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Differential systems involving impulses. (English) Zbl 0539.34001
Lecture Notes in Mathematics. 954. Berlin-Heidelberg-New York: Springer-Verlag. vii, 102 p. DM 19.80; $ 8.30 (1982).
From the authors’ preface: “An attempt is made to unify the results from several research papers published during the last fifteen years. Chapter 2 contains results on existence and uniqueness (for \(Dx=F(t,x)+G(t,x)Du\), \(Dx\), \(Du\) denote distributional derivatives, \(u\) is a function of bounded variation and \(Du\) can be viewed as a Stieltjes measure which has the effect of instantaneously changing the state). In Chapter 3 fixed point theorems and generalized integral inequalities are employed to derive results on stability and asymptotic equivalence. Chapter 4 provides a study for systems
\[ Dy=A(t)yDu+f(t,y)+g(t,y)Du\quad u(t)=t+\sum^{\infty}_{k=1}a_k H_k(t),\]
\(H_k(t)=0,\ t<t_k\), \(H_k(t)=1,\ t\geq t_ k\). Chapter 5 deals with extensions of Lyapunov’s second method to the study of impulse systems.”

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A37 Ordinary differential equations with impulses
34D20 Stability of solutions to ordinary differential equations
34E99 Asymptotic theory for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations