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Fixed points and additive \({\rho}\) -functional equations
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In this paper, we solve the additive \({\rho}\)-functional equations$$\begin{aligned} f(x+y)-f(x)-f(y)= & {} \rho(2f(\frac{x+y}{2})-f(x)-f(y)), \\ 2f(\frac{x+y}{2})-f(x)-f(y)= & {} \rho(f(x+y)-f(x)-f(y)), \end{aligned}$$ where \({\rho}\) is a fixed non-Archimedean number or a fixed real or complex number with \({\rho \neq 1}\). Using the fixed point method, we prove the Hyers–Ulam stability of the above additive \({\rho}\)-functional equations in non-Archimedean Banach spaces and in Banach spaces.KeywordsHyers–Ulam stabilityadditive \({\rho}\)-functional equationfixed pointnon-Archimedean normed spaceBanach spaceMathematics Subject Classification39B5239B6247H1012J2546S1047S10References1.L. Aiemsomboon and W. Sintunavaat, On new stability results for generalized Cauchy functional equations on groups by using Brzdęk’s fixed point theorem. J. Fixed Point Theory Appl., to appear.2.Aoki T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2, 64–66 (1950)MathSciNetCrossRefMATHGoogle Scholar3.Bahyrycz A., Brzdęk J., Jabłońska E., Olko J.: On functions that are approximate fixed points almost everywhere and Ulam's type stability. J. Fixed Point Theory Appl. 17, 659–668 (2015)MathSciNetCrossRefMATHGoogle Scholar4.Bahyrycz A., Piszczek M.: Hyperstability of the Jensen functional equation. Acta Math. Hungar. 142, 353–365 (2014)MathSciNetCrossRefMATHGoogle Scholar5.Balcerowski M.: On the functional equations related to a problem of Z. Boros and Z. Daróczy. Acta Math. Hungar. 138, 329–340 (2013)MathSciNetCrossRefMATHGoogle Scholar6.L. Cădariu and V. Radu, Fixed points and the stability of Jensen's functional equation. J. Inequal. Pure Appl. Math. 4 (2003), Art. ID 4, 7 pages.7.L. Cădariu and V. Radu, On the stability of the Cauchy functional equation: A fixed point approach. In: Iteration Theory (ECIT ’02), Grazer Math. Ber. 346, Karl-Franzens-Univ. Graz, Graz, 2004, 43–52.8.L. Cădariu and V. Radu, Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008 (2008), Art. ID 749392, 15 pages.9.Chahbi A., Bounader N.: On the generalized stability of d'Alembert functional equation. J. Nonlinear Sci. Appl. 6, 198–204 (2013)MathSciNetMATHGoogle Scholar10.Daróczy Z., Maksa G.: A functional equation involving comparable weighted quasi-arithmetic means. Acta Math. Hungar. 138, 147–155 (2013)MathSciNetCrossRefMATHGoogle Scholar11.Diaz J., Margolis B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74, 305–309 (1968)MathSciNetCrossRefMATHGoogle Scholar12.M. E. Gordji and Th. M. Rassias, Fixed points and generalized stability for quadratic and quartic mappings in C*-algebras. J. Fixed Point Theory Appl. 17 (2015), 703–715.13.Eskandani G. Z., Gǎvruta P.: Hyers-Ulam-Rassias stability of pexiderized Cauchy functional equation in 2-Banach spaces. J. Nonlinear Sci. Appl. 5, 459–465 (2012)MathSciNetMATHGoogle Scholar14.Fechner W.: On some functional inequalities related to the logarithmic mean. Acta Math. Hungar. 128, 31–45 (2010)MathSciNetCrossRefMATHGoogle Scholar15.Gǎvruta P.: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)MathSciNetCrossRefMATHGoogle Scholar16.Ghazanfari A. G., Alizadeh Z.: On approximate ternary m-derivations and \({\sigma}\)-homomorphisms. J. Fixed Point Theory Appl. 17, 625–640 (2015)MathSciNetCrossRefMATHGoogle Scholar17.Gilányi A.: Eine zur Parallelogrammgleichung äquivalente Ungleichung. Aequationes Math. 62, 303–309 (2001)MathSciNetCrossRefGoogle Scholar18.Gilányi A.: On a problem by K. Nikodem. Math. Inequal. Appl. 5, 707–710 (2002)MathSciNetMATHGoogle Scholar19.Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)MathSciNetCrossRefMATHGoogle Scholar20.G. Isac and Th. M. Rassias, Stability of \({\psi}\) -additive mappings: Applications to nonlinear analysis. Int. J. Math. Math. Sci. 19 (1996), 219–228.21.Miheţ D., Radu V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)MathSciNetCrossRefMATHGoogle Scholar22.Moslehian M. S., Sadeghi G.: A Mazur-Ulam theorem in non-Archimedean normed spaces. Nonlinear Anal. 69, 3405–3408 (2008)MathSciNetCrossRefMATHGoogle Scholar23.C. Park, Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras. Fixed Point Theory Appl. 2007 (2007), Art. ID 50175, 15 pages.24.C. Park, Generalized Hyers-Ulam stability of quadratic functional equations: A fixed point approach. Fixed Point Theory Appl. 2008 (2008), Art. ID 493751, 9 pages.25.Park C.: Orthogonal stability of a cubic-quartic functional equation. J. Nonlinear Sci. Appl. 5, 28–36 (2012)MathSciNetMATHGoogle Scholar26.Prager W., Schwaiger J.: A system of two inhomogeneous linear functional equations. Acta Math. Hungar. 140, 377–406 (2013)MathSciNetCrossRefMATHGoogle Scholar27.Radu V.: The fixed point alternative and the stability of functional equations. Fixed Point Theory 4, 91–96 (2003)MathSciNetMATHGoogle Scholar28.Th. M. Rassias, On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), 297–300.29.Ravi K., Thandapani E., SenthilKumar B. V.: Solution and stability of a reciprocal type functional equation in several variables. J. Nonlinear Sci. Appl. 7, 18–27 (2014)MathSciNetMATHGoogle Scholar30.S. M. Ulam, A Collection of Mathematical Problems. Interscience Publ., New York, 1960.31.Zaharia C.: On the probabilistic stability of the monomial functional equation. J. Nonlinear Sci. Appl. 6, 51–59 (2013)MathSciNetMATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsChoonkil Park1Dong Yun Shin2Jung Rye Lee3Email author1.Department of Mathematics, Research Institute for Natural SciencesHanyang UniversitySeoulSouth Korea2.Department of MathematicsUniversity of SeoulSeoulSouth Korea3.Department of MathematicsDaejin UniversityKyeonggiSouth Korea About this article CrossMark Print ISSN 1661-7738 Online ISSN 1661-7746 Publisher Name Springer International Publishing About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s11784-016-0282-3_Fixed points and additive $${\rho}", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s11784-016-0282-3_Fixed points and additive $${\rho}", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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