Geodesic
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 $f$
Archivum mathematicum, Tome 38 (2002) no. 2, pp. 129-148

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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.
Classification : 34B15, 47N20
Keywords: nonlinear boundary value problem; existence; lower and upper functions; $\alpha $-condensing operator; Borsuk antipodal theorem; Leray-Schauder degree; homotopy 
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     author = {Stan\v{e}k, Svatoslav},
     title = {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 $f$},
     journal = {Archivum mathematicum},
     pages = {129--148},
     publisher = {mathdoc},
     volume = {38},
     number = {2},
     year = {2002},
     mrnumber = {1909594},
     zbl = {1087.34007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2002__38_2_a4/}
}
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Staněk, Svatoslav. 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 $f$. Archivum mathematicum, Tome 38 (2002) no. 2, pp. 129-148. http://geodesic.mathdoc.fr/item/ARM_2002__38_2_a4/