Geodesic
Additive $\rho$--functional inequalities
Park, Choonkil1
1 Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea
Journal of nonlinear sciences and its applications, Tome 7 (2014) no. 5, p. 296-310.

Voir la notice de l'article provenant de la source International Scientific Research Publications

In this paper, we solve the additive $\rho$-functional inequalities
$\|f(x + y) - f(x) - f(y)\| \leq \| \rho( 2f (\frac{ x + y}{ 2}) - f(x) - f(y) ) \|, \qquad (1)$
;
$\|2f (\frac{ x + y}{ 2}) - f(x) - f(y)\| \leq \| \rho(f(x + y) - f(x) - f(y) ) \|, \qquad (2)$
; where $\rho$ is a fixed non-Archimedean number with $|\rho|1$ or $\rho$ is a fixed complex number with $|\rho|1$. Using the direct method, we prove the Hyers-Ulam stability of the additive $\rho$-functional inequalities (1) and (2) in non-Archimedean Banach spaces and in complex Banach spaces, and prove the Hyers-Ulam stability of additive $\rho$-functional equations associated with the additive $\rho$-functional inequalities (1) and (2) in non-Archimedean Banach spaces and in complex Banach spaces.
DOI : 10.22436/jnsa.007.05.02
Classification : 46S10, 39B62, 39B52, 47S10, 12J25
Keywords: Hyers-Ulam stability, additive \(\rho\)-functional equation, additive \(\rho\)-functional inequality, non-Archimedean normed space, Banach space.

Park, Choonkil 1

1 Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea 
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Park, Choonkil. Additive \(\rho\)--functional inequalities. Journal of nonlinear sciences and its applications, Tome 7 (2014) no. 5, p. 296-310. doi : 10.22436/jnsa.007.05.02. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.007.05.02/

[1] T. Aoki On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, Volume 2 (1950), pp. 64-66

[2] Bahyrycz, A.; Piszczek, M. Hyperstability of the Jensen functional equation, Acta Math. Hungar., Volume 142 (2014), pp. 353-365

[3] Balcerowski, M. On the functional equations related to a problem of Z. Boros and Z. Daróczy, Acta Math. Hungar., Volume 138 (2013), pp. 329-340

[4] Daróczy, Z.; Gy. Maksa A functional equation involving comparable weighted quasi-arithmetic means, Acta Math. Hungar., Volume 138 (2013), pp. 147-155

[5] Fechner, W. Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math., Volume 71 (2006), pp. 149-161

[6] Fechner, W. On some functional inequalities related to the logarithmic mean, Acta Math. Hungar., Volume 128 (2010), pp. 31-45

[7] P. Găvruţa A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., Volume 184 (1994), p. 431-43

[8] A. Gilányi Eine zur Parallelogrammgleichung äquivalente Ungleichung, Aequationes Math., Volume 62 (2001), pp. 303-309

[9] A. Gilányi On a problem by K. Nikodem, Math. Inequal. Appl., Volume 5 (2002), pp. 707-710

[10] D. H. Hyers On the stability of the linear functional equation, Proc. Natl. Acad. Sci. U.S.A., Volume 27 (1941), pp. 222-224

[11] Moslehian, M. S.; Sadeghi, Gh. A Mazur-Ulam theorem in non-Archimedean normed spaces, Nonlinear Anal.- TMA, Volume 69 (2008), pp. 3405-3408

[12] Park, C.; Cho, Y.; M. Han Functional inequalities associated with Jordan-von Neumann-type additive functional equations , J. Inequal. Appl., Article ID 41820 , Volume 2007 (2007), pp. 1-13

[13] Prager, W.; Schwaiger, J. A system of two inhomogeneous linear functional equations, Acta Math. Hungar., Volume 140 (2013), pp. 377-406

[14] Rassias, Th. M. On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., Volume 72 (1978), pp. 297-300

[15] J. Rätz On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math., Volume 66 (2003), pp. 191-200

[16] Ulam, S. M. A Collection of the Mathematical Problems, Interscience Publ. , New York, 1960

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