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

Abstract

Abstract 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: D α c v ( t ) + h ( t , v ( t ) ) = 0 , 0 < t < 1 , v ″ ( 0 ) = v ‴ ( 0 ) = 0 , v ′ ( 0 ) = v ( 1 ) = β ∫ 0 1 v ( s ) d s , $$ \begin{aligned} &{}^{c}D^{\alpha }v(t) + h \bigl(t,v(t) \bigr) = 0,\quad 0< t< 1, \\ &v''(0) = v'''(0) = 0, \\ &v'(0) = v(1) = \beta \int _{0}^{1} v(s) \,ds, \end{aligned} $$ where 3 < α < 4 $3<\alpha <4$ , 0 < β < 2 $0<\beta <2$ , D α c ${}^{c}D^{\alpha }$ 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.

Keywords

• Controlled b-Branciari metric type spaces
• Fixed point
• Singular fractional differential equation
• Integral boundary conditions
• Green’s function