Weighted shadowing for delay differential equations (Q2085567)

The paper considers the nonlinear delay differential equation \[ \displaystyle{x'(t) = L(t)x_t +f(t,x_t)}\tag{1} \] where \(L:R\times C(-r,0;R^d)\mapsto R^d\) is continuous, bounded linear and \(f\) is continuous and globally Lipschitz. For this equation \(w\)-shadowing is defined as follows. Equation (1) is called \(w\)-shadowing on \((\sigma,\infty)\) if there exists \(K>0\) such that if \(\delta>0\) and \(y:[\sigma-r,\infty)\mapsto R^d\) is a \(C^1\) function on \([\sigma,\infty)\) satisfying \[ \displaystyle{w(t)|y'(t)-L(t)y_t-f(t,y_t)|\leq\delta\;,\;t\geq\sigma}\tag{2} \] then (1) has a solution \(x\) on \([\sigma,\infty)\) such that \[ \displaystyle{w(t)\|x_t-y_t\|\leq K\delta\;,\;t\geq\sigma} \] The function \(y\) satisfying (2) is called \(w\)-weighted \(\delta\)-pseudo-solution of (1). Let further \((T(t,s))_{t\geq s}\) be the solution operator of the linear equation \[ \displaystyle{x'(t)=L(t)x_t}\tag{3} \] defined as \(T(t,s)\phi:=x_t(s,\phi)\), \(\phi\in C(-r,0;R^d)\) and assume existence of a family of bounded projection \((P(t))_{t\in R}\) on \(C(-r,0;R^d)\) with the following properties i) \(P(t)\in C\) is continuous; ii) \(P(t)T(t,s) = T(t,s)P(s)\) whenever \(t,s\in R\), \(t\geq s\); the linear operator \(T(t,s)|_{\ker\;P(s)}\ker P(s)\mapsto\ker P(t)\) is onto and invertible whenever \(t,s\in R\), \(t\geq s\). Define next \(T(t,s):=(T(t,s)|_{\ker\;P(t)})^{-1}\) for \(t\leq s\) such that \(T(t,s)|_{\ker\;P(s)} \ker P(s)\mapsto\ker P(t)\) is a linear bounded operator for \(t\leq s\) and let \(Q(t):=I-P(t)\), \(t\in R\). Using the notions above, the main result of the paper gives sufficient conditions for \(w\)-shadowing of (1).