A Peano theorem for reflexive Banach spaces via compact embedding

In infinite-dimensional Banach spaces, the classical Peano theorem fails, and continuity of the vector field does not ensure local solvability of the Cauchy problem x ′ ( t ) = f ( t , x ( t ) ) . This paper proves a Peano existence theorem for reflexive Banach spaces by embedding the problem in the associated Kuelbs–Hilbert completion, in the sense of [48] . The approach relies on the compactness of the induced canonical inclusion map, and on a Kurzweil–Henstock integral framework formulated through Henstock conditions, which extend the classical Carathéodory conditions. In this setting, local solutions are obtained in the Kuelbs–Hilbert completion for a broad class of initial-value problems. At the same time, the case of continuous coefficients follows as an immediate corollary.