| Title:
|
Linear FDEs in the frame of generalized ODEs: variation-of-constants formula (English) |
| Author:
|
Collegari, Rodolfo |
| Author:
|
Federson, Márcia |
| Author:
|
Frasson, Miguel |
| Language:
|
English |
| Journal:
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Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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68 |
| Issue:
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4 |
| Year:
|
2018 |
| Pages:
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889-920 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
functional differential equation |
| Keyword:
|
variation-of-constants formula |
| MSC:
|
34K06 |
| MSC:
|
34K40 |
| idZBL:
|
Zbl 07031687 |
| idMR:
|
MR3881886 |
| DOI:
|
10.21136/CMJ.2018.0023-17 |
| . |
| Date available:
|
2018-12-07T06:16:51Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147511 |
| . |
| Reference:
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