Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions. (English) Zbl 1152.39318

Lee, Young-Su; Chung, Soon-Yeong

Appl. Math. Lett. 21, No. 7, 694-700 (2008).

Summary: Making use of the fundamental solution of the heat equation we reformulate and prove the stability theorem of a special case of the Euler-Lagrange-Rassias functional equation in the spaces of tempered distributions and Fourier hyperfunctions.

MSC:

39B52	Functional equations for functions with more general domains and/or ranges
39B82	Stability, separation, extension, and related topics for functional equations

Keywords:

quadratic functional equation; stability; distributions; heat kernel; Gauss transform

References:

[1]	Bae, J.-H., On the stability of 3-dimensional quadratic functional equation, Bull. Korean Math. Soc., 37, 477-486 (2000) · Zbl 0967.39011
[2]	Baker, J. A., Functional equations, tempered distributions and Fourier transforms, Trans. Amer. Math. Soc., 315, 57-68 (1989) · Zbl 0674.39010
[3]	Bouikhalene, B.; Elqorachi, E., Ulam-Gavruta-Rassias stability of the Pexider functional equation, Internat. J. Appl. Math. Stat., 7, 27-39 (2007)
[4]	Chung, J., Stability of functional equations in the spaces of distributions and hyperfunctions, J. Math. Anal. Appl., 286, 177-186 (2003) · Zbl 1033.39025
[5]	Chung, J., Generalized Pompeiu equation in distributions, Appl. Math. Lett., 19, 485-490 (2006) · Zbl 1090.39013
[6]	Chung, J.; Chung, S.-Y.; Kim, D., The stability of Cauchy equations in the space of Schwartz distributions, J. Math. Anal. Appl., 295, 107-114 (2004) · Zbl 1053.39043
[7]	Chung, S.-Y., Reformulation of some functional equations in the space of Gevrey distributions and regularity of solutions, Aequationes Math., 59, 108-123 (2000) · Zbl 0945.39013
[8]	Gavruta, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184, 431-436 (1994) · Zbl 0818.46043
[9]	Gavruta, P., An answer to a question of John M. Rassias concerning the stability of Cauchy equation, Hadronic Math. Ser., 67-71 (1999)
[10]	Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci., 27, 222-224 (1941) · Zbl 0061.26403
[12]	Park, C. G., Stability of an Euler-Lagrange-Rassias type additive mapping, Internat. J. Appl. Math. Stat., 7, 101-111 (2007)
[13]	Rassias, J. M., On approximation of approximately linear mappings by linear mappings, J. Funct. Anal., 46, 126-130 (1982) · Zbl 0482.47033
[14]	Rassias, J. M., On approximation of approximately linear mappings by linear mappings, Bull. Sci. Math., 108, 445-446 (1984) · Zbl 0599.47106
[15]	Rassias, J. M., Solution of a problem of Ulam, J. Approx. Theory, 57, 268-273 (1989) · Zbl 0672.41027
[16]	Rassias, J. M., On the stability of the Euler-Lagrange functional equation, Chinese J. Math., 20, 185-190 (1992) · Zbl 0753.39003
[17]	Rassias, J. M., Solution of a stability problem of Ulam, Discuss. Math., 12, 95-103 (1992) · Zbl 0779.47005
[18]	Rassias, J. M., Complete solution of the multi-dimensional problem of Ulam, Discuss. Math., 14, 101-107 (1994) · Zbl 0819.39012
[19]	Rassias, J. M., On the stability of the non-linear Euler-Lagrange functional equation in real normed linear spaces, J. Math. Phys. Sci., 28, 231-235 (1994) · Zbl 0840.46024
[20]	Rassias, J. M., On the stability of the general Euler-Lagrange functional equation, Demonstratio Math., 29, 755-766 (1996) · Zbl 0884.47040
[21]	Rassias, J. M., Solution of the Ulam stability problem for Euler-Lagrange quadratic mappings, J. Math. Anal. Appl., 220, 613-639 (1998) · Zbl 0928.39014
[22]	Rassias, J. M., On the stability of the multi-dimensional Euler-Lagrange functional equation, J. Indian Math. Soc., 66, 1-9 (1999) · Zbl 1141.39310
[23]	Rassias, J. M., On some approximately quadratic mappings being exactly quadratic, J. Indian Math. Soc., 69, 155-160 (2002) · Zbl 1104.39300
[24]	Ravi, K.; Arunkumar, M., On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation, Internat. J. Appl. Math. Stat., 7, 143-156 (2007)
[25]	Sibaha, M. A.; Bouikhalene, B.; Elqorachi, E., Ulam-Gavruta-Rassias stability for a linear functional equation, Internat. J. Appl. Math. Stat., 7, 157-168 (2007)