Existence, uniqueness and abstract approach to hyers–ulam stability in banach lattice algebras and an application

Abstract

The abstract equation of the form u = Ku · L(Fu) is investigated in this paper. By applying a fixed point theorem for the product of operators K and A = L(F) defined on a Banach lattice algebra E, we obtain existence and uniqueness results of fixed points of the operator T = K · A. Moreover, we state a sufficient condition on the spectral radius of a majorant linear mapping of L under which the equation u = T u has the L-Hyers–Ulam stability. As an application, the obtained results are used to prove existence and uniqueness of solutions and Hyers–Ulam stability of a (p1, p2,…, pn)-Laplacian hybrid fractional differential system. An example is also constructed to illustrate the main results. This work contains many new ideas, and gives a unified approach applicable to several types of differential and integral equations.