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A functional equation having monomials as solutions. (English) Zbl 1191.39026
For vector spaces $$X,Y$$ and $$f: X\to Y$$ consider the functional equations
$f(2x+y)+f(2x-y)=2^{n-2}[f(x+y)+f(x-y)+6f(x)]$ for $$n=1,2,3$$ and
$f(2x+y)+f(2x-y)+6f(y)=4[f(x+y)+f(x-y)+6f(x)].$ General solutions of the above equation as well as their stability are investigated. The stability results are obtained using both the direct and the fixed point methods.

##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations
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##### References:
 [1] J.-H. Bae, W.-G. Park, On a cubic and a Jensen-quadratic equations, Abstract Appl. Anal. 2007 (2007) (Article ID 45179). [2] L. Cădariu, V. Radu, Fixed points and the stability of Jensen’s functional equation, J. Inequal. Pure Appl. Math. 4 (2003) (Article 4). [3] Cădariu, L.; Radu, V., On the stability of the Cauchy functional equation: a fixed point approach, Grazer math. ber., 346, 43-52, (2004) [4] Chung, J.K.; Sahoo, P.K., On the general solution of a quartic functional equation, Bull. Korean math. soc., 40, 565-576, (2003) · Zbl 1048.39017 [5] Găvruta, P., A generalization of the hyers – ulam – rassias stability of approximately additive mappings, J. math. anal. appl., 184, 431-436, (1994) · Zbl 0818.46043 [6] Hyers, D.H., On the stability of the linear functional equation, Proc. nat. acad. sci., 27, 222-224, (1941) · Zbl 0061.26403 [7] Jun, K.-W.; Kim, H.-M., The generalized hyers – ulam – rassias stability of a cubic functional equation, J. math. anal. appl., 274, 867-878, (2002) · Zbl 1021.39014 [8] Jung, S.-M., A fixed point approach to the stability of isometries, J. math. anal. appl., 329, 2, 879-890, (2007) · Zbl 1153.39309 [9] S.-M. Jung, A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl. 2007 (2007) 9 (Article ID 57064). [10] Jung, S.-M.; Kim, T.-S., A fixed point approach to the stability of the cubic functional equation, Boletin de la sociedad matematica mexicana, 3-12, 1, 51-57, (2006) · Zbl 1133.39028 [11] Jung, S.-M.; Kim, T.-S.; Lee, K.-S., A fixed point approach to the stability of quadratic functional equation, Bull. Korean math. soc., 43, 3, 531-541, (2006) · Zbl 1113.39031 [12] S.-M. Jung, Z.-H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. 2008 (2008) (Article ID 732086). [13] Margolis, B.; Dias, J.B., A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. am. math. soc., 126, 305-309, (1968) · Zbl 0157.29904 [14] C. Park, Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach, Fixed Point Theory Appl. 2008 (2008) (Article ID 493751). · Zbl 1146.39048 [15] C. Park, Stability of the Cauchy-Jensen functional equation in $$C^\ast$$-algebras: a fixed point approach, Fixed Point Theory Appl. 2008 (2008) (Article ID 872190). [16] C. Park, Fixed points and stability of the Cauchy functional equation in $$C^\ast$$-algebras, Fixed Point Theory and Appl. 2009 (2009) (Article ID 809232). [17] Radu, V., The fixed point alternative and the stability of functional equations, Fixed point theory, 4, 91-96, (2003) · Zbl 1051.39031 [18] Rassias, Th.M., On the stability of linear mappings in Banach spaces, Proc. am. math. soc., 72, 297-300, (1978) · Zbl 0398.47040 [19] I.A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian). [20] Ulam, S.M., A collection of mathematical problems, (1968), Interscience Publishers New York · Zbl 0086.24101
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