Hyers-Ulam-Rassias stability of some additive fuzzy set-valued functional equations with the fixed point alternative. (English) Zbl 1470.39069

Summary: Let \(Y\) be a real separable Banach space and let \(\left(\mathcal{K}_C \left(Y\right), d_\infty\right)\) be the subspace of all normal fuzzy convex and upper semicontinuous fuzzy sets of \(Y\) equipped with the supremum metric \(d_\infty\). In this paper, we introduce several types of additive fuzzy set-valued functional equations in \(\left(\mathcal{K}_C \left(Y\right), d_\infty\right)\). Using the fixed point technique, we discuss the Hyers-Ulam-Rassias stability of three types additive fuzzy set-valued functional equations, that is, the generalized Cauchy type, the Jensen type, and the Cauchy-Jensen type additive fuzzy set-valued functional equations. Our results can be regarded as important extensions of stability results corresponding to single-valued functional equations and set-valued functional equations, respectively.


39B82 Stability, separation, extension, and related topics for functional equations
26E50 Fuzzy real analysis
47H10 Fixed-point theorems
Full Text: DOI


[1] Ulam, S. M., Problems in Modern Mathematics (1960), New York, NY, USA: Wiley, New York, NY, USA · 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) · Zbl 0061.26403
[3] Aoki, T., On the stability of the linear transformation in Banach spaces, Journal of the Mathematical Society of Japan, 2, 64-66 (1950) · Zbl 0040.35501
[4] Moslehian, M. S.; Rassias, T. M., Stability of functional equations in non-Archimedean spaces, Applicable Analysis and Discrete Mathematics, 1, 2, 325-334 (2007) · Zbl 1257.39019
[5] Rassias, T. M., On the stability of the linear mapping in Banach spaces, Proceedings of the American Mathematical Society, 72, 2, 297-300 (1978) · Zbl 0398.47040
[6] Găvruţa, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, Journal of Mathematical Analysis and Applications, 184, 3, 431-436 (1994) · Zbl 0818.46043
[7] Jung, S. M., Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis (2011), Berlin, Germany: Springer, Berlin, Germany · Zbl 1221.39038
[8] Radu, V., The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4, 1, 91-96 (2003) · Zbl 1051.39031
[9] Ciepliński, K., Applications of fixed point theorems to the Hyers-Ulam stability of functional equations-a survey, Annals of Functional Analysis, 3, 1, 151-164 (2012) · Zbl 1252.39032
[10] Mirmostafaee, A. K.; Moslehian, M. S., Fuzzy versions of Hyers-Ulam-Rassias theorem, Fuzzy Sets and Systems, 159, 6, 720-729 (2008) · Zbl 1178.46075
[11] Mirmostafaee, A. K.; Mirzavaziri, M.; Moslehian, M. S., Fuzzy stability of the Jensen functional equation, Fuzzy Sets and Systems, 159, 6, 730-738 (2008) · Zbl 1179.46060
[12] Jang, S. Y.; Lee, J. R.; Park, C.; Shin, D. Y., Fuzzy stability of Jensen-type quadratic functional equations, Abstract and Applied Analysis, 2009 (2009) · Zbl 1167.39015
[13] Lee, J. R.; Jang, S.-Y.; Park, C.; Shin, D. Y., Fuzzy stability of quadratic functional equations, Advances in Difference Equations, 2010 (2010) · Zbl 1192.39021
[14] Mohiuddine, S. A.; Alotaibi, A., Fuzzy stability of a cubic functional equation via fixed point technique, Advances in Difference Equations, 2012, 48 (2012) · Zbl 1288.39012
[15] Park, C., A fixed point approach to the fuzzy stability of an additive-quadratic-cubic functional equation, Fixed Point Theory and Applications, 2009 (2009) · Zbl 1187.39049
[16] Nikodem, K.; Popa, D., On single-valuedness of set-valued maps satisfying linear inclusions, Banach Journal of Mathematical Analysis, 3, 1, 44-51 (2009) · Zbl 1163.26353
[17] Lu, G.; Park, C., Hyers-Ulam stability of additive set-valued functional equations, Applied Mathematics Letters, 24, 8, 1312-1316 (2011) · Zbl 1220.39030
[18] Park, C.; O’Regan, D.; Saadati, R., Stability of some set-valued functional equations, Applied Mathematics Letters. An International Journal of Rapid Publication, 24, 11, 1910-1914 (2011) · Zbl 1236.39034
[19] 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, 81 (2012) · Zbl 1283.39010
[20] Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions. Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, 580 (1977), Berlin, Germany: Springer, Berlin, Germany · Zbl 0346.46038
[21] Diamond, P.; Kloeden, P., Metric Spaces of Fuzzy Sets: Theory and Applications (1994), Singapore: World Scientific, Singapore · Zbl 0873.54019
[22] Inoue, H., A strong law of large numbers for fuzzy random sets, Fuzzy Sets and Systems, 41, 3, 285-291 (1991) · Zbl 0737.60003
[23] Nikodem, K., K-Convex and K-Concave Set-Valued Functions (1989), Krakow, Poland: Zeszyty Naukowe, Politech, Krakow, Poland · Zbl 0642.39006
[24] Diaz, J. B.; Margolis, B., A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bulletin of the American Mathematical Society, 74, 305-309 (1968) · Zbl 0157.29904
[25] Gordji, M. E.; Park, C.; Savadkouhi, M. B., The stability of a quartic type functional equation with the fixed point alternative, Fixed Point Theory, 11, 2, 265-272 (2010) · Zbl 1208.39036
[26] Gajda, Z., On stability of the Cauchy equation on semigroups, Aequationes Mathematicae, 36, 1, 76-79 (1988) · Zbl 0658.39006
[27] Cădariu, L.; Radu, V., Fixed points and the stability of Jensen’s functional equation, Journal of Inequalities in Pure and Applied Mathematics, 4, 1, article 4 (2003) · Zbl 1043.39010
[28] Jung, S. M., Hyers-Ulam-Rassias stability of Jensen’s equation and its application, Proceedings of the American Mathematical Society, 126, 11, 3137-3143 (1998) · Zbl 0909.39014
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.