Connectedness of solution space for ODE

Question

Given an explicit first order IVP $y'=f(x,y)$, $x\in\mathbb{R}$, $f$ : continuous, can we say that the solution space of this problem is a connected set?

(Say we are working on $L_p$-normed space)

Answer 1

A theorem due to Hellmuth Kneser states that, for a fixed initial condition $y(\xi) = \eta$, if $x > \xi$ is such that there exists a solution to the IVP defined (at least) on $[x, \xi]$ then the set $$ S(x) = \{\, y(x) : y(\cdot)\text{ is a solution of the IVP defined on }[\xi,x]\,\} $$ is a continuum in $\mathbb{R}^n$. See the discussion If an IVP does not enjoy uniqueness, then there are infinitely many solutions.

Hukuhara showed that the set $S$ of solutions defined on $[\xi,x]$ is a continuum in the Banach space $C([\xi,x])$ of continuous functions with supremum norm, and Aronszajn proved that $S$ is even the intersection of a decreasing sequence of compact absolute retracts.

There is a monograph: Smaïl Djebali, Lech Górniewicz, Abdelghani Ouahab, Solution Sets for Differential Equations and Inclusions, De Gruyter Series in Nonlinear Analysis and Applications, 118, Walter de Gruyter, 2012.