Czechoslovak Mathematical Journal, Vol. 62, No. 4, pp. 1033-1053, 2012

On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

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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