We consider the existence of nontrivial solutions of the boundary-value problems for nonlinear fractional differential equations
where λ > 0 is a parameter, 1 < α ≤ 2, η ∈ (0, 1), \(\beta \in \mathbb{R} = \left({-\infty, +\infty} \right) \), βη α−1 ≠ 1, D α is a Riemann–Liouville differential operator of order α, \(f:\left(0, 1 \right) \times \mathbb{R} \to \mathbb{R} \) is continuous, f may be singular for t = 0 and/or t = 1, and q(t) : [0, 1] → [0, +∞) We give some sufficient conditions for the existence of nontrivial solutions to the formulated boundary-value problems. Our approach is based on the Leray–Schauder nonlinear alternative. In particular, we do not use the assumption of nonnegativity and monotonicity of f essential for the technique used in almost all available literature.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 9, pp. 1211–1219, September, 2010.
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Guo, Y. Solvability of boundary-value problems for nonlinear fractional differential equations. Ukr Math J 62, 1409–1419 (2011). https://doi.org/10.1007/s11253-011-0439-6
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DOI: https://doi.org/10.1007/s11253-011-0439-6