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In this paper, we first introduce the notions of $(2,\beta)$-Banach spaces and we will reformulate the fixed point theorem \cite[Theorem 1]{22} in this space. We also show that this theorem is a very efficient and convenient tool for proving the new hyperstability results of the general linear equation in $(2,\beta)$-Banach spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. Our results are improvements and generalizations of the main results of Piszczek \cite{p21}, Brzd\k{e}k \cite{18,brz180} and Bahyrycz et al. \cite{ba1} in $(2,\beta)$-Banach spaces.
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참고문헌 (42)
참고문헌 신청
T. Aoki / 1950 / On the stability of the linear transformation in Banach spaces / J. Math. Soc. Japan 2:64~66
A. Bahyrycz / 2014 / Hyperstability of the Jensen functional equation / Acta Math. Hungar. 142(2):353~365
D. G. Bourgin / 1949 / Approximately isometric and multiplicative transformations on continuous function rings / Duke Math. J. 16:385~397
D. G. Bourgin / 1951 / Classes of transformations and bordering transformations / Bull. Amer. Math. Soc. 57:223~237
J. Brzdek / 2013 / Remarks on hyperstability of the Cauchy functional equation / Aequationes Math. 86(3):255~267
A. Bahyrycz / 2014 / Hyperstability of the Jensen functional equation / Acta Math. Hungar. 142(2):353~365
D. G. Bourgin / 1949 / Approximately isometric and multiplicative transformations on continuous function rings / Duke Math. J. 16:385~397
D. G. Bourgin / 1951 / Classes of transformations and bordering transformations / Bull. Amer. Math. Soc. 57:223~237
J. Brzdek / 2013 / Remarks on hyperstability of the Cauchy functional equation / Aequationes Math. 86(3):255~267
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