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 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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 $f$},
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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/

[1] Cabada A., Pouso R. L.: Existence results for the problem $(\Phi (u^{\prime }))^{\prime }=f(t,u,u^{\prime })$ with nonlinear boundary conditions. Nonlinear Anal. 35 (1999), 221–231. | MR

[2] Deimling K.: Nonlinear Functional Analysis. Springer-Verlag, Berlin Heidelberg, 1985. | MR | Zbl

[3] Kiguradze I. T.: Boundary Value Problems for Systems of Ordinary Differential Equations. In “Current Problems in Mathematics: Newest Results", Vol. 30, 3–103, Moscow 1987 (in Russian); English transl.: J. Soviet Math. 43 (1988), 2340–2417. | MR | Zbl

[4] Kiguradze I. T., Lezhava N. R.: Some nonlinear two-point boundary value problems. Differentsial’nyie Uravneniya 10 (1974), 2147–2161 (in Russian); English transl.: Differ. Equations 10 (1974), 1660–1671. | MR

[5] Kiguradze I. T., Lezhava N. R.: On a nonlinear boundary value problem. Funct. theor. Methods in Differ. Equat., Pitman Publ. London 1976, 259–276. | MR | Zbl

[6] Staněk S.: Two-point functional boundary value problems without growth restrictions. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 37 (1998), 123–142. | MR | Zbl

[7] Thompson H. B.: Second order ordinary differential equations with fully nonlinear two point boundary conditions. Pacific. J. Math. 172 (1996), 255–277. | MR | Zbl

[8] Thompson H. B.: Second order ordinary differential equations with fully nonlinear two point boundary conditions II. Pacific. J. Math. 172 (1996), 279–297. | MR | Zbl

[9] Wang M. X., Cabada A., Nieto J. J.: Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions. Ann. Polon. Math. 58 (1993), 221–235. | MR | Zbl