The study of a separated boundary value problem for a fractional equation involving a derivative of lower order at the Schauder fixed point

Abstract

In this study, we explore a new category of separated boundary value problems for non-linear fractional differential equations where the non-linear term f relies on a lower-order fractional derivative given by: cD0+ αx(t)=f(t,x(t),cD0+ βx(t)) , tϵ[0,T],1<α≤2 ,0<β≤1
subject to separated fractional boundary conditions :
a1x(0)+b1(cD0+ γx(0))=c1,a2x(T)+b2(cD0+ γx(T))=c2 ,0<γ<1
Here cD0+ α, is the Caputo fractional derivative, f: [0,T] ×R2→ Ris a continuous function, and ai, bi, ci ,i = 1, 2 are real constants satisfying: a1≠ 0 and T > 0.
The aim of this work is to study the existence and uniqueness of solutions to the separated fractional boundary value problem using the Schauder fixed point theorem, which is a topological theorem that asserts a relatively compact map has a fixed point that is not necessarily unique. Therefore, it is not necessary to establish estimates on the function, but rather on its compactness.
To conclude, an example is presented to illustrate the application of this theorem.