@article{Kumar2023, 
author = {M. Sathish Kumar and M. Deepa and J Kavitha and V. Sadhasivam},
title = {Existence theory of fractional order three-dimensional differential system at resonance},
year = {2023},
journal = {Mathematical Modelling and Control},
volume = {3},
number = {2},
pages = {127-138},
keywords = {resonance, fractional differential equation, coincidence degree theory},
url = {https://www.sciopen.com/article/10.3934/mmc.2023012},
doi = {10.3934/mmc.2023012},
abstract = {This paper deals with three-dimensional differential system of nonlinear fractional order problem                                 D                                    0                              +                                                          α                          υ        (        ϱ        )        =        f        (        ϱ        ,        ω        (        ϱ        )        ,                  ω                      ′                          (        ϱ        )        ,                  ω                      ′            ′                          (        ϱ        )        ,        .        .        .        ,                  ω                      (            n            −            1            )                          (        ϱ        )        )        ,                ϱ        ∈        (        0        ,        1        )        ,                                      D                                    0                              +                                                          β                          ν        (        ϱ        )        =        g        (        ϱ        ,        υ        (        ϱ        )        ,                  υ                      ′                          (        ϱ        )        ,                  υ                      ′            ′                          (        ϱ        )        ,        .        .        .        ,                  υ                      (            n            −            1            )                          (        ϱ        )        )        ,                ϱ        ∈        (        0        ,        1        )        ,                                      D                                    0                              +                                                          γ                          ω        (        ϱ        )        =        h        (        ϱ        ,        ν        (        ϱ        )        ,                  ν                      ′                          (        ϱ        )        ,                  ν                      ′            ′                          (        ϱ        )        ,        .        .        .        ,                  ν                      (            n            −            1            )                          (        ϱ        )        )        ,                ϱ        ∈        (        0        ,        1        )        ,            with the boundary conditions,                       υ        (        0        )        =                  υ                      ′                          (        0        )        =        .        .        .        =                  υ                      (            n            −            2            )                          (        0        )        =        0        ,                          υ                      (            n            −            1            )                          (        0        )        =                  υ                      (            n            −            1            )                          (        1        )        ,                            ν        (        0        )        =                  ν                      ′                          (        0        )        =        .        .        .        =                  ν                      (            n            −            2            )                          (        0        )        =        0        ,                          ν                      (            n            −            1            )                          (        0        )        =                  ν                      (            n            −            1            )                          (        1        )        ,                            ω        (        0        )        =                  ω                      ′                          (        0        )        =        .        .        .        =                  ω                      (            n            −            2            )                          (        0        )        =        0        ,                          ω                      (            n            −            1            )                          (        0        )        =                  ω                      (            n            −            1            )                          (        1        )        ,            where        D                  0                  +                            α        ,      D                  0                  +                            β        ,      D                  0                  +                            γ       are the standard Caputo fractional derivative,    n  −  1  <  α  ,  β  ,  γ  ≤  n  ,    n  ≥  2 and we derive sufficient conditions for the existence of solutions to the fraction order three-dimensional differential system with boundary value problems via Mawhin's coincidence degree theory, and some new existence results are obtained. Finally, an illustrative example is presented.}
}