We consider the existence of solutions for the following Hadamard-type fractional differential equations:

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 \(\beta _{2}<\alpha -1\)).

我们考虑以下Hadamard型分数阶微分方程的解的存在：

其中\（2 <\ alpha \ leq 3 \），\（0 <\ beta _ {1} \ leq \ alpha -2 <\ beta _ {2} \ leq \ alpha -1 \），\（f：J \ times \ mathbb {R} ^ {3} \ rightarrow \ mathbb {R} \）满足q-Carathéodory条件，\（q，g_ {1}，g_ {2}：J \ rightarrow \ mathbb {R} ^ {+} \）是非负数，其中\（J = [1，+ \ infty）\）。非线性项f取决于低阶\（\ beta _ {1} \），\（\ beta _ {2} \）的分数导数，这会增加额外的复杂度以验证解的存在。在我们的问题中发生的奇点与左端点\（t = 1 \ ）上C（1，+ \ infty）\）中的\（{} ^ {H} D ^ {\ beta _ {2}} u \ ）（如果\（\ beta _ {2} <\ alpha -1 \））。