Hyers-Ulam Stability of a Generalized Second-Order Nonlinear Differential Equation ()
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 
They
proved 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 
is
such 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 the
Equation (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 
and
that 
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 the
Equation (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 
and
be 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 on
If 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 condition
in 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.