### Absolutely continuous functions of two variables in the sense of Carathéodory.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

We establish new efficient conditions sufficient for the unique solvability of the initial value problem for two-dimensional systems of linear functional differential equations with monotone operators.

Some Wintner and Nehari type oscillation criteria are established for the second-order linear delay differential equation.

We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing...

Nonimprovable sufficient conditions for the solvability and unique solvability of the problem $${u}^{\text{'}}\left(t\right)=F\left(u\right)\left(t\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2.0em}{0ex}}u\left(a\right)-\lambda u\left(b\right)=h\left(u\right)$$ are established, where $F:\to $ is a continuous operator satisfying the Carathèodory conditions, $h:\to R$ is a continuous functional, and $\lambda \in $.

Nonimprovable, in a certain sense, sufficient conditions for the unique solvability of the boundary value problem $${u}^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),\phantom{\rule{2.0em}{0ex}}u\left(a\right)+\lambda u\left(b\right)=c$$ are established, where $\ell :C\left(\right[a,b];R)\to L\left(\right[a,b];R)$ is a linear bounded operator, $q\in L\left(\right[a,b];R)$, $\lambda \in {R}_{+}$, and $c\in R$. The question on the dimension of the solution space of the homogeneous problem $${u}^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right),\phantom{\rule{2.0em}{0ex}}u\left(a\right)+\lambda u\left(b\right)=0$$ is discussed as well.

**Page 1**