HYERS–ULAM STABILITY, WELL-POSEDNESS, AND DATA DEPENDENCE FOR FISHER-TYPE CONTRACTIVE MAPPINGS
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Abstract
In this paper, we study Hyers–Ulam stability, well-posedness, and data dependence for mappings satisfying a Fisher-type inequality. Assuming that the mapping has a fixed point , we first show that the Fisher-type inequality implies the unique-preimage property . This structural property yields the global residual estimate , , where is the Fisher parameter. Consequently, Hyers–Ulam stability and well-posedness follow in an arbitrary metric space, without compactness, completeness, or continuity assumptions. Data dependence is treated separately: if a perturbed mapping has a fixed point and satisfies for all , then . Combining these abstract estimates with the fixed point theorems in [10], we obtain existence, uniqueness, convergence, and stability conclusions in boundedly compact and -orbitally compact metric spaces, and in complete metric spaces under an additional asymptotic condition. Finally, we present an example of a discontinuous mapping, together with numerical illustrations.
Keywords
Fixed point, Fisher-type contraction, well-posedness, Hyers--Ulam stability, data dependence.
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References
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