We consider the existence of solutions for the following Hadamard-type fractional differential equations: {DαHu(t)+q(t)f(t,u(t),HDβ1u(t),HDβ2u(t))=0,1<t<+∞,u(1)=0,Dα−2Hu(1)=∫1+∞g1(s)u(s)dss,Dα−1Hu(+∞)=∫1+∞g2(s)u(s)dss,\documentclass[12pt]{minimal}
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\begin{document}$$ \textstyle\begin{cases} {}^{H}D^{\alpha }u(t)+q(t)f(t,u(t), {}^{H}D^{\beta _{1}}u(t),{}^{H}D^{ \beta _{2}}u(t))=0,\quad 1< t< +\infty , \\ u(1)=0, \\ {}^{H}D^{\alpha -2}u(1)=\int ^{+\infty }_{1}g_{1}(s)u(s)\frac{ds}{s}, \\ {}^{H}D^{\alpha -1}u(+\infty )=\int ^{+\infty }_{1}g_{2}(s)u(s) \frac{ds}{s}, \end{cases} $$\end{document} where 2<α≤3\documentclass[12pt]{minimal}
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\begin{document}$2<\alpha \leq 3$\end{document}, 0<β1≤α−2<β2≤α−1\documentclass[12pt]{minimal}
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\begin{document}$0<\beta _{1}\leq \alpha -2<\beta _{2}\leq \alpha -1$\end{document}, f:J×R3→R\documentclass[12pt]{minimal}
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\begin{document}$f:J \times \mathbb{R}^{3}\rightarrow \mathbb{R}$\end{document} satisfies the q-Carathéodory condition, q,g1,g2:J→R+\documentclass[12pt]{minimal}
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\begin{document}$q,g_{1},g_{2}:J\rightarrow \mathbb{R}^{+}$\end{document} are nonnegative, where J=[1,+∞)\documentclass[12pt]{minimal}
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\begin{document}$J=[1,+\infty )$\end{document}. Nonlinear term f is dependent on the fractional derivative of lower order β1\documentclass[12pt]{minimal}
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\begin{document}$\beta _{1}$\end{document}, β2\documentclass[12pt]{minimal}
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\begin{document}$\beta _{2}$\end{document}, which creates additional complexity to verify the existence of solutions. The singularity occurring in our problem is associated with Dβ2Hu∈C(1,+∞)\documentclass[12pt]{minimal}
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\begin{document}${}^{H}D^{\beta _{2}}u\in C(1,+\infty )$\end{document} at the left endpoint t=1\documentclass[12pt]{minimal}
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\begin{document}$t=1$\end{document} (if β2<α−1\documentclass[12pt]{minimal}
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\begin{document}$\beta _{2}<\alpha -1$\end{document}).
