## Remarks on the conditional quadratic and the conditional D’Alembert functional equations on particular normed spaces.(English)Zbl 1470.39052

Summary: Let $$X$$ be a real normed space with dimension greater than 2 and let $$f$$ be a real functional defined on $$X$$. Applying some ideas from the studies made on the conditional Cauchy functional equation on the restricted domain of the vectors of equal norm and the isosceles orthogonal vectors, the conditional quadratic equation and the D’Alembert one, namely, $$\| x \| = \| y \| \Rightarrow f(x + y) + f(x - y) = 2 f(x) + 2 f(y)$$ and $$\| x \| = \| y \| \Rightarrow f(x + y) + f(x - y) = 2 f(x) f(y)$$, have been studied in this paper, in order to describe their solutions. Particular normed spaces are introduced for this aim.

### MSC:

 39B52 Functional equations for functions with more general domains and/or ranges
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### References:

 [1] Alsina, C.; Garcia Roig, J. L., On a conditional cauchy equation on rhombuses, Functional Analysis, Approximation Theory and Numerical Analysis, 5-7 (1994), River Edge, NJ, USA: World Scientific Publishing Company, River Edge, NJ, USA · Zbl 0877.39015 [2] Gudder, S.; Strawther, D., Orthogonally additive and orthogonally increasing functions on vector spaces, Pacific Journal of Mathematics, 58, 2, 427-436 (1975) · Zbl 0311.46015 [3] Rätz, J., On orthogonally additive mappings, Aequationes Mathematicae, 28, 1-2, 35-49 (1985) · Zbl 0569.39006 [4] Szabó, Gy., $$\phi$$-orthogonally additive mappings I, Acta Mathematica Hungarica, 58, 1-2, 101-111 (1991) · Zbl 0763.46022 [5] Paganoni, L.; Rätz, J., Conditional functional equations and orthogonal additivity, Aequationes Mathematicae, 50, 1-2, 135-142 (1995) · Zbl 0876.39008 [6] Szabó, Gy., A conditional cauchy equation on normed spaces, Publicationes Mathematicae Debrecen, 42, 3-4, 265-271 (1993) · Zbl 0807.39010 [7] James, R. C., Orthogonality in normed linear spaces, Duke Mathematical Journal, 12, 291-302 (1945) · Zbl 0060.26202 [8] Szabó, Gy., Isosceles orthogonally additive mappings and inner product spaces, Publicationes Mathematicae Debrecen, 46, 3-4, 373-384 (1995) · Zbl 0865.46012
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