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Dichotomies for generalized ordinary differential equations and applications. (English) Zbl 1396.34034
The paper develops a basic theory of ordinary and exponential dichotomies for generalized linear differential equations of the form \[ x(t)=x(t_0)+\int_{t_0}^t \text{d}[A(s)]x(s), \] where \(A:J\subset\mathbb R\to {\mathcal L}(X)\) is a function with locally bounded variation, and solutions taking values in a Banach space \(X\).
The results are analogues of the classical theory presented in W. A. Coppel’s book [Dichotomies in stability theory. Springer, Cham (1978; Zbl 0376.34001)], and include the relation between the exponential dichotomy and the existence of a unique bounded solution to the inhomogeneous equation \[ x(t)=x(t_0)+\int_{t_0}^t \text{d}[A(s)]x(s)+f(t)-f(t_0). \]
In the second part of the paper, the authors show that their results are also applicable to measure differential equations and impulsive differential equations, which represent a special case of generalized ODEs. Reviewer’s remarks:
1) A correct reference for Theorem 4.1 is [Š. Schwabik, Math. Bohem. 125, No. 4, 431–454 (2000; Zbl 0974.34057)]; the finite-dimensional proof from [Š. Schwabik, Generalized ordinary differential equations. Singapore: World Scientific (1992; Zbl 0781.34003), Theorem 6.17] no longer works in the context of Banach spaces.
2) Condition (A3) on page 3158 might be simplified, since \(I+\lim_{r\to t+}\int_t^r G(s)\,\text{d}u(s)=I+G(t)\Delta^+u(t)\).

34D09 Dichotomy, trichotomy of solutions to ordinary differential equations
45A05 Linear integral equations
34G10 Linear differential equations in abstract spaces
34A36 Discontinuous ordinary differential equations
Full Text: DOI
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