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On an antiperiodic type boundary value problem for first order linear functional
differential equations

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*Robert Hakl, Alexander Lomtatidze and Jiri Sremr*

**Address.** Robert Hakl, Mathematical Institute, Czech Academy of Sciences,
Zizkova 22, 616 62 Brno, CZECH REPUBLIC
Alexander Lomtatidze, Department of Mathematical Analysis, Faculty of
Science, Masaryk University, Janackovo nam. 2a, 662 95 Brno, CZECH REPUBLIC

Jiri Sremr, Department of Mathematical Analysis, Faculty of Science,
Masaryk University, Janackovo nam. 2a, 662 95 Brno, CZECH REPUBLIC

**E-mail:** hakl@ipm.cz, bacho@math.muni.cz,
sremr@math.muni.cz

**Abstract.** Nonimprovable, in a certain sense, sufficient conditions
for the unique solvability of the boundary value problem $$ u'(t)=\ell(u)(t)+q(t),\qquad
u(a)+\lambda u(b)=c $$ are established, where $\ell: C([a,b];R)\to L([a,b];
R)$ is a linear bounded operator, $q\in L([a,b];R)$, $\lambda\in R_+$,
and $c\in R$. The question on the dimension of the solution space of the
homogeneous problem $$ u'(t)=\ell(u)(t),\qquad u(a)+\lambda u(b)=0 $$ is
discussed as well.

**AMSclassification.** 34K06, 34K10.

**Keywords.** Linear functional differential equation, antiperiodic
type BVP, solvability and unique solvability.