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Zbl 1012.34014
Eloe, Paul W.; Gao, Yang
The method of quasilinearization and a three-point boundary value problem.
(English)
[J] J. Korean Math. Soc. 39, No.2, 319-330 (2002). ISSN 0304-9914

The authors apply the method of quasilinearization to the differential equation $$x''(t)=f(t,x(t)) \tag 1$$ with the linear boundary conditions $$x(0)=a,\quad x(1)=x(1/2), \tag 2$$ or with the nonlinear boundary conditions $$x(0)=a,\quad x(1)=g(x(1/2)). \tag 3$$ Here, $f$ and $g$ are supposed to be continuous functions. The authors assume that there are lower and upper solutions to problem (1), (2) or (1), (3) and that $f_x, f_{xx}$ are continuous and $f_x>0$, $f_{xx}\ge 0$ on $[0,1]\times \bbfR$. In the case of problem (1), (3), they additionaly assume that $g', g''$ are continuous and $0\le g' <1$, $g''\le 0$ on $\bbfR$. Then they prove the existence of a monotone sequence of lower solutions and of a monotone sequence of upper solutions to problem (1), (2) or (1), (3). Both sequences converge to the unique solution to the problem under consideration.
[Irena Rachuunková (Olomouc)]
MSC 2000:
*34B10 Multipoint boundary value problems
34B15 Nonlinear boundary value problems of ODE

Keywords: quasilinearization; quadratic sequence; boundary value problem; nonlinear boundary conditions

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