## On Some Properties of Linear Mapping Induced by Linear Descriptor Differential Equation

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Zhuk, Serhiy

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In this paper we introduce linear mapping D from WnF\subset Ln into Lm\times
Rm, induced by linear differential equation d/dt
Fx(t)-C(t)x(t)=f(t),Fx(t_0)=f_0. We prove that D is closed dense defined
mapping for any m\times n-matrix F. Also adjoint mapping D* is constructed and
its domain WmF is described. Some kind of so-called "integration by parts"
formula for vectors from WnF, WmF is suggested. We obtain a necessary and
sufficient condition for existence of generalized solution of equation
Dx=(f,f_0). Also we find a sufficient criterion for closureness of the R(D) in
Lm\times Rm which is formulated in terms of transparent conditions for blocks
of matrix C(t). Some examples are supplied to illustrate obtained results.

Comment: Reported at int.conf.MAA-2007,Odessa, Ukraine (http://maa2007.onu.edu.ua/), submitted to Nonlinear Oscillations (http://www.springerlink.com/content/108782/), 8 pages

Comment: Reported at int.conf.MAA-2007,Odessa, Ukraine (http://maa2007.onu.edu.ua/), submitted to Nonlinear Oscillations (http://www.springerlink.com/content/108782/), 8 pages

##### Keywords

Mathematics - Classical Analysis and ODEs, Mathematics - Optimization and Control