Periodic solutions for a class of impulsive delay differential equations
In this paper, the author considers a 2-dimensional \(T\)-periodic linear delay differential equation (DDE) \[ x'(t) +A(t)x(t)+b_1h(t)Bx(t-r)=0,\quad t\ge t_0, \] for the particular matrices \[ A(t)=\left[\begin{matrix} a_1(t)&-a_1(t)\\ -a_2(t)&a_2(t) \end{matrix}\right], \qquad B=\left[\begin{matrix} 1&1\\ 1&1 \end{matrix}\right] \] with \(a_1(t),a_2(t),h(t)\) \(T\)-periodic, \(b_1\in\mathbb{R}, r>0\), with possible additive impulses at times \(t_k=t_0+kT (k\in\mathbb{N})\). By a linear change of variables \(y(t)=Cx(t)\) (for a convenient \(2\times 2\) constant matrix \(C\)), the original system can be transformed into a system of a linear homogeneous ODE \(y_2'(t)+a(t)y_2(t)=0\) and a DDE \(y_1'(t)+b(t)y_1(t-r)=-c(t)\), where \(c(t)=(a_1(t)+a_2(t))y_2(t)\). Under certain conditions, constant impulses can be added to \(y_i(t)\, (i=1,2)\), in order to obtain \(T\)-periodic positive solutions of the system. Examples are presented, where such periodic solutions do and do not exist. For the entire collection see [Zbl 1479.34005].
scientific article; zbMATH DE number 7615937
scientific article