Nonlinear systems with a small matrix as the coefficient of the derivative
Consider the set of differential equations: (1) \(A(t,\epsilon)u^.=f_ 1(t,u,v,\epsilon)\); \(v^.=f_ 2(t,u,v,\epsilon)\) where A(t,\(\epsilon)\) is a nondegenerate, symmetric \(\ell\times \ell\) matrix; \(u,f_ 1\in R^{\ell}\); \(v,f_ 2\in R^ m\). For equation (1), the data satisfy various conditions, the most important are: the family \(\{\) A(t,\(\epsilon)\}\) converge to zero in \(L^ 2\) when \(\epsilon\to \epsilon_ 0\); continuity and Lipschitz conditions on the vector functions \(f_ 1,f_ 2\); the families \(\{f_ 1(t,u,v,\epsilon)\}\), \(\{f_ 2(t,u,v,\epsilon)\}\) converge respectively to the vector functions \(\hat f{}_ 1(t,u,v)\), \(\hat f{}_ 2(t,u,v)\); \(\hat f{}_ 1,\hat f_ 2\) have continuous partial derivatives with respect to t,u,v. This paper is a study of the convergence in \(L^ 2\) of the solutions of (1). In particular, the convergence conditions of the solution of (1) to the solution of the degenerate system: (2) \(0=\hat f_ 1(t,u,v)\); \(v^.=\hat f_ 2(t,u,v)\) are given. The considered problem is a generalization of the classical problem with A(t,\(\epsilon)\) reduced to a parameter (cf. Tikhonov, Pontrjagin, Vasileva, Mishchenko, etc...).
scientific article; zbMATH DE number 3873771