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Singular nonlinear multipoint conjugate boundary value problems. (English) Zbl 0903.34016
The authors prove the existence of solutions for the multipoint conjugate boundary value problem $$y^{(n)} = f(x,y), \quad x \in [0,1]\setminus \{a_1,a_2,\dots,a_k\},\tag 1$$ $$y^{(j)} (a_i) = 0, \quad 0 \leq j \leq n_i-1,\ 1 \leq i \leq k,\tag 2$$ where $0 = a_1 < a_2 < \dots < a_k = 1$ are fixed and $f(x,y)$ has a singularity at $y=0$. The approach of the authors is to construct a sequence of perturbations of $f$ which terms lack the singularity of $f$ and to apply a fixed point theorem to the respective boundary value problems. It is shown that the obtained sequence of iterates converges to a solution to (1), (2).

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