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An initial value problem and boundary value problems for systems of ordinary differential equations. Vol. I: Linear theory. (Nachal’naya i kraevye zadachi dlya sistem obyknovennykh differentsial’nykh uravnenij. Tom I: Linejnaya teoriya.) (Russian) Zbl 0956.34505
Tbilisi: Metsniereba, 216 p. (1997).
This monograph is devoted to the theory of initial and boundary value problems for systems of ordinary linear differential equations. It is divided into six chapters. Chapter I is devoted to the classical theory of the Cauchy problem, the question of well-posedness of the Cauchy problem and to the Lyapunov stability of solutions. Chapter II contains sufficient conditions for unique solvability and so-called \(H\)-well-posedness of the Cauchy problem with a weight for linear differential systems whose coefficients have nonintegrable singularities. Sufficient conditions for unique solvability and well-posedness of general boundary value problems are given in Chapter III. These results are applied to multipoint boundary value problems in Chapter IV. There are also given (in some sense optimal) criteria for unique solvability of the perturbed Cauchy-Nicoletti problem. The sign property of the solutions of this problem is investigated and an algorithm for construction of a solution is given. In Chapter V necessary and sufficient conditions for uniqueness of solution of two-point linear boundary value problems are formulated. Chapter VI is devoted to the investigation of the existence of solutions that are periodic and bounded on \(\mathbf R\) or \(\mathbf R_+\), of linear differential systems with coefficients locally integrable on \(\mathbf R\). The book is intended for scientists and students who specialize in the field of differential equations.

34A30 Linear ordinary differential equations and systems
34A40 Differential inequalities involving functions of a single real variable
34B05 Linear boundary value problems for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations