Previous |  Up |  Next

Article

Keywords:
functional differential equation; boundary value problem; periodic problem
Summary:
Consider boundary value problems for a functional differential equation $$\begin {cases} x^{(n)}(t) =(T^+x)(t)-(T^-x)(t)+f(t),&t\in [a,b],\\ l x=c, \end {cases} $$ 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.
References:
[1] Azbelev, N. V., Maksimov, V. P., Rahmatullina, L. F.: Introduction to the Theory of Functional Differential Equations. Nauka, Moskva (1991), Russian. MR 1144998
[2] Lomtatidze, A., Mukhigulashvili, S.: On periodic solutions of second order functional differential equations. Mem. Differ. Equ. Math. Phys. 5 (1995), 125-126. Zbl 0866.34054
[3] Lomtatidze, A., Mukhigulashvili, S.: On a two-point boundary value problem for second order functional differential equations. II. Mem. Differ. Equ. Math. Phys. 10 (1997), 150-152. Zbl 0939.34511
[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). MR 1846770 | Zbl 0994.34052
[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). MR 2088497 | Zbl 1048.34004
[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. MR 1875885 | Zbl 1011.34061
[7] Lasota, A., Opial, Z.: Sur les solutions periodiques des equations differentielles ordinaires. Ann. Pol. Math. 16 (1964), 69-94. MR 0170072 | Zbl 0142.35303
[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. MR 1942637 | Zbl 1017.34065
[9] Mukhigulashvili, S. V.: On the unique solvability of the Dirichlet problem for a second-order linear functional-differential equation. Differ. Equ. 40 (2004), 515-523. DOI 10.1023/B:DIEQ.0000035789.25629.86 | MR 2153646 | Zbl 1080.34050
[10] Mukhigulashvili, S.: On a periodic boundary value problem for second-order linear functional differential equations. Bound. Value Probl. 3 (2005), 247-261. MR 2202215 | Zbl 1106.34039
[11] Mukhigulashvili, S.: On a periodic boundary value problem for third order linear functional differential equations. Nonlinear Anal., Theory Methods Appl. 66 (2007), 527-535. DOI 10.1016/j.na.2005.11.046 | MR 2279544 | Zbl 1157.34340
[12] Mukhigulashvili, S.: On a periodic boundary value problem for forth order linear functional differential equations. Georgian Math. J. 14 (2007), 533-542. MR 2352323
[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. MR 2355659 | Zbl 1174.34049
[14] Hakl, R., Mukhigulashvili, S.: On one estimate for periodic functions. Georgian Math. J. 12 (2005), 97-114. MR 2136888 | Zbl 1081.26010
[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.
Partner of
EuDML logo