Banach Spaces — Selected Topics. Lect. Notes Math. 485 (1975).

[9] Dunford, N. and J. T. Schwartz: Linear Operators. Vol. I: General Theory, 3rd ed. New York: Int. Publ. 1966.

[10] Heinz, H.-P.: Theorems of Ascoli type involving measures of noncompactness. Nonlin. Anal. 5 (1981), 277 – 286.

[11] Heinz, H.-P.: On the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions. Nonlin. Anal. 7 (1983), 1351 – 1371.

[12] Krasnoselskiĭ, M. A. and S. G. Kreĭn: On the theory of ordinary differential equations in Banach spaces (in Russian). Voronezh. Gos. Univ. Trudy, Sem. Funktsional. Anal. 2 (1956), 3 – 23.

[13] Kunze, M. and G. Schlüchtermann: Strongly generated Banach spaces and measures of noncompactness. Math. Nachr. 191 (1998), 197 – 214.

[14] Mönch, H.: Boundary value problems for nonlinear ordinary differential equations in Banach spaces. Nonlin. Anal. 4 (1980), 985 – 999.

[15] Mönch, H. and G.-F. von Harten: On the Cauchy problem for ordinary differential equations in Banach spaces. Arch. Math. (Basel) 39 (1982), 153 – 160.

[16] Nussbaum, R. D.: A generalization of the Ascoli theorem and an application to functional differential equations. J. Math. Anal. Appl. 35 (1971), 600 – 610.

[17] Sadovskiĭ, B. N.: On the local solvability of ordinary differential equations in Banach space (in Russian). Problemy Mat. Anal. Slozh. Sistem 3 (1968), 232 – 343.

[18] Väth, M.: Ideal Spaces. Lect. Notes Math. 1664 (1997).

[19] Väth, M.: Complete continuity of some nonlinear Volterra operator in Banach spaces. Z. Anal. Anw. 17 (1998), 23 – 35.

Received 11.05.1998