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On solvability of nonlinear boundary value problems for the equation $(x'+g(t,x,x'))'=f(t,x,x')$
with one-sided growth restrictions on $f$

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*Svatoslav Stanek*

**Address.** Department of Mathematical Analysis, Faculty of Science,
Palacky University, Tomkova 40, 779 00 Olomouc, CZECH REPUBLIC
**E-mail:** stanek@risc.upol.cz

**Abstract.** We consider boundary value problems for second order
differential equations of the form $(x'+g(t,x,x'))'=f(t,x,x')$ with the
boundary conditions $r(x(0),x'(0),x(T)) + \varphi(x)=0$, $w(x(0),x(T),x'(T))+
\psi(x)=0$, where $g,r,w$ are continuous functions, $f$ satisfies the local
Carath\'eodory conditions and $\varphi, \psi$ are continuous and nondecreasing
functionals. Existence results are proved by the method of lower and upper
functions and applying the degree theory for $\alpha$-condensing operators.

**AMSclassification.** 34B15.

**Keywords.** Nonlinear boundary value problem, existence, lower
and upper functions, $\alpha$-condensing operator, Borsuk antipodal theorem,
Leray-Schauder degree, homotopy.