On Peano’s Theorem in Banach Spaces

We show that if X is a separable Banach space (or more generally a Banach with an infinite-dimensional separable quotient) then there is a continuous mapping f : X → X such that the autonomous differential equation x′ = f(x) has no solution at any point. In order to put our results into context, let us start by formulating the classical theorem of Peano. Theorem 1. (Peano) Let X = R, f : R×X → X be a continuous mapping, t0 ∈ R, x0 ∈ X. Then the ordinary differential equation x′ = f(t, x) (1) together with an initial condition x(t0) = x0 (2) has a solution on some open interval containing t0. Using an infinite dimensional Banach space X = c0, Dieudonne [D] constructed a counterexample to Theorem 1. Many counterexamples in various infinite dimensional Banach spaces followed, e.g. [LL], [B], [H], [Y], [G1], and [C] for every nonreflexive Banach space. Finally, Godunov in [G3] proved that Theorem 1 is false in every infinite dimensional Banach space. More precisely, for every infinite dimensional Banach space X, t0 ∈ R, x0 ∈ X, there exists a continuous mapping f : R × X → X, such that there exists no solution to the differential equation x′ = f(t, x), that satisfies x(t0) = x0. Yet, the above construtions are in fact showing the failure of condition (2), and the constructed examples have many solutions on intervals not containing the given time t0. Moreover, Lasota and Yorke [LY] (see also Vidossich [V]) proved that for every Banach space X, and every initial condition x(t0) = x0, the set of all continuous mappings f : R × X → X, such that x′ = f(t, x), x(t0) = x0 has a solution that exists on the whole real line, is a generic set. More precisely, putting the topology of uniform convergence on the space of all continuous mappings f : R×X → X, the set of mappings admitting a solution has a complement of first Baire category. In view of these resuts, it is natural to consider a weaker form of Peano’s theorem in infinite dimensional Banach spaces. Theorem 2. (Peano-weak form) Let X = R, f : R×X → X be a continuous mapping. Then the ordinary differential equation x′ = f(t, x) (3) has a solution on some open interval. Showing the failure of this theorem in infinite dimensional Banach spaces is clearly a harder problem. In [G2], Godunov constructed a counterexample to Theorem 2 in the Hilbert space. Finally, Shkarin [S] proved that Theorem 2 fails for every Banach space X that has a complemented subspace with an unconditional Schauder basis. (To be more precise, Shkarin’s result is even stronger as it contains a precise quantitative information on the modulus of continuity of f -we refer the reader to [S]). Shkarin’s result applies to many ”classical” Banach spaces, such as Lp 1 ≤ p < ∞, or C[0, 1]. However, there exist separable reflexive Banach spaces that contain no unconditional basic sequence ([F]). Similarly, the classical nonseparable Banach space `∞ is also not covered by [S] (because all its complemented subspaces are again isomorphic to `∞, a result of Rosenthal [LT]). The main result of this note, Theorem 8, states that if X is a Banach space with an infinite-dimensional separable quotient (in particular every separable Banach space, of course) then Theorem 2 fails to be true for some continuous mapping f . A slightly stronger result holds, namely there is a continuous mapping f : X → X such that the autonomous differential equation x′ = f(x) has no solution at any point. We note that the question whether every Banach space has a separable quotient is one of the outstanding problems of the Banach space theory. It is known to hold in all reasonable classes of Banach spaces, such as reflexive, weakly compactly generated, C(K) where K is a compact Date: June 2009. 2000 Mathematics Subject Classification. 34A34, 34G20, 46B20, 46B26, 46G05.