Systems of differential equations of alternately retarded and advanced type (Q1359563)

The global asymptotic behavior and the oscillation of solutions of the following system of equations with piecewise constant argument is investigated: \[ x'(t)+Ax(t)+B(g(t))=f(t)\quad\text{ for}\quad t>0,\qquad x(0)=x_0. \] Here \(A\) and \(B\) are \(r\times r\) matrices, \(x\) is an \(r\)-vector, \(f(t)\) is a locally integrable function on \([0,\infty)\), \(g(t)\) is a piecewise constant function defined by \(g(t)=np\) for \(t\in[np-l,(n+1)p-l]\), \(n=\pm1,\pm2,\dots \), \(p\) and \(l\) are positive constants satisfying \(p>l\). The peculiarity of this system is that the argument deviation \(\tau(t)=t-g(t)\) is negative in \([np-l,np)\) and positive in \((np,(n+1)p-l]\), hence the system is called of alternatively advanced and retarded type. For such a system, the changes of sign in the argument lead to interesting periodic properties and to complications in the asymptotic and oscillatory behaviour of the solutions. The following results are derived: In the case when \(A\) is an \(r\times r\) zero matrix a closed formula for the solution is provided. It is shown that under some restriction on the matrix \(B\) and the function \(f\) every solution tends to zero as \(t\to\infty\). Necessary and sufficient conditions are given for the oscillation of all solutions when \(f\equiv 0\). For the case when \(A\) is nonsingular, a formula for the solution is provided, conditions are stated under which every solution converges to zero as \(t\to\infty\) and oscillatory results are obtained. The results are applied to analyse the global behavior of the solutions of a damped spring-mass system subject to an external piecewise constant force.