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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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