Abstract
We are concerned with the existence of solutions for the singular fractional boundary value problem \(^{c}\kern-1pt D^{\alpha}u = f(t,u)\), u(0)+u(1)=0, u′(0)=0, where α∈(1,2), f∈C([0,1]×(ℝ∖{0})) and lim x→0 f(t,x)=∞ for all t∈[0,1]. Here, \(^{c}\kern-1pt D\) 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.
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The research of the second author was supported by grants PrF-2011-022 and PrF-2012-017.
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O’Regan, D., Staněk, S. Fractional boundary value problems with singularities in space variables. Nonlinear Dyn 71, 641–652 (2013). https://doi.org/10.1007/s11071-012-0443-x
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DOI: https://doi.org/10.1007/s11071-012-0443-x