Geodesic
On solvability sets of boundary value problems for linear functional differential equations
Mathematica Bohemica, Tome 136 (2011) no. 2, pp. 145-154

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Consider boundary value problems for a functional differential equation $$\begin {cases} x^{(n)}(t) =(T^+x)(t)-(T^-x)(t)+f(t),\in [a,b],\\ l x=c, \end {cases} $$ where $T^{+},T^{-}\colon \bold C[a,b]\to \bold L[a,b]$ are positive linear operators; $l\colon \bold {AC}^{n-1}[a,b]\to \mathbb {R}^n$ is a linear bounded vector-functional, $f\in \bold L[a,b]$, $c\in \mathbb {R}^n$, $n\ge 2$. \endgraf 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 $\|T^{\pm }\|_{\bold C\to \bold L}={\mathcal T}^{\pm }$ 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.
Consider boundary value problems for a functional differential equation $$\begin {cases} x^{(n)}(t) =(T^+x)(t)-(T^-x)(t)+f(t),\in [a,b],\\ l x=c, \end {cases} $$ where $T^{+},T^{-}\colon \bold C[a,b]\to \bold L[a,b]$ are positive linear operators; $l\colon \bold {AC}^{n-1}[a,b]\to \mathbb {R}^n$ is a linear bounded vector-functional, $f\in \bold L[a,b]$, $c\in \mathbb {R}^n$, $n\ge 2$. \endgraf 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 $\|T^{\pm }\|_{\bold C\to \bold L}={\mathcal T}^{\pm }$ 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.
DOI : 10.21136/MB.2011.141577
Classification : 34K06, 34K10, 34K13
Keywords: functional differential equation; boundary value problem; periodic problem 
Bravyi, Eugene. On solvability sets of boundary value problems for linear functional differential equations. Mathematica Bohemica, Tome 136 (2011) no. 2, pp. 145-154. doi: 10.21136/MB.2011.141577
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