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Fractional boundary value problems with singularities in space variables. (English) Zbl 1268.34023
Summary: We are concerned with the existence of solutions for the singular fractional boundary value problem $$^{c}D^{\alpha}u = f(t,u)$$, $$u(0)+u(1)=0$$, $$u'(0)=0$$, where $$\alpha\in (1,2)$$, $$f\in C ([0,1] \times (\mathbb{R}\setminus \{0\}))$$ and $$\lim_{x\to 0} f(t,x)= \infty$$ for all $$t\in [0,1]$$. Here, $$^{c}D$$ is the Caputo fractional derivative. Increasing solutions of the problem vanish at points of $$(0,1)$$, that is, they “pass through” the singularity of $$f$$ inside of $$(0,1)$$. The results are based on combining regularization and sequential techniques with a nonlinear alternative. In limit processes, the Vitali convergence theorem is used.

MSC:
 34A08 Fractional ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations
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