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On the initial value problems of first order impulsive differential systems. (English) Zbl 0914.65073

Consider the initial value problems for the first-order impulsive ordinary differential equations: \(u'=f(t, u, u)\) for \(t=t_i\) and \(\triangle u| _{t=t_i}=I_i(u(t_i))\), \(i=1,\dots,m\), \(u(0)=x_0\), where \(f=(f_1,\dots,f_k)\in C[J\times\mathbb{R}^k\times\mathbb{R}^k, \mathbb{R}^k], J=[0, T]\) \((T>0)\), \(0<t_1<\dots<t_m<T\), \(I_i\in C[\mathbb{R}^k, \mathbb{R}^k]\), \(\triangle u| _{t=t_i}= u(t^+_i-u^-_i)\), \(i=1,\dots,m; m\in \mathbb{R}^k\). The existence and iterative approximation of minimax quasi-solutions for this class of systems is proved by using monotone iterative methods.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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References:

[1] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov,Theory of Impulsive Differential Equations, World Sci. Publishing Co., Singapore (1989). · Zbl 0719.34002
[2] M. Khavanin and V. Lakshmikantham, The method of mixed monotone and first order differential systems,Nonlinear Anal. TMA,10, 9 (1986), 873–877. · Zbl 0611.34016
[3] L. H. Erbe and X. Liu, Quasi-solutions of nonlinear impulsive equations in abstract cones,Appl. Anal.,34, 2 (1989), 231–250. · Zbl 0662.34015
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