Two (multi) point nonlinear Lyapunov systems --- existence and uniqueness
The authors consider a system of nonlinear differential equations of the form (1) \(T'(t)=A(t)T(t)+T(t)B(t)+F(t,T(t))\), \(a\leq t\leq b\), and a set of multi-point boundary conditions written as (2) \(MT(a)+NT(b)=\alpha\) or \(\sum^ n_{i=1}M_ iT(t_ i)=\alpha\) with \(a=t_ 1<t_ 2<\cdots<t_ n=b\). In (1) and (2) \(A(t)\), \(B(t)\) and \(T(t)\) stand for square matrices of order \(n\) whose entires are continuous functions of \(t\) while \(M,N\) and \(M_ i\) are given constant matrices of order \(n\). Their aim is two-fold: i) To establish the general solution of the system (1) in terms of the fundamental matrices and, ii) To discuss the existence of the solution to the boundary value problem (1)-(2). Their results are based on a theorem (theorem 2.2) which claims that any solution of (1) is of the form \(T(t)=YCZ^*+\overline T(t)\), where \(\overline T(t)\) is a particular solution to (1), \(C\) is an arbitrary constant square matrix while \(Y\) and \(Z\) stand for the solutions to the homogeneous systems \(Y'=AY\) and \(Z'=B^*Z\). Here \((*)\) denote the transpose of the complex conjugate matrix. Since this theorem is obviously incorrect because \(F(t,YCZ^*+\overline T)\) is not identically equal to \(F(t,\overline T)\), all of their main results are wrong.
scientific article; zbMATH DE number 91558