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We consider the existence of solutions for the following Hadamard-type fractional differential equations: $$ \textstyle\begin{cases} {}^{H}D^{\alpha }u(t)+q(t)f(t,u(t), {}^{H}D^{\beta _{1}}u(t),{}^{H}D^{ \beta _{2}}u(t))=0,\quad 1< t< +\infty , \\ u(1)=0, \\ {}^{H}D^{\alpha -2}u(1)=\int ^{+\infty }_{1}g_{1}(s)u(s)\frac{ds}{s}, \\ {}^{H}D^{\alpha -1}u(+\infty )=\int ^{+\infty }_{1}g_{2}(s)u(s) \frac{ds}{s}, \end{cases} $$where $2<\alpha \leq 3$, $0<\beta _{1}\leq \alpha -2<\beta _{2}\leq \alpha -1$, $f:J \times \mathbb{R}^{3}\rightarrow \mathbb{R}$satisfies the q-Carathéodory condition, $q,g_{1},g_{2}:J\rightarrow \mathbb{R}^{+}$are nonnegative, where $J=[1,+\infty )$. Nonlinear term f is dependent on the fractional derivative of lower order $\beta _{1}$, $\beta _{2}$, which creates additional complexity to verify the existence of solutions. The singularity occurring in our problem is associated with ${}^{H}D^{\beta _{2}}u\in C(1,+\infty )$at the left endpoint $t=1$(if $\beta _{2}<\alpha -1$).

Existence of solutions for integral boundary value problems of singular Hadamard-type fractional differential equations on infinite interval