Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

Collegari, Rodolfo, Federson, Márcia, and Frasson, Miguel. "Linear FDEs in the frame of generalized ODEs: variation-of-constants formula." Czechoslovak Mathematical Journal 68.4 (2018): 889-920. <http://eudml.org/doc/294474>.

@article{Collegari2018,
abstract = {We present a variation-of-constants formula for functional differential equations of the form \[ \dot\{y\}=\{\mathcal \{L\}\}(t)y\_t+f(y\_t,t), \quad y\_\{t\_0\}=\varphi , \] where $\{\mathcal \{L\}\}$ is a bounded linear operator and $\varphi $ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application $t\mapsto f(y_t,t)$ is Kurzweil integrable with $t$ in an interval of $\mathbb \{R\}$, for each regulated function $y$. This means that $t\mapsto f(y_t,t)$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J. Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type \[ \frac\{\{\rm d\}x\}\{\{\rm d\}\tau \} = D[A(t)x],\quad x(t\_0)=\widetilde\{x\} \] and the solutions of the perturbed Cauchy problem \[ \frac\{\{\rm d\}x\}\{\{\rm d\}\tau \} = D[A(t)x+F(x,t)], \quad x(t\_0)=\widetilde\{x\}. \] Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form \[ \dot\{y\}=\{\mathcal \{L\}\}(t)y\_t, \quad y\_\{t\_0\}=\varphi , \] where $\mathcal \{L\}$ is a bounded linear operator and $\varphi $ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs.},
author = {Collegari, Rodolfo, Federson, Márcia, Frasson, Miguel},
journal = {Czechoslovak Mathematical Journal},
keywords = {functional differential equation; variation-of-constants formula},
language = {eng},
number = {4},
pages = {889-920},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Linear FDEs in the frame of generalized ODEs: variation-of-constants formula},
url = {http://eudml.org/doc/294474},
volume = {68},
year = {2018},
}

TY - JOUR
AU - Collegari, Rodolfo
AU - Federson, Márcia
AU - Frasson, Miguel
TI - Linear FDEs in the frame of generalized ODEs: variation-of-constants formula
JO - Czechoslovak Mathematical Journal
PY - 2018
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 68
IS - 4
SP - 889
EP - 920
AB - We present a variation-of-constants formula for functional differential equations of the form \[ \dot{y}={\mathcal {L}}(t)y_t+f(y_t,t), \quad y_{t_0}=\varphi , \] where ${\mathcal {L}}$ is a bounded linear operator and $\varphi $ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application $t\mapsto f(y_t,t)$ is Kurzweil integrable with $t$ in an interval of $\mathbb {R}$, for each regulated function $y$. This means that $t\mapsto f(y_t,t)$ may admit not only many discontinuities, but it can also be highly oscillating and yet, we are able to obtain a variation-of-constants formula. Our main goal is achieved via theory of generalized ordinary differential equations introduced by J. Kurzweil (1957). As a matter of fact, we establish a variation-of-constants formula for general linear generalized ordinary differential equations in Banach spaces where the functions involved are Kurzweil integrable. We start by establishing a relation between the solutions of the Cauchy problem for a linear generalized ODE of type \[ \frac{{\rm d}x}{{\rm d}\tau } = D[A(t)x],\quad x(t_0)=\widetilde{x} \] and the solutions of the perturbed Cauchy problem \[ \frac{{\rm d}x}{{\rm d}\tau } = D[A(t)x+F(x,t)], \quad x(t_0)=\widetilde{x}. \] Then we prove that there exists a one-to-one correspondence between a certain class of linear generalized ODE and the Cauchy problem for a linear functional differential equations of the form \[ \dot{y}={\mathcal {L}}(t)y_t, \quad y_{t_0}=\varphi , \] where $\mathcal {L}$ is a bounded linear operator and $\varphi $ is a regulated function. The main result comes as a consequence of such results. Finally, because of the extent of generalized ODEs, we are also able to describe the variation-of-constants formula for both impulsive FDEs and measure neutral FDEs.
LA - eng
KW - functional differential equation; variation-of-constants formula
UR - http://eudml.org/doc/294474
ER -