Solvability of the periodic problem for higher-order linear functional differential equations
The author studies the solvability of the periodic functional differential equation \[ x^{(n)} = T^+ x - T^- x + f, \] where solutions are real-valued functions \(t \in \mathbb{R} \mapsto x(t)\) with a locally integrable weak derivative of order \(n\), of the same fixed period \(\tau\) than the locally integrable function \(t \in \mathbb{R} \mapsto f(t)\). The operators \(T^+\) and \(T^-\) are linear and positive, e.g. defined by \((Tx)(t) = k(t) x(t- \sigma(t))\) for essentially bounded and nonnegative \(\tau\)-periodic functions \(k\) and \(\sigma\). The question of the unique solvability of such systems has been fully characterized in the past for the sets of operators \(T^+\) and \(T^-\) of fixed integral norms \[ \| T\|_1 = \int_0^{\tau} (T \mathbf{1})(t)\,dt. \] The author extends the analysis to operators with essentially bounded values, subject to pointwise constraints, i.e. based on \[ \| T\|_{\infty} = \sup_t T \mathbf{1}(t). \] Necessary and sufficient conditions for the unique solvability are derived. As in the integral case, these conditions depend on values which change with the order \(n\); these values are explicited in terms of the Favard and Stechkin constants.
scientific article; zbMATH DE number 6485712
scientific article