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**Fixed points and stability of an additive functional equation of \(n\)-Apollonius type in \(C^{\ast}\)-algebras.**
*(English)*
Zbl 1159.47032

This article deals with Ulam’s stability property for the following additive functional equation of \(n\)-Appolonius type:

\[ \sum_{i=1}^n f(z - z_i) + \frac1n \sum_{1 \leq i < j \leq n} f(x_i + x_j) - n f\bigg(z - \frac1{n^2} \sum_{i=1}^n x_i\bigg) = 0,\eqno(1) \] where the unknown function \(f(\cdot)\) is a map between two \(C^*\)-algebras \(A\) and \(B\). In other words, the following solvability problem is studied: if \(f(\cdot)\) is a solution of the inhomogeneous equation

\[ \sum_{i=1}^n f(z - z_i) + \frac1n \sum_{1 \leq i < j \leq n} f(x_i + x_j) - n f\bigg(z - \frac1{n^2} \sum_{i=1}^n x_i\bigg) = \phi(z,x_1,\dots,x_n)\eqno(2) \]

with a “small” (in a suitable sense) \(\phi(z,x_1,\dots,x_n)\), then (2) has a solution \(f(\cdot)\) that is close to an exact solution for (1). Two cases are considered: when \(f(\cdot)\) is closed to a homomorphism between \(A\) and \(B\) (i.e., the difference \(f(xy) - f(x)f(y)\) is small) and, in the case \(B = A\), when \(f(\cdot)\) is closed to an generalized derivation (i.e., \(f(xyz) - f(xy)z + xf(y)z - xf(yz)\) is small); in both cases, it is assumed that \(f(x^*) - f(x)^*\) is small. The main tool for the construction of the corresponding solutions is the Banach–Cacciopoli fixed point principle. The question of the existence of solutions to (1) that are homomorphisms or generalized derivations is not discussed, although it is proved that the solutions to (1) are additive.

\[ \sum_{i=1}^n f(z - z_i) + \frac1n \sum_{1 \leq i < j \leq n} f(x_i + x_j) - n f\bigg(z - \frac1{n^2} \sum_{i=1}^n x_i\bigg) = 0,\eqno(1) \] where the unknown function \(f(\cdot)\) is a map between two \(C^*\)-algebras \(A\) and \(B\). In other words, the following solvability problem is studied: if \(f(\cdot)\) is a solution of the inhomogeneous equation

\[ \sum_{i=1}^n f(z - z_i) + \frac1n \sum_{1 \leq i < j \leq n} f(x_i + x_j) - n f\bigg(z - \frac1{n^2} \sum_{i=1}^n x_i\bigg) = \phi(z,x_1,\dots,x_n)\eqno(2) \]

with a “small” (in a suitable sense) \(\phi(z,x_1,\dots,x_n)\), then (2) has a solution \(f(\cdot)\) that is close to an exact solution for (1). Two cases are considered: when \(f(\cdot)\) is closed to a homomorphism between \(A\) and \(B\) (i.e., the difference \(f(xy) - f(x)f(y)\) is small) and, in the case \(B = A\), when \(f(\cdot)\) is closed to an generalized derivation (i.e., \(f(xyz) - f(xy)z + xf(y)z - xf(yz)\) is small); in both cases, it is assumed that \(f(x^*) - f(x)^*\) is small. The main tool for the construction of the corresponding solutions is the Banach–Cacciopoli fixed point principle. The question of the existence of solutions to (1) that are homomorphisms or generalized derivations is not discussed, although it is proved that the solutions to (1) are additive.

Reviewer: Peter Zabreiko (Minsk)

### MSC:

47H10 | Fixed-point theorems |

46H99 | Topological algebras, normed rings and algebras, Banach algebras |

### Keywords:

delay equations; dual semigroup; sun-star-calculus; Lipschitz perturbations; principle of linearized stability; center manifold; Hopf bifurcation; physiologically structured populations
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\textit{F. Moradlou} et al., Abstr. Appl. Anal. 2008, Article ID 672618, 13 p. (2008; Zbl 1159.47032)

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