Eloe, P.; Hristova, S. G. Method of the quasilinearization for nonlinear impulsive differential equations with linear boundary conditions. (English) Zbl 1029.34010 Electron. J. Qual. Theory Differ. Equ. 2002, Paper No. 10, 14 p. (2002). The authors consider the nonlinear impulsive differential equation \[ x'= f(t, x(t))\quad\text{for }t\in [0,T],\quad t\neq \tau_k,\tag{1} \]\[ x(\tau_k+ 0)= I_k(x(\tau_k)),\quad k= 1,2,\dots, p,\tag{2} \] where \(\tau_{k+1}> \tau_k\), \(k= 1,2,\dots, p-1\); \(\tau_k\in (0,T)\), with the linear boundary value condition \[ Mx(0)- Nx(T)= c,\tag{3} \] where \(f: [0, T]\times \mathbb{R}\to \mathbb{R}\), \(I_k: \mathbb{R}\to \mathbb{R}\), \(k= 1,2,\dots, p\), and \(c\), \(M\), \(N\) are constants.If (1)–(3) has a lower and an upper solution, sufficient conditions for the existence of a solution to (1)–(3) between these solutions are derived. Reviewer: Stepan Kostadinov (Plovdiv) Cited in 5 Documents MSC: 34A37 Ordinary differential equations with impulses 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:impulsive differential equations; linear boundary value problem; quasi-linearization; quadratic convergence PDFBibTeX XMLCite \textit{P. Eloe} and \textit{S. G. Hristova}, Electron. J. Qual. Theory Differ. Equ. 2002, Paper No. 10, 14 p. (2002; Zbl 1029.34010) Full Text: DOI EuDML EMIS