On the existence of square-integrable solutions for systems with weak nonlinearity
The paper considers the problem of structural stability of systems under disturbance of coefficients having small \(L^2(\mathbb R)\)-norm. The following main result gives conditions that guarantee that, for every solution of the perturbed system, there exists a solution of the original system which is close to the former in the \(L^2(\mathbb R)\)-norm. Suppose that the system \[ \dot x=X(t,x),\quad x\in \mathbb R^n, \quad t\in \mathbb R,\tag{1} \] has a continuous right-hand side \(X(t,x)\). In addition suppose that all its partial derivatives in \(x\) are continuous and \[ |X(t,x)|\leq M,\quad \left|{\partial X\over \partial x}(t,x)\right|\leq M. \] Assume also that \(X(t,0)\in L^2(\mathbb R)\). Let \(x(t,x_0)\) be the solution of this system with initial data \((0,x_0)\). By definition, put \(P(t,x_0){:}={\partial\over \partial x}x(t,x(t,x_0))\), and suppose that the system \[ \dot x= P(t,x_0)x \] is hyperbolic for any \(x_0\). Consider the perturbed system \[ \dot x= X(t,x)+Y(t,x).\tag{2} \] Under these assumptions, it is claimed that, for any \(\varepsilon>0\), there exists \(\delta>0\) such that, if the mapping \(Y(t,x):\mathbb R^{n+1}\rightarrow \mathbb R^n\) is continuous, all its partial derivatives with respect to \(x\) are continuous, and, in addition, if \[ \|Y(t,\omega(t))\|_{L^2}\leq \delta,\quad \left|{\partial X\over \partial x}(t,x)\right|\leq\delta, \] where \(\omega(t)\) is an arbitrary solution of system (1), then, for any solution \(x\) of system (1), there exists a solution \(y\) of system (2) such that \(\|x(t)-y(t)\|_{L^2}\leq\varepsilon\).
scientific article; zbMATH DE number 6109207
scientific article