Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 128479, 15 pagesdoi:10.1155/2011/128479

Research Article

The Stability of Some Differential Equations

M. M. Pourpasha,¹ Th. M. Rassias,² R. Saadati,³ and S. M. Vaezpour⁴

¹Department of Mathematics, Chalous Branch, Islamic Azad University (IAU), Chalous 46615, Iran²Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece³Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran 14778, Iran⁴Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran

Received 17 October 2011; Accepted 17 November 2011

Academic Editor: Alexander P. Seyranian

Copyright © 2011 M. M. Pourpasha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We generalize the results obtained by Jun and Min (2009) and use fixed point method to obtain the stability of the functional equation , for a class of functions of a vector space into a Banach space where is an involution. Then we obtain the stability of the differential equations of the form .

1. Introduction and Preliminary

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms.

The stability concept that was introduced by Rassias’ theorem [2] in 1978 provided a large influence to a number of mathematicians to develop the notion of what is known today by the term Hyers-Ulam-Rassias stability of the linear mapping. Since then, the stability of several functional equations has been extensively investigated by several mathematicians, see [3–5]. They have many applications in Information Theory, Physics, Economic Theory, and Social and Behavior Sciences.

In 1996, Isac and Rassias [6] were the first to use the fixed point methods to investigate the Hyers-Ulam-Rassias stability.

Let be a set. A function is called a generalized metric on if and only if satisfies

(1) , if and only if ,

(2) , for all ,

(3) , for all .

Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity.

We now introduce one of fundamental results of fixed point theory. For the proof, refer to [7]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [8].

Theorem 1.1. Let be a generalized complete metric space. Assume that is a strictly contractive operator with the Lipschitz constant . If there exists a nonnegative integer such that for some , then the followings are true: (a) the sequence converges to a fixed point of ,(b) is the unique fixed point of in (c) if , then

2. Stability of the Generalized Functional Equations

The stability problem for a general equation of the form was investigated by Cholewa [9] in 1984. Indeed, Cholewa proved the superstability of the above equation under some additional assumptions on the functions and spaces involved.

Recently, Jung and Min [10] applied the fixed point method to the investigate the stability of functional equation

In this section, we generalized the Jun and Min’s results and use fixed point approach to obtain the stability of the functional equation for a class of functions of a vector space into a Banach space where is an involution.

Theorem 2.1. Let and be a vector space over and a Banach space over , respectively. Let be a Banach space over . Assume that is a bounded linear transformation, whose norm is denoted by , satisfying for all and there exists a real number with for all . Moreover, assume that is a given function satisfying for all . If and a function satisfies the inequality for any , then there exists a unique solution of (2.3) such that

Proof. First, we denote by the set of all functions and by d the generalized metric on defined as By a similar method used at the proof of [4, Theorem 3.1], we can show that is a generalized complete metric space. Now, let us define an operator by for every . We assert that is strictly contractive on . Given , let be an arbitrary constant with , that is, for each . By (2.5), (2.6), (2.10), and (2.11), we have for every . Then, from (2.9) we have for any , where is the Lipschitz constant with . Thus, is strictly contractive.Now, we claim that . Replacing by and by in (2.7), then it follows from (2.6) and (2.10) that for every . Then, Now, it follows from Theorem 1.1(a) that there exists a function which is a fixed point of , such that From Theorem 1.1(c), we get which implies the validity of (2.8). According to Theorem 1.1(b), is the unique fixed point of with .We now assert that for all and . Indeed, it follows from (2.4), (2.5), (2.6), (2.7), and (2.10) that for any . Then, it follows from (2.4), (2.5), (2.6), (2.10), and (2.17) that for all , which proves the validity of (2.17).Finally, we prove that for any . Since is continuous as a bounded linear transformation, it follows from (2.15) and (2.17) that for all , which implies that is a solution of  (2.7).

Corollary 2.2. Let and be a vector space over and a Banach space over , respectively, and let be a Banach space over . Assume that is a bounded linear transformation, whose norm is denoted by , satisfying condition (2.4) and that there exists a real number satisfying condition (2.5). If and a function satisfies the inequality for all and for some nonnegative real constants and , then there exists a unique solution of 1.2 such that for all .

Example 2.3. Assume that , and consider the Banach spaces and , where we define for all . Let and be fixed complex numbers with , and let be a linear transformation defined by Then it is easy to show that satisfies condition (2.13).If and are complex numbers satisfying for all , then Thus, we get which implies the boundedness of the linear transformation .On the other hand, we obtain for any , then we have If the function satisfies the inequality for all and for some , then Corollary 2.2 (with and ) implies that there exists a unique function such that for all and for any .

3. Stability of the Generalized Differential Equations

Let be a normed space, and let be an open interval. Assume that for any function satisfying the differential inequality for all and for some , there exists a solution of the differential equation such that for any , where is an expression of only. Then, we say that the above differential equation has the Hyers-Ulam stability.

If the above statement is also true when we replace and by and , where are functions not depending on and explicitly, then we say that the corresponding differential equation has the Hyers-Ulam-Rassias stability (or the generalized Hyers-Ulam stability).

We may apply these terminologies for other differential equations. For more detailed definitions of the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability, refer to [11, 12].

In 1998, Alsina and Ger investigated the Hyers-Ulam stability of differential equations. They proved in [13] that if a differentiable function satisfies the differential inequality , where is an open subinterval of , then there exists a differentiable function satisfying and for any .

Alsina and Ger’s results have been generalized by Takahasi et al. [14]. They proved that the Hyers-Ulam stability holds for the Banach space-valued differential equation (see also [15]).

Recently, Takahasi et al. also proved the Hyers-Ulam stability of linear differential equations of first order, , where is a continuous function, and they also proved the Hyers-Ulam stability of linear differential equations of other types (see [16–18]).

In this section, for a bounded and continuous function , we will adopt the idea of Cdariu and Radu [19, 20] and prove the Hyers-Ulam-Rassias stability as well as the Hyers-Ulam stability of the differential equations of the form

Theorem 3.1. For given real numbers and with , let be a closed interval and choose . Let and be positive constants with . Assume that is a continuous function which satisfies a Lipschitz condition for any and . If a continuously differentiable function satisfies for all , where are continuous functions in which and is a continuous function with for each , then there exists a unique continuous function such that (consequently, is a solution to (2.15)) and for all .

Proof. Let us define a set of all continuous functions by and introduce a generalized metric on as follows: By a similar method used at the proof of [4, Theorem 3.1], we assert that is complete. Let be a Cauchy sequence in .Then, for any , there exists an integer such that for all . It further follows from (3.10) that Equation (3.11) implies that is a Cauchy sequence in . Since is complete, converges for each . Thus, we can define a function by Let increase to infinity, then by (3.11) we have Since is bounded on , converges uniformly to . Hence, is continuous and .Further, considering (3.10) and (3.13), then Then, the Cauchy sequence converges to in . Hence, is complete.Now, define the operator by for all . (Indeed, according to the Fundamental Theorem of Calculus, is continuously differentiable on , since and are continuous functions. Hence, we may conclude that .) We prove that is strictly contractive on . For any , let be an arbitrary constant with , then, by (2.15), we have for any . It then follows from (3.4), (3.6), (3.10), (3.15), and (3.16) that for all . Then, . Hence, we can conclude that for any (note that ). It follows from (3.9) and (3.15) that for an arbitrary , there exists a constant with for all , since and are bounded on and . Thus, (3.10) implies that Therefore, according to Theorem 1.1(a), there exists a continuous function such that in and , that is, satisfies (3.7) for every . For any , since and are bounded on and , there exists a constant such that for any . Hence, we have for all , that is, . Hence, in view of Theorem 1.1(b), we conclude that is the unique continuous function with the property (3.7).On the other hand, it follows from (3.5) that for all . If we integrate each term in the above inequality from to , then we obtain for any . Thus, by (3.6) and (3.15), we get for each , which implies that Finally, Theorem 1.1(c) and (3.24) implys that which means that inequality (3.24) holds true for all .

Now, we prove the last theorem for unbounded intervals. Also we show that Theorem 3.1 is also true if is replaced by an unbounded interval such as , , or .

Theorem 3.2. For given real numbers and , let denote either , , or . Set either for or for or is a fixed real number if . Let and be positive constants with . Assume that is a continuous function which satisfies Lipschitz condition (3.4) for any and . If a continuously differentiable function satisfies for all , where is a continuous function and is a continuous function satisfying condition (3.6) for each , then there exists a unique continuous function such that for all .

Proof. We prove for . The other cases can be proved similarly. For any , we define . (We set for and for ). By Theorem 3.1, there exists a unique continuous function such that for all . The uniqueness of implies that, if , then For any , define as Moreover, define a function by and we assert that is continuous. For an arbitrary , we choose the integer . Then, belongs to the interior of and there exists an such that for all with . Since is continuous at , so is . That is, is continuous at for any .We will now show that satisfies (3.8) for all . For an arbitrary , we choose the integer . Then, it holds that and it follows from (3.28) and (3.32) that since for any . Then, from (3.30) and (3.32) we have Since for every , by (3.29) and (3.32), we have for any .Finally, we show that is unique. Let be another continuous function which satisfies (3.8), with in place of , for all . Suppose is an arbitrary real number. Since the restrictions and both satisfy (3.7) and (3.8) for all , the uniqueness of implies that as required.

Corollary 3.3. Given and , let denote a closed ball of radius and centered at , that is, , and let be a continuous function which satisfies a Lipschitz condition (3.4) for all and , where is a constant with . If a continuously differentiable function satisfies the differential inequality for all and for some , then there exists a unique continuous function satisfying (3.7) and for any .

Example 3.4. We choose positive constants and with . For a positive number , let be a closed interval. Given a polynomial , we assume that a continuously differentiable function satisfies for all . If we set and , then the above inequality has the identical form. Moreover, we obtain for each , since for all . By Theorem 3.1, there exists a unique continuous function such that for any .

Example 3.5. Let be a constant greater than 1 and choose a constant with . Given an interval and a polynomial , suppose is a continuously differentiable function satisfying for all . If we set , then we have for any . By Theorem 3.2, there exists a unique continuous function with for any .

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