# zbMATH — the first resource for mathematics

A new approach to Hyers-Ulam stability of $$r$$-variable quadratic functional equations. (English) Zbl 1458.39023
Summary: In this paper, we investigate the general solution of a new quadratic functional equation of the form $\sum_{1\leq i<j<k \leq r} \phi (l_i+l_j+l_k)=(r-2) \sum_{i=1,i \neq j}^r \phi (l_i+l_j) + \tfrac{-r^2+3r-2}{2} \sum_{i=1}^r\phi (l_i).$ We prove that a function admits, in appropriate conditions, a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.
##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 47H10 Fixed-point theorems
##### Keywords:
quadratic mapping; fixed-point method; Ulam stability
Full Text:
##### References:
 [1] Ulam, S. M., Problems in Modern Mathematics, Science Editions (1964), New York: Wiley, New York · Zbl 0137.24201 [2] Hyers, D. H., On the stability of the linear functional equation, Proceedings of the National Academy of Sciences of the United States of America, 27, 222-224 (1941) · JFM 67.0424.01 [3] Jung, S. M., On the Hyers-Ulam-Rassias stability of a quadratic functional equation, Journal of Mathematical Analysis and Applications, 232, 384-393.1 (1999) [4] Kenary, H. A.; Rezaei, H.; Gheisari, Y.; Park, C., On the stability of set-valued functional equations with the fixed point alternative, Fixed Point Theory and Applications, 2012 (2012) [5] Mirmostafaee, A. K.; Moslehian, M. S., Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy sets and systems, 159, 720-729.1 (2008) · Zbl 1178.46075 [6] Moslehian, M. S.; Rassias, T. M., Stability of functional equations in non-Archimedean spaces, Applicable Analysis and Discrete Mathematics, 1, 325-334 (2007) · Zbl 1257.39019 [7] Nikodem, K.; Popa, D., On single-valuedness of set-valued maps satisfying linear inclusions, Banach Journal of Mathematical Analysis, 3, 44-51 (2009) · Zbl 1163.26353 [8] Rassias, T. M., On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 297-300.1 (1978) [9] Radu, V., The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4, 91-96 (2003), 1 · Zbl 1051.39031 [10] Hyers, D. H.; Isac, G.; Rassias, T. M., Stability of Functional Equations in Several Variables (1998), Boston: Birkhäuser, Boston · Zbl 0907.39025 [11] Bodaghi, A., Intuitionistic fuzzy stability of the generalized forms of cubic and quartic functional equations, Journal of Intelligent & Fuzzy Systems, 30, 2309-2317 (2016) · Zbl 1361.39013 [12] Jung, S. M., Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis (2011), New York: Springer, New York · Zbl 1221.39038 [13] Cieplinski, K., On the generalized Hyers-Ulam stability of multi-quadratic mappings, Computers & Mathematcs with Applications, 62, 3418-3426.1 (2011) · Zbl 1236.39025 [14] Jung, S. M., On the Hyers-Ulam stability of the functional equations that have the quadratic property, Journal of Mathematical Analysis and Applications, 222, 126-137.1 (1998) [15] Lee, Y. W., On the stability of a quadratic Jensen type functional equation, Journal of Mathematical Analysis and Applications, 270, 590-601.1 (2002) [16] Park, C. G., On the Hyers-Ulam-Rassias stability of generalized quadratic mappings in Banach modules, Journal of Mathematical Analysis and Applications, 291, 214-223.1 (2004) · Zbl 1046.39023 [17] Park, C.; Kenary, H. A.; Rassias, T. M., Hyers-Ulam-Rassias stability of the additive-quadratic mappings in non- Archimedean Banach spaces, Journal of Inequalities and Applications, 2012 (2012) · Zbl 1279.39021 [18] Shen, H. Y.; Lan, Y. Y., On the general solution of a quadratic functional equation and its Ulam stability in various abstract spaces, Journal of Nonlinear Sciences and Applications, 7, 368-378 (2014) · Zbl 1312.39027 [19] Dineshkumar, C.; Udhayakumar, R.; Vijayakumar, V.; Nisar, K. S., A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systems, Chaos, Solitons & Fractals, article 110472 (2020) [20] Vijayakumar, V.; Udhayakumar, R., Results on approximate controllability for non-densely defined Hilfer fractional differential system with infinite delay, Chaos, Solitons & Fractals, 139, article 110019 (2020) [21] Kavitha, K.; Vijayakumar, V.; Udhayakumar, R., Results on controllability of Hilfer fractional neutral differential equations with infinite delay via measures of noncompactness, Chaos, Solitons & Fractals, 139, article 110035 (2020) [22] Mohan Raja, M.; Vijayakumar, V.; Udhayakumar, R.; Zhou, Y., A new approach on the approximate controllability of fractional differential evolution equations of order 1
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.