Existence of solutions for a class of discontinuous differential equations in \(\mathbb{R}^n\)
The author investigates the Cauchy problem \[ x_i'= g_i(t, q_{i1}(x_1(t)), q_{i2}(x_2(t)),\dots, q_{in}(x_n(t))),\quad x_i(0)= 0,\quad i= 1,2,\dots, n.\tag{1} \] It is assumed that \(g_i: [0,\infty)\times \mathbb{R}^n\to [a_i,b_i]\) \((0\not\in[a_i, b_i])\) satisfies the Carathéodory conditions and \(q_{ij}:\Delta_{ij}\to \overline{\mathbb{R}}\) are measurable \((i,j= 1,2,\dots, n)\), where \(\Delta_{ij}= [0,\infty)\) for \(a_i>0\) and \(\Delta_{ij}= (-\infty,0]\) if \(a_i< 0\). Under the above assumptions it is proved that problem (1) has an absolutely continuous solution on the interval \([0,\infty)\). A scalar case is also considered. Interesting examples illustrating the obtained results are given. The results extend several ones obtained earlier by D. C. Biles, A. Bressan and P. Binding.
scientific article; zbMATH DE number 1295269