# zbMATH — the first resource for mathematics

Hyers-Ulam stability of linear functional differential equations. (English) Zbl 1319.34120
Dealing with delay differential equations of the form $y^{(n)}(t)=g(t)\,y(t-\tau)+h(t)\text{ \;on \;}[0,b]$ where $$\tau>0$$, the notion of Hyers-Ulan stability is first introduced and then investigated via different methods.
Popular approachs, such as, iteraction method and fixed point method, are used to obtain the stability results. Moreover, the advantages and disadvantages of each method are discussed. However, the most inovative approach in this paper relies on an open mapping theorem. The latter appears not only in the study of the aforementioned equation, but also in results involving more general linear functional differential equations with constant and multiple delays.

##### MSC:
 34K06 Linear functional-differential equations 34K20 Stability theory of functional-differential equations 34K27 Perturbations of functional-differential equations
Full Text:
##### References:
 [1] András, S.; Mészáros, A., Ulam-Hyers stability of elliptic partial differential equations in Sobolev spaces, Appl. Math. Comput., 229, 131-138, (2014), (MR3159861) · Zbl 1364.35086 [2] Aoki, T., On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2, 64-66, (1950), (MR0040580) · Zbl 0040.35501 [3] Banach, S., Sur LES opérations dans LES ensembles abstraits et leur application aux équation intégrales, Fund. Math., 3, 133-181, (1922) · JFM 48.0201.01 [4] Bourgin, D. G., Classes of transformations and bordering transformations, Bull. Amer. Math. Soc., 57, 223-237, (1951), (MR0042613) · Zbl 0043.32902 [5] Brillouet-Belluot, N.; Brzdȩk, J.; Ciepliński, K., On some recent development in Ulam’s type stability, Abstr. Appl. Anal., (2012), Article ID 716936, 41 pp. (MR2999925) · Zbl 1259.39019 [6] Brzdȩk, J.; Cădariu, L.; Ciepliński, K., Fixed point theory and the Ulam stability, J. Funct. Spaces, (2014), Article ID 829419, 16 pp. (MR3251598) · Zbl 1314.39029 [7] Brzdȩk, J.; Ciepliński, K.; Leśniak, Z., On Ulam’s type stability of the linear equation and related issues, Discrete Dyn. Nat. Soc., (2014), Article ID 536791, 14 pp. (MR3256293) [8] Ciepliński, K., Applications of fixed point theorems to the Hyers-Ulam stability of functional equations - a survey, Ann. Funct. Anal., 3, 1, 151-164, (2012), (MR2903276) · Zbl 1252.39032 [9] Gselmann, E., Stability properties in some classes of second order partial differential equations, Results Math., 65, 95-103, (2014), (MR3162431) · Zbl 1307.35096 [10] Hatori, O.; Kobayasi, K.; Miura, T.; Takagi, H.; Takahasi, S. E., On the best constant of Hyers-Ulam stability, J. Nonlinear Convex Anal., 5, 387-393, (2004), (MR2111613) · Zbl 1079.39025 [11] Huang, J.; Jung, S.-M.; Li, Y., On Hyers-Ulam stability of nonlinear differential equations, Bull. Korean Math. Soc., (2015), in press [12] J. Huang, Y. Li, On the stability of linear differential operators, in preparation. [13] Hyers, D. H., On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27, 222-224, (1941), (MR0004076) · JFM 67.0424.01 [14] Jung, S.-M., Hyers-Ulam stability of linear differential equations of first order (II), Appl. Math. Lett., 19, 854-858, (2006), (MR2240474) · Zbl 1125.34328 [15] Jung, S.-M., Hyers-Ulam stability of linear partial differential equations of first order, Appl. Math. Lett., 22, 70-74, (2009), (MR2484284) · Zbl 1163.39308 [16] Jung, S.-M., A fixed point approach to the stability of differential equations $$y^\prime = F(x, y)$$, Bull. Malays. Math. Sci. Soc., 33, 2, 47-56, (2010), (MR2603339) · Zbl 1184.26012 [17] Jung, S.-M., Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, Springer Optim. Appl., vol. 48, (2011), Springer New York, (MR2790773) · Zbl 1221.39038 [18] Jung, S.-M.; Brzdȩk, J., Hyers-Ulam stability of the delay equation $$y^\prime(t) = \lambda y(t - \tau)$$, Abstr. Appl. Anal., (2010), Article ID 372176, 10 pp. (MR2746009) · Zbl 1210.34108 [19] Li, Y., Hyers-Ulam stability of linear differential equations $$y'' = \lambda^2 y$$, Thai J. Math., 8, 2, 215-219, (2010), (MR2763684) · Zbl 1230.34012 [20] Li, Y.; Shen, Y., Hyers-Ulam stability of linear differential equations of second order, Appl. Math. Lett., 23, 306-309, (2010), (MR2565196) · Zbl 1188.34069 [21] Miura, T.; Miyajima, M.; Takahasi, S.-E., Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr., 258, 90-96, (2003), (MR2000046) · Zbl 1039.34054 [22] Radu, V., The fixed point alternative and the stability of functional equations, Fixed Point Theory, 4, 1, 91-96, (2003), (MR2031824) · Zbl 1051.39031 [23] Rassias, Th. M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72, 297-300, (1978), (MR0507327) · Zbl 0398.47040 [24] Rus, I. A., Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 10, 2, 305-320, (2009), (MR2569004) · Zbl 1204.47071 [25] Takagi, H.; Miura, T.; Takahasi, S.-E., Essential norms and stability constants of weighted composition operators on $$C(X)$$, Bull. Korean Math. Soc., 40, 4, 583-591, (2003), (MR2018640) · Zbl 1060.47033 [26] Ulam, S. M., A collection of mathematical problems, (1960), Interscience New York, (MR0120127) · Zbl 0086.24101 [27] Zheng, Z., Theory of functional differential equations, (1994), Anhui Education Press, (in Chinese)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.