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 $f$

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.