Solvability of a periodic type boundary value problem

for first order scalar functional differential equations

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.