3-variable Jensen \(\rho\)-functional inequalities and equations
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Authors
Gang Lu
- Department of Mathematics, School of Science, Shenyang University of Technology, Shenyang 110870, P. R. China.
Qi Liu
- Department of Mathematics, School of Science, Shenyang University of Technology, Shenyang 110870, P. R. China.
Yuanfeng Jin
- Department of Mathematics, Yanbian University, Yanji 133001, P. R. China.
Jun Xie
- Department of Mathematics, School of Science, Shenyang University of Technology, Shenyang 110870, P. R. China.
Abstract
In this paper, we introduce and investigate Jensen \(\rho\)-functional inequalities associated with the following
Jensen functional equations
\[f(x + y + z) + f(x + y - z) - 2f(x) - 2f(y) = 0,\\
f(x + y + z) - f(x - y - z) - 2f(y) - 2f(z) = 0.\]
We prove the Hyers-Ulam-Rassias stability of the Jensen \(\rho\)-functional inequalities in complex Banach spaces
and prove the Hyers-Ulam-Rassias stability of the Jensen \(\rho\)-functional equations associated with the \(\rho\)-
functional inequalities in complex Banach spaces.
Share and Cite
ISRP Style
Gang Lu, Qi Liu, Yuanfeng Jin, Jun Xie, 3-variable Jensen \(\rho\)-functional inequalities and equations, Journal of Nonlinear Sciences and Applications, 9 (2016), no. 12, 5995--6003
AMA Style
Lu Gang, Liu Qi, Jin Yuanfeng, Xie Jun, 3-variable Jensen \(\rho\)-functional inequalities and equations. J. Nonlinear Sci. Appl. (2016); 9(12):5995--6003
Chicago/Turabian Style
Lu, Gang, Liu, Qi, Jin, Yuanfeng, Xie, Jun. "3-variable Jensen \(\rho\)-functional inequalities and equations." Journal of Nonlinear Sciences and Applications, 9, no. 12 (2016): 5995--6003
Keywords
- Jensen functional inequalities
- Hyers-Ulam-Rassias stability
- complex Banach spaces.
MSC
References
-
[1]
T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64--66
-
[2]
J. Aczél, J. Dhombres, Functional equations in several variables, With applications to mathematics, information theory and to the natural and social sciences, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (1989)
-
[3]
L. Cădariu, V. Radu, Fixed points and the stability of Jensen's functional equation, JIPAM. J. Inequal. Pure Appl. Math., 4 (2003), 7 pages
-
[4]
I. S. Chang, M. Eshaghi Gordji, H. Khodaei, H. M. Kim, Nearly quartic mappings in \(\beta\)-homogeneous F-spaces, Results Math., 63 (2013), 529--541
-
[5]
Y. J. Cho, C. K. Park, R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett., 23 (2010), 1238--1242
-
[6]
Y. J. Cho, R. Saadati, Y.-O. Yang, Approximation of homomorphisms and derivations on Lie \(C^*\)-algebras via fixed point method, J. Inequal. Appl., 2013 (2013), 9 pages
-
[7]
P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76--86
-
[8]
J. B. Diaz, B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), 305--309
-
[9]
A. Ebadian, N. Ghobadipour, T. M. Rassias, M. Eshaghi Gordji, Functional inequalities associated with Cauchy additive functional equation in non-Archimedean spaces, Discrete Dyn. Nat. Soc., 2011 (2011), 14 pages
-
[10]
P. Găvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431--436
-
[11]
D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U. S. A., 27 (1941), 222--224
-
[12]
D. H. Hyers, G. Isac, T. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc., Boston (1998)
-
[13]
G. Isac, T. M. Rassias, On the Hyers-Ulam stability of \(\psi\)-additive mappings, J. Approx. Theory, 72 (1993), 131--137
-
[14]
V. Lakshmikantham, S. Leela, J. Vasundhara Devi, Theory of fractional dynamic systems, Cambridge Scientific Publishers, (2009)
-
[15]
S.-B. Lee, J.-H. Bae, W.-G. Park, On the stability of an additive functional inequality for the fixed point alternative, J. Comput. Anal. Appl., 17 (2014), 361--371
-
[16]
G. Lu, C. K. Park, Hyers-Ulam Stability of Additive Set-valued Functional Equations, Appl. Math. Lett., 24 (2011), 1312--1316
-
[17]
C. K. Park, Y. S. Cho, M.-H. Han, Functional inequalities associated with Jordan-von Neumann-type additive functional equations, J. Inequal. Appl., 2007 (2007), 13 pages
-
[18]
T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297--300
-
[19]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46 (1982), 126--130
-
[20]
J. M. Rassias, On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math., 108 (1984), 445--446
-
[21]
J. M. Rassias, Solution of a problem of Ulam, J. Approx. Theory, 57 (1989), 268--273
-
[22]
J. M. Rassias, Complete solution of the multi-dimensional problem of Ulam, Discuss. Math., 14 (1994), 101--107
-
[23]
T. M. Rassias, Functional equations and inequalities, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht (2000)
-
[24]
T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Math. Appl., 62 (2000), 23--130
-
[25]
T. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl., 251 (2000), 264--284
-
[26]
J. M. Rassias, Refined Hyers-Ulam approximation of approximately Jensen type mappings, Bull. Sci. Math., 131 (2007), 89--98
-
[27]
J. M. Rassias, M. J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl., 281 (2003), 516--524
-
[28]
S. M. Ulam, Problems in modern mathematics (Chapter VI), Wiley,, New York (1960)
-
[29]
T. Z. Xu, J. M. Rassias, W. X. Xu,, A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces, Int. J. Phys. Sci., 6 (2011), 313--324