A fixed point approach to the solution of singular fractional differential equations with integral boundary conditions

Abstract

AbstractIn this article, we first demonstrate a fixed point result under certain contraction in the setting of controlledb-Branciari metric type spaces.Thereafter, we specifically consider a following boundary value problem (BVP) for a singular fractional differential equation of order α:$$ \begin{aligned} &{}^{c}D^{\alpha }v(t) + h \bigl(t,v(t) \bigr) = 0,\quad 0< t< 1, \\ &v''(0) = v'''(0) = 0, \\ &v'(0) = v(1) = \beta \int _{0}^{1} v(s) \,ds, \end{aligned} $$Dαcv(t)+h(t,v(t))=0,0<t<1,v″(0)=v‴(0)=0,v′(0)=v(1)=β∫01v(s)ds,where$3<\alpha <4$3<α<4,$0<\beta <2$0<β<2,${}^{c}D^{\alpha }$Dαcis the Caputo fractional derivative andhmay be singular at$v = 0$v=0.Eventually, we investigate the existence and uniqueness of solutions of the aforementioned boundary value problem of orderαvia a fixed point problem of an integral operator.