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On the Ulam stability of a class of Banach space valued linear differential equations of second order
Advances in Difference Equations volume 2014, Article number: 294 (2014)
Abstract
Let E be a complex Banach space. We prove the Ulam stability of a class of Banach space valued second order linear differential equations , where , with for each ; I denotes an open interval in ℝ, λ is a fixed positive real number. Moreover, we also provide some applications of our results.
1 Introduction
At present, the Ulam stability (Hyers-Ulam stability or Hyers-Ulam-Rassias stability) is one of the most active research topics in the theory of functional equations. The study of such stability problems for functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to this question for Banach spaces. Afterwards, Rassias [3] generalized the result of Hyers [2] for linear mappings in which the Cauchy difference is allowed to be unbounded, and this work has great influence on the development of this type of stability theory of functional equations. Since then, the stability problems for various functional equations have been extensively studied. For more details, the reader is referred to [4, 5].
In 1993, Obloza [6] initiated the investigation on the Hyers-Ulam stability of differential equations. Later, Alsina and Ger [7] proved that the Hyers-Ulam stability of the differential equation holds. Specifically, for a given , if f is a differentiable function on an open subinterval I into ℝ with for all , then there exists a differentiable function such that and satisfying for all . Soon after, the above result was generalized by Miura and Takahasi et al. [8–10]. Up to the present, the Hyers-Ulam stability or Hyers-Ulam-Rassias stability of the first order and higher order linear differential equations have been widely and extensively investigated by many authors [11–20].
Several years ago, Jung together with Kim and Rassias [21, 22] discussed the general solution of the complex-valued Chebyshev differential equation and studied its Hyers-Ulam stability. Hereafter, Miura et al. [23] further improved the stability result of Chebyshev differential equation. Inspired by the work of Miura et al., we will start the following work.
Let E be a complex Banach space with a norm . Unless otherwise stated, let , , denote the open interval of ℝ. In addition, we denote by and () the set of all E-valued continuous functions on I and the set of all strongly (two times) differentiable functions which have a continuous (second) derivative on I, respectively. In a more general framework, we shall investigate the Ulam stability of the following second order linear differential equation:
where (here denotes the set of all positive real numbers), with for each , λ is a fixed positive real number, .
2 Main results
For a n-order E-valued differential equation
we say that it has the Hyers-Ulam stability or it is stable in the sense of Hyers-Ulam sense if for a given and a n times strongly differentiable function satisfying for all , then there exists an exact solution h of this equation such that for all , where depends only on ϵ, and . More generally, if ϵ and are replaced by two control functions , respectively, we say that the above differential equation has the Hyers-Ulam-Rassias stability or it is stable in the sense of Hyers-Ulam-Rassias.
The following theorem is the main result of this paper.
Theorem 2.1 Let be a function such that is integrable on the interval for each with . Suppose that satisfies the differential inequality
for all . Moreover, assume that the equation
has a solution on , where corresponds to . Then there exists such that h satisfies Eq. (1) and
for all , where is an arbitrary fixed point.
Proof Firstly, we take the solution of Eq. (3) and make an appropriate substitution for the variable x. Obviously, the condition , implies that the inverse function exists on I.
Now, we define the map by for each . Obviously, . Then we can obtain
Moreover, since , we get for each . Thus, it follows from Eq. (3) that
which implies that for each .
Therefore, we have
for all . Note that for each . Based on the inequality (2) and the preceding equality, we can obtain
for all .
We set , . Then we have
It follows that . Multiplying both sides of the previous equality by , , we obtain
By integrating both sides of the above equality from (here is an arbitrary fixed point in J) to s with respect to τ, , it follows that
Furthermore, we can infer that
for all .
In view of , for each , we have
Analogously, by integrating both sides of the last equality from to t with respect to s, we conclude that
Define the map by
Clearly, . Therefore, we can infer that
and thus
From the equalities (7) and (8), it is easy to see that for each . Furthermore, we can infer from Eq. (6) that
for each . Then, by Eq. (5), we can obtain
Finally, we define the map by
Therefore, we get
Furthermore, we can obtain
According to the derivative of the inverse function, it follows that
Moreover, in view of together with being a solution of Eq. (3), we can infer from Eq. (12) that
Then it follows from Eqs. (8), (10), (11), and (13) that
Hence, we get
for all . This means that is a solution of Eq. (1).
Recall that for each , namely, for each . By Eq. (9), we can obtain
where . □
Remark 1 In Theorem 2.1, the arbitrariness of does not mean that the right-hand side of the inequality (4) may be arbitrarily changed with respect to an appropriate function h, because one can see from Eq. (7) that the desired function h depends on the choice of . Furthermore, we can conclude from Eq. (4) that for an arbitrary fixed , there exists an appropriate solution h of Eq. (1) such that h can be used to approximate the function f that satisfies the inequality (2), and the error can be estimated by the control function of the right-hand side of Eq. (4).
The following result associated with the Hyers-Ulam stability of Eq. (1) is a direct consequence of Theorem 2.1.
Corollary 2.2 Let be a finite interval, i.e., and let be a given number. Suppose that satisfies the differential inequality
for all . Moreover, assume that Eq. (3) has a solution on , where corresponds to . Then there exists such that h satisfies Eq. (1) and
for all .
Proof According to Theorem 2.1, it suffices to verify that the inequality (15) holds for all . By Eqs. (5) and (9), we have
for all . □
Remark 2 In Corollary 2.2, since is an arbitrary fixed point, the right-hand side of the equality (15) can be further improved. In fact, since is arbitrary, if we take the midpoint of the interval , i.e., putting , then we can obtain a better upper bound of the equality (15). Furthermore, the inequality (15) can be improved as
3 Applications
In this section, some practical examples are given to illustrate the main results proposed in the previous section.
Example 1 Let E be a complex Banach space. Then the differential equation
has the Hyers-Ulam stability when , where . More precisely, for each , if satisfies the following differential inequality:
for all , then there exists such that h satisfies Eq. (17) and
for all .
According to Corollary 2.2, we take , . Obviously, we have for each . Consider the equation . It is easy to know that is a solution on . Clearly, it is easy to check that satisfies the condition of Corollary 2.2, and hence there exists such that h satisfies Eq. (17) and the inequality (18).
Remark 3 In Example 1, if , , (here ℕ denotes the set of all natural numbers), Eq. (17) will degenerate into the Chebyshev differential equation. Therefore, the main results obtained in [23] will be included in Example 1 as a special case.
Example 2 Let E be a complex Banach space. Then the differential equation
has the Hyers-Ulam-Rassias stability when , where . Specifically, for each , if satisfies the following differential inequality:
for all , then there exists such that h satisfies Eq. (19) and
for all , where is an arbitrary fixed point in I.
Let , . Clearly, for each . Consider the differential equation . It can easily be verified that () is a solution on . For simplicity, we take . Therefore, is an appropriate substitution for the variable x. According to Theorem 2.1, there exists such that h satisfies Eq. (19). Moreover, it follows from Eq. (9) that
for all . This implies that the inequality (20) holds.
Next, we shall further consider a more general example as a complement of Example 2.
Example 3 Let E be a complex Banach space. For each , assume that a function satisfies the following differential inequality:
for all , where . Then there exists such that
and
for all , where is an arbitrary fixed point in I.
Let , . Consider the differential equation . Based on the theory of ordinary differential equations, we see that is a solution on , where corresponds to . Therefore, one can see that is an appropriate substitution for the variable x. By Theorem 2.1, there exists such that the equality (21) holds and
for all , where .
References
Ulam SM: Problems in Modern Mathematics. Wiley, New York; 1960.
Hyers DH: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27: 222-224. 10.1073/pnas.27.4.222
Rassias TM: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72: 297-300. 10.1090/S0002-9939-1978-0507327-1
Jung SM: Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, Berlin; 2011.
Rassias TM: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 2000, 62: 23-130. 10.1023/A:1006499223572
Obloza M: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat. 1993, 13: 259-270.
Alsina C, Ger R: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 1998, 2: 373-380.
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. 10.3836/tjm/1255958187
Miura T: On the Hyers-Ulam stability of a differentiable map. Sci. Math. Jpn. 2002, 55: 17-24.
Takahasi SE, Miura T, Miyajima S:On the Hyers-Ulam stability of the Banach space-valued differential equation. Bull. Korean Math. Soc. 2002, 39: 309-315. 10.4134/BKMS.2002.39.2.309
Abdollahpour MR, Najati A: Stability of linear differential equations of third order. Appl. Math. Lett. 2011, 24: 1827-1830. 10.1016/j.aml.2011.04.043
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. 10.1016/j.amc.2010.09.062
Jung SM: Hyers-Ulam stability of linear differential equations of first order. Appl. Math. Lett. 2004, 17: 1135-1140. 10.1016/j.aml.2003.11.004
Jung SM: Hyers-Ulam stability of linear differential equations of first order (II). Appl. Math. Lett. 2006, 19: 854-858. 10.1016/j.aml.2005.11.004
Jung SM: Hyers-Ulam stability of linear differential equations of first order (III). J. Math. Anal. Appl. 2005, 311: 139-146. 10.1016/j.jmaa.2005.02.025
Li Y, Shen Y: Hyers-Ulam stability of linear differential equations of second order. Appl. Math. Lett. 2010, 23: 306-309. 10.1016/j.aml.2009.09.020
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. 10.1016/S0022-247X(03)00458-X
Popa D, Raşa I: On the Hyers-Ulam stability of the linear differential equation. J. Math. Anal. Appl. 2011, 381: 530-537. 10.1016/j.jmaa.2011.02.051
Popa D, Raşa I: Hyers-Ulam stability of the linear differential operator with non-constant coefficients. Appl. Math. Comput. 2012, 219: 1562-1568. 10.1016/j.amc.2012.07.056
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. 10.1016/j.jmaa.2003.12.044
Jung SM, Kim B: Chebyshev’s differential equation and its Hyers-Ulam stability. Differ. Equ. Appl. 2009, 1: 199-207.
Jung SM, Rassias TM: Approximation of analytic functions by Chebyshev functions. Abstr. Appl. Anal. 2011., 2011: Article ID 432961
Miura T, Yakahasi SE, Hayata T, Tanahashi K: Stability of the Banach space valued Chebyshev differential equation. Appl. Math. Lett. 2012, 25: 1976-1979. 10.1016/j.aml.2012.03.012
Acknowledgements
This work was supported by ‘Qing Lan’ Talent Engineering Funds by Tianshui Normal University. The second author acknowledges the support of the Humanity and Social Science Youth Foundation of Ministry of Education of China (no. 13YJC630012), and the Specially Commissioned Project of the Capital University of Economics and Business. The third author acknowledges the support of the National Natural Science Foundation of China (no. 11226268).
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Shen, Y., Chen, W. & Lan, Y. On the Ulam stability of a class of Banach space valued linear differential equations of second order. Adv Differ Equ 2014, 294 (2014). https://doi.org/10.1186/1687-1847-2014-294
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DOI: https://doi.org/10.1186/1687-1847-2014-294