In this paper, we are interested in the existence and Hyers-Ulam stability for the abstract equation in Banach lattices and application. In the application, we transform the given fractional differential system of (p₁,p₂,...,pn)-Laplacian Hilfer equations into an equivalent integral equation. Then we establish sufficient conditions and employ the fixed point index arguments and spectral theory to obtain new results on the existence and Hyers-Uulam stability. Examples illustrating the main theoritical results are also constructed. This work contains several new ideas, and gives a unified approach applicable to many types of initial value systems involving (p₁,p₂,...,pn)-Laplacian type operators