| Summary: | Many researchers are interested in the existence theory of solutions to fractional differential equations. In the literature, existence results have been obtained by using a variety of fixed point problems, including the fixed-point problems of Lefschetz, Kleene, Tychonoff, and Banach. In this article, we propose a generalized version of the contraction principle in the context of controlled rectangular metric space. With this result, we address the existence and uniqueness results for the following fractional-order differential equations.1. The nonlinear multi-term fractional delay differential equation L(D)ζ(ϖ)=σ(ϖ,ζ(ϖ),ζ(ϖ−τ)),ϖ∈J=[0,T],T>0;ζ(ϖ)=σ̄(ϖ),ϖ∈[−τ,0]. where, L(D)=γwcDδw+γw−1cDδw−1+...+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3....w),γw≠0,0≤δ0<δ1<δ2....δw−1<δw<1, and cDδ denotes the Caputo fractional derivative of order δ.2. The Caputo type fractional differential equation cDδζ(ϖ)=σ(ϖ,ζ(ϖ)),5<δ≤6,ϖ∈[0,T], with ζ(0)=−ζ(T),ζ′(0)=−ζ′(T),ζ′′(0)=−ζ′′(T),ζ′′′(0)=−ζ′′′(T),ζ4(0)=−ζ4(T),ζ5(0)=−ζ5(T)
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