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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))(1)

with the linear boundary conditions

x(0)=a,x(1)=x(1/2),(2)

or with the nonlinear boundary conditions

x(0)=a,x(1)=g(x(1/2))·(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 0 on [0,1]×. In the case of problem (1), (3), they additionaly assume that g ' ,g '' are continuous and 0g ' <1, g '' 0 on . 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.

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