Extended existence results for FDEs with nonlocal conditions

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point $0\le c<d$ and fixed points at the end of the interval $[0,d]$. For $i=0,1$, the boundary conditions are as follows: $a_i,b_i>0$, $a_{0}p(c)=-b_{0}p(d),a_1{p}^{{}^{\prime }}(c)=-b_1{p}^{{}^{\prime }}(d)$. Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixed-point theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved.