Solving existence results in multi-term fractional differential equations via fixed points

Abstract

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)