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Boundary value problems for systems of linear functional differential equations. (English) Zbl 1161.34300
Folia Facultatis Scientiarum Naturalium Universitatis Masarykianae Brunensis. Mathematica 12. Brno: Masaryk University (ISBN 80-210-3106-9/pbk). 108 p. (2003).
This monograph consists of two chapters. In Chapter 1, the boundary value problem \[ \frac{dx(t)}{dt}=p(x)(t)+q(t),\qquad l(x)=c\tag{1} \] is considered on the interval \([a,b]\), where \(p:C([a,b];\mathbb R^n)\to L([a,b];\mathbb R^n)\) and \(l:C([a,b];\mathbb R^n)\to \mathbb R^n\) are linear bounded operators, \(q\in L([a,b];\mathbb R^n)\), and \(c\in R^n\). In § 1.1, there are established general theorems on the Fredholmity of \((1)\) and Green’s formula. § 1.2 deals with theorems on differential inequalities. Based on results from § 1.2, in § 1.3 there are established effective criteria on the unique solvability of problem \((1)\). § 1.4 is devoted to the well–posedness of \((1)\). All the results are concretized for systems of equations with deviating arguments and for special cases of boundary conditions (initial condition, multi-point boundary condition, and periodic condition).
In Chapter 2, there is considered the system of equations \[ \frac{dx(t)}{dt}=p(x)(t)+q(t)\tag{2} \] on the real axis. In § 2.1, there are established effective criteria guaranteeing the existence and uniqueness of \(\omega-\)periodic solutions of \((2)\), provided \(p:C_{\omega}(\mathbb R^n)\to L_{\omega}(\mathbb R^n)\) is a linear bounded operator and \(q\in L_{\omega}(\mathbb R^n)\). In § 2.2, there are proved theorems on the existence and uniqueness of a bounded solution of \((2)\), assuming that \(p:C_{loc}(\mathbb R;\mathbb R^n)\to L_{loc}(\mathbb R;\mathbb R^n)\) is a linear continuous operator and \(q\in L_{loc}(\mathbb R;\mathbb R^n)\). Both in § 2.1 and § 2.2, the results obtained are concretized for systems of equations with deviating arguments.
Reviewer: Robert Hakl (Brno)

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34K10 Boundary value problems for functional-differential equations