A Proposal for the International Systematization of Saints' Legends and Sagen

Translation with and introduction by the editor

Application of Green's operator to quadratic variational problems

We use Green's function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space $L_2$ of quadratically summable functions. We get the necessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert-Schmidt operator in $L_2$ with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1) a variational problem with deviating argument, and 2) the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar's axis)

A condition on delay for differential equations with discrete state-dependent delay

Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Analysis: Theory, Methods and Applications, 70 (11) (2009), 3978-3986] is developed. We propose and study a state-dependent analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions $C$. The dynamical system is constructed in $C$ and the existence of a compact global attractor is proved

Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions

Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional assumption on the state-dependent delay function to guarantee the well posedness. For the constructed dynamical system we study the long-time asymptotic behavior and prove the existence of a compact global attractor