Prescribed asymptotic behavior of nonlinear dynamic equations under impulsive perturbations (Q6554395)

This paper presents an attractive investigation of the problem of asymptotic integration, addressing a topic rich in historical context and important theoretical and practical applications. The main framework of the study is to obtain the asymptotic behavior of the solutions of the second-order nonlinear impulsive dynamic equation of the form\N\[\N\begin{cases} (p(t)x^\Delta)^\Delta+q(t)x^\sigma=f(t,x)& t\neq t_k ,\\\N\Delta_0 x+p_kx=G_k(x) & t= t_k ,\\\N\Delta_0 (p(t)x^\Delta)+q_kx^\sigma+r_k x^\Delta=F_k(x) &t= t_k. \end{cases}\tag{1}\N\]\NUsing recently constructed principal and nonprincipal solutions, they successfully demonstrated that these nonlinear impulsive dynamic equations can exhibit prescribed asymptotic behavior at infinity. Namely, they show that there is a solution \(x\) of \((1)\) such that\N\[\Nx(t)=a\,v(t)+b\,u(t)+o(\psi(t)) \quad t\to\infty,\N\]\Nwhere \(u\) and \(v\) are the principal and nonprincipal solutions of the associated homogeneous impulsive equation, and \(\psi\) is a function given in terms of \(u\) and/or \(v\). Moreover, they obtain the necessary compactness criteria for the space under consideration. They support the theoretical findings in the paper by providing illustrative examples and graphical images. Their findings regarding asymptotic behavior have valuable applications for boundary value problems on half-line and point out potential ways for future research. Overall, this paper makes a significant contribution to the impulsive dynamic equations literature and is recommended for researchers interested in the asymptotic behavior of differential and dynamic equations.