# zbMATH — the first resource for mathematics

On the Ulam stability of a class of Banach space valued linear differential equations of second order. (English) Zbl 1417.34141
Summary: Let $$E$$ be a complex Banach space. We prove the Ulam stability of a class of Banach space valued second order linear differential equations $$p(x)y'' (x) + q(x)y' (x) + \lambda y(x) = 0$$, where $$p \in C^1 [I, \mathbb R ^+]$$, $$q \in C[I, \mathbb R]$$ with $$p' (x) = 2q(x)$$ for each $$x \in I$$; $$I$$ denotes an open interval in $$\mathbb R$$, $$\lambda$$ is a fixed positive real number. Moreover, we also provide some applications of our results.

##### MSC:
 34G10 Linear differential equations in abstract spaces 34D20 Stability of solutions to ordinary differential equations
Full Text:
##### References:
 [1] Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1960. [2] Hyers DH: On the stability of the linear functional equation.Proc. Natl. Acad. Sci. USA 1941, 27:222-224. · Zbl 0061.26403 [3] Rassias TM: On the stability of the linear mapping in Banach spaces.Proc. Am. Math. Soc. 1978, 72:297-300. · Zbl 0398.47040 [4] Jung SM: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, Berlin; 2011. · Zbl 1221.39038 [5] Rassias TM: On the stability of functional equations and a problem of Ulam.Acta Appl. Math. 2000, 62:23-130. · Zbl 0981.39014 [6] Obloza M: Hyers stability of the linear differential equation.Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13:259-270. · Zbl 0964.34514 [7] Alsina C, Ger R: On some inequalities and stability results related to the exponential function.J. Inequal. Appl. 1998, 2:373-380. · Zbl 0918.39009 [8] Miura T, Takahasi SE, Choda H: On the Hyers-Ulam stability of real continuous function valued differentiable map.Tokyo J. Math. 2001, 24:467-476. · Zbl 1002.39039 [9] Miura T: On the Hyers-Ulam stability of a differentiable map.Sci. Math. Jpn. 2002, 55:17-24. · Zbl 1025.47041 [10] Takahasi SE, Miura T, Miyajima S:On the Hyers-Ulam stability of the Banach space-valued differential equation[InlineEquation not available: see fulltext.]. Bull. Korean Math. Soc. 2002, 39:309-315. · Zbl 1011.34046 [11] Abdollahpour MR, Najati A: Stability of linear differential equations of third order.Appl. Math. Lett. 2011, 24:1827-1830. · Zbl 1235.34162 [12] Cîmpean DS, Popa D: On the stability of the linear differential equation of higher order with constant coefficients.Appl. Math. Comput. 2010, 217:4141-4146. · Zbl 1211.34065 [13] Jung SM: Hyers-Ulam stability of linear differential equations of first order.Appl. Math. Lett. 2004, 17:1135-1140. · Zbl 1061.34039 [14] Jung SM: Hyers-Ulam stability of linear differential equations of first order (II).Appl. Math. Lett. 2006, 19:854-858. · Zbl 1125.34328 [15] Jung SM: Hyers-Ulam stability of linear differential equations of first order (III).J. Math. Anal. Appl. 2005, 311:139-146. · Zbl 1087.34534 [16] Li Y, Shen Y: Hyers-Ulam stability of linear differential equations of second order.Appl. Math. Lett. 2010, 23:306-309. · Zbl 1188.34069 [17] Miura T, Miyajima S, Takahasi SE: A characterization of Hyers-Ulam stability of first order linear differential operators.J. Math. Anal. Appl. 2003, 286:136-146. · Zbl 1045.47037 [18] Popa D, Raşa I: On the Hyers-Ulam stability of the linear differential equation.J. Math. Anal. Appl. 2011, 381:530-537. · Zbl 1222.34069 [19] Popa D, Raşa I: Hyers-Ulam stability of the linear differential operator with non-constant coefficients.Appl. Math. Comput. 2012, 219:1562-1568. · Zbl 1368.34075 [20] Takahasi SE, Takagi H, Miura T, Miyajima S: The Hyers-Ulam stability constants of first order linear differential operators.J. Math. Anal. Appl. 2004, 296:403-409. · Zbl 1074.47022 [21] Jung SM, Kim B: Chebyshev’s differential equation and its Hyers-Ulam stability.Differ. Equ. Appl. 2009, 1:199-207. · Zbl 1181.34013 [22] Jung, SM; Rassias, TM, Approximation of analytic functions by Chebyshev functions, No. 2011 (2011) [23] Miura T, Yakahasi SE, Hayata T, Tanahashi K: Stability of the Banach space valued Chebyshev differential equation.Appl. Math. Lett. 2012, 25:1976-1979. · Zbl 1257.34043
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.