ABSTRACT

In this paper we have established the stability of a generalized nonlinear second-order differential equation in the sense of Hyers and Ulam. We also have proved the Hyers-Ulam stability of Emden-Fowler type equation with initial conditions.

1. Introduction

In 1940 Ulam posed the basic problem of the stability of functional equations: Give conditions in order for a linear mapping near an approximately linear mapping to exist [1]. The problem for approximately additive mappings, on Banach spaces, was solved by Hyers [2]. The result obtained by Hyers was generalized by Rassias [3].

After then, many mathematicians have extensively investigated the stability problems of functional equations (see [4-6]).

Alsina and Ger [7] were the first mathematicians who investigated the Hyers-Ulam stability of the differential equation Theyproved that if a differentiable function satisfies for all then there exists a differentiable function satisfying for any such that for all. This result of alsina and Ger has been generalized by Takahasi et al [8] to the case of the complex Banach space valued differential equation

Furthermore, the results of Hyers-Ulam stability of differential equations of first order were also generalized by Miura et al. [9], Jung [10] and Wang et al. [11].

Motivation of this study comes from the work of Li [12] where he established the stability of linear differential equation of second order in the sense of the Hyers and Ulam Li and Shen [13] proved the stability of nonhomogeneous linear differential equation of second order in the sense of the Hyers and Ulam while Gavruta et al. [14] proved the Hyers-Ulam stability of the equation with boundary and initial conditions.

The author in his study [15] estabilshed the HyersUlam stability of the equations of the second order

and

with the initial conditions

In this paper we investigate the Hyers-Ulam stability of the following nonlinear differential equation of second order

(1)

with the initial condition

(2)

where

Moreover we investigate the Hyers-Ulam stability of the Emden-Fowler nonlinear differential equation of second order

(3)

with the initial condition

(4)

where

and is bounded in.

Definition 1.1 We will say that the Equation (1) has the Hyers-Ulam stability if there exists a positive constant with the following property:

For every, if

(5)

with the initial condition (2), then there exists a solution of the Equation (1), such that

.

Definition 1.2 We say that Equation (3) has the HyersUlam stability with initial conditions (4) if there exists a positive constant with the following property:

For every, if

(6)

and, then there exists some

satisfying and

, such that.

2. Main Results on Hyers-Ulam Stability Theorem 2.1 If issuch that

and

then the Equation (1) is stable in the sense of Hyers and Ulam.

Proof. Let and be a twice continuously differentiable real-valued function on We will show that there exists a function satisfying Equation (1) such that

where is a constant that never depends on nor on Since is a continuous function on then it keep its sign on some interval

Without loss of generality assume that on Assume that satisfies the inequation (5) with the initial conditions (2) and that

From the inequality (5) we have

(7)

Since on and then by Mean Value Theorem in. Multiplying the inequality (7) by and then integrating from to, we obtain

Since we get that

Therefore

Hence for all Obviously, satisfies the Equation (1) and the zero initial condition (2) such that

Hence the Equation (1) has the Hyers-Ulam stability with initial condition (2).

Remark 2.1 Suppose that satisfies the inequality (5) with the initial condition (2). If

then, if

we can similarly show that the Equation (1) has the Hyers-Ulam stability with initial condition (2).

Theorem 2.2 Suppose that is a twice continuously differentiable function and.

If then the Equation (3) is stable in the sense of Hyers and Ulam.

Proof. Let and be a twice continuously differentiable real-valued function on We will show that there exists a function satisfying Equation (3) such that

where is a constant that never depends on nor on Since is a continuous function on then it keeps its sign on some interval Without loss of generality assume that on

Suppose that satisfies the inequation (6) with the initial conditions (4) and that

We have

(8)

Since in then, Multiplying the inequality (8) by and integrating, we obtain

By hypothesis, so we get that

Therefore

Hence for all Clearly, the zero function satisfies theEquation (1) and the zero initial condition (2) such that

Hence the Equation (3) has the Hyers-Ulam stability with initial condition (4).

Remark 2.2 Suppose that satisfies the inequality (6) with the initial condition (4). If

then, if

we can similarly show that the Equation (3) has the Hyers-Ulam stability with initial condition (4).

Example 2.2 Consider the equation

(9)

and the inequality

(10)

where

It should be noted that for a given satisfies the inequation (10) and the conditions of the Theorem 2.2. Therefore the Equation (9) has the HyersUlam stability.

3. A Special Case of Equation (3)

Consider the special case (when) of the Equation (3)

(11)

with the initial conditions

(12)

and the inequation

(13)

where

Theorem 3.1 Assume that is a twice continuously differentiable function and Then, If the Equation (11) is stable in the sense of Hyers and Ulam.

Proof. Assume that andthat is a twice continuously differentiable real-valued function on We will show that there exists a function satisfying Equation (11) such that

where is a constant that never depends on nor on Since is a continuous function on then it keeps its sign on some interval Without loss of generality assume that on Suppose that satisfies the inequation (13) with the initial conditions (12).

We have

(14)

Applying the Mean Value Theorem to the function on the interval we find that in. Multiplying the inequality (14) by and then integrating we obtain

If, we obtain the inequality

Therefore

Thus for all The zero solution of theEquation (11) with the zero initial condition (12) such that

Hence the Equation (11) has the Hyers-Ulam stability with initial condition (12).

Remark 3.1 Assume that satisfies the inequality (13) with the initial condition (12). If then, if we can similarly obtain the Hyers-Ulam stability criterion for the Equation (11) has with initial condition (12).

Remark 3.2 It should be noted that if on and hence on then in the proofs of Theorem 2.1, 2.2 and 3.1, we can multiply by the inequation (7) (and (8), (14)) to obtain the inequality

Then we can apply the same argument used above to get sufficient criteria for the Hyers-Ulam stability for the Equations (1), (3) and (11).

Example 3.1 Consider the equation

(15)

and the inequality

(16)

where

First it should be noted that for a given, satisfies the inequation (16) and the conditions of the Theorem 3.1. Therefore the Equation (15) has the Hyers-Ulam stability.

4. An Additional Case On Hyers-Ulam Stability

In this section we consider the Hyers-Ulam stability of the following equation

(17)

with the initial condition

(18)

where

and is continuous for such that

Using an argument similar to that used in [16], we can prove the following Theorem:

Theorem 4.1 Suppose that is a twice continuously differentiable function.

If then the problem (17), (18) is stable in the Hyers-Ulam sense.

Proof. Let andbe a twice continuously differentiable real-valued function on satisfying the inequality

(19)

We will show that there exists a function

Satisfying Equation (18) such that

where is a constant that doesn’t depend on nor onIf we integrate the inequality (19) with respect to we should obtain

(20)

It is clear that is a solution of the Equation (21)

(21)

satisfying the zero initial condition

(22)

Now, let’s estimate the difference

Since the function satisfies the Lipschitz condition, and from the inequality (20) we have

From which it follows that

where Hence the problem (17), (18) has the Hyers-Ulam stability.

Remark 4.1 Notice that if satisfies Lipschitz conditionin the region then the Emden-Fowler nonlinear differential equation is Hyers-Ulam stable with zero initial condition.

5. Acknowledgements

The author thanks the anonymous referees for helpful comments and suggestions.

REFERENCES