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Linear integral equations in the space of regulated functions. (English) Zbl 0941.45001

Summary: We investigate systems of linear integral equations in the space 𝔾 L n of n-vector valued functions which are regulated on the closed interval [0,1] (i.e.  such that can have only discontinuities of the first kind in [0,1]) and left-continuous in the corresponding open interval (0,1)· In particular, we are interested in systems of the form

x(t)-A(t)x(0)- 0 1 B(t,s)[dx(s)]=f(t),

where f𝔾 L n , the columns of the n×n-matrix valued function A belong to 𝔾 L n , the entries of B(t,.) have a bounded variation on [0,1] for any t[0,1] and the mapping t[0,1]B(t,.) is regulated on [0,1] and left-continuous on (0,1) as the mapping with values in the space of n×n-matrix valued functions of bounded variation on [0,1]· The integral stands for the Perron-Stieltjes one treated as the special case of the Kurzweil-Henstock integral.

In particular, we prove basic existence and uniqueness results for the given equation and obtain the explicit form of its adjoint equation. A special attention is paid to the Volterra (causal) type case. It is shown that in that case the given equation possesses a unique solution for any right-hand side from 𝔾 L n , and its representation by means of resolvent operators is given.

MSC:
45B05Fredholm integral equations
45D05Volterra integral equations