@article{Ghasemi2022, 
author = {Majid Ghasemi and Mahnaz Khanehgir and Reza Allahyari and Hojjatollah Amiri Kayvanloo},
title = {Positive solutions of infinite coupled system of fractional differential equations in the sequence space of weighted means},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {2},
pages = {2680-2694},
keywords = {Darbo's theorem, difference sequence space of weighted means, infinite system of boundary value problems, measure of noncompactness},
url = {https://www.sciopen.com/article/10.3934/math.2022151},
doi = {10.3934/math.2022151},
abstract = {We first discuss the existence of solutions of the infinite system of    (  n  −  1  ,  n  )-type semipositone boundary value problems (BVPs) of nonlinear fractional differential equations         {                                        D                                          0                +                                                    α                                            u            i                    (          ρ          )          +          η                      f            i                    (          ρ          ,          v          (          ρ          )          )          =          0          ,                          ρ          ∈          (          0          ,          1          )          ,                                                  D                                          0                +                                                    α                                            v            i                    (          ρ          )          +          η                      g            i                    (          ρ          ,          u          (          ρ          )          )          =          0          ,                          ρ          ∈          (          0          ,          1          )          ,                                                  u            i                          (              j              )                                (          0          )          =                      v                          i                                      (              j              )                                (          0          )          =          0          ,                          0          ≤          j          ≤          n          −          2          ,                                                  u                          i                                (          1          )          =          ζ                      ∫            0            1                                u            i                    (          ϑ          )          d          ϑ          ,                                v                          i                                (          1          )          =          ζ                      ∫            0            1                                v            i                    (          ϑ          )          d          ϑ          ,                          i          ∈                      N                    ,                        in the sequence space of weighted means        c    0    (      W    1    ,      W    2    ,  Δ  ), where    n  ≥  3,    α  ∈  (  n  −  1  ,  n  ],    η  ,  ζ are real numbers,    0  <  η  <  α  ,        D                  0        +                    α       is the Riemann-Liouville's fractional derivative, and        f    i    ,      g    i    ,    i  =  1  ,  2  ,  …, are semipositone and continuous. Our approach to the study of solvability is to use the technique of measure of noncompactness. Then, we find an interval of    η such that for each    η lying in this interval, the system of    (  n  −  1  ,  n  )-type semipositone BVPs has a positive solution. Eventually, we demonstrate an example to show the effectiveness and usefulness of the obtained result.}
}