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Zbl 1055.34033
Khan, Rahmat Ali
The generalized method of quasilinearization and nonlinear boundary value problems with integral boundary conditions. (English)
[J] Electron. J. Qual. Theory Differ. Equ. 2003, Paper No. 19, 15 p., electronic only (2003). ISSN 1417-3875/e

From the text: We study the method of quasilinearization for the nonlinear boundary value problem with integral boundary conditions $$x''(t)= f(t,x),\ t\in J=[0,1],$$ $$x(0)-k_1x'(0)= \int^1_0h_1\bigl(x(s)\bigr)ds, \ x(1)+ k_2x'(1)= \int^1_0h_2\bigl(x(s)\bigr)ds,\tag 1$$ where $f:J \times \bbfR\to\bbfR$ and $h_i:\bbfR\to \bbfR$, $i=1,2,$ are continuous functions and $k_i$ are nonnegative constants.\par We obtain a monotone sequence of iterates converging uniformly and rapidly to a solution of the second-order nonlinear boundary value problem with nonlinear integral boundary conditions.

MSC 2000:: *34B15 Nonlinear boundary value problems of ODE
34A45 Theoretical approximation of solutions of ODE

Keywords: quasilinearization; integral boundary value problem; upper and lower solutions; rapid convergence

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