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Alternative Contraction Principle and Alternative Jensen and Jensen Type Mappings

John Michael Rassias

Abstract



In 1940 and 1960 (and 1964) S. M. Ulam ([11],[12]) proposed the general Ulam stability problem: " When is it true that by slightly changing the hypotheses of a theorem one can still assert that the thesis of the theorem remains true or approximately true ?" Almost all proofs on the Ulam stability of additive functional equations have employed the well-known Hyers (direct) method established by D. H. Hyers [2] in 1941. On the other hand, V. Radu [6] in 2003 noticed that a fixed point alternative method is very important for the solution of the Ulam problem. Then, L. Cadariu and V. Radu [1] investigated the stability of the Jensen functional equation. In 2003-2004 M. J. Rassias and the author ( [7]-[9]) of this paper investigated the Ulam stability of Jensen and Jensen type mappings by applying the Hyers method. An example of a stable functional equation with the Hyers method not to work, has recently been published by Z. Kaizer and Zs. Pales ([3],[5]). In this paper we establish the Ulam stability of alternative Jensen and Jensen type mappings as well as alternative additive mappings of two forms, via an alternative contraction principle. This kind of stability problems can be applied in stochastic analysis, actuarial and financial mathematics.

Keywords


Alternative contraction principle, Alternative Jensen and Jensen type, Alternative additive mapping of two forms, Hyers (direct) method, Ulam stability

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