It is shown that a self-adjoint block matrix multivalued linear operator (linear relation) is still self-adjoint under diagonally dominant block matrix perturbations. The results obtained generalize the corresponding one for linear operators and relax some of the required conditions. Then, we give some perturbation theorems for matrix linear relations in Banach spaces based on a resolvent approach. Fur-thermore, we apply the obtained results to investigate the existence of solutions for a system of degenerate partial differential equations.
Ammar et al. (2025) studied this question.