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Solvability of a periodic type boundary value problem

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for first order scalar functional differential equations

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*Robert Hakl, Alexander Lomtatidze and Jiri Sremr*

**Address.**

R. Hakl

Mathematical Institute, Czech Academy of Sciences

Zizkova 22, 616 62 Brno, Czech Republic

A. Lomtatidze

Department of Mathematical Analysis, Faculty of Science,

Masaryk University

Janackovo nam. 2a, 662 95 Brno, Czech Republic

J. Sremr

Mathematical Institute, Czech Academy of Sciences

Zizkova 22, 616 62 Brno, Czech Republic

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

**Abstract.**

Nonimprovable sufficient conditions

for the solvability and unique solvability

of the problem

$$

u'(t)=F(u)(t)\,,\qquad u(a)-\lambda u(b)=h(u)

$$

are established,

where $F:C([a,b]; R)\to L([a,b]; R)$ is

a continuous operator satisfying the Carath\`eodory conditions,

$h:C([a,b]; R)\to R$ is a continuous functional, and $\lambda\in R_+$.

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

**Keywords. **Functional differential equation, periodic type

boundary value problem, solvability, unique solvability.