Nonuniqueness theorem for a singular Cauchy-Nicoletti problem
The paper deals with the Cauchy-Nicoletti BVP \[ x'=f(t,x),\quad x_i(t_i)=x_{0i}, \tag{1} \] with\(f(t,x)=(f_1(t,x_1,\dots,x_n),\dots, f_n(t,x_1,\dots,x_n)),\) \(x_{0i}\in \mathbb{R},\) \(t_i\in [a,A]\subset \mathbb{R}\), \(i=1,\dots, n.\) For \(\sigma_1,\dots, \sigma_p \in [a,A]\) and \(J=[a,A]\setminus \{ \sigma_1,\dots, \sigma_p\},\) \(f\) is supposed to satisfy the Carathéodory conditions on \(J\times \mathbb{R}^n\) and can have time singularities at \(t=\sigma_1,\dots, t=\sigma_p.\) The author investigates the nonuniqueness of (1). A general theorem ensuring the existence of two different solutions is given. The proof is based on suitable Lyapunov functions. The main result is illustrated by corollaries and examples.
scientific article; zbMATH DE number 2163676