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Method of the quasilinearization for nonlinear impulsive differential equations with linear boundary conditions. (English) Zbl 1029.34010

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.

MSC:

34A37 Ordinary differential equations with impulses
34B15 Nonlinear boundary value problems for ordinary differential equations
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