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

Kalaivani Chandran, Kalpana Gopalan, Sumaiya Tasneem Zubair, Thabet Abdeljawad
2021 Advances in Difference Equations  
AbstractIn 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: $$ \begin{aligned} &{}^{c}D^{\alpha }v(t) + h \bigl(t,v(t) \bigr) = 0,\quad 0< t< 1, \\ &v"(0) = v"'(0) = 0, \\ &v'(0) = v(1) = \beta \int _{0}^{1} v(s) \,ds, \end{aligned} $$ D α c v ( t ) + h ( t , v ( t )
more » ... ) = 0 , 0 < t < 1 , v ″ ( 0 ) = v ‴ ( 0 ) = 0 , v ′ ( 0 ) = v ( 1 ) = β ∫ 0 1 v ( s ) d s , where $3<\alpha <4$ 3 < α < 4 , $0<\beta <2$ 0 < β < 2 , ${}^{c}D^{\alpha }$ D α c is the Caputo fractional derivative and h may be singular at $v = 0$ 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.
doi:10.1186/s13662-021-03225-y fatcat:mywfnipqz5g65eo3xlrr7nsnza