Solution of some irregular functional equations and their stability

الموضوعات :

Y. Sayyari ¹ (Department of Mathematics, Sirjan University of Technology, P. O. Box 78137-33385, Sirjan, Iran)
M. Dehghanian ² (Department of Mathematics, Sirjan University of Technology, P. O. Box 78137-33385, Sirjan, Iran)
Sh. Nasiri ³ (Department of Computer Engineering, Sirjan University of Technology P. O. Box 78137-33385, Sirjan, Iran)

تاريخ الإرسال : 07 الثلاثاء , ربيع الثاني, 1444 تاريخ التأكيد : 03 الثلاثاء , جمادى الثانية, 1444 تاريخ الإصدار : 07 الخميس , جمادى الأولى, 1444

الکلمات المفتاحية: Hyers-Ulam stability, Additive functional equation, unital algebra,

ملخص المقالة :

‎In this note‎, ‎we study the following functional equations‎:‎\begin{align*}‎‎&L(L(p ,r)+L(‎q‎,r)+p + q ,r)+L(L( p, r)+ p , r)+L(q, r )=0,\\‎‎&L(L( p , r )+ p + q+e, r )+L( p, r)=L( p + q , r )+ p L(q , r)‎\end{align*}‎‎and‎ ‎$‎L( p , q )=L(\zeta p , q), \vert \zeta\vert <1‎$,‎without any regularity assumption for all $ p , q , r \in A$, where $L:A^2\rightarrow A$ is defined by ‎$‎L( p , q ):=g( p + q )-g( p )-g( q )‎$‎ for all $ p , q\in A$. Also, we find general solutions of the above functional equations on algebras, unital algebras and real numbers, respectively. Finally, we investigate the stability of those functional equations in algebras and unital algebras, respectively.

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