Existence and uniqueness of nonlocal boundary conditions for Hilfer–Hadamard-type fractional differential equations

Abstract

Abstract In this paper, we use some fixed point theorems in Banach space for studying the existence and uniqueness results for Hilfer–Hadamard-type fractional differential equations D α , β H x ( t ) + f ( t , x ( t ) ) = 0 $$ {}_{\mathrm{H}}D^{\alpha ,\beta }x(t)+f\bigl(t,x(t)\bigr)=0 $$ on the interval ( 1 , e ] $(1,e]$ with nonlinear boundary conditions x ( 1 + ϵ ) = ∑ i = 1 n − 2 ν i x ( ζ i ) , H D 1 , 1 x ( e ) = ∑ i = 1 n − 2 σ i H D 1 , 1 x ( ζ i ) . $$ x(1+\epsilon )=\sum_{i=1}^{n-2}\nu _{i}x(\zeta _{i}),\qquad {}_{\mathrm{H}}D^{1,1}x(e)= \sum_{i=1}^{n-2} \sigma _{i}\, {}_{\mathrm{H}}D^{1,1}x( \zeta _{i}). $$

Keywords

• Existence
• Uniqueness
• Nonlinear boundary value problems
• Hilfer–Hadamard type
• Fractional differential equation and fractional calculus