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