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Positive solutions of four-point singular boundary value problems. (English) Zbl 1152.34016

Summary: Existence of \(C^1\) positive solutions for a class of second-order nonlinear singular equations of the type
\[ -x''(t)+\lambda x'(t)= f(t,x(t)), \quad t\in(0,1), \]
subject to four-point boundary conditions of the type
\[ x(0)= ax(\eta), \quad x(1)= bx(\delta), \qquad 0<\eta\leq\delta<1, \]
is established. Existence of \(C^1\)-positive solution is proved by means of the upper and lower solutions method. Examples show the validity of our results. Finally, the method of quasilinearization is developed to approximate the solution. We show that under suitable conditions on f, there exists a sequence of solutions of linear problems that converges monotonically and quadratically to the solution of the original nonlinear problem.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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