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 [
After then, many mathematicians have extensively investigated the stability problems of functional equations (see [4-6]).
Alsina and Ger [
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 [
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. [
Motivation of this study comes from the work of Li [
Li and Shen [
while Gavruta et al. [
with boundary and initial conditions.
The author in his study [
and
with the initial conditions 
In this paper we investigate the Hyers-Ulam stability of the following nonlinear differential equation of second order
with the initial condition
where
Moreover we investigate the Hyers-Ulam stability of the Emden-Fowler nonlinear differential equation of second order
with the initial condition
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
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
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
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
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
and the inequality
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.
) of the Equation (3)
is a twice continuously differentiable function and 
Then, If 
the Equation (11) is stable in the sense of Hyers and Ulam.
and
that 
is a twice continuously differentiable real-valued function on 
We will show that there exists a function 
satisfying Equation (11) such that
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).
on the interval 
we find that 
in
. Multiplying the inequality (14) by 
and then integrating we obtain
, we obtain the inequality
for all 
The zero solution 
of the
Equation (11) with the zero initial condition (12) such 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).
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
, 
satisfies the inequation (16) and the conditions of the Theorem 3.1. Therefore the Equation (15) has the Hyers-Ulam stability.
is continuous for 
such that
is a twice continuously differentiable function.
then the problem (17), (18) is stable in the Hyers-Ulam sense.
Let 
and
be a twice continuously differentiable real-valued function on 
satisfying the inequality
is a constant that doesn’t depend on 
nor on
If we integrate the inequality (19) with respect to 
we should obtain
is a solution of the Equation (21)
satisfies the Lipschitz condition, and from the inequality (20) we have
Hence the problem (17), (18) has the Hyers-Ulam stability.
satisfies Lipschitz condition
in the region 
then the Emden-Fowler nonlinear differential equation 
is Hyers-Ulam stable with zero initial condition.