On the Ulam stability of a class of Banach space valued linear differential equations of second order
© Shen et al.; licensee Springer. 2014
Received: 18 September 2014
Accepted: 12 November 2014
Published: 25 November 2014
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.
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  concerning the stability of group homomorphisms. Hyers  gave a first affirmative partial answer to this question for Banach spaces. Afterwards, Rassias  generalized the result of Hyers  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  initiated the investigation on the Hyers-Ulam stability of differential equations. Later, Alsina and Ger  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.  further improved the stability result of Chebyshev differential equation. Inspired by the work of Miura et al., we will start the following work.
where (here denotes the set of all positive real numbers), with for each , λ is a fixed positive real number, .
2 Main results
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.
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.
which implies that for each .
for all .
for all .
for all . This means that is a solution of Eq. (1).
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.
for all .
for all . □
In this section, some practical examples are given to illustrate the main results proposed in the previous section.
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  will be included in Example 1 as a special case.
for all , where is an arbitrary fixed point in I.
for all . This implies that the inequality (20) holds.
Next, we shall further consider a more general example as a complement of Example 2.
for all , where is an arbitrary fixed point in I.
for all , where .
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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