A note on existence theorem of Peano. (Q2897380)

The paper is concerned with the infinite dimensional initial value problem \( u_t(t,x) = f(t,u(t,x))\), \(u(0,x) = x\) where the right-hand side must not be Lipschitzian. The author shows under the assumption of the continuity of the \(m\)-vector function \(f\) in \(\mathbb {R}^{m+1}\) and under certain other restrictive suppositions, that the above initial value problem has for any positive constants \(T\), \(R\) and \(p\in [1,\infty )\) a solution \(w(t,x)\in C(\bar I_T,L^p(B_R))\cap C^1(I_T,L^p(B_R))\), where \(I_T=(-T,T)\) and \(B_R\) is an open ball in \(\mathbb {R}^m\) with radius \(R\) and the center at the origin. Moreover, he derives that to any \(\varepsilon >0\) there exists a closed set \(M_{\varepsilon }\subset B_R\) such that \(\mu (B_R\setminus M_{\varepsilon })<\varepsilon \) and \(w(t,x)\in C(M_{\varepsilon },C(\bar I_T))\), where \(\mu \) denotes the standard Lebesgue measure in \(B_R\).