Quasilinearization method and nonlocal singular three point boundary value problems

R. A. Khan, Centre for Advanced Mathematics and Physics, NUST, Rawalpindi, Pakistan

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 17., pp. 1-13.

Abstract: The method of upper and lower solutions and quasilinearization for nonlinear singular equations of the type
$$-x''(t) +\lambda x'(t)= f(t,x(t)),\,t\in (0,1),$$ subject to nonlocal three-point boundary conditions $$x(0)=\delta x(\eta),\quad x(1)=0, \quad 0<\eta < 1,$$ are developed. Existence of a $C^{1}$ positive solution is established. A monotone sequence of solutions of linear problems converging uniformly and rapidly to a solution of the nonlinear problem is obtained.

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