  In this work, we investigate a static and spherically symmetric
Bardeen-Kiselev black hole with cosmological constant which is a solution of
the Einstein-non-linear Maxwell field equations along with a quintessential
field. We compute the quasinormal frequencies for Bardeen-Kiselev black
hole(BH) with cosmological constant due to electromagnetic and gravitational
perturbations. By varying the BH parameters, we discuss the behaviour of both
real and imaginary parts of the BH quasinormal frequencies and compare
frequencies with Reissner-Nordstr\"om-de Sitter BH surrounded by quintessence
(RN-dSQ). Interestingly, it shows that the response of Bardeen-Kiselev BH with
cosmological constant and RN-dSQ under electromagnetic perturbations are
different when the charge parameter $q$, the state parameter $w$ and the
normalization factor $c$ are varied, but for the gravitational perturbations,
the response of Bardeen-Kiselev BH with cosmological constant and RN-dSQ are
different only when the charge parameter $q$ is varied. Therefore, compared
with the gravitational perturbations, the electromagnetic perturbations can be
used to understand nonlinear and linear electromagnetic fields in curved
spacetime separately. Another interesting observation is that due to the
presence of quintessence, the electromagnetic perturbations around the
Bardeen-Kiselev BH with cosmological constant damps faster and oscillates
slowly, and for the gravitational perturbations, the quasinormal mode decays
slowly and oscillates slowly. We also study the reflection and transmission
coefficients along with absorption cross section for the Bardeen-Kiselev BH
with cosmological constant, it shows that the transmission coefficients will
increase due to the presence of quintessence.
