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A Fixed Point Approach to Hyers-Ulam-Rassias Stability of Nonlinear Differential Equations

Maher Nazmi Qarawani

Department of Mathematics, Al-Quds Open University, Salfit, West-Bank, Palestine

Abstract: In this paper we use the fixed point approach to obtain sufficient conditions for Hyers-Ulam-Rassias stability of nonlinear differential. Some illustrative examples are given.

Keywords: hyers-ulam-rassias stability, fixed point, nonlinear differential equations

Citation: Maher Nazmi Qarawani, A Fixed Point Approach to Hyers-Ulam-Rassias Stability of Nonlinear Differential Equations. American Journal of Applied Mathematics and Statistics, 3 (6). 226-232. January 2015. doi:10.12691/ajams-3-6-3

1. Introduction

The objective of this article is to investigate the Hyers-Ulam-Rassias Stability for the nonlinear differential equation

[picture] | (1)

and the perturbed nonlinear differential equation of second order

[picture] | (2)

by fixed point method under assumptions: are continuous, and that

[picture] | (3)

[picture] | (4)

Suppose that there is such that if then

[picture] | (5)

where as and

Furthermore, we assume that there is a positive constant such that and with

[picture] | (6)

In 1940, Ulam [1] posed the stability problem of functional equations. In the talk, Ulam discussed a problem concerning the stability of homomorphisms. A significant breakthrough came in 1941, when Hyers [2] gave a partial solution to Ulam's problem. During the last two decades very important contributions to the stability problems of functional equations were given by many mathematicians (see [3-11]). More than twenty years ago, a generalization of Ulam's problem was proposed by replacing functional equations with differential equations: The differential equation has the Hyers-Ulam stability if for given and a function such that

there exists a solution of the differential equation such that

and

The first step in the direction of investigating the Hyers-Ulam stability of differential equations was taken by Obloza (see [12,13]). Thereafter, Alsina and Ger [14] have studied the Hyers-Ulam stability of the linear differential equation . The Hyers-Ulam stability problems of linear differential equations of first order and second order with constant coefficients were studied in the papers ([15,16]) by using the method of integral factors. The results given in [17,18,19] have been generalized by Popa and Rasa [20,21] for the linear differential equations of nth order with constant coefficients. In addition to above-mentioned studies, several authors have studied the Hyers-Ulam stability for differential equations of first and second order (see 22-26). The Hyers-Ulam-Rassias Stability by Fixed Point Technique for Half-linear Differential Equations with Unbounded Delay has been established by Qarawani [27]. in [28] has used fixed point theory to establish Liapunov stability for functional differential equations. Some researchers have used the fixed point approach to investigate the Hyers-Ulam stability for differential equations [e.g. [29,30]].

Definition 1 Let

on where . We say that equation (1.2) ( or (1.1) with has the Hyers-Ulam-Rassias (HUR) stability with respect to if there exists a positive constant with the following property: For each , if

[picture] | (7)

then there exists some of the equation (4) such that .

Theorem 1 The Contraction Mapping Principle.

Let be a complete metric space and let If there is a constant such that for each pair we have then there is one and only one point with

2. Main Results On Hyers-Ulam-Rassias Stability

Theorem 2 Suppose that satisfies the inequality (1) with small initial condition . Let be a continuous function such that

[picture] | (8)

If (3)-(6) hold, then the solution of (1) is stable in the sense of Hyers-Ulam-Rassias.

Proof. Let be the space of all continuous functions from and define the set by

Then, equipped with the supremum metric , is a complete metric space. Now suppose that (3) holds. For and find appropriate constants and such that

Multiplying both sides of (1) by and then integrating once with respect to yields

[picture] | (9)

Now, we multiply Eq. (9) by and integrate with respect to to obtain

Define by

[picture] | (10)

It is clear that for , is continuous. Let with , for some positive constant . Then there is a with Since as then we can find a constant such that

Then using (3),(4) in the definition of , we have

Since as we can choose a number sufficiently small such that on and with

[picture] | (11)

Then from (4) we obtain

which implies that

To see that is a contraction under the supremum metric, let then

From this and in view of (4) and (11) we get the estimate

Thus, by the contraction mapping principle, has a unique fixed point, say in which solves (1) and is bounded.

Next we show that the solution is stable in Hyers-Ulam-Rassias. From the inequality (7) we get

[picture] | (12)

Multiplying the inequality (12) by we obtain

Or equivalently, we have

Integrate the last inequality from to and then multiply the obtained inequality by to get

Integrating again with respect to we have

Hence from (8), (20) we infer that To show that is stable we estimate the difference

Thus

which means that (7) holds true (with) for all .

Example 1 Consider the differential equation

Let us estimate the integrals

and for all we obtain

Since then

Therefore, we take which tends to zero as

Now, if we set then we have

Let us take . Then for the corresponding coefficients by (1.3), we can choose small positive constants such that

and so

Thus, all the conditions of Theorem (3.1) are satisfied, hence the Eq. (3.6) is HUR stable for

Theorem 3 Suppose that satisfies the inequality (7) with small initial condition . Let be a continuous function such that

[picture] | (13)

If (3)-(7) hold, then the solution of (2) is stable in the sense of Hyers-Ulam-Rassias.

Proof. Define where is the supremum metric. Thenis a complete metric space.

Now suppose that (3) holds. For and we find constants and so that

Applying the same approach used in Theorem 1 we define by

Then from (4) we obtain

which implies that

To see that is a contraction under the supremum metric, let then

From this and using (4) and (11) we get the estimate

Thus, by the contraction mapping principle, has a unique fixed point, say in which solves (1) and is bounded.

Next we show that the solution is stable in Hyers-Ulam-Rassias. From the inequality (7) we get

[picture] | (14)

Multiplying the inequality (14) by we obtain

Or equivalently, we have

Integrating the last inequality from to and then multiplying the obtained inequality by we get

Integrating again with respect to we have

From the definition ofand in view of (20), we infer that Now, to show that is stable we estimate the difference

Thus

which completes the proof.

Example 2 Consider the nonlinear differential equation

One can similarly, as in Example 1 establish the validity of conditions (1.3)-(1.6). So, to establish the stability of this equation, it remains to estimate the integral

Let us take and .

Then for these coefficients by (3), we can choose small positive constants such that

From which it follows that

Hence the conditions of Theorem 2 are satisfied.

3. Conclusion

We have obtained two theorems which provide the sufficient conditions for the Hyers-Ulam-Rassias Stability of solutions of two nonlinear differential equations. To illustrate the results we provided two examples satisfying the assumptions of the two proved theorems.

References

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