Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition - Imed Bachar; Habib Mâagli; Hassan Eltayeb (Journal of Function Spaces)
This paper deals with the following boundary value problem Dαut=ft,ut,t∈0,1,u0=u1=Dα−3u0=u′1=0, where 3<α≤4,Dα is the Riemann-Liouville fractional derivative, and the nonlinearity f, which could be singular at both t=0 and t=1, is required to be continuous on 0,1×ℝ satisfying a mild Lipschitz assumption. Based on the Banach fixed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution u satisfying ut≤cωt,for t∈0,1 and c>0, where ωt≔tα−21−t2.