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Hyers-Ulam stability of the Cauchy functional equation on square-symmetric groupoids. (English) Zbl 0980.39022
The author investigates the stability of the functional equation \[ f(x\diamond y)=f(x)\ast f(y) \text{ }(x,y \in X) \] where \(f:X \rightarrow Y\) and \((X, \diamond)\), \((Y, \ast)\) are groupoids with square-symmetric operations, i.e., operations \(\diamond\) and \(\ast\) satisfying \((x_1\diamond x_2)\diamond (x_1\diamond x_2)= (x_1\diamond x_1)\diamond (x_2 \diamond x_2)\) and \((y_1\ast y_2)\ast (y_1\ast y_2)= (y_1\ast y_1)\ast (y_2 \ast y_2)\) for all \(x_1,x_2 \in X\) and \(y_1,y_2 \in Y\), respectively. The results generalize the classical theorem of D. H. Hyers [Proc. Nat. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] on the stability of the Cauchy functional equation.

MSC:
39B72 Systems of functional equations and inequalities
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