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Abstract. Let Ex, E2 be two Banach spaces, and let f: Ex-* E2 be a mapping, that is "approximately linear". S. M. Ulam posed the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam's problem.

Abstract. Let Ex, E2 be two Banach spaces, and let f: Ex-* E2 be a mapping, that is "approximately linear". S. M. Ulam posed the problem: "Give conditions in order for a linear mapping near an approximately linear mapping to exist". The purpose of this paper is to give an answer to Ulam's problem.

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We prove the generalized Hyers-Ulam stability of the Cauchy functional equation f (x + y) = f (x) + f (y) and the quadratic functional equation f (x + y) + f (x − y) = 2f (x) + 2f (y) in non-Archimedean normed spaces.

The Hyers-Ulam stability of mappings is in development and several authors have remarked interesting applications of this theory to various mathematical problems. In this paper some applications in nonlinear analysis are presented, especially in fixed point theory. These kinds of applications seem not to have ever been remarked before by other authors.