In this article, we first demonstrate a fixed point result under certain contraction in the setting of controlled b-Branciari metric type spaces.

• Thereafter, we specifically consider a following boundary value problem (BVP) for a singular fractional differential equation of order α:

  $$ \begin{aligned} &{}^{c}D^{\alpha }v(t) + h \bigl(t,v(t) \bigr) = 0,\quad 0< t< 1, \\ &v''(0) = v'''(0) = 0, \\ &v'(0) = v(1) = \beta \int _{0}^{1} v(s) \,ds, \end{aligned} $$

  where \(3<\alpha <4\), \(0<\beta <2\), \({}^{c}D^{\alpha }\) is the Caputo fractional derivative and h may be singular at \(v = 0\).

• Eventually, we investigate the existence and uniqueness of solutions of the aforementioned boundary value problem of order α via a fixed point problem of an integral operator.

在本文中，我们首先证明受控b -Branciari度量类型空间的设置在一定收缩下的不动点结果。

• 此后，我们具体考虑阶数为α的奇异分数阶微分方程的以下边值问题（BVP）  ：

  $$ \ begin {aligned}＆{} ^ {c} D ^ {\ alpha} v（t）+ h \ bigl（t，v（t）\ bigr）= 0，\ quad 0 <t <1，\ \＆v''（0）= v'''（0）= 0，\\＆v'（0）= v（1）= \ beta \ int _ {0} ^ {1} v（s）\，ds ，\ end {aligned} $$

  其中\（3 <\ alpha <4 \），\（0 <\ beta <2 \），\（{} ^ {c} D ^ {\ alpha} \）是Caputo分数导数，并且h在\（v = 0 \）。

• 最后，我们通过积分算子的不动点问题，研究了上述α阶边值问题解的存在性和唯一性。