# New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations

Mathematica Bohemica (2002)

• Volume: 127, Issue: 4, page 509-524
• ISSN: 0862-7959

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## Abstract

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The nonimprovable sufficient conditions for the unique solvability of the problem ${u}^{\text{'}}\left(t\right)=\ell \left(u\right)\left(t\right)+q\left(t\right),\phantom{\rule{2.0em}{0ex}}u\left(a\right)=c,$ where $\ell \phantom{\rule{0.222222em}{0ex}}C\left(I;ℝ\right)\to L\left(I;ℝ\right)$ is a linear bounded operator, $q\in L\left(I;ℝ\right)$, $c\in ℝ$, are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator $\ell$ is not of Volterra’s type with respect to the point $a$.

## How to cite

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Hakl, Robert, Lomtatidze, Alexander, and Půža, Bedřich. "New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations." Mathematica Bohemica 127.4 (2002): 509-524. <http://eudml.org/doc/249031>.

@article{Hakl2002,
abstract = {The nonimprovable sufficient conditions for the unique solvability of the problem $u^\{\prime \}(t)=\ell (u)(t)+q(t),\qquad u(a)=c,$ where $\ell \: C(I;\mathbb \{R\})\rightarrow L(I;\mathbb \{R\})$ is a linear bounded operator, $q\in L(I;\mathbb \{R\})$, $c\in \mathbb \{R\}$, are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator $\ell$ is not of Volterra’s type with respect to the point $a$.},
author = {Hakl, Robert, Lomtatidze, Alexander, Půža, Bedřich},
journal = {Mathematica Bohemica},
keywords = {linear functional differential equations; differential equations with deviating arguments; initial value problems; linear functional-differential equations; differential equations with deviating arguments; initial value problems},
language = {eng},
number = {4},
pages = {509-524},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations},
url = {http://eudml.org/doc/249031},
volume = {127},
year = {2002},
}

TY - JOUR
AU - Hakl, Robert
AU - Lomtatidze, Alexander
AU - Půža, Bedřich
TI - New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations
JO - Mathematica Bohemica
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 127
IS - 4
SP - 509
EP - 524
AB - The nonimprovable sufficient conditions for the unique solvability of the problem $u^{\prime }(t)=\ell (u)(t)+q(t),\qquad u(a)=c,$ where $\ell \: C(I;\mathbb {R})\rightarrow L(I;\mathbb {R})$ is a linear bounded operator, $q\in L(I;\mathbb {R})$, $c\in \mathbb {R}$, are established which are different from the previous results. More precisely, they are interesting especially in the case where the operator $\ell$ is not of Volterra’s type with respect to the point $a$.
LA - eng
KW - linear functional differential equations; differential equations with deviating arguments; initial value problems; linear functional-differential equations; differential equations with deviating arguments; initial value problems
UR - http://eudml.org/doc/249031
ER -

## References

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1. Introduction to the Theory of Functional Differential Equations, Nauka, Moskva, 1991. (Russian) (1991) MR1144998
2. Optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations, Czechoslovak Math. J (to appear). (to appear) MR1923257
3. A note on the Fredholm property of boundary value problems for linear functional differential equations, Mem. Differential Equations Math. Phys. 20 (2000), 133–135. (2000) Zbl0968.34049MR1789344
4. On multi-point boundary value problems for systems of functional differential and difference equations, Mem. Differential Equations Math. Phys. 5 (1995), 1–113. (1995) MR1415806
5. On boundary value problems for systems of linear functional differential equations, Czechoslovak Math. J. 47 (1997), 341–373. (1997) MR1452425
6. Differential and Integral Equations: Boundary Value Problems and Adjoints, Academia, Praha, 1979. (1979) MR0542283

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