On the existence and uniqueness of solutions of ODEs with discontinuous right-hand sides and Sobolev orthogonal systems of functions

In this paper we introduce the concept of a solution of the Cauchy problem for a system of ordinary differential equations of the form $y'(x)=f(x,y)$, $ y(0)=y_0, \quad 0\le x\le 1$ in which the right-hand side of $f=(f_1,\ldots,f_m)$ is not necessarily continuous in the domain of its definition $G\subset\mathbb{R}^{m+1}$. The problems of the existence and uniqueness of the solution of the Cauchy problem are considered. In order to define the concept of a solution of the Cauchy problem for the equation, we introduced the class $AC^m[0,1]$ consisting of all the absolutely continuous vector-valued functions $y=y(x)=(y_1(x),\ldots,y_m(x))$ defined on $[0,1]$. A vector-valued function $y\in AC^m[0,1]$ is called a solution of the Cauchy problem if $y'(x)=f(x,y(x))$ holds for almost all $x\in[0,1]$ and satisfies condition $y(0)=y_0$. When considering questions related to the existence and uniqueness of the Cauchy problem in the sense of the above definition, systems of functions orthonormal in the sense of Sobolev and generated by a given system $\{\varphi_k(x)\}_{k=0}^\infty$ orthonormal in the weighted Lebesgue space $L_\rho^2(0,1)$ with weight $\rho=\rho(x)$ play a key role.