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The method of quasilinearization and a three-point boundary value problem. (English) Zbl 1012.34014
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.

34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
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