Existence, uniqueness, and comparison results for a differential equation with discontinuous nonlinearities
The following initial value problem (IVP) is studied: \(x'=q(x)g(t,x)\), \(x(0)=x_0\). It is assumed that \(q:\mathbb{R}\to (0,\infty)\) and \(g:J\times\mathbb{R}\to\mathbb{R}\) \((J=[0,T])\). Moreover, \(q\) is measurable, essentially bounded and \(1/q\) is locally essentially bounded while \(t\to g(t,z)\) is measurable, \(\limsup_{y\to z-} g(t,y)\leq g(t,z)=\lim_{y\to z+} g(t,y)\) and \(|g(t,z)|\leq p(t)\psi(|z|)\), where \(p\) is nonnegative, Lebesgue integrable on \(J\) and \(\psi:\mathbb{R}_+\to(0,\infty)\) is assumed to be increasing and \(\int^\infty_0 dt/\psi(t)=\infty\). It is shown that under the above assumptions (IVP) has a solution. Under some additional assumptions a few uniqueness results are proved. The dependence on initial data is also discussed.
scientific article; zbMATH DE number 930189