Rezolventa Linearne Diferencijalne Jednačine


Let $\dot{x}=A(t)x$ be a homogeneous linear differential equation, where $A(t)$ is a continuous operator-valued function defined on an interval $T$, taking values in the space $L(R^{n};R^{n})$. The resolvent of this equation is the operator-valued function $R(t,\tau)$, defined on the square $T\times T$, satisfying conditions $$ R_{t}(t,au)=A(t)\circ R(t,au),\quad R(au,au)=I, $$ where $I\in L(R^{n};R^{n})$ is the identity operator. A connection between the resolvent and the Cauchy function is deduced. In particular, the resolvent is used for finding partial derivatives of the Cauchy function.