On solvability sets of boundary value problems for linear functional differential equations

Eugene Bravyi

Mathematica Bohemica (2011)

  • Volume: 136, Issue: 2, page 145-154
  • ISSN: 0862-7959

Abstract

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Consider boundary value problems for a functional differential equation x ( n ) ( t ) = ( T + x ) ( t ) - ( T - x ) ( t ) + f ( t ) , t ∈ [ a , b ] , l x = c , where T + , T - : 𝐂 [ a , b ] → 𝐋 [ a , b ] are positive linear operators; l : 𝐀𝐂 n - 1 [ a , b ] → ℝ n is a linear bounded vector-functional, f ∈ 𝐋 [ a , b ] , c ∈ ℝ n , n ≥ 2 . Let the solvability set be the set of all points ( 𝒯 + , 𝒯 - ) ∈ ℝ 2 + such that for all operators T + , T - with ∥ T ± ∥ 𝐂 → 𝐋 = 𝒯 ± the problems have a unique solution for every f and c . A method of finding the solvability sets are proposed. Some new properties of these sets are obtained in various cases. We continue the investigations of the solvability sets started in R. Hakl, A. Lomtatidze, J. Šremr: Some boundary value problems for first order scalar functional differential equations. Folia Mathematica 10, Brno, 2002.

How to cite

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Bravyi, Eugene. "On solvability sets of boundary value problems for linear functional differential equations." Mathematica Bohemica 136.2 (2011): 145-154. <http://eudml.org/doc/196949>.

@article{Bravyi2011,
abstract = {Consider boundary value problems for a functional differential equation \[\{\left\lbrace \begin\{array\}\{ll\} x^\{(n)\}(t) =(T^+x)(t)-(T^-x)(t)+f(t),&t\in [a,b],\\ l x=c, \end\{array\}\right.\} \] where $T^\{+\},T^\{-\}\colon \mathbf \{C\}[a,b]\rightarrow \mathbf \{L\}[a,b]$ are positive linear operators; $l\colon \mathbf \{AC\}^\{n-1\}[a,b]\rightarrow \mathbb \{R\}^n$ is a linear bounded vector-functional, $f\in \mathbf \{L\}[a,b]$, $c\in \mathbb \{R\}^n$, $n\ge 2$. Let the solvability set be the set of all points $(\{\mathcal \{T\}\}^+,\{\mathcal \{T\}\}^-)\in \mathbb \{R\}_2^+$ such that for all operators $T^\{+\}$, $T^\{-\}$ with $\Vert T^\{\pm \}\Vert _\{\mathbf \{C\}\rightarrow \mathbf \{L\}\}=\{\mathcal \{T\}\}^\{\pm \}$ the problems have a unique solution for every $f$ and $c$. A method of finding the solvability sets are proposed. Some new properties of these sets are obtained in various cases. We continue the investigations of the solvability sets started in R. Hakl, A. Lomtatidze, J. Šremr: Some boundary value problems for first order scalar functional differential equations. Folia Mathematica 10, Brno, 2002.},
author = {Bravyi, Eugene},
journal = {Mathematica Bohemica},
keywords = {functional differential equation; boundary value problem; periodic problem; functional differential equation; boundary value problem; periodic problem},
language = {eng},
number = {2},
pages = {145-154},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On solvability sets of boundary value problems for linear functional differential equations},
url = {http://eudml.org/doc/196949},
volume = {136},
year = {2011},
}

TY - JOUR
AU - Bravyi, Eugene
TI - On solvability sets of boundary value problems for linear functional differential equations
JO - Mathematica Bohemica
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 136
IS - 2
SP - 145
EP - 154
AB - Consider boundary value problems for a functional differential equation \[{\left\lbrace \begin{array}{ll} x^{(n)}(t) =(T^+x)(t)-(T^-x)(t)+f(t),&t\in [a,b],\\ l x=c, \end{array}\right.} \] where $T^{+},T^{-}\colon \mathbf {C}[a,b]\rightarrow \mathbf {L}[a,b]$ are positive linear operators; $l\colon \mathbf {AC}^{n-1}[a,b]\rightarrow \mathbb {R}^n$ is a linear bounded vector-functional, $f\in \mathbf {L}[a,b]$, $c\in \mathbb {R}^n$, $n\ge 2$. Let the solvability set be the set of all points $({\mathcal {T}}^+,{\mathcal {T}}^-)\in \mathbb {R}_2^+$ such that for all operators $T^{+}$, $T^{-}$ with $\Vert T^{\pm }\Vert _{\mathbf {C}\rightarrow \mathbf {L}}={\mathcal {T}}^{\pm }$ the problems have a unique solution for every $f$ and $c$. A method of finding the solvability sets are proposed. Some new properties of these sets are obtained in various cases. We continue the investigations of the solvability sets started in R. Hakl, A. Lomtatidze, J. Šremr: Some boundary value problems for first order scalar functional differential equations. Folia Mathematica 10, Brno, 2002.
LA - eng
KW - functional differential equation; boundary value problem; periodic problem; functional differential equation; boundary value problem; periodic problem
UR - http://eudml.org/doc/196949
ER -

References

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  4. Lomtatidze, A., Mukhigulashvili, S., Some two-point boundary value problems for second-order functional-differential equations, Folia Facult. Scien. Natur. Univ. Masar. Brunensis, Brno (2000). (2000) Zbl0994.34052MR1846770
  5. Hakl, R., Lomtatidze, A., Šremr, J., Some boundary value problems for first order scalar functional differential equations, Folia Facult. Scien. Natur. Univ. Masar. Brunensis, Brno (2002). (2002) Zbl1048.34004MR2088497
  6. Hakl, R., Lomtatidze, A., Půža, B., On periodic solutions of first order nonlinear functional differential equations of non-Volterra's type, Mem. Differ. Equ. Math. Phys. 24 (2001), 83-105. (2001) Zbl1011.34061MR1875885
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  8. Hakl, R., Lomtatidze, A., Půža, B., New optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations, Math. Bohem. 127 (2002), 509-524. (2002) Zbl1017.34065MR1942637
  9. Mukhigulashvili, S. V., 10.1023/B:DIEQ.0000035789.25629.86, Differ. Equ. 40 (2004), 515-523. (2004) Zbl1080.34050MR2153646DOI10.1023/B:DIEQ.0000035789.25629.86
  10. Mukhigulashvili, S., On a periodic boundary value problem for second-order linear functional differential equations, Bound. Value Probl. 3 (2005), 247-261. (2005) Zbl1106.34039MR2202215
  11. Mukhigulashvili, S., 10.1016/j.na.2005.11.046, Nonlinear Anal., Theory Methods Appl. 66 (2007), 527-535. (2007) Zbl1157.34340MR2279544DOI10.1016/j.na.2005.11.046
  12. Mukhigulashvili, S., On a periodic boundary value problem for forth order linear functional differential equations, Georgian Math. J. 14 (2007), 533-542. (2007) MR2352323
  13. Šremr, J., Solvability conditions of the Cauchy problem for two-dimensional systems of linear functional differential equations with monotone operators, Math. Bohem. 132 (2007), 263-295. (2007) Zbl1174.34049MR2355659
  14. Hakl, R., Mukhigulashvili, S., On one estimate for periodic functions, Georgian Math. J. 12 (2005), 97-114. (2005) Zbl1081.26010MR2136888
  15. Bravyi, E. I., On the solvability of the periodic boundary value problem for a linear functional differential equation, Bulletin of Udmurt Univ. Mathematics, Mechanics, Computer Science. Izhevsk 3 (2009), 12-24 Russian. (2009) MR2384909

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