Initial value problems for the second order mixed monotone type of impulsive differential equations in Banach spaces. (English) Zbl 0964.34008
The authors consider initial value problems to the impulsive differential equation of the form
\[
u''- f(x, u,u')=\theta,\quad x\in J,\quad x\neq x_i,\quad \Delta u|_{x=x_i}= I(u(x_i)),\quad \Delta u'|_{x= x_i}=\widetilde I(u(x_i)),
\]
\[ i= 1,2,\dots, m\quad u(0)= w_0,\quad u'(0)= w_1, \] where \(f\) is a continuous function of its arguments, \(J= [0,1]\), \(I_i\) and \(\widetilde I_i\) are impulsive sources (continuous functions) propagating the impulsive effect, \(0< x_1< x_2<\cdots< x_m< 1\), are given fixed points. The authors use the coupled fixed-point theorem for mixed monotone condensing operators to obtain an interesting existence and uniqueness theorem.
\[ i= 1,2,\dots, m\quad u(0)= w_0,\quad u'(0)= w_1, \] where \(f\) is a continuous function of its arguments, \(J= [0,1]\), \(I_i\) and \(\widetilde I_i\) are impulsive sources (continuous functions) propagating the impulsive effect, \(0< x_1< x_2<\cdots< x_m< 1\), are given fixed points. The authors use the coupled fixed-point theorem for mixed monotone condensing operators to obtain an interesting existence and uniqueness theorem.
Reviewer: Drumi Bainov (Sofia)
MSC:
34A37 | Ordinary differential equations with impulses |
47H10 | Fixed-point theorems |
34L30 | Nonlinear ordinary differential operators |
34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
34G10 | Linear differential equations in abstract spaces |
References:
[1] | Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002 |
[2] | Shihsen, Chang; Yihai, Ma, Coupled fixed points for mixed monotone condensing operators and an existence theorem of the solutions for a class of functional equations arising in dynamic programming, J. Math. Anal. Appl., 160, 468-479 (1991) · Zbl 0753.47029 |
[3] | Dajun, Guo, Initial value problems for nonlinear second order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl., 200, 1-13 (1996) · Zbl 0851.45012 |
[4] | Dajun, Guo; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045 |
[5] | Dajun, Guo; Lakshmikantham, V., Coupled fixed points of nonlinear operators with application, Nonlinear Appl., 11, 623-632 (1987) · Zbl 0635.47045 |
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