# Article

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Keywords:
functional differential equation; boundary value problem; differential inequality; solution set
Summary:
Consider the homogeneous equation $$u'(t)=\ell (u)(t)\qquad \mbox {for a.e. } t\in [a,b]$$ where $\ell \colon C([a,b];\Bbb R)\to L([a,b];\Bbb R)$ is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.
References:
[1] Azbelev, N. V., Maksimov, V. P., Rakhmatullina, L. F.: Introduction to the Theory of Functional-Differential Equations. Russian. English summary Moskva: Nauka (1991). MR 1144998 | Zbl 0725.34071
[2] Bravyi, E., Hakl, R., Lomtatidze, A.: Optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations. Czech. Math. J. 52 (2002), 513-530. DOI 10.1023/A:1021767411094 | MR 1923257 | Zbl 1023.34055
[3] Domoshnitsky, A.: Maximum principles and nonoscillation intervals for first order Volterra functional differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 15 (2008), 769-813. MR 2469306
[4] Domoshnitsky, A., Maghakyan, A., Shklyar, R.: Maximum principles and boundary value problems for first-order neutral functional differential equations. J. Inequal. Appl. (2009), 26 pp. Article ID 141959 MR 2551738 | Zbl 1181.34083
[5] Hakl, R., Lomtatidze, A., Půža, B.: On nonnegative solutions of first order scalar functional differential equations. Mem. Differ. Equ. Math. Phys. 23 (2001), 51-84. MR 1873258 | Zbl 1009.34058
[6] Hakl, R., Lomtatidze, A., Stavroulakis, I. P.: On a boundary value problem for scalar linear functional differential equations. Abstr. Appl. Anal. (2004), 45-67. DOI 10.1155/S1085337504309061 | MR 2058792 | Zbl 1073.34078
[7] Hakl, R., Lomtatidze, A., Šremr, J.: Some Boundary Value Problems for First Order Scalar Functional Differential Equations. Folia Facultatis Scientiarum Naturalium Universitatis Masarykianae Brunensis. Mathematica 10. Brno: Masaryk University (2002). Zbl 1048.34004
[8] Lomtatidze, A., Opluštil, Z.: On nonnegative solutions of a certain boundary value problem for first order linear functional differential equations. Electron. J. Qual. Theory Differ. Equ. Paper No. 16, 21 pp (2003). MR 2170484 | Zbl 1072.34066
[9] Lomtatidze, A., Opluštil, Z., Šremr, J.: Nonpositive solutions to a certain functional differential inequality. Nonlinear Oscil. (N. Y.) 12 (2009), 474-509 DOI 10.1007/s11072-010-0090-4 | MR 2641285
[10] Lomtatidze, A., Opluštil, Z., Šremr, J.: On a nonlocal boundary value problem for first order linear functional differential equations. Mem. Differ. Equ. Math. Phys. 41 (2007), 69-85. MR 2391943 | Zbl 1215.34076
[11] Lomtatidze, A., Opluštil, Z., Šremr, J.: Solvability conditions for a nonlocal boundary value problem for linear functional differential equations. Fasc. Math. 41 (2009), 81-96. MR 2529035 | Zbl 1207.34079
[12] Opluštil, Z.: New solvability conditions for a non-local boundary value problem for nonlinear functional differential equations. Nonlin. Osc. 11 (2008), 384-406 DOI 10.1007/s11072-009-0038-8 | MR 2512754
[13] Opluštil, Z., Šremr, J.: On a non-local boundary value problem for linear functional differential equations. Electron. J. Qual. Theory Differ. Equ. (2009), Paper No. 36, 13 pp. MR 2511289 | Zbl 1183.34105
[14] Rontó, A., Pylypenko, V., Dilna, D.: On the unique solvability of a non-local boundary value problem for linear functional differential equations. Math. Model. Anal. 13 (2008), 241-250. DOI 10.3846/1392-6292.2008.13.241-250 | MR 2418224 | Zbl 1162.34053

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