Boundary Value Problems for Delay Differential Systems

• Published in: Advances in difference equations 2010-01, Vol.2010 (1), p.1-20
• Main Authors: Boichuk, A., Diblík, J., Khusainov, D., Růžičková, M.
• Format: Article
• Language: English
• Subjects: Analysis; Boundary conditions; Boundary value problems; Criteria; Delay; Difference and Functional Equations; Difference equations; Differential equations; Functional Analysis; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrices; Ordinary Differential Equations; Partial Differential Equations; Recent Trends in Differential and Difference Equations; Research Article; Vectors (mathematics)
• ISSN: 1687-1839; 1687-1847
• DOI: 10.1155/2010/593834

Conditions are derived of the existence of solutions of linear Fredholm's boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay, assuming that these solutions satisfy the initial and boundary conditions. Utilizing a delayed matrix exponential and a method of pseudoinverse by Moore-Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a differential system with a single delay. As an example of application of the results derived, the problem of bifurcation of solutions of boundary-value problems for systems of ordinary differential equations with a small parameter and with a finite number of measurable delays of argument is considered.