On boundary value problems for systems of linear functional-differential equations. (English) Zbl 0930.34047

The authors consider the linear system (1) \(\dot {x}(t)=p(x)(t)+q(t),\) (2) \(\ell (x)=c_0,\) with \(q\in L^1([a,b],\mathbb{R}^n),\) \(c_0\in \mathbb{R}^n,\) \(p: C([a,b],\mathbb{R}^n)\to L^1([a,b],\mathbb{R}^n)\) and \(\ell : C([a,b],\mathbb{R}^n)\to \mathbb{R}^n\) are linear bounded operators such that there is \(\eta \in L^1([a,b],\mathbb{R})\) such that \(\|p(x)(t)\|\leq \eta (t)\|x\|_C\) holds for all \(t\in [a,b]\) and \(x\in C([a,b],\mathbb{R}^n).\) For a given positive integer \(k,\) a nonnegative integer \(m\) and fixed \(t_0\in [a,b]\) arbitrarily chosen, they define \(p^m(x)(t)=\int_{t_0}^t p(p^{m-1}(x))(s)\text{d}s\) (where \(p^0(x)(t)\equiv x(t)\)) and \(\Lambda_k= \ell (p^0(E)+\dots +p^{k-1}(E))\) (where \(E\) is the unit matrix). Moreover, if \(\det (\Lambda_k)\neq 0,\) they put \(p^{k,0}(x)(t)=x(t)\) and \(p^{k,m}(x)(t)=p^m(x)(t) - [p^0(E)(t)+\dots +p^{m-1}(E)(t)] \Lambda_k^{-1}.\) One of the main results is the following theorem on existence and uniqueness of a solution to (1),(2):
If there exist positive integers \(k\) and \(m,\) a nonnegative integer \(m_0\) and a matrix \(A=(a_{i j})\in \mathbb{R}^{n\times n}\) such that \(r(A)<1\) holds for its spectral radius, \(\det (\Lambda_k)\neq 0\) and \(\|p_i^{k,m}(x)\|_C\leq \sum_{j=1}^{n} a_{i j} \|p_j^{k,m_0}(x)\|_C\) holds for any component \(p_i^{k,m}(x),\) \(i=1,\dots ,n,\) of the vector-valued function \(p^{k,m}(x),\) then the given problem (1),(2) possesses a unique solution for any \(q\in L^1([a,b],\mathbb{R}^n)\) and any \(c_0\in \mathbb{R}^n.\)
As corollaries several efficient criteria for the unique solvability of the given problem are obtained. In the case that \(p\) is a Volterra operator, necessary conditions for the unique solvability are given. Furthermore, theorems concerning continuous dependence of the solution on the coefficients are proved and applications to various boundary value problems for systems of the form \(\dot {x}(t)=P(t)x(\tau (t))+q(t)\) are given.
Reviewer: M.Tvrdý (Praha)


34K10 Boundary value problems for functional-differential equations
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