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Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 1033-1053, 2012
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On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

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Alexander Domoshnitsky, Robert Hakl, Bedřich Půža

* Alexander Domoshnitsky*, Department of Mathematics and Computer Science, The Ariel University Center of Samaria, 44837 Ariel, Israel, e-mail: ` adom@ariel.ac.il`; * Robert Hakl*, Institute of Mathematics, Academy of Sciences of the Czech Republic, Branch in Brno, Žižkova 22, 616 62 Brno, Czech Republic, e-mail: ` hakl@ipm.cz`; * Bedřich Půža*, Institute of Mathematics, Academy of Sciences of the Czech Republic, Branch in Brno, Žižkova 22, 616 62 Brno, Czech Republic, e-mail: ` puza@math.muni.cz`

**Abstract:** Consider the homogeneous equation

$$u'(t)=\ell(u)(t)\qquadfor a.e. t\in[a,b]$$

where $\ell C([a,b];\Bbb R)\to L([a,b];\Bbb R)$ 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.

**Keywords:** functional differential equation, boundary value problem, differential inequality, solution set

**Classification (MSC 2010):** 34K06, 34K10

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