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In this paper, we study a fractional differential equation $$^cD^α _{0+} u(t) + f(t, u(t)) = 0, \ t ∈ (0, +∞)$$ satisfying the boundary conditions: $$u′(0)=0,\ \lim\limits_{t→+∞} \ ^cD^{α−1}_{0+} u(t) = g(u),$$ where $1<α\leq 2,$ $^cD^α_{0+}$ is the standard Caputo fractional derivative of order $α.$ The main tools used in the paper is a contraction principle in the Banach space and the fixed point theorem due to D. O’Regan. Under a compactness criterion, the existence of solutions is established.
}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/aam/20615.html} }In this paper, we study a fractional differential equation $$^cD^α _{0+} u(t) + f(t, u(t)) = 0, \ t ∈ (0, +∞)$$ satisfying the boundary conditions: $$u′(0)=0,\ \lim\limits_{t→+∞} \ ^cD^{α−1}_{0+} u(t) = g(u),$$ where $1<α\leq 2,$ $^cD^α_{0+}$ is the standard Caputo fractional derivative of order $α.$ The main tools used in the paper is a contraction principle in the Banach space and the fixed point theorem due to D. O’Regan. Under a compactness criterion, the existence of solutions is established.