Conditional stability for piecewise continuous solutions of differential equations
Let the \(n\times n\) matrix \(A(t)\) be continuous and \(h(t)\) and \(k(t)\) be two scalar continuous and positive functions on \(I=[a,\infty)\). We say that the linear system \[ x'= A(t)x\tag{1} \] has an \((h,k)\) dichotomy if and only if there exist a fundamental matrix \(\Phi(t)\) of (1), a projection \(P\) and a constant \(c>0\) such that \[ |\Phi(t) P\Phi^{-1}(s)|\leq c{h(t)\over h(s)},\quad (t\geq s\geq a), \] \[ |\Phi(t)(I- P)\Phi^{-1}(s)|\leq c{k(s)\over k(t)},\quad (s\geq t\geq a). \] Consider the impulsive differential system on \(I\), \[ \begin{aligned} y' & = A(t)y+ b(t)+ f(t,y),\quad t\neq t_i,\\ \Delta y(t_i) & = g(t_i,y(t_i)),\quad y(t_i)= y(t^-_i),\quad t=t_i\tag{2}\end{aligned} \] with fixed impulse times \(\{t_i\}^\infty_{i=1}\), \(t_i< t_{i+1}\), \(\lim_{i\to\infty} t_i=\infty\) and \(\Delta y(t_i)= y(t^+_i)- y(t_i)\). As usual \(y(t^+_i)\) (resp. \(y(t^-_i)\)) denotes the lateral right (resp. left) limit of \(y(t)\) as \(t\to t_i\). Denote \(h_+= h\) and \(h_-={1\over k}\) and define \(V_\pm=\{x|x:I\to\mathbb{R}^n\), \({x(t)\over h_\pm(t)}\) bounded on \(I\}\). If (1) has an \((h,k)\) dichotomy and \(b(t)\) is a continuous vector function on \(I\), then the author gives sufficient conditions concerning only \(h\), \(k\), \(f\) and \(g\) which guarantee the existence of a one-to-one continuous correspondence between the solutions \(x_\pm\) of the system \(x'= A(t)x+ b(t)\) belonging to \(V_\pm\) and the solutions \(y_\pm\) of (2) belonging to \(V_\pm\). Moreover, if \(\lim_{t\to\infty} {1\over h(t)} \Phi(t)P= 0\) then we have the following formulae \[ y_\pm= x_\pm+h\cdot o(1)\quad\text{as }t\to\infty. \] Several applications to asymptotic integration theory are done, establishing some results of Levinson type [see e.g. \textit{N. Levinson}, Duke Math. J. 15, 111-126 (1948; Zbl 0040.19402)].
scientific article; zbMATH DE number 1011029