Fixed points and approximately heptic mappings in non-Archimedean normed spaces

The stability problems concerning group homomorphisms was raised by Ulam [?] in ????and affirmatively answered for Banach spaces by Hyers [?] in the next year. Hyers’ theoremwas generalized by Aoki [?] for additive mappings and by Rassias [?] for linear mappingsby considering an unbounded Cauchy difference. In ????, a generalization of the Rassiastheorem was obtained by G?vruta [?] by replacing the unbounded Cauchy difference by ageneral control function.In ????, Radu [?] proposed a new method for obtaining the existence of exact solutionsand error estimations, based on the fixed point alternative (see also [?, ?]).Let (X, d) be a generalized metric space. An operator T : X → X satisfies a Lipschitz conditionwith the Lipschitz constant L if there exists a constant L ≥ ? such that d(Tx,Ty) ≤Ld(x, y) for all x, y ∈ X. If the Lipschitz constant L is less than ?, then the operator T iscalled a strictly contractive operator. Note that the distinction between the generalizedmetric and the usual metric is that the range of the former is permitted to include theinfinity. We recall the following theorem by Margolis and Diaz.Using the fixed point method, we investigate the stability of the system of additive, quadratic and quartic functional equations with constant coefficients in non-Archimedean normed spaces. Also, we give an example to show that some results in the stability of functional equations in (Archimedean) normed spaces are not valid in non-Archimedean normed spaces