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nonlinear boundary value problem; existence; lower and upper functions; $\alpha $-condensing operator; Borsuk antipodal theorem; Leray-Schauder degree; homotopy
We consider boundary value problems for second order differential equations of the form $(x^{\prime }+g(t,x,x^{\prime }))^{\prime }=f(t,x,x^{\prime })$ with the boundary conditions $r(x(0),x^{\prime }(0),x(T)) + \varphi (x)=0$, $w(x(0),x(T),x^{\prime }(T))+ \psi (x)=0$, where $g,r,w$ are continuous functions, $f$ satisfies the local Carathéodory 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.
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