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On Hyers-Ulam stability of generalized Wilson’s equation. (English) Zbl 1082.39022

Let \(K\) be a compact subgroup of a locally compact group \(G\). Let \(\omega_K\) be a normalized Haar measure of \(K\). A homeomorphism \(\varphi:G\to G\) is called a morphism of \(G\) if \(\varphi\) is a homomorphism or \(\varphi\) is an antihomomorphism (i.e. \(\varphi(xy)=\varphi(y)\varphi(x)\), \(x,y\in G\)). Let \(\Phi\) be a finite group of morphisms of \(G\) which are \(K\)-invariant (i.e. \(\varphi(K)\subset K\), \(\varphi\in\Phi\)). Let \(| \Phi| \) be the number of elements of \(\Phi\).

The functional equation \[ \sum_{\varphi\in\Phi}\int_Kf\bigl(xk\varphi(y)k^{-1}\bigr)d\omega_K(k)=| \Phi| f(x)g(y),\quad x,y\in G\tag{1} \] is studied for continuous complex-valued functions \(f,g\) defined on \(G\) such that \(f\) satisfies the Kannappan type condition \[ \int_K\int_Kf\bigl(zkxk^{-1}hyh^{-1}\bigr)d\omega_K(k)d\omega_K(h) =\int_K\int_Kf\bigl(zkyk^{-1}hxh^{-1}\bigr)d\omega_K(k)d\omega_K(h).\tag{2} \] Notice that for a commutative group \(G\) and \(\phi=\{id_G\}\) (1) reduces to the functional equation of Cauchy type \(f(xy)=f(x)g(y)\) and for \(\Phi=\{id_G,\sigma\}\), where \(\sigma\) is the involution of \(G\), (1) reduces to \(f(xy)+f\bigl(x\sigma(y)\bigr)=2f(x)g(y)\), which is the equation of Wilson type.
The main result is the following Theorem: Let \(\delta>0\) and \(f,g:G\to{\mathbb C}\) be continuous functions such that \(f\) satisfies (2) and \[ \Bigl| \sum_{\varphi\in\Phi}\int_Kf\bigl(xk\varphi(y)k^{-1}\bigr)d\omega_K(k)-| \Phi| f(x)g(y)\Bigr| \leq\delta,\quad x,y\in G. \] Then
(i) \(f,g\) are bounded, or
(ii) \(f\) is unbounded and \(g\) satisfies the equation \[ \sum_{\varphi\in\Phi}\int_Kg\bigl(xk\varphi(y)k^{-1}\bigr)d\omega_K(k)=| \Phi| g(x)g(y),\quad x,y\in G,\tag{3} \] or
(iii) \(g\) is unbounded, \(f\) satisfies (1) and if \(f\neq 0\), then \(g\) is a solution of (3).
Numerous particular cases of this result are given.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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